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  author  = {N. B. Budanur and B. Hof},
  title   = {Heteroclinic path to spatially localized chaos in pipe flow},
  journal = {J. Fluid Mech.},
  year    = {2017},
  volume  = {827},
  doi     = {10.1017/jfm.2017.516},
}

@Article{BudMez12,
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  title   = {Geometry of the ergodic quotient reveals coherent structures in flows},
  journal = {Physica D},
  year    = {2012},
  volume  = {241},
  pages   = {1255--1269},
  note    = {\arXiv{1204.2050}},
  doi     = {10.1016/j.physd.2012.04.006},
}

@Article{BuMoMe12,
  Title                    = {{Applied Koopmanism}},
  Author                   = {Budi{\v s}i{\'c}, M. and Mohr, R. M. and Mezi{\'c}, I.},
  Journal                  = {Chaos},
  Year                     = {2012},
  Note                     = {\arXiv{1206.3164}},
  Pages                    = {047510},
  Volume                   = {22},
  DOI                      = {10.1063/1.4772195},
  Numpages                 = {33}
}

@Article{burke2009,
  author =  {Burke, J. and Houghton, S. M. and Knobloch, E.},
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  volume =  {80},
  pages =   {036202},
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@Article{BurKno06,
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  Author                   = {Burke, J. and Knobloch, E.},
  Journal                  = {Phys. Rev. E},
  Year                     = {2006},
  Pages                    = {056211},
  Volume                   = {73},
  DOI                      = {10.1103/PhysRevE.73.056211}
}

@Article{BurKno07,
  Title                    = {Homoclinic snaking: structure and stability},
  Author                   = {Burke, J. and Knobloch, E.},
  Journal                  = {Chaos},
  Year                     = {2007},
  Pages                    = {037102},
  Volume                   = {17},
  DOI                      = {10.1063/1.2746816}
}

@Article{Busse04,
  Title                    = {Visualizing the dynamics of the onset of turbulence},
  Author                   = {F. H. Busse},
  Journal                  = {Science},
  Year                     = {2004},
  Pages                    = {1574--1575},
  Volume                   = {305}
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@Unpublished{ByMaGr14,
  Title                    = {Unstable and marginally stable solutions of reaction diffusion systems with weakly broken continuous symmetries},
  Author                   = {G. Byrne, C. Marcotte and R. O. Grigoriev},
  Note                     = {To be submitted},
  Year                     = {2014}
}

@InProceedings{ByMaGr14a,
  author    = {G. Byrne and C. D. Marcotte and R. O. Grigoriev},
  title     = {Exact coherent structures and dynamics of cardiac tissue},
  booktitle = {{Proc. of 8\textsuperscript{th} Conference of the European Study Group on Cardiovascular Oscillations, Trento, Italy}},
  year      = {2014},
  publisher = {IEEE},
  doi       = {10.1109/esgco.2014.6847523},
}

@Article{BZMmaw00,
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  volume   = {85},
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  abstract = {MAWs of various periods P occur in phase chaotic states.
The pair of MAWs cease to exist via a saddle-node bifurcation. For
periods beyond this bifurcation, incoherent near-MAW evolve toward
defects.},
  doi      = {10.1103/physrevlett.85.86},
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@Misc{C77,
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  year      = {2007},
  address   = {New York},
}

@Book{Canuto88,
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@Article{CarCha04,
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  Year                     = {2004},
  Note                     = {\arXiv{astro-ph/0305186}},
  Pages                    = {291--325},
  Volume                   = {13},
  DOI                      = {10.1142/S0218271804004542}
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@Article{carl97int,
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  volume  = {56},
  pages   = {6687},
  doi     = {10.1103/physreve.56.6687},
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@Book{CartanMF,
  Title                    = {La m\'ethode du rep\`ere mobile, la th\'eorie des groupes continus, et les espaces g\'en\'eralis\'es},
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  Publisher                = {Hermann},
  Year                     = {1935},
  Address                  = {Paris},
  Series                   = {{Expos\'es} de {G\'eom\'etrie}},
  Volume                   = {5}
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@Article{CB92,
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  Year                     = {1992},
  Pages                    = {511--527},
  Volume                   = {234}
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@Article{CB97,
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  journal  = {J. Fluid Mech.},
  year     = {1997},
  volume   = {344},
  pages    = {137--153},
  abstract = {The manifold of Nagata steady solutions is
   explored in the parameter space of the problem and their
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   lead to time-periodic solutions whose properties do not
   differ much from those of the steady solutions except that
   the amplitude varies in time. In some cases travelling wave
   solutions which are asymmetric with respect to the midplane
   of the layer are found as quaternary states of flow.
   Similarities with longitudinal vortices recently observed in
   experiments are discussed.},
  doi      = {10.1017/s0022112097005818},
}

@Incollection{CBconverg,
  author    = {R. Artuso and H. H. Rugh and P. Cvitanovi{\'c}},
  title     = {Why does it work?},
  booktitle = {{Chaos: Classical and Quantum}},
  publisher = {Niels Bohr Inst.},
  year      = {2017},
  editor    = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
  address   = {Copenhagen},
  url       = {http://ChaosBook.org/paper.shtml#converg},
}

@Incollection{CBook:appendApplic,
  author    = {P. Cvitanovi{\'c} and G. Vattay},
  title     = {Appendix: {Transport of vector fields}},
  booktitle = {{Chaos: Classical and Quantum}},
  publisher = {Niels Bohr Inst.},
  year      = {2011},
  editor    = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
  address   = {Copenhagen},
  url       = {http://ChaosBook.org/paper.shtml#appendApplic},
}

@Incollection{CBtrace,
  author    = {P. Cvitanovi{\'c}},
  title     = {Trace formulas},
  booktitle = {Chaos: Classical and Quantum},
  publisher = {Niels Bohr Inst.},
  year      = {2017},
  editor    = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
  address   = {Copenhagen},
  url       = {http://ChaosBook.org/paper.shtml#trace},
}

@Article{CCKFNRS12,
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  Journal                  = {Nature},
  Year                     = {2012},
  Pages                    = {51--56},
  Volume                   = {487},
  DOI                      = {10.1038/nature11129}
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@Article{CCW05,
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  Year                     = {2005},
  Pages                    = {3551--3560},
  Volume                   = {133}
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@Article{CE89,
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    depend on the size of the interval.},
  doi      = {10.1016/0167-2789(93)90168-z},
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@Article{CEEksgl93,
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  pages   = {203--214},
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@Book{cellwolf,
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  Year                     = {1994},
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@Incollection{CERRS,
  Title                    = {Pinball scattering},
  Author                   = {B. Eckhardt and G. Russberg and P. Cvitanovi\'c and P. E. Rosenqvist and P. Scherer},
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@Article{CGS92,
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  Journal                  = {Chaos},
  Year                     = {1992},
  Pages                    = {85--90},
  Volume                   = {2},
  DOI                      = {10.1063/1.165902}
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@Article{champneys2008,
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@Article{Champneys98,
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  Pages                    = {158--186},
  Volume                   = {112},
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@Article{chand03tf,
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  volume   = {181},
  pages    = {171},
  abstract = {Use wavelets to extract local frequencies and use the
    major frequencies to identify the characteristics of phase space
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  doi      = {10.1016/s0167-2789(03)00117-9},
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@Article{chang94,
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  Year                     = {1994},
  Pages                    = {103--136},
  Volume                   = {26}
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@TechReport{channelflow,
  author      = {J. F. Gibson},
  title       = {{Channelflow}: {A} spectral {Navier-Stokes} simulator in {C}++},
  institution = {U. New Hampshire},
  year        = {2017},
  note        = {{\tt {Channelflow.org}}},
  publisher   = {J. F. Gibson},
  url         = {http://Channelflow.org},
}

@Misc{channelflowDat,
  Title                    = {Database of invariant solutions of plane {Couette} flow},
  Author                   = {J. F. Gibson},
  Note                     = {{\tt {Channelflow.org/database}}},
  Year                     = {2013},
  Institution              = {U. New Hampshire},
  Publisher                = {J. F. Gibson}
}

@Article{CHASCH14,
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  year    = {2014},
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  pages   = {506--517},
  doi     = {10.1017/jfm.2014.150},
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@Article{chate94,
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  volume   = {7},
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    linearly stale stable plane waves separated by localized objects with
    a well defined dynamics. In the transition reginon, asymptotic states
    with an irregular, frozen spatial structure are shown to occur. The
    bistable region beyond the Benjamin-Feir stable region is of the same
    spatiotemporal intermittency type with phase turbulence as the},
  doi      = {10.1088/0951-7715/7/1/007},
}

@Article{ChaWri81,
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  Year                     = {1981},
  Pages                    = {1419--1433},
  Volume                   = {23},
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@Article{ChePuShr99,
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  note    = {\arXiv{chao-dyn/9905027}},
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@Article{cherh00,
  Title                    = {Nonlinear equilibrium states in plane {Couette} flow: Continuation from {P}oiseuille flow to {Couette} flow},
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  Year                     = {2000},
  Pages                    = {69--80},
  Volume                   = {74},
  ISSN                     = {0179-9614}
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@Article{cherh96,
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  Year                     = {1996},
  Pages                    = {317--320},
  Volume                   = {36},
  DOI                      = {10.1007/978-94-009-0297-8_89}
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@Article{Cherh97,
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  volume  = {342},
  pages   = {159--177},
  doi     = {10.1017/s0022112097005661},
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@Article{CheSu90,
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  pages   = {275--92},
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@Article{chfield,
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  year    = {2000},
  volume  = {288},
  pages   = {61},
  note    = {\arXiv{nlin.CD/0001034}},
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@Article{CHHM98,
  author  = {Cendra, H. and Holm, D. D. and Hoyle, M. J. W. and Marsden, J. E.},
  journal = {J. Math. Phys.},
  title   = {The {Maxwell-Vlasov} equations in {Euler-Poincar\'e} form},
  year    = {1998},
  pages   = {3138--3157},
  volume  = {39},
  doi     = {10.1063/1.532244},
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@Article{Chillingworth2000,
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  year    = {2000},
  volume  = {15},
  pages   = {101--137},
  doi     = {10.1080/02681110050032119},
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@Article{choi94,
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  year    = {1994},
  volume  = {262},
  pages   = {75},
  doi     = {10.1017/s0022112094000431},
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@Book{Chossat94,
  author    = {P. Chossat and G. Iooss},
  publisher = {Springer},
  title     = {{The {Taylor-Couette} Problem}},
  year      = {1994},
  address   = {New York},
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@Book{ChossLaut00,
  title     = {{Methods in Equivariant Bifurcations and Dynamical Systems}},
  publisher = {World Scientific},
  year      = {2000},
  author    = {P. Chossat and R. Lauterbach},
  address   = {Singapore},
  isbn      = {978-981-02-3828-5},
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@Book{ChPiWa89,
  Title                    = {Group Representation Theory for Physicists},
  Author                   = {Chen, J.-Q. and Ping, J. and Wang, F.},
  Publisher                = {World Scientific},
  Year                     = {1989},
  Address                  = {Singapore},
  DOI                      = {10.1142/0262}
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@Article{ChriPol96,
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  year    = {1996},
  volume  = {9},
  pages   = {1623--1640},
  doi     = {10.1088/0951-7715/9/6/014},
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@Article{ChriPol97,
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  year    = {1997},
  volume  = {109},
  pages   = {32--41},
  doi     = {10.1016/s0167-2789(97)00156-5},
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@Article{Christiansen97,
  Title                    = {Hopf's last hope: {Spatiotempora}l chaos in terms of unstable recurrent patterns},
  Author                   = {F. Christiansen and P. Cvitanovi{\'c} and V. Putkaradze},
  Journal                  = {Nonlinearity},
  Year                     = {1997},
  Addendum                 = {\arXiv{chao-dyn/9606016}},
  Pages                    = {55--70},
  Volume                   = {10},
  DOI                      = {10.1088/0951-7715/10/1/004}
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@Article{ChrPol95,
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  year    = {1995},
  volume  = {51},
  pages   = {3811},
  note    = {\arXiv{chao-dyn/9411005}},
  doi     = {10.1103/physreve.51.r3811},
}

@Article{ChSaKa14,
  Title                    = {On the equivalence of dynamically orthogonal and bi-orthogonal methods: {Theory} and numerical simulations},
  Author                   = {M. Choi and T. P. Sapsis and G. E. Karniadakis},
  Journal                  = {J. Comput. Physics},
  Year                     = {2014},
  Pages                    = {1--20},
  Volume                   = {270},
  DOI                      = {10.1016/j.jcp.2014.03.050}
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@Article{ChTuBa16,
  Title                    = {Turbulent-laminar patterns in shear flows without walls},
  Author                   = {Chantry, M. and Tuckerman, L. S. and Barkley, D.},
  Journal                  = {J. Fluid M.},
  Year                     = {2016},
  Pages                    = {R8},
  Volume                   = {791},
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  Author                   = {C. Crowley and M. Krygier and D. Borrero-Echeverry and R. Grigoriev and M. Schatz},
  Booktitle                = {68\textsuperscript{th} Annual Meeting of the APS Division of Fluid Dynamics},
  Year                     = {2015},
  Series                   = {Bull. Am. Phys. Soc},
  Volume                   = {60},
  URL                      = {http://meetings.aps.org/link/BAPS.2015.DFD.D29.2}
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@Misc{CNS,
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@Phdthesis{Coff76,
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  School                   = {City Univ. of New York, NY.},
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@Book{Cornwell97,
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@Phdthesis{Crofts07thesis,
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  Year                     = {1989},
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@Book{DasBuch,
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  Year                     = {2017},
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@Incollection{DasBuchMirror,
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  year      = {2017},
  editor    = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
  address   = {Copenhagen},
  url       = {http://ChaosBook.org/paper.shtml#discrete},
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@Article{DauBer12,
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  Pages                    = {145--157},
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@Article{DegHaWa13,
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@Article{deLMeAvHo12,
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@Article{Dembo1982,
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@Article{DeMeMe14,
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@InCollection{DeNa12,
  author    = {Deguchi, K. and Nagata, M.},
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@Book{Dennis96,
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@Book{devnmap,
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@Article{DhoZhi13,
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@Book{Diacu96,
  Title                    = {Celestial Encounters: The Origins of Chaos and Stability},
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  Booktitle                = {Statistical and Particle Physics},
  Year                     = {1983},
  Address                  = {Edinburgh},
  Editor                   = {K. C. Bowler and A. J. McKane},
  Organization             = {Proc. 26\textsuperscript{th} Scottish Universities Summer School in Physics},
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  Publisher                = {SUSSP Publ.}
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@Article{FeKaSh82,
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  Journal                  = {Physica D},
  Year                     = {1982},
  Pages                    = {370--386},
  Volume                   = {5},
  DOI                      = {10.1016/0167-2789(82)90030-6}
}

@Article{FelCou98,
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  Pages                    = {3873--3889},
  Volume                   = {57},
  DOI                      = {10.1103/PhysRevD.57.3873}
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@Article{FelsOlver98,
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  Pages                    = {161--213},
  Volume                   = {51}
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@Article{FelsOlver99,
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  Pages                    = {127--208},
  Volume                   = {55}
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@Techreport{fftwmanual,
  Title                    = {{FFT}W 3.0 User's Manual},
  Author                   = {M. Frigo and S.G. Johnson},
  Institution              = {fftw.org},
  Year                     = {2006},
  Publisher                = {M. Frigo and S.G. Johnson},
  URL                      = {http://www.fftw.org/fftw3.pdf}
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@Article{FKYY83,
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@Article{focusPOT,
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  Year                     = {2013},
  Pages                    = {1--4},
  Volume                   = {726},
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@Article{Foiasks88,
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  Pages                    = {197},
  Volume                   = {67}
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@Article{FowlerCLE82,
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  volume  = {4},
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@Article{FowlerCLE83,
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@Article{franzosi-2004,
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  pages   = {036218},
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}

@Article{FrCv11,
  Title                    = {Reduction of continuous symmetries of chaotic flows by the method of slices},
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  Year                     = {2012},
  Pages                    = {2074--2084},
  Volume                   = {17},
  DOI                      = {10.1016/j.cnsns.2011.07.007}
}

@InProceedings{Fried87,
  Title                    = {Lefschetz formulas for flows},
  Author                   = {Fried, D.},
  Booktitle                = {The Lefschetz Centennial Conference},
  Year                     = {1987},
  Address                  = {Providence, RI},
  Editor                   = {D. Sundararaman and A. Verjovsky and S. Gitler},
  Pages                    = {41--43},
  Publisher                = {AMS},
  Volume                   = {58},
  DOI                      = {10.1090/conm/058.3}
}

@Article{Frigo05,
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  pages   = {216--231},
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}

@Book{frisch,
  Title                    = {Turbulence},
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  Year                     = {1996},
  Address                  = {Cambridge, UK}
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@Misc{FWweb,
  Title                    = {Exact coherent structures},
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  Year                     = {2006},
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@Book{GaQu95,
  Title                    = {{Lyapunov} Matrix Equation in System Stability and Control},
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  Year                     = {1995},
  Address                  = {New York}
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  abstract = {A trace formula is derived for the
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      differential equations describing noisy time-continuous
      nonlinear dynamical systems. In the weak-noise limit, the
      trace formula provides estimations of the eigenvalues of
      the Fokker-Planck operator on the basis of the
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      the effect of noise on a periodic orbit is given in terms
      of the period and the derivative of the period with
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  doi      = {10.1023/A:1013167928166},
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  Pages                    = {74--94},
  Volume                   = {51},
  Keywords                 = {dynamical systems, Koopman}
}

@Book{gatermannHab,
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@Book{gauge,
  title     = {Chaos and Gauge Field Theory},
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@Unpublished{GGJLF10,
  Title                    = {Connecting curves for dynamical systems},
  Author                   = {R. Gilmore and J.-M. Ginoux and T. Jones and C. Letellier and U. S. Freitas},
  Note                     = {\arXiv{1003.1703}},
  Year                     = {2010}
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@Unpublished{GHC08a,
  Title                    = {Periodic orbits of plane {Couette} flow},
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  Note                     = {In preparation},
  Year                     = {2010}
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@Article{GHCV08,
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  volume  = {621},
  pages   = {365--376},
  doi     = {10.1017/S0022112008005065},
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@Article{GHCW07,
  Title                    = {Visualizing the geometry of state space in plane {Couette} flow},
  Author                   = {J. F. Gibson and J. Halcrow and P. Cvitanovi{\'c}},
  Journal                  = {J. Fluid Mech.},
  Year                     = {2008},
  Pages                    = {107--130},
  Volume                   = {611},
  DOI                      = {10.1017/S002211200800267X}
}

@Article{GiacoOtto05,
  Title                    = {New bounds for the {Kuramoto-Sivashinsky} equation},
  Author                   = {L. Giacomelli and F. Otto},
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  Year                     = {2005},
  Pages                    = {297--318},
  Volume                   = {58}
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@Article{GibMcCLE82,
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  year    = {1982},
  volume  = {5},
  pages   = {108--122},
  doi     = {10.1016/0167-2789(82)90053-7},
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@Article{GibMcCLE83,
  Title                    = {The real and complex {Lorenz} equations},
  Author                   = {Gibbon, J. D. and McGuinness, M. J.},
  Journal                  = {Proc. R. Irish. Acad., Ser. A},
  Year                     = {1983},
  Pages                    = {17--22},
  Volume                   = {83}
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@Article{GiBr13,
  Title                    = {Spanwise-localized solutions of planar shear flows},
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  Journal                  = {J. Fluid Mech.},
  Year                     = {2014},
  Note                     = {\arXiv{1304.6323}},
  Pages                    = {25--61},
  Volume                   = {745},
  DOI                      = {10.1017/jfm.2014.89}
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@Techreport{GibsonMovies,
  Title                    = {Movies of plane {Couette}},
  Author                   = {J. F. Gibson and P. Cvitanovi{\'c}},
  Institution              = {Georgia Inst. of Technology},
  Year                     = {2015},
  Publisher                = {Center for Nonlinear Science},
  URL                      = {http://ChaosBook.org/tutorials}
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@Phdthesis{GibsonPhD,
  Title                    = {Dynamical systems models of wall-bounded, shear-flow turbulence},
  Author                   = {J. F. Gibson},
  School                   = {Cornell University},
  Year                     = {2002}
}

@Article{GiChLiPo12,
  Title                    = {Covariant {Lyapunov} vectors},
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  Year                     = {2013},
  Pages                    = {254005},
  Volume                   = {46},
  DOI                      = {10.1088/1751-8113/46/25/254005}
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@Unpublished{GiLe04,
  Title                    = {Existence of relative periodic orbits near relative equilibria},
  Author                   = {Ginzburg, V. and {Lerman}, E.},
  Note                     = {\arXiv{math/0402264}},
  Year                     = {2004}
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@Article{ginelli-2007-99,
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@Article{GioAdr98,
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  Pages                    = {3864},
  Volume                   = {81},
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@Article{GLEM07,
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  volume  = {79},
  pages   = {016305},
  note    = {\arXiv{0704.3781}},
  doi     = {10.1103/physreve.79.016305},
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@Book{glend94,
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  Publisher                = {Cambridge Univ. Press},
  Year                     = {1994},
  Address                  = {Cambridge},
  Key                      = {QA372 1994}
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@Book{GL-Gil07b,
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  Annote                   = {resonant response leading to algebraic growth for small times. It occurs when the eigenvalues of the Orr-Sommerfeld equation coincide with the eigenvalues of the homogeneous operator for the streamwise velocity component. The resonance has been investigated numerically for plane Couette flow. The aspect ratio of the most rapidly growing wave increases with the Reynolds number, with its spanwise wavelength approaching a constant value of about 3 channel heights.}
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@Techreport{Halc04,
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  Year                     = {2004},
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@Phdthesis{HalcrowThesis,
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  Year                     = {2008},
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@Book{haleinf84,
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@Article{HAMANE05,
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@Book{hamer,
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@Book{hao90b,
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@Book{hao98,
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@InProceedings{HDB92,
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@Book{henonrtb2,
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  title   = {Very large-scale motion in the outer layer},
  journal = {Phys. Fluids},
  year    = {1999},
  volume  = {11},
  pages   = {417--422},
  doi     = {10.1063/1.869889},
}

@Article{kirby_reconstructing_1992,
  author   = {M. Kirby and D. Armbruster},
  title    = {Reconstructing phase space from {PDE} simulations},
  journal  = {Z. Angew. Math. Phys.},
  year     = {1992},
  volume   = {43},
  pages    = {999--1022},
  abstract = {We propose the {Karhunen-Lo\'eve} {(K-L)} decomposition as a tool
    to analyze complex spatio-temporal structures in {PDE} simulations
    in terms of concepts from dynamical systems theory. Taking the {Kuramoto-Sivashinsky}
    equation as a model problem we discuss the {K-L} decomposition for
    4 different values of its bifurcation parameter a. We distinguish
    two modes of using the {K-L} decomposition: As an analytic and synthetic
    tool respectively. Using the analytic mode we find unstable fixed
    points and stable and unstable manifolds in a parameter regime with
    structurally stable homoclinic orbits (a=17.75). Choosing the data
    for a {K-L} analysis carefully by restricting them to certain burst
    events, we can analyze a more complicated intermittent regime at
    a=68. We establish that the spatially localized oscillations around
    a so called `strange' fixed point which are considered as fore-runners
    of spatially concentrated zones of turbulence are in fact created
    by a very specific limit cycle (a=83.75) which, for a=87, bifurcates
    into a modulated traveling wave. Using the {K-L} decomposition synthetically
    by determining an optimal Galerkin system, we present evidence that
    the {K-L} decomposition systematically destroys dissipation and leads
    to blow up solutions.},
  doi      = {10.1007/bf00916425},
}

@Inproceedings{KK05,
  author    = {G. Kawahara and S. Kida and M. Nagata},
  title     = {Unstable periodic motion in plane {Couette} system: {The} skeleton of turbulence},
  booktitle = {{One Hundred Years of Boundary Layer Research}},
  year      = {2005},
  address   = {Dordrecht},
  publisher = {Kluwer},
  doi       = {10.1007/978-1-4020-4150-1_40},
}

@Article{KKR71,
  author  = {Kim, H.T. and Kline, S.J. and Reynolds, W.C.},
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  volume  = {50},
  pages   = {133--160},
  doi     = {10.1017/s0022112071002490},
}

@Article{kleinman10,
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  journal = {Phys. Fluids},
  year    = {2010},
  volume  = {22},
  pages   = {116103},
  doi     = {10.1063/1.3507317},
}

@InProceedings{Kleiser80,
  author       = {L. Kleiser and U. Schumann},
  title        = {Treatment of incompressibility and boundary conditions in 3-{D} numerical spectral simulations of plane channel flows},
  booktitle    = {Proc. 3\textsuperscript{rd} GAMM Conf. Numerical Methods in Fluid Mechanics},
  year         = {1980},
  editor       = {E. Hirschel},
  pages        = {165--173},
  address      = {Viewweg, Braunschweig},
  organization = {GAMM},
  doi          = {10.1007/978-3-322-86146-7_17},
}

@Article{klewicki95,
  Title                    = {Viscous sublayer flow visualizations at {$R_{\theta} \sim 1500000$}},
  Author                   = {Klewicki, J. C. and Metzger, M. M. and Kelner, E. and E. M. Thurlow},
  Journal                  = {Phys. Fluids},
  Year                     = {1995},
  Pages                    = {857},
  Volume                   = {7}
}

@Article{KLH94,
  author  = {G. Kreiss and A. Lundbladh and D. S. Henningson},
  title   = {Bounds for threshold amplitudes in subcritical shear flows},
  journal = {J. Fluid Mech.},
  year    = {1994},
  volume  = {270},
  pages   = {175--198},
  doi     = {10.1017/s0022112094004234},
}

@Article{KMM87,
  author  = {J. Kim and P. Moin and R. Moser},
  title   = {Turbulence statistics in fully developed channel flow at low {Reynolds} number},
  journal = {J. Fluid Mech.},
  year    = {1987},
  volume  = {177},
  pages   = {133--166},
  doi     = {10.1017/s0022112087000892},
}

@Article{knobloch2015,
  author =  {Knobloch, E.},
  title =   {Spatial localization in dissipative systems},
  journal = {Ann. Math.. Rev. Condens. Matter Phys.},
  year =    {2015},
  volume =  {6},
  pages =   {325--359},
  doi =     {10.1146/annurev-conmatphys-031214-014514}
}

@Article{KNSks90,
  author  = {I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel},
  title   = {Back in the saddle again: a computer assisted study of the {Kuramoto-Sivashinsky} equation},
  journal = {SIAM J. Appl. Math.},
  year    = {1990},
  volume  = {50},
  pages   = {760--790},
  doi     = {10.1137/0150045},
}

@Article{Koch89,
  author  = {W. Koch},
  title   = {Direct resonances in {Orr-Sommerfeld} problems},
  journal = {Acta Mechanica},
  year    = {1986},
  volume  = {59},
  pages   = {11},
  doi     = {10.1007/bf01177057},
}

@Article{kolm91,
  author  = {A. N. Kolmogorov},
  title   = {The local structure of turbulence in incompressible viscous fluid for very large {Reynolds} numbers},
  journal = {Proc. R. Soc. Lond. A},
  year    = {1991},
  volume  = {434},
  pages   = {9--13},
}

@Article{kook89,
  author  = {H-T. Kook and J. D. Meiss},
  title   = {Periodic orbits for reversible symplectic mappings},
  journal = {Physica D},
  year    = {1989},
  volume  = {35},
  pages   = {65--86},
  doi     = {10.1016/0167-2789(89)90096-1},
}

@Article{kooknewt,
  author   = {H-T Kook and J. D. Meiss},
  title    = {Application of {N}ewton's method to {L}agrangian mappings},
  journal  = {Physica D},
  year     = {1989},
  volume   = {36},
  pages    = {317--326},
  abstract = {An algorithm of {N}ewton's method is presented to find
    periodic orbits for {L}agrangian mappings. The method is based on
    block-diagonalization of the Hessian Matrix of the action function.},
  doi      = {10.1016/0167-2789(89)90087-0},
}

@InProceedings{KraOsi00,
  author    = {B. Krauskopf and H. M. Osinga},
  title     = {Investigating torus bifurcations in the forced {Van der Pol} oscillator},
  booktitle = {{Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems}},
  year      = {2000},
  editor    = {E. Doedel and L. Tuckerman},
  pages     = {199--208},
  address   = {New York},
  publisher = {Springer},
  isbn      = {9781461212089},
}

@Article{Kras04,
  author   = {I. Krasikov and G. J. Rodgers and C. E. Tripp},
  title    = {Growing random sequences},
  journal  = {J. Phys. A},
  year     = {2004},
  volume   = {37},
  pages    = {2365--2370},
  abstract = {The random sequence has been generalized both in the form of addition and in the probability distributions. Critical parameter values are discovered around which the system has very different behavior.},
  doi      = {10.1088/0305-4470/37/6/026},
}

@Article{KreEck12,
  Title                    = {Periodic orbits near onset of chaos in plane {Couette} flow},
  Author                   = {T. Kreilos and B. Eckhardt},
  Journal                  = {Chaos},
  Year                     = {2012},
  Pages                    = {047505},
  Volume                   = {22},
  DOI                      = {10.1063/1.4757227}
}

@PhdThesis{Kreilos14,
  author = {Kreilos, T.},
  title =  {{Turbulence Transition in Shear Flows and Dynamical Systems Theory}},
  school = {Philipps-Universit{\"a}t Marburg},
  year =   {2014},
  note =   {http://archiv.ub.uni-marburg.de/diss/z2014/0356/pdf/dtk.pdf}
}

@Article{KrGiSc16,
  Title                    = {Localized travelling waves in the asymptotic suction boundary layer},
  Author                   = {T. Kreilos and J. F. Gibson and T. M. Schneider},
  Journal                  = {J. Fluid Mech.},
  Year                     = {2016},
  Pages                    = {R3},
  Volume                   = {795},
  DOI                      = {10.1017/jfm.2016.242}
}

@Article{KRSR67,
  author  = {Kline, S. J. and {R}eynolds, W. C. and Schraub, F. A. and Runstadler, P. W.},
  title   = {The structure of turbulent boundary layers},
  journal = {J. Fluid Mech.},
  year    = {1967},
  volume  = {30},
  pages   = {741--773},
  doi     = {10.1017/s0022112067001740},
}

@Article{kruskal62,
  author   = {M. Kruskal},
  title    = {Asymptotic theory of {Hamiltonian} and other systems with all solutons nearly periodic},
  journal  = {J. Math. Phys.},
  year     = {1962},
  volume   = {3},
  pages    = {806},
  abstract = {Asymptotic expansion to all orders. A systematic way to construct the adiabatic invariants.},
  doi      = {10.1063/1.1724285},
}

@Article{kruskal65,
  author =  {N. J. Zabusky and M. D. Kruskal},
  title =   {Interaction of ``solitons'' in a collisionless plasma and the recurrence of initial states},
  journal = {Phys. Rev. Lett.},
  year =    {1965},
  volume =  {15},
  pages =   {240--243},
  doi =     {10.1103/PhysRevLett.15.240}
}

@Article{KrVeSchEck12,
  Title                    = {Edge states for the turbulence transition in the asymptotic suction boundary layer},
  Author                   = {Kreilos, T. and Veble, G. and Schneider, T. M. and Eckhardt, B.},
  Journal                  = {J. Fluid M.},
  Year                     = {2013},
  Note                     = {\arXiv{1209.0593}},
  Pages                    = {100--122},
  Volume                   = {726},
  DOI                      = {10.1017/jfm.2013.212}
}

@Article{KrZaEc14,
  author  = {T. Kreilos and S. Zammert and B. Eckhardt},
  title   = {Comoving frames and symmetry-related motions in parallel shear flows},
  journal = {J. Fluid Mech.},
  year    = {2014},
  volume  = {751},
  pages   = {685--697},
  doi     = {10.1017/jfm.2014.305},
}

@Article{ks04com,
  author   = {A.-K. Kassam and L. N. Trefethen},
  title    = {Fourth-order time stepping for stiff {PDE}s},
  journal  = {SIAM J. Sci. Comp.},
  year     = {2005},
  volume   = {26},
  pages    = {1214--1233},
  abstract = {A contour integral method is presented to evaluate
    accurately the matrix functions with removable singularities.},
  doi      = {10.1137/s1064827502410633},
}

@Article{kschang86,
  author   = {H.-C. Chang},
  title    = {Travelling waves on fluid interfaces: {N}ormal form analysis of the {K}uramoto-{S}ivashisky equation},
  journal  = {Phys. Fluids},
  year     = {1986},
  volume   = {29},
  pages    = {3142},
  abstract = {The lowest order normal form analysis is give to the KSe and the approximate analytical wavelength-amplitude and wavespeed-amplitude relation are given. They agree quite well with the numerical calculations.},
  doi      = {10.1063/1.865965},
}

@Article{ksgreene88,
  author  = {J. M. Greene and J.-S. Kim},
  title   = {The steady states of the {Kuramoto-Sivashinsky} equation},
  journal = {Physica D},
  year    = {1988},
  volume  = {33},
  pages   = {99--120},
  doi     = {10.1016/s0167-2789(98)90013-6},
  abtract = {Use Fourier modes to discuss the energy transport from long wavelength modes to short ones. The generation of steady states with periodic boundary condition is studied systematically with a bifurcation analysis. Their stability is investigated and the scaling in the limit of large system size is presented.},
}

@Article{ksgrim91,
  Title                    = {The non-existence of a certain class of travelling wave solutions of the {Kuramoto-Sivashinsky} equation},
  Author                   = {R. Grimshaw and A. P. Hooper},
  Journal                  = {Physica D},
  Year                     = {1991},
  Pages                    = {231--238},
  Volume                   = {50},
  DOI                      = {10.1016/0167-2789(91)90177-B}
}

@Article{ksham95,
  author  = {S. Bouquet},
  title   = {Hamiltonian structure and integrability of the stationary {Kuramoto-Sivashinsky} equation},
  journal = {J. Math. Phys.},
  year    = {1995},
  volume  = {36},
  pages   = {1242},
  doi     = {10.1063/1.531118},
}

@Article{kshooper88,
  Title                    = {Travelling wave solutions of the {Kuramoto-Sivashinsky} equation},
  Author                   = {A. P. Hooper and R. Grimshaw},
  Journal                  = {Wave Motion},
  Year                     = {1988},
  Pages                    = {405--420},
  Volume                   = {10},
  DOI                      = {10.1016/0165-2125(88)90045-5}
}

@Article{kskent92,
  author   = {P. Kent and J. Elgin},
  title    = {Travelling-waves of the {Kuramoto-Sivashinsky} equation: period-multiplying bifurcations},
  journal  = {Nonlinearity},
  year     = {1992},
  volume   = {5},
  pages    = {899--919},
  abstract = {Some properties of the KSe are discussed. A unified
    argument about the connections and periodic orbits is given. In
    particular, the K-bifurcation of the system is conjectured to arises
    in the 1:n resonances of a fixed point.},
  doi      = {10.1088/0951-7715/5/4/004},
}

@Article{kstroy89,
  Title                    = {The existence of steady solutions of the {Kuramoto-Sivashinsky} equation},
  Author                   = {W. C. Troy},
  Journal                  = {J. Diff. Eqn.},
  Year                     = {1989},
  Pages                    = {269--313},
  Volume                   = {82},
  Abstract                 = {For c=1, the existence of two periodic orbits and two
                 heteroclinic orbits are proved and suggestions for
                 proving the existence of more complicated orbits are
                 given.},
  DOI                      = {10.1016/0022-0396(89)90134-4}
}

@Article{kstroy92,
  Title                    = {Steady solutions of the {Kuramoto-Sivashinsky} equation for small wave speed},
  Author                   = {J. Jones and W. C. Troy and A. D. MacGillivary},
  Journal                  = {J. Diff. Eqn.},
  Year                     = {1992},
  Pages                    = {28--55},
  Volume                   = {96},
  Abstract                 = {For small c>0, at least one odd periodic solution
                 exists. For any c>0, there existes no monotone bounded
                 travelling wave solution. The normal form analysis may
                 not be able to give correct results to all orders.},
  DOI                      = {10.1016/0022-0396(92)90143-B}
}

@Article{ksyang97,
  Title                    = {On travelling-wave solutions of the {Kuramoto-Sivashinsky} equation},
  Author                   = {T.-S. Yang},
  Journal                  = {Physica D},
  Year                     = {1997},
  Pages                    = {25--42},
  Volume                   = {110},
  DOI                      = {10.1016/S0167-2789(97)00121-8}
}

@Article{KT84,
  author  = {L. Kadanoff and C. Tang},
  title   = {Escape rate from strange repellers},
  journal = {Proc. Natl. Acad. Sci. USA},
  year    = {1984},
  volume  = {81},
  pages   = {1276--1279},
  doi     = {10.1073/pnas.81.4.1276},
}

@Article{ku,
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  journal  = {Progr. Theor. Phys.},
  year     = {1976},
  volume   = {55},
  pages    = {365},
  abstract = {The KSe is derived through the CGLe near the bifurcation
    point. Some scaling argument is used to sort out the domiant terms.},
  doi      = {10.1143/ptp.55.356},
}

@Article{KuKuKnMo04,
  Title                    = {Catastrophe theoretic classification of nonlinear oscillators},
  Author                   = {A. Yu. Kuznetsova and A. P. Kuznetsov and C. Knudsen and E. Mosekilde},
  Journal                  = {Int. J. Bif. Chaos},
  Year                     = {2004},
  Pages                    = {1241--1266},
  Volume                   = {14},
  DOI                      = {10.1142/S0218127404009995}
}

@Book{kura84tur,
  Title                    = {Chemical Oscillations, Waves and Turbulence},
  Author                   = {Y. Kuramoto},
  Publisher                = {Springer},
  Year                     = {1984},
  Address                  = {Berlin}
}

@Article{kuturb78,
  author   = {Y. Kuramoto},
  title    = {Diffusion-induced chaos in reaction systems},
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  year     = {1978},
  volume   = {64},
  pages    = {346--367},
  abstract = {Phase turbulence and amplitude turbulence
                are named and distinguished from a dynamical systems point
                of view. The prototyped equations are derived.},
  doi      = {10.1143/ptps.64.346},
}

@Article{KZ,
  author  = {L. Kleiser and T. A. Zang},
  title   = {Numerical simulation of transition in wall-bounded shear flows},
  journal = {Ann. Rev. Fluid Mech.},
  year    = {1991},
  volume  = {23},
  pages   = {495--537},
  doi     = {10.1146/annurev.fluid.23.1.495},
}

@Book{Lax02,
  title     = {{Functional Analysis}},
  publisher = {Wiley},
  year      = {2002},
  author    = {P. D. Lax},
  address   = {New York},
  isbn      = {9781118626740},
}

@Article{Lagha07,
  Title                    = {Modeling of plane {Couette} flow. {II. On} the spreading of a turbulent spot},
  Author                   = {Lagha, M.},
  Journal                  = {Phys. Fluids},
  Year                     = {2007},
  Pages                    = {4108},
  Volume                   = {19},
  DOI                      = {10.1063/1.2793143}
}

@Article{Lagha07b,
  Title                    = {Modeling transitional plane {Couette} flow},
  Author                   = {M. Lagha and P. Manneville},
  Journal                  = {Eur. Phys. J. B},
  Year                     = {2007},
  Pages                    = {433--447},
  Volume                   = {58},
  DOI                      = {10.1140/epjb/e2007-00243-y}
}

@Article{LagMann07,
  Title                    = {Modeling of plane {Couette} flow. {I. Large} scale flow around turbulent spots},
  Author                   = {Lagha, M. and {Manneville}, P.},
  Journal                  = {Phys. Fluids},
  Year                     = {2007},
  Pages                    = {4105},
  Volume                   = {19},
  DOI                      = {10.1063/1.2768946}
}

@Article{LaKo89,
  author  = {D. P. Lathrop and E. J. Kostelich},
  title   = {Characterization of an experimental strange attractor by periodic orbits},
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  year    = {1989},
  volume  = {40},
  pages   = {4028--4031},
  doi     = {10.1103/physreva.40.4028},
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@Article{lamb98,
  author  = {J. S. W. Lamb and J. A. G. Roberts},
  title   = {Time reversal symmetry in dynamical systems: {A} survey},
  journal = {Physica D},
  year    = {1998},
  volume  = {112},
  pages   = {1},
  doi     = {10.1016/s0167-2789(97)00199-1},
}

@Article{lanCvit07,
  Title                    = {Unstable recurrent patterns in {Kuramoto-Sivashinsky} dynamics},
  Author                   = {Lan, Y. and Cvitanovi{\'c}, P.},
  Journal                  = {Phys. Rev. E},
  Year                     = {2008},
  Addendum                 = {\arXiv{0804.2474}},
  Pages                    = {026208},
  Volume                   = {78},
  DOI                      = {10.1103/PhysRevE.78.026208}
}

@Article{landau44,
  author  = {L. D. Landau},
  title   = {On the problem of turbulence},
  journal = {Akad. Nauk. Doklady},
  year    = {1944},
  volume  = {44},
  pages   = {339},
  doi     = {10.1016/b978-0-08-010586-4.50057-2},
}

@Book{Landau59c,
  Title                    = {Quantum Mechanics},
  Author                   = {L.D. Landau and E.M. Lifshitz},
  Publisher                = {Pergamon Press},
  Year                     = {1959},
  Address                  = {Oxford}
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@Article{lanmaw03,
  author  = {Y. Lan and N. Garnier and P. Cvitanovi{\'c}},
  title   = {Stationary modulated-amplitude waves in the 1{D} complex {Ginzburg-Landau} equation},
  journal = {Physica D},
  year    = {2004},
  volume  = {188},
  pages   = {193--212},
  doi     = {10.1016/s0167-2789(03)00289-6},
}

@Phdthesis{LanThesis,
  Title                    = {Dynamical systems approach to $1-d$ spatiotemporal chaos -- {A} cyclist's view},
  Author                   = {Y. Lan},
  School                   = {School of Physics, Georgia Inst. of Technology},
  Year                     = {2004},
  Address                  = {Atlanta},
  Note                     = {{\tt ChaosBook.org/projects/theses.html}}
}

@Article{lanVar1,
  author  = {Y. Lan and P. Cvitanovi{\'c}},
  title   = {Variational method for finding periodic orbits in a general flow},
  journal = {Phys. Rev. E},
  year    = {2004},
  volume  = {69},
  pages   = {016217},
  note    = {\arXiv{nlin.CD/0308008}},
  doi     = {10.1103/physreve.69.016217},
}

@Book{lapack,
  Title                    = {{LAPACK} Users' Guide},
  Author                   = {Anderson, E. and Bai, Z. and Bischof, C. and Blackford, S. and Demmel, J. and Dongarra, J. and Du Croz, J. and Greenbaum, A. and Hammarling, S. and McKenney, A. and Sorensen, D.},
  Publisher                = {Society for Industrial and Applied Mathematics},
  Year                     = {1999},
  Address                  = {Philadelphia},
  Edition                  = {Third},
  ISBN                     = {0-89871-447-8 (paperback)}
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  year    = {1992},
  volume  = {46},
  pages   = {6390},
  doi     = {10.1103/physreva.46.6390},
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@Book{latt,
  Title                    = {Quantum Fields on the Computer},
  Author                   = {M. Creutz},
  Publisher                = {World Scientific},
  Year                     = {1992},
  Address                  = {Singapore}
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@Article{lattref,
  author   = {B. I. Henry and S. D. Watt and S. L. Wearne},
  title    = {A lattice refinement scheme for finding periodic orbits},
  journal  = {Anziam J.},
  year     = {2000},
  volume   = {42E},
  pages    = {C735--C751},
  abstract = {A lattice refinement scheme is proposed based on the
    linear approximation of a given map. Jacobian matrix is not
    calculated directly but inferred from the neighboring points.},
  doi      = {10.21914/anziamj.v42i0.619},
}

@Book{Lau05,
  Title                    = {Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World},
  Author                   = {Lautrup, B.},
  Publisher                = {{CRC Press}},
  Year                     = {2011},
  Address                  = {Boca Raton, FL}
}

@Article{lau92,
  Title                    = {The cocoon bifurcations in three-dimensional systems with two fixed points},
  Author                   = {Y.-T. Lau},
  Journal                  = {Int. J. Bifur. Chaos},
  Year                     = {1992},
  Pages                    = {543--558},
  Volume                   = {2},
  DOI                      = {10.1142/S0218127492000690}
}

@Incollection{lauw86,
  Title                    = {Two-dimensional iterative maps},
  Author                   = {H. A. Lauwerier},
  Booktitle                = {Chaos},
  Publisher                = {Princeton Univ. Press},
  Year                     = {1986},
  Address                  = {Princeton},
  Editor                   = {V. Holden},
  Pages                    = {58--96},
  ISBN                     = {9780691610535}
}

@Article{LCC06,
  Title                    = {Variational method for locating invariant tori},
  Author                   = {Y. Lan and C. Chandre and P. Cvitanovi{\'c}},
  Journal                  = {Phys. Rev. E},
  Year                     = {2006},
  Note                     = {\arXiv{nlin.CD/0508026}},
  Pages                    = {046206},
  Volume                   = {74}
}

@Article{lee01,
  author  = {K. Lee and L. Cortelezzi and J. Kim and J. L. Speyer},
  title   = {Application of reduced order controller to turbulent flows for drag reduction},
  journal = {Phys. Fluids},
  year    = {2001},
  volume  = {13},
  pages   = {1321},
  doi     = {10.1063/1.1359420},
}

@Article{Lega:holes,
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  journal = {Physica D},
  year    = {2001},
  volume  = {269},
  pages   = {152--153},
  doi     = {10.1016/s0167-2789(01)00174-9},
}

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  author    = {A. Lundbladh and and S. Henningson and S. C. Reddy},
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  Title                    = {Applications of {Lie} Groups to Differential Equations},
  Author                   = {P. J. Olver},
  Publisher                = {Spring-Verlag},
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  Series                   = {GTM},
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@Article{LiHuCa10,
  author  = {Libert, A.-S. and Hubaux, C. and Carletti, T.},
  title   = {{The Global Symplectic Integrator}: an efficient tool for stability studies of dynamical systems. {Application} to the {Kozai} resonance in the restricted three-body problem},
  journal = {Mon. Not. R. Astron. Soc.},
  year    = {2011},
  volume  = {414},
  pages   = {659--667},
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@Article{LiMa83,
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  pages   = {97--135},
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  Title                    = {The Theory of Hydrodynamic Stability},
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@Unpublished{LipCvi07,
  Title                    = {Optimal resolution of the state space of a chaotic flow in presence of noise},
  Author                   = {D. Lippolis and P. Cvitanovi\'c},
  Note                     = {In preparation, 2011}
}

@Article{LipCvi08,
  Title                    = {How well can one resolve the state space of a chaotic map?},
  Author                   = {D. Lippolis and P. Cvitanovi\'c},
  Journal                  = {Phys. Rev. Lett.},
  Year                     = {2010},
  Note                     = {\arXiv{0902.4269}},
  Pages                    = {014101},
  Volume                   = {104},
  DOI                      = {10.1103/PhysRevLett.104.014101}
}

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  author = {J.-G. Liu and J. Liu and R. L. Pego},
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  year   = {2005},
  url    = {https://arxiv.org/abs/math/0502549},
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@Article{lixigraham06,
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  journal = {J. Fluid Mech},
  year    = {2006},
  volume  = {565},
  pages   = {353--362},
  doi     = {10.1017/s0022112006002138},
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@Article{lixigraham07,
  author  = {W. Li and M.D. Graham},
  title   = {Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow},
  journal = {Phys. Fluids},
  year    = {2007},
  volume  = {19},
  number  = {083101},
  pages   = {1--15},
  doi     = {10.1063/1.2748443},
}

@Incollection{Llibre15,
  Title                    = {The averaging theory for computing periodic orbits},
  Author                   = {Llibre, J.},
  Booktitle                = {{Central Configurations, Periodic Orbits, and Hamiltonian Systems}},
  Publisher                = {Springer},
  Year                     = {2015},
  Address                  = {Basel},
  Pages                    = {1--104},
  Series                   = {Advanced Courses in Mathematics - CRM Barcelona},
  DOI                      = {10.1007/978-3-0348-0933-7_1},
  ISBN                     = {978-3-0348-0932-0},
  Language                 = {English}
}

@Book{lls,
  Title                    = {Systems of Quasilinear Equations and Their Applications to Gas Dynamics},
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  Year                     = {1983},
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@Book{LM94,
  Title                    = {Chaos, Fractals, and Noise; Stochastic Aspects of Dynamics},
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  Publisher                = {Springer},
  Year                     = {1994},
  Address                  = {Berlin}
}

@Article{LoCaCoPeGo11,
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  Author                   = {Lombardi, M. and Caulfield, C.P. and Cossu, C. and Pesci, A.I. and Goldstein, R.E.},
  Journal                  = {J. Fluid Mech.},
  Year                     = {2011},
  Pages                    = {184--206},
  Volume                   = {671},
  DOI                      = {10.1017/S0022112010005574}
}

@Article{lop05rel,
  Title                    = {Relative periodic solutions of the {Complex Ginzburg-Landau} equation},
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  Journal                  = {SIAM J. Appl. Dyn. Syst.},
  Year                     = {2006},
  Note                     = {\arXiv{nlin/0408018}},
  Pages                    = {1042--1075},
  Volume                   = {4},
  DOI                      = {10.1137/040618977}
}

@Article{lopez98,
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  doi =     {S10.1006/jcph.1997.5872}
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@Article{LopMar00,
  Title                    = {Determining the self-rotation number following a {Neimark-Sacker} bifurcation in the periodically forced {Taylor-Couette} flow},
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  Year                     = {2000},
  Pages                    = {61--74},
  Volume                   = {51},
  DOI                      = {10.1007/PL00001507},
  Keywords                 = {Key words. Floquet theory, self-rotation number, parametric excitation, quasiperiodic flow, Taylor--Couette flow.}
}

@Article{LopMarq05,
  author  = {J. M. Lopez and F. Marques},
  title   = {Finite aspect ratio {Taylor-Couette} flow: {Shil'nikov} dynamics of 2-tori},
  journal = {Physica D},
  year    = {2005},
  volume  = {211},
  pages   = {168--191},
  doi     = {10.1016/j.physd.2005.08.011},
}

@Article{LorentzDiff,
  Title                    = {Transport properties of the {Lorentz} gas in terms of periodic orbits},
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  Year                     = {1995},
  Note                     = {\arXiv{chao-dyn/9312003}},
  Pages                    = {113--120},
  Volume                   = {6},
  DOI                      = {10.1016/0960-0779(95)80018-C}
}

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  Journal                  = {SIAM J. Sci. Comput.},
  Year                     = {2012},
  Addendum                 = {\arXiv{1006.0428}},
  Pages                    = {B24--B43},
  Volume                   = {34},
  DOI                      = {10.1137/100784734}
}

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  Title                    = {Spatiotemporal dynamics in {2D Kolmogorov} flow over large domains},
  Author                   = {Lucas, D. and Kerswell, R. R.},
  Journal                  = {J. Fluid Mech.},
  Year                     = {2014},
  Note                     = {\arXiv{1308.3356}},
  Pages                    = {518--554},
  Volume                   = {750},
  DOI                      = {10.1017/jfm.2014.270}
}

@Article{LucKer14,
  Title                    = {Recurrent flow analysis in spatiotemporally chaotic 2-dimensional {Kolmogorov} flow},
  Author                   = {Lucas, D. and Kerswell, R. R.},
  Journal                  = {Phys. Fluids},
  Year                     = {2015},
  Note                     = {\arXiv{1406.1820}},
  Pages                    = {518--554},
  Volume                   = {27},
  DOI                      = {10.1063/1.4917279}
}

@Proceedings{Lumley_1989,
  Title                    = {Whither Turbulence? Turbulence at the Crossroads: Proceedings, Workshop Cornell U., {March} 22-24, 1989},
  Year                     = {1990},
  Address                  = {New York},
  Editor                   = {Lumley, J. L.},
  Publisher                = {Springer},
  Volume                   = {357},
  DOI                      = {10.1007/3-540-52535-1}
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@Article{lumley67,
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  Author                   = {A. Lundbladh and A. Johansson},
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  Year                     = {1991},
  Pages                    = {499--516},
  Volume                   = {229}
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  Year                     = {1968},
  Pages                    = {197--221},
  Volume                   = {19},
  Abstract                 = {The Liapunov characteristic numbers are defined by investigating the limiting volume of infinitesimal size. Their existence is proved by construction and application of the ergodic theorem. At a.e. every point, the tangent space is divided into stable and unstable subspace corresponding the positive and negative Liapunov numbers. If the eigenvalues are non-degenerate, the Liyapunov vectors are well defined a.e.}
}

@Article{Lyons13,
  author   = {Lyons, L.},
  journal  = {Contemporary Physics},
  title    = {{Bayes and Frequentism}: a particle physicist's perspective},
  year     = {2013},
  pages    = {1--16},
  volume   = {54},
  abstract = {In almost every scientific field, an experiment involves
    collecting data and then analysing it. The analysis stage will often
    consist in trying to extract some physical parameter and estimating
    its uncertainty; this is known as Parameter Determination. An example
    would be the determination of the mass of the top quark, from data
    collected from high energy proton--proton collisions. A different aim
    is to choose between two possible hypotheses. For example, are data
    on the recession speed s of distant galaxies proportional to their
    distance d, or do they fit better to a model where the expansion of
    the Universe is accelerating? There are two fundamental approaches to
    such statistical analyses -- Bayesian and Frequentist. This article
    discusses the way they differ in their approach to probability, and
    then goes on to consider how this affects the way they deal with
    Parameter Determination and Hypothesis Testing. The examples are
    taken from everyday life and from particle physics.},
  doi      = {10.1080/00107514.2012.756312},
}

@Book{lyschm,
  title     = {{{Lyapunov-Schmidt} Methods in Nonlinear Analysis and Applications}},
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  address   = {Berlin},
  doi       = {10.1007/978-94-017-2122-6},
}

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  Title                    = {A low-dimensional model for simulating 3-dimensional cylinder flow},
  Author                   = {X. Ma and G.E. Karniadakis},
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  Year                     = {2002},
  Pages                    = {181--190},
  Volume                   = {458}
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  Title                    = {Measuring shape with topology},
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  Year                     = {2012},
  Pages                    = {073516},
  Volume                   = {53},
  DOI                      = {10.1063/1.4737391}
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  Author                   = {Mallet, J.-L.},
  Publisher                = {Oxford Univ. Press},
  Year                     = {2002},
  Address                  = {Oxford}
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@Book{man90b,
  Title                    = {Dissipative structures and weak turbulence},
  Author                   = {P. Manneville},
  Publisher                = {Academic Press},
  Year                     = {1990},
  Address                  = {Boston}
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@Book{mandel,
  Title                    = {The Fractal Geometry of Nature},
  Author                   = {B. B. Mandelbrot},
  Publisher                = {WH Freeman Co.},
  Year                     = {1982},
  Address                  = {New York},
  ISBN                     = {9780716711865}
}

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  Title                    = {Spots and turbulent domains in a model of transitional plane {Couette} flow},
  Author                   = {P. Manneville},
  Journal                  = {Theor. Comp. Fluid Dyn.},
  Year                     = {2004},
  Pages                    = {169--181},
  Volume                   = {18},
  DOI                      = {10.1007/s00162-004-0142-4}
}

@Article{Mann08,
  Title                    = {Understanding the sub-critical transition to turbulence in wall flows},
  Author                   = {P. Manneville},
  Journal                  = {Pramana},
  Year                     = {2008},
  Pages                    = {1009--1021},
  Volume                   = {70},
  DOI                      = {10.1007/s12043-008-0105-1}
}

@Unpublished{Mann14,
  Title                    = {On the transition to turbulence of wall-bounded flows in general, and plane {Couette} flow in particular},
  Author                   = {Manneville, P.},
  Note                     = {\arXiv{1403.6374}},
  Year                     = {2014}
}

@Article{marcio03,
  Title                    = {Topological characterization of spatial-temporal chaos},
  Author                   = {M. Gameiro and W. Kalies and K. Mischaikow},
  Journal                  = {Phys. Rev. E},
  Year                     = {2004},
  Pages                    = {035203},
  Volume                   = {70},
  Abstract                 = {Use the Betti number to characterize the spatiotemporal
    dynamics of a system. A lyapunov exponent is suggested to measure the
    space as well as temporal complexity of the evolution.},
  DOI                      = {10.1103/PhysRevE.70.035203}
}

@PhdThesis{Marcotte2016,
  author = {C. D. Marcotte},
  title =  {Understanding sustained spiral chaos using non-chaotic solutions of a simple model of atrial excitation},
  school = {Georgia Institute of Technology},
  year =   {2016}
}

@Article{Marcus84,
  Title                    = {Simulation of {Taylor-Couette} flow. {Part} 1. {Numerical} methods and comparison with experiment},
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  Year                     = {1984},
  Pages                    = {45--64},
  Volume                   = {146},
  DOI                      = {10.1017/S0022112084001762}
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@Article{MarLopBla04,
  Title                    = {Bifurcations in systems with {Z2} spatio-temporal and {O(2)} spatial symmetry},
  Author                   = {F. Marques and J. M. Lopez and H. M. Blackburn},
  Journal                  = {Physica D},
  Year                     = {2004},
  Pages                    = {247--276},
  Volume                   = {189},
  DOI                      = {10.1016/j.physd.2003.09.041}
}

@Book{MarRat99,
  Title                    = {Introduction to Mechanics and Symmetry},
  Author                   = {Marsden, J. E. and Ratiu, T. S.},
  Publisher                = {Springer},
  Year                     = {1999},
  Address                  = {New York}
}

@Book{Marsd92,
  Title                    = {Lectures on Mechanics},
  Author                   = {Marsden, J. E.},
  Publisher                = {Cambridge Univ. Press},
  Year                     = {1992},
  Address                  = {Cambridge}
}

@Book{marsdenbb,
  Title                    = {The {H}opf Bifurcation and its Applications},
  Author                   = {J. E. Marsden and M. McCracken},
  Publisher                = {Springer},
  Year                     = {1976}
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  author   = {C. C. Martens and M. J. Davis and G. S. Ezra},
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  title    = {Local Frequency analysis of chaotic motion in multidimensional systems: energy transport and bottlenecks in planar {OCS}},
  year     = {1987},
  pages    = {519},
  volume   = {142},
  abstract = {local frequency analysis is used and action-angle
    coordinate is parametrized by the frequency ratio. The transport
    zones and partial barriers are identified according to the},
  doi      = {10.1016/0009-2614(87)80655-3},
}

@Article{Marw08,
  Title                    = {A historical review of recurrence plots},
  Author                   = {N. Marwan},
  Journal                  = {Eur. Phys. J. Spec. Top.},
  Year                     = {2008},
  Pages                    = {3--12},
  Volume                   = {164},
  DOI                      = {10.1140/epjst/e2008-00829-1}
}

@Article{MaSiRo73,
  Title                    = {Statistical mechanics of classical systems},
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  Pages                    = {423--437},
  Volume                   = {8}
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  pages   = {207--263},
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@Book{MathWalk73,
  Title                    = {Mathematical Methods of Physics},
  Author                   = {J. Mathews and R. L. Walker},
  Publisher                = {W. A. Benjamin},
  Year                     = {1970},
  Address                  = {Reading, MA},
  ISBN                     = {9780805370027}
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  volume  = {261},
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@Article{MC96pha,
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  journal = {Physica D},
  year    = {1996},
  volume  = {96},
  pages   = {30--46},
  abtract = {instability of plane waves},
  doi     = {10.1016/0167-2789(96)00045-0},
}

@Article{McKean74,
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  year    = {1972},
  volume  = {25},
  pages   = {225--246},
  doi     = {10.1002/cpa.3160250302},
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@Article{McKeon04,
  Title                    = {Further observations on the mean velocity in fully-developed pipe flow},
  Author                   = {B. J. McKeon and J. Li and W. Jiang and J. F. Morrison and A. J. Smits},
  Journal                  = {J. Fluid Mech.},
  Year                     = {2004},
  Pages                    = {135--147},
  Volume                   = {501}
}

@Article{Mcoh03,
  author   = {M. van Hecke},
  title    = {Coherent and incoherent structures in systems described by the 1{D CGLE}: {Experiments} and identification},
  journal  = {Physica D},
  year     = {2003},
  volume   = {174},
  pages    = {134},
  abstract = {Give an overview of the most important properties of the
    coherent structures and address the question how such structures may
    be identified in the experimental space-time data set.},
  doi      = {10.1016/s0167-2789(02)00687-5},
}

@Article{mcord86,
  Title                    = {Uniqueness of connection orbits in the equation $y^{(3)}=y^2-1$},
  Author                   = {C. K. McCord},
  Journal                  = {J. Math. Anal. Appl.},
  Year                     = {1986},
  Pages                    = {584--592},
  Volume                   = {114}
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@Book{Mees81,
  title     = {{Dynamics of Feedback Systems}},
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@Article{MEF04,
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@Article{Mehta05,
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@Article{mellibovsky11,
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  volume  = {670},
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@Article{mellibovsky12,
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  year    = {2012},
  volume  = {709},
  pages   = {149--190},
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@Article{MelMes15,
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  Journal                  = {J. Fluid Mech.},
  Year                     = {2015},
  Pages                    = {R1},
  Volume                   = {779},
  DOI                      = {10.1017/jfm.2015.440}
}

@Article{mercader2010,
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@Article{Meseguer2009,
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  Journal                  = {Phys. Rev. E},
  Year                     = {2009},
  Pages                    = {036309},
  Volume                   = {79},
  DOI                      = {10.1103/PhysRevE.79.036309}
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@Article{MeseguerANM07,
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@Article{MFE04,
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@Article{MFE04b,
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  Year                     = {2005},
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  Volume                   = {4},
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}

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characteristic structure of the system, and the local minima of
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the period.},
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}

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@Phdthesis{MitchellThesis13,
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  Author                   = {Mitchell Jr., R.},
  School                   = {School of Physics, Georgia Inst. of Technology},
  Year                     = {2013},
  Address                  = {Atlanta},
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}

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  Pages                    = {89--111},
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@Article{MMAM09,
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  Year                     = {2009},
  Pages                    = {046315},
  Volume                   = {80},
  DOI                      = {10.1103/PhysRevE.80.046315}
}

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@InProceedings{Moffatt90,
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  editor    = {J. L. Lumley},
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@Book{Moon87,
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  Editor                   = {C. S. Wells},
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  pages   = {467--521},
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@Article{MorrGree80,
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@Article{Morrison02,
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@Article{Morrison04,
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@Article{moser96,
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@Article{moser96a,
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@Article{MoserPF99,
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  Year                     = {1999},
  Pages                    = {943--946},
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@Book{mosersr,
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@Book{mpc,
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@Book{MuKe05,
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  author =    {T. Mullin and R. R. Kerswell},
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@Article{Mul10,
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  Pages                    = {1--24},
  Volume                   = {43}
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@Incollection{MuPe06,
  Title                    = {Recent observations of the transition to turbulence in a pipe},
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  Booktitle                = {IUTAM Symposium on Laminar-Turbulent Transition},
  Publisher                = {Springer},
  Year                     = {2006},
  Address                  = {New York},
  Editor                   = {Govindarajan, R.},
  Pages                    = {45--55},
  Series                   = {Fluid Mechanics and Its Applications},
  Volume                   = {78},
  DOI                      = {10.1007/1-4020-4159-4_5}
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  volume  = {86},
  pages   = {2018},
  abtract = {Ordered hole-defect dynamics happens when an unstable hole invade a plane wave state and periodically nucleates defects from which new holes are born. The interaction between the holes and a self-disordered background are essential for the occurrence of spatiotemporal chaos in hole-defect states.},
  doi     = {10.1103/physrevlett.86.2018},
}

@Article{N90,
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  Pages                    = {519--527},
  Volume                   = {217},
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  Year                     = {1996},
  Pages                    = {231--243},
  Volume                   = {307}
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  Pages                    = {2023--2025},
  Volume                   = {55}
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@Article{NaBu83,
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  Year                     = {1983},
  Pages                    = {1--26},
  Volume                   = {135}
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@Article{NagDeg13,
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  Journal                  = {J. Fluid Mech.},
  Year                     = {2013},
  Volume                   = {735},
  DOI                      = {10.1017/jfm.2013.515}
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  Author                   = {Nambu, Y.},
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  Pages                    = {2405--2414},
  Volume                   = {7},
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@Incollection{Narasimha89,
  Title                    = {The utility and drawbacks of traditional approaches},
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  Booktitle                = {Whither Turbulence? Turbulence at the Cross-Road},
  Publisher                = {Springer},
  Year                     = {1989},
  Address                  = {Berlin},
  Editor                   = {J. Lumley},
  Pages                    = {13--48}
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  Pages                    = {389--440},
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@Article{Neimark59,
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  Year                     = {1986},
  Pages                    = {49--58},
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@Article{newcgl2,
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  Year                     = {1986},
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@Book{newton1687pnpm,
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@Article{nhouse78,
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@Article{NiHa91,
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  Year                     = {1991},
  Pages                    = {6410--6413},
  Volume                   = {43},
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@Article{NiHa92,
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  Journal                  = {Z. Phys. B},
  Year                     = {{1992}},
  Pages                    = {261--262},
  Volume                   = {{89}},
  Abstract                 = {We show that the geometrical phase defined for the dissipative systems
    [1] is invariant under a unitary transformation. This implies for
    lasers an invariance of the geometrical phase for different choices
    of the reference frequency.},
  DOI                      = {10.1007/BF01318153}
}

@Article{NiHa92a,
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  Year                     = {1992},
  Pages                    = {2109--2112},
  Volume                   = {68},
  Abstract                 = {We show that the geometrical {(Berry)} phases discovered in Hamiltonian
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    of vectors for nonlinear dissipative systems with cyclic attractors.
    If the nonlinear dissipative systems possess a certain kind of asymptotic
    solution defined in this Letter, the phase and amplitude accumulation
    of a geometrical type can be defined. Detuned one- and two-photon
    lasers showing periodic intensity pulsations are taken as examples
    of such systems.}
}

@Article{NingHakenCLE90,
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@Article{Nizette2001,
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  Year                     = {2001},
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}

@InProceedings{NK04,
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  Series                   = {Proceedings of the Tenth European Turbulence Conference}
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@Article{NLSper77,
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  Abstract                 = {A nonlinear {Schr\"odinger} equation is found with growth and damping terms in the weakly nonlinear regime. An exact solitary solution is found for the equation.}
}

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}

@Article{noack03,
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  Pages                    = {335--363},
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  Pages                    = {89--92},
  Volume                   = {77},
  DOI                      = {10.1007/BF01456821}
}

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  Volume                   = {19},
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}

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}

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}

@Techreport{octave,
  Title                    = {{GNU Octave}: a high-level language for numerical computation},
  Author                   = {J. Eaton},
  Institution              = {University of Wisconsin},
  Year                     = {2007},
  Publisher                = {J. Eaton},
  URL                      = {http://www.octave.org}
}

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@Book{OlvSha06,
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@TechReport{openpipeflow,
  author      = {Willis, A. P.},
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  note        = {{\tt {Openpipeflow.org}}},
  publisher   = {A. P. Willis},
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}

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  Year                     = {1907},
  Pages                    = {69--138},
  Volume                   = {27}
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@Article{Orszag71,
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  pages   = {689--703},
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@Article{ott90a,
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@Article{Pal61,
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@Article{Palacios07,
  Title                    = {Heteroclinic cycles},
  Author                   = {A. Palacios},
  Journal                  = {Scholarpedia},
  Year                     = {2007},
  Pages                    = {2352},
  Volume                   = {2}
}

@Book{Pan96,
  title     = {{Incompressible Flow}},
  publisher = {Wiley},
  year      = {1996},
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  address   = {New York},
  doi       = {10.1002/9781118713075},
}

@Book{Pan97,
  Title                    = {Self-sustaining mechanisms of wall turbulence},
  Editor                   = {R. L. Panton},
  Publisher                = {Computational Mechanics Publications},
  Year                     = {1997},
  Address                  = {Southhampton}
}

@Article{ParLau86,
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}

@Article{Pasq87,
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  volume  = {2},
  pages   = {195--226},
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@Article{PC88,
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  Journal                  = {Phys. Rev. Lett.},
  Year                     = {1988},
  Pages                    = {2729--2732},
  Volume                   = {61},
  DOI                      = {10.1103/PhysRevLett.61.2729}
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@Article{PCar,
  Title                    = {Group theory for {Feynman} diagrams in non-{Abelian} gauge theories},
  Author                   = {Cvitanovi\'c, P.},
  Journal                  = {Phys. Rev. D},
  Year                     = {1976},
  Pages                    = {1536--1553},
  Volume                   = {14},
  DOI                      = {10.1103/PhysRevD.14.1536}
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@Book{PCgr,
  title     = {{Group Theory: Birdtracks, Lie's and Exceptional Groups}},
  publisher = {Princeton Univ. Press},
  year      = {2004},
  author    = {Cvitanovi{\'c}, P.},
  address   = {Princeton NJ},
  url       = {https://press.princeton.edu/titles/8839.html},
}

@Article{pchaot,
  Title                    = {Exploring chaotic motion through periodic orbits},
  Author                   = {D. Auerbach and P. Cvitanovi{\'c} and J.-P. Eckmann and G. Gunaratne and I. Procaccia},
  Journal                  = {Phys. Rev. Lett.},
  Year                     = {1987},
  Pages                    = {23},
  Volume                   = {58},
  Abstract                 = {Use near recurrence to extract periodic orbits
         and their stability eigenvalues.}
}

@Misc{pcopencourse,
  author = {Cvitanovi\'c, P.},
  title =  {Nonlinear dynamics 1 \& 2: {Geometry} of chaos},
  url =    {http://chaosbook.org/course1/about.html}
}

@Article{PeaHal13,
  Title                    = {Lagrangian coherent structures: {The} hidden skeleton of fluid flows},
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  Journal                  = {Physics Today},
  Year                     = {2013},
  Number                   = {2},
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  Volume                   = {66},
  DOI                      = {10.1063/PT.3.1886}
}

@Article{peckham90,
  author  = {Peckham, B. B.},
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  year    = {1990},
  volume  = {3},
  pages   = {261--280},
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}

@Article{PenDai07,
  Title                    = {A potential-flow, deformable-body model for fluid-structure interactions with compact vorticity: application to animal swimming measurements},
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  Journal                  = {Experiments in Fluids, to appear},
  Year                     = {2007},
  URL                      = {http://dabiri.caltech.edu/people/dabiri.html}
}

@Book{Penr04,
  Title                    = {The Road to Reality: A Complete Guide to the Laws of the Universe},
  Author                   = {R. Penrose},
  Publisher                = {A. A. Knopf},
  Year                     = {2005},
  Address                  = {New York}
}

@Article{percgl,
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  year    = {1999},
  volume  = {40},
  pages   = {884},
  doi     = {10.1063/1.532692},
}

@Book{perg,
  Title                    = {An introduction to ergodic theory},
  Author                   = {P. Walters},
  Publisher                = {Speringer-Verlag},
  Year                     = {1981},
  Address                  = {New York}
}

@Book{perkob,
  Title                    = {Differential Equations and Dynamical Systems},
  Author                   = {L. Perko},
  Publisher                = {Springer},
  Year                     = {1991},
  Address                  = {New York}
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@Article{PetCorBol06,
  Title                    = {Symbolic dynamics of coupled map lattices},
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  Year                     = {2006},
  Pages                    = {034105},
  Volume                   = {96},
  DOI                      = {10.1103/PhysRevLett.96.034105}
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@Article{PetCorBol07,
  Title                    = {Deconstructing spatiotemporal chaos using local symbolic dynamics},
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  Journal                  = {Phys. Rev. Lett.},
  Year                     = {2007},
  Pages                    = {214101},
  Volume                   = {99},
  DOI                      = {10.1103/PhysRevLett.99.214101}
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@Article{Peterhof1999,
  Title                    = {All-optical clock recovery using multisection distributed-feedback lasers},
  Author                   = {D. Peterhof and B. Sandstede},
  Journal                  = {J. Nonlinear Sci.},
  Year                     = {1999},
  Pages                    = {575--613},
  Volume                   = {9}
}

@Article{Petr40,
  Title                    = {Application of {Galerkin}'s method to the problem of the stability of the flow of a viscous fluid},
  Author                   = {Petrov, G.I.},
  Journal                  = {Prikl. Mat. Mekh.},
  Year                     = {1940},
  Pages                    = {3--12},
  Volume                   = {4}
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@InProceedings{petsc-efficient,
  Title                    = {Efficient management of parallelism in object oriented numerical software libraries},
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  Booktitle                = {Modern Software Tools in Scientific Computing},
  Year                     = {1997},
  Editor                   = {E. Arge and A. M. Bruaset and H. P. Langtangen},
  Pages                    = {163--202},
  Publisher                = {Birkh{\"{a}}user Press}
}

@Techreport{petsc-user-ref,
  Title                    = {{PETS}c Users Manual},
  Author                   = {S. Balay and K. Buschelman and V. Eijkhout and W. D. Gropp and D. Kaushik and M. G. Knepley and L. C. McInnes and B. F. Smith and H. Zhang},
  Institution              = {Argonne National Laboratory},
  Year                     = {2004},
  Number                   = {ANL-95/11 - Revision 2.1.5}
}

@Misc{petsc-web-page,
  Title                    = {{PETSc} web page},
  Author                   = {S. Balay and K. Buschelman and W. D. Gropp and D. Kaushik and M. G. Knepley and L. C. McInnes and B. F. Smith and H. Zhang},
  Note                     = {www.mcs.anl.gov/petsc},
  Year                     = {2001}
}

@Article{PexMull06,
  author  = {J. Peixinho and T. Mullin},
  title   = {Decay of turbulence in pipe flow},
  journal = {Phys. Rev. Lett.},
  year    = {2006},
  volume  = {96},
  pages   = {094501},
  doi     = {10.1103/physrevlett.96.094501},
}

@Article{PexMull07,
  Title                    = {Finite-amplitude thresholds for transition in pipe flow},
  Author                   = {J. Peixinho and T. Mullin},
  Journal                  = {J. Fluid Mech.},
  Year                     = {2007},
  Pages                    = {169--178},
  Volume                   = {582}
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@Book{Peyret02,
  Title                    = {Spectral Methods for Incompressible Flows},
  Author                   = {R. Peyret},
  Publisher                = {Springer},
  Year                     = {2002},
  Address                  = {New York}
}

@Incollection{Pfenninger61,
  Title                    = {Transition in the inlet length of tubes at high Reynolds numbers},
  Author                   = {W. Pfenninger},
  Booktitle                = {Boundary Layer and Flow Control},
  Publisher                = {Pergamon},
  Year                     = {1961},
  Address                  = {Oxford, UK},
  Editor                   = {G. V. Lachmann},
  Pages                    = {970--980},
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}

@Book{PG97,
  Title                    = {Chaos, Scattering and Statistical Mechanics},
  Author                   = {P. Gaspard},
  Publisher                = {Cambridge Univ. Press},
  Year                     = {1997},
  Address                  = {Cambridge}
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@Article{PhysToday04,
  Title                    = {New experiments set the scale for the onset of turbulence in pipe flow},
  Author                   = {R. Fitzgerald},
  Journal                  = {Physics Today},
  Year                     = {2004},
  Number                   = {2},
  Pages                    = {21--23},
  Volume                   = {57}
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@Article{PhysWorld04,
  author  = {Barenghi, C. F.},
  journal = {Physics World},
  title   = {Turbulent transition for fluids},
  year    = {2004},
  number  = {12},
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  volume  = {17},
  doi     = {10.1088/2058-7058/17/12/25},
}

@Article{pimsimp,
  author  = {P. Moresco and S. P. Dawson},
  title   = {The {PIM}-simplex method: an extension of the {PIM}-triple method to saddles with an arbitrary number of expanding directions},
  journal = {Physica D},
  year    = {1999},
  volume  = {126},
  pages   = {38--48},
  doi     = {10.1016/s0167-2789(98)00234-6},
}

@Article{pimstag,
  author   = {D. Sweet and H. E. Nusse and J. A. Yorke},
  title    = {Stagger-and-Step Method: Detecting and Computing Chaotic Saddles in Higher Dimensions},
  journal  = {Phys. Rev. Lett.},
  year     = {2001},
  volume   = {86},
  pages    = {2261},
  abstract = {This is a statistical extension of the PIM method. Here,
    the representative points are selected according to the so-called
    Exponential Stagger Distribution about the current point, instead of
    the points on the line or vertices of a simplex. This is a brutal
    force method while the authors claim that it is quite efficient.},
  doi      = {10.1103/physrevlett.86.2261},
}

@Article{pimyk,
  Title                    = {A procedure for finding numerical trajectories on chaotic saddles},
  Author                   = {H. E. Nusse and J. A. Yorke},
  Journal                  = {Physica D},
  Year                     = {1989},
  Pages                    = {137},
  Volume                   = {36},
  Abstract                 = {{PIM} (proper interior maximum) method is proposed to
    find chaotic trajectories on a nonattracting set with one-d unstable
    manifold. By partitioning a straight line straddling the stable
    manifolds, the point with longer and longer escaping time is found
    in a sequel until a sufficiently small interval is reached when the
    point is considered to be very close to the stable manifolds. After
    that, evolving and restraining the line element results in a
    NUMERICAL orbit is obtained.}
}

@Article{pizza2012chaotic,
  Title                    = {Chaotic dynamics in premixed hydrogen/air channel flow combustion},
  Author                   = {Pizza, G. and Frouzakis, C. E. and Mantzaras, J.},
  Journal                  = {Combustion Theory and Modelling},
  Year                     = {2012},
  Pages                    = {275--299},
  Volume                   = {16}
}

@Article{PlSiFi91,
  Title                    = {An investigation of chaotic {Kolmogorov} flows},
  Author                   = {Platt, N. and Sirovich, L. and Fitzmaurice, N.},
  Journal                  = {Phys. Fluids A},
  Year                     = {1991},
  Pages                    = {681--696},
  Volume                   = {3},
  DOI                      = {10.1063/1.858074}
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@Article{PoBaGr88,
  author  = {A. Politi and R. Badii and P. Grassberger},
  title   = {On the geometric structure of non-hyperbolic attractors},
  journal = {J. Phys. A},
  year    = {1988},
  volume  = {15},
  pages   = {L763},
  doi     = {10.1088/0305-4470/21/15/004},
}

@Article{Poinc1896,
  Title                    = {Sur les solutions p\'eriodiques et le principe de moindre action},
  Author                   = {H. Poincar\'e},
  Journal                  = {C. R. Acad. Sci. Paris},
  Year                     = {1896},
  Pages                    = {915--918},
  Volume                   = {123}
}

@Book{poincare,
  Title                    = {Les m{\'e}thodes nouvelles de la m{\'e}chanique c{\'e}leste},
  Author                   = {H. Poincar{\'e}},
  Publisher                = {Guthier-Villars},
  Year                     = {1899},
  Address                  = {Paris},
  Note                     = {For a very readable exposition of Poincar{\'e}'s work and the development of the dynamical systems theory up to 1920's see \refref{JBG97}.}
}

@Book{poinper,
  Title                    = {New Methods in Celestial Mechanics},
  Author                   = {H. Poincar{\'e}},
  Publisher                = {Springer},
  Year                     = {1992},
  Address                  = {New York},
  Series                   = {History of Modern Physics and Astronomy},
  Volume                   = {13}
}

@Article{Poiseuille1844,
  Title                    = {Recherches exp{\'e}rimentales sur le mouvement des liquides dans les tubes de tr{\`e}s-petits diam{\`e}tres},
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  Pages                    = {961},
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@Article{PoKno05,
  Title                    = {Dynamics in the 1:2~{s}patial resonance with broken reflection symmetry},
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  Journal                  = {Physica D},
  Year                     = {2005},
  Pages                    = {318--344},
  Volume                   = {201},
  DOI                      = {10.1016/j.physd.2005.01.001}
}

@Article{pomeau80,
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  Year                     = {1980},
  Pages                    = {189},
  Volume                   = {74},
  DOI                      = {10.1007/BF01197757}
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@Book{popebook,
  Title                    = {Turbulent Flows},
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  Publisher                = {Cambridge Univ. Press},
  Year                     = {2000},
  Address                  = {Cambridge}
}

@Article{postgal89,
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  year    = {1998},
  volume  = {35},
  pages   = {941--972},
  doi     = {10.1137/s0036142995296096},
}

@Article{PP83,
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  pages   = {573--591},
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@Article{PP90,
  Title                    = {Zeta functions and the periodic orbit structure of hyperbolic dynamics},
  Author                   = {W. Parry and M. Pollicott},
  Journal                  = {Ast\'erisque},
  Year                     = {1990},
  Pages                    = {268},
  Volume                   = {187-188}
}

@Article{Prants14,
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  Journal                  = {Eur. Phys. J. Spec. Top.},
  Year                     = {2014},
  Pages                    = {2723--2743},
  Volume                   = {223},
  DOI                      = {10.1140/epjst/e2014-02288-5}
}

@Article{PrBrPo93,
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  volume =  {5},
  pages =   {762--764},
  doi =     {10.1063/1.858659}
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@Article{pre88top,
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@Book{Press:96,
  Title                    = {Numerical Recipes in Fortran},
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  Year                     = {1996},
  Address                  = {Cambridge}
}

@Article{Pringle07,
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  volume  = {99},
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@Article{Pringle09,
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  volume  = {367},
  pages   = {457--472},
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@Article{Pringle2010,
  Title                    = {Using nonlinear transient growth to construct the minimal seed for shear flow turbulence},
  Author                   = {Pringle, C. C. T. and Kerswell, R. R.},
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  Year                     = {2010},
  Pages                    = {154502},
  Volume                   = {105},
  DOI                      = {10.1103/PhysRevLett.105.154502}
}

@Article{PSD01,
  Title                    = {Detecting unstable periodic orbits in chaotic continuous-time dynamical systems},
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  Journal                  = {Phys. Rev. E},
  Year                     = {2001},
  Pages                    = {026214},
  Volume                   = {64},
  Abstract                 = {Change the stability of the fixed points on the
    Poincar{\'e} section to find periodic orbits.}
}

@Article{PSDB00,
  Title                    = {Theory and applications of the systematic detection of unstable periodic orbits in dynamical systems},
  Author                   = {D. Pingel and P. Schmelcher and F. K. Diakonos and O. Biham},
  Journal                  = {Phys. Rev. E},
  Year                     = {2000},
  Note                     = {\arXiv{nlin.CD/0006011}},
  Pages                    = {2119},
  Volume                   = {62}
}

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@Article{PWK15,
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  year    = {2015},
  volume  = {27},
  pages   = {064102},
  doi     = {10.1063/1.4922183},
}

@Article{PXBFB14,
  Title                    = {Redistribution of kinetic energy in turbulent flows},
  Author                   = {Pumir, A. and Xu, H. and Boffetta, G. and Falkovich, G. and Bodenschatz, E.},
  Journal                  = {Phys. Rev. X},
  Year                     = {2014},
  Pages                    = {041006},
  Volume                   = {4},
  DOI                      = {10.1103/PhysRevX.4.041006}
}

@Article{R1883,
  author  = {O. Reynolds},
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  year    = {1883},
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  pages   = {935--982},
  doi     = {10.1098/rspl.1883.0018},
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@Article{R1894,
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  volume  = {186},
  pages   = {123--161},
  doi     = {10.1098/rspl.1894.0075},
}

@Article{RaCoRi14,
  Title                    = {Relative periodic orbits in plane {Poiseuille} flow},
  Author                   = {S. Rawat and C. Cossu and F. Rincon},
  Journal                  = {Compt. Rend. M\'ecanique},
  Year                     = {2014},
  Pages                    = {485--489},
  Volume                   = {342},
  DOI                      = {10.1016/j.crme.2014.05.008}
}

@Article{rademacher2005,
  author  = {Rademacher, J. D. M.},
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  year    = {2005},
  volume  = {218},
  pages   = {390--443},
  doi     = {10.1016/j.jde.2005.03.016},
}

@Article{Radziunas2006,
  author  = {M. Radziunas},
  title   = {Numerical bifurcation analysis of the traveling wave model of multisection semiconductor lasers},
  journal = {Physica D},
  year    = {2006},
  volume  = {213},
  pages   = {98--112},
  doi     = {10.1016/j.physd.2005.11.003},
}

@Article{RaHaAb96,
  author  = {Rauh, A. and Hannibal, L. and Abraham, N. B.},
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  year    = {1996},
  volume  = {99},
  pages   = {45--58},
  doi     = {10.1016/s0167-2789(96)00129-7},
}

@Article{Rand82,
  author  = {Rand, D.},
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  year    = {1982},
  volume  = {79},
  pages   = {1--3},
  doi     = {10.1007/bf02416564},
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@Article{RaOsSeSi82,
  Title                    = {Universal transition from quasiperiodicity to chaos in dissipative systems},
  Author                   = {D. Rand and S. Ostlund and J. Sethna and E. Siggia},
  Journal                  = {Phys. Rev. Lett.},
  Year                     = {1982},
  Pages                    = {132--135},
  Volume                   = {49},
  DOI                      = {10.1103/PhysRevLett.49.132}
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@Article{RCHR15,
  Title                    = {On the self-sustained nature of large-scale motions in turbulent {Couette} flow},
  Author                   = {Rawat, S. and Cossu, C. and Hwang, Y. and Rincon, F.},
  Journal                  = {J. Fluid Mech.},
  Year                     = {2015},
  Pages                    = {515--540},
  Volume                   = {782},
  DOI                      = {10.1017/jfm.2015.550}
}

@Article{Recke1998,
  author  = {L. Recke and D. Peterhof},
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  year    = {1998},
  volume  = {144},
  pages   = {233--262},
  doi     = {10.1006/jdeq.1997.3379},
}

@Article{Recke1998a,
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  pages   = {94--101},
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@Article{Recke2010,
  author  = {L. Recke and A. Samoilenko and A. Teplinsky and V. Tkachenko and S. Yanchuk},
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  year    = {2011},
  volume  = {31},
  pages   = {847--875},
  note    = {\arXiv{1108.5990}},
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}

@Article{reimann1,
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@Article{reimann2,
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@Article{Rem03,
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  volume  = {35},
  pages   = {229},
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@Article{Rem06,
  Title                    = {On boundary conditions for incompressible {Navier-Stokes} problems},
  Author                   = {Rempfer, D.},
  Journal                  = {Appl. Mech. Rev.},
  Year                     = {2006},
  Pages                    = {107--125},
  Volume                   = {59},
  DOI                      = {10.1115/1.2177683}
}

@Article{ReSaTkYa11,
  author  = {Recke, L. and {Samoilenko}, A. and {Tkachenko}, V. and {Yanchuk}, S.},
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  year    = {2012},
  volume  = {11},
  pages   = {771--800},
  doi     = {10.1137/110846750},
}

@Book{Richter10,
  Title                    = {Semiclassical Theory of Mesoscopic Quantum Systems},
  Author                   = {Richter, K.},
  Publisher                = {Springer},
  Year                     = {2000},
  Address                  = {Berlin},
  DOI                      = {10.1007/BFb0109634},
  ISBN                     = {9783662156506},
  URL                      = {http://www.physik.uni-regensburg.de/forschung/richter/richter/pages/research/springer-tracts-161.pdf}
}

@Article{RiMeAv16,
  Title                    = {Emergence of spatio-temporal dynamics from exact coherent solutions in pipe flow},
  Author                   = {P. Ritter and F. Mellibovsky and M. Avila},
  Journal                  = {New J. Phys.},
  Year                     = {2016},
  Pages                    = {083031},
  Volume                   = {18},
  DOI                      = {10.1088/1367-2630/18/8/083031}
}

@Article{Rincon07,
  author   = {Rincon, F.},
  title    = {On the existence of two-dimensional nonlinear steady states in plane {Couette} flow},
  journal  = {Phys. Fluids},
  year     = {2007},
  volume   = {19},
  pages    = {4105},
  note     = {\arXiv{0706.1165}},
  abstract = {It is not possible to obtain the nonlinear plane
    Couette flow solutions reported by Cherhabili and
    Ehrenstein [Eur. J.  Mech.  B/Fluids 14, 667 (1995)] using
    their Poiseuille-Couette homotopy. We also demonstrate that
    the steady solutions obtained by Mehta and Healey [Phys.
    Fluids 17, 094108 (2005)] for small symmetry-preserving
    perturbations are influenced by an artefact of the modified
    system of equations used in their paper.},
  doi      = {10.1063/1.2753982},
}

@Book{Risken96,
  Title                    = {The {F}okker-{P}lanck Equation},
  Author                   = {H. Risken and H. Haken},
  Publisher                = {Springer},
  Year                     = {1996},
  Address                  = {New York}
}

@Incollection{RMChaosHist,
  author    = {R. Mainieri and P. Cvitanovi{\'c}},
  title     = {A brief history of chaos},
  booktitle = {{Chaos: Classical and Quantum}},
  publisher = {Niels Bohr Inst.},
  year      = {2017},
  editor    = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
  address   = {Copenhagen},
  url       = {http://ChaosBook.org/paper.shtml#appendHist},
}

@Article{Robinson91,
  author  = {S. K. Robinson},
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  year    = {1991},
  volume  = {23},
  pages   = {601--639},
  doi     = {10.1146/annurev.fluid.23.1.601},
}

@Techreport{Robinson91b,
  Title                    = {The Kinematics of turbulent boundary layer structure},
  Author                   = {S. K. Robinson},
  Institution              = {NASA},
  Year                     = {1991},
  Note                     = {103859}
}

@Article{ROC07,
  author  = {F. Rincon and G. I. Ogilvie and C. Cossu},
  title   = {On self-sustaining processes in {Rayleigh}-stable rotating plane Couette flows and subcritical transition to turbulence in accretion disks},
  journal = {Astron. Astrophys.},
  year    = {2007},
  volume  = {463},
  pages   = {817--832},
  doi     = {10.1051/0004-6361:20066544},
}

@Article{Rolland14,
  author  = {Rolland, J.},
  title   = {Turbulent spot growth in plane {Couette} flow: statistical study and formation of spanwise vorticity},
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  year    = {2014},
  volume  = {46},
  pages   = {015512},
  note    = {\arXiv{1401.3586}},
  doi     = {10.1088/0169-5983/46/1/015512},
}

@Article{Romanov73,
  author  = {V. A. Romanov},
  title   = {Stability of plane-parallel {Couette} flow},
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  year    = {1973},
  volume  = {7},
  pages   = {137--146},
  doi     = {10.1007/bf01078886},
}

@Article{Rome69,
  author  = {Rome, H.},
  title   = {A direct solution to the linear variance equation of a time-invariant linear system},
  journal = {IEEE Trans. Automatic Control},
  year    = {1969},
  volume  = {14},
  pages   = {592--593},
  doi     = {10.1109/tac.1969.1099271},
}

@Article{RoSa00,
  author  = {Roweis, S. T. and Saul, L. K.},
  title   = {Nonlinear dimensionality reduction by locally linear embedding},
  journal = {Science},
  year    = {2000},
  volume  = {290},
  pages   = {2323--2326},
  doi     = {10.1126/science.290.5500.2323},
}

@Article{ross,
  author  = {O. E. R{\"{o}}ssler},
  title   = {An equation for continuous chaos},
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  year    = {1976},
  volume  = {57},
  pages   = {397},
  doi     = {10.1016/0375-9601(76)90101-8},
}

@Article{roth99,
  author  = {D. Rothstein and E. Henry and J. P. Gollub},
  title   = {Persistent patterns in transient chaotic fluid mixing},
  journal = {Nature},
  year    = {1999},
  volume  = {401},
  pages   = {770--772},
}

@Article{Roupas11,
  author  = {Z. Roupas},
  title   = {Phase space geometry and chaotic attractors in dissipative {Nambu} mechanics},
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  year    = {2012},
  volume  = {45},
  pages   = {195101},
  note    = {\arXiv{1110.0766}},
  doi     = {10.1088/1751-8113/45/19/195101},
}

@Article{RP67,
  author  = {W. C. Reynolds and M. C. Potter},
  title   = {Finite-amplitude instability of parallel shear flows},
  journal = {J. Fluid Mech.},
  year    = {1967},
  volume  = {27},
  pages   = {465},
  doi     = {10.1017/s0022112067000485},
}

@Article{RPM,
  Title                    = {Stabilization of unstable procedures: the recursive projection method},
  Author                   = {G. M. Shroff and H. B. Keller},
  Journal                  = {SIAM J. Numer. Anal.},
  Year                     = {1993},
  Pages                    = {1099--1120},
  Volume                   = {30},
  DOI                      = {10.1137/0730057},
}

@Article{RSBH98,
  Title                    = {On stability of streamwise streaks and transition thresholds in plane channel flows},
  Author                   = {S. C. Reddy and P. J. Schmid and J. S. Baggett and D. S. Henningson},
  Journal                  = {J. Fluid Mech.},
  Year                     = {1998},
  Pages                    = {269--303},
  Volume                   = {365}
}

@Book{rtb,
  Title                    = {Theory of Orbits},
  Author                   = {V. Szebehely},
  Publisher                = {Academic Press},
  Year                     = {1967},
  Address                  = {New York}
}

@Article{rue04ne,
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  journal  = {Physics Today},
  year     = {2004},
  volume   = {57},
  number   = {5},
  pages    = {48--53},
  abstract = {New ideas of nonequilibrium physics are introduced. Emphasize the SRB measure approach and energy fluctuation theorem.},
  doi      = {10.1063/1.1768674},
}

@Article{ruelext,
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  volume  = {87},
  pages   = {287--302},
  doi     = {10.1007/bf01218566},
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  pages   = {167},
  doi     = {10.1007/bf01893621},
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@Book{ruelle,
  Title                    = {Statistical Mechanics, Thermodynamic Formalism},
  Author                   = {D. Ruelle},
  Publisher                = {Addison-Wesley},
  Year                     = {1978},
  Address                  = {Reading, MA}
}

@Article{Ruelle76,
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  year    = {1976},
  volume  = {34},
  pages   = {231--242},
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@Article{ruelle86,
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  journal  = {J. Stat. Phys.},
  year     = {1986},
  volume   = {44},
  pages    = {281--292},
  abstract = {For a class of differentiable dynamical systems (called Axiom
A systems) it has been shown by Pollicott and the author that correlation
functions have Fourier transforms which are meromorphic in a strip. The
poles (or resonances) are, however, not easy to locate. This note reviews
the results which are known and discusses a simple model where the
position of resonances can be estimated. The effect of noise is also
discussed.},
  doi      = {10.1007/bf01011300},
}

@Article{ruelle86a,
  author  = {D. Ruelle},
  title   = {Resonances of chaotic dynamical systems},
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  year    = {1986},
  volume  = {56},
  pages   = {405--407},
  doi     = {10.1103/physrevlett.56.405},
}

@Techreport{Ruelle95,
  author      = {Ruelle, D.},
  title       = {Functional determinants related to dynamical systems and the thermodynamic formalism},
  institution = {Inst. Hautes Etud. Sci.},
  year        = {1995},
  number      = {IHES-P-95-30},
  address     = {Bures-sur-Yvette},
  url         = {http://cds.cern.ch/record/281769},
}

@Article{RVVCMG97,
  Title                    = {Review article: {The} dynamics of optically pumped molecular lasers. {On} its relation with the {Lorenz - Haken model}},
  Author                   = {Rold{\'a}n , E. and {de Valc{\'a}rcel}, G. J. and {Vilaseca}, R. and {Corbal{\'a}n}, R. and {Mart{\'{\i}}nez}, V. J. and {Gilmore}, R.},
  Journal                  = {Quantum Semiclass. Opt.},
  Year                     = {1997},
  Pages                    = {R1},
  Volume                   = {9},
  DOI                      = {10.1088/1355-5111/9/1/001}
}

@Article{Saad1986,
  Title                    = {{GMRES}: {A} generalized minimal residual algorithm for solving nonsymmetric linear systems},
  Author                   = {Saad, Y. and Schultz, M. H.},
  Journal                  = {SIAM J. Sci. Stat. Comput.},
  Year                     = {1986},
  Pages                    = {856--869},
  Volume                   = {7},
  DOI                      = {10.1137/0907058}
}

@Article{Sacker65,
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@Article{SalEck15,
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  Journal                  = {Phys. Fluids},
  Year                     = {2015},
  Pages                    = {045109},
  Volume                   = {27},
  DOI                      = {10.1063/1.4917281}
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@Article{SALOHO11,
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  year    = {2011},
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  pages   = {193--204},
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@Article{Samoilenko2005,
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@Article{SaOuGoHa11,
  Title                    = {Neutrally buoyant particle dynamics in fluid flows: {Comparison} of experiments with {Lagrangian} stochastic models},
  Author                   = {T. P. Sapsis and N. T. Ouellette and J. P. Gollub and G. Haller},
  Journal                  = {Phys. Fluids},
  Year                     = {2011},
  Pages                    = {093304},
  Volume                   = {23},
  DOI                      = {10.1063/1.3632100}
}

@Article{SapLerm09,
  Title                    = {Dynamically orthogonal field equations for continuous stochastic dynamical systems},
  Author                   = {Sapsis, T. P. and Lermusiaux, P. F. J.},
  Journal                  = {Phys. D},
  Year                     = {2009},
  Pages                    = {2347--2360},
  Volume                   = {238},
  DOI                      = {10.1016/j.physd.2009.09.017}
}

@Article{SapMaj13,
  Title                    = {Blending modified {Gaussian} closure and non-{Gaussian} reduced subspace methods for turbulent dynamical systems},
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  Year                     = {2013},
  Pages                    = {1039--1071},
  Volume                   = {23},
  DOI                      = {10.1007/s00332-013-9178-1}
}

@Article{SapMajMQDPhysD,
  Title                    = {A statistically accurate modified quasilinear {Gauss}ian closure for uncertainty quantification in turbulent dynamical systems},
  Author                   = {Sapsis, T. P. and Majda, A. J.},
  Journal                  = {Phys. D},
  Year                     = {2013},
  Pages                    = {34--45},
  Volume                   = {252},
  DOI                      = {10.1016/j.physd.2013.02.009}
}

@Article{SapMajPhysD13,
  Title                    = {Blending reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state},
  Author                   = {Sapsis, T. P. and Majda, A. J.},
  Journal                  = {Phys. D},
  Year                     = {2013},
  Pages                    = {61--76},
  Volume                   = {258},
  DOI                      = {10.1016/j.physd.2013.05.004}
}

@Article{Sapsis13,
  Title                    = {Attractor local dimensionality, nonlinear energy transfers and finite-time instabilities in unstable dynamical systems with applications to two-dimensional fluid flows},
  Author                   = {Sapsis, T. P.},
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  Year                     = {2013},
  Volume                   = {469},
  DOI                      = {10.1098/rspa.2012.0550}
}

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@Article{Sartorius1998,
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  Year                     = {1998},
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@Article{SB95,
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@Article{SCD07,
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  year    = {2010},
  volume  = {9},
  pages   = {1--33},
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}

@Unpublished{SCD09b,
  Title                    = {Recurrent spatio-temporal structures of translationally invariant {Kuramoto-Sivashinsky} flow},
  Author                   = {Siminos, E. and Cvitanovi{\'c}, P. and Davidchack, R. L.},
  Note                     = {In preparation},
  Year                     = {2011}
}

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  year    = {2006},
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  pages   = {041103},
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}

@Article{SchEckVoll07,
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  year    = {2007},
  volume  = {75},
  pages   = {066313},
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}

@Article{SchEckYor07,
  author  = {T. M. Schneider and B. Eckhardt and J. Yorke},
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  volume  = {99},
  pages   = {034502},
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@Phdthesis{Schmi99,
  Title                    = {Transition to turbulence in linearly stable shear flows},
  Author                   = {A. Schmiegel},
  School                   = {Philipps-Universit{\"a}t Marburg},
  Year                     = {1999},
  Note                     = {Available on {\tt archiv.ub.uni-marburg.de/diss/z2000/0062}}
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  Title                    = {Experimental observation of nonlinear traveling waves in turbulent pipe flow},
  Author                   = {B. Hof and C. W. H. van Doorne and J. Westerweel and F. T. M. Nieuwstadt and H. Faisst and B. Eckhardt and H. Wedin and R. R. Kerswell and F. Waleffe},
  Journal                  = {Science},
  Year                     = {2004},
  Pages                    = {1594--1598},
  Volume                   = {305},
  Abstract                 = {Transition to turbulence in pipe flow is one of the most
    fundamental and longest-standing problems in fluid dynamics.
    Stability theory suggests that the flow remains laminar for all flow
    rates, but in practice pipe flow becomes turbulent even at moderate
    speeds. This transition drastically affects the transport efficiency
    of mass, momentum, and heat. On the basis of the recent discovery of
    unstable traveling waves in computational studies of the
    Navier-Stokes equations and ideas from dynamical systems theory, a
    model for the transition process has been suggested. We report
    experimental observation of these traveling waves in pipe flow,
    confirming the proposed transition scenario and suggesting that the
    dynamics associated with these unstable states may indeed capture the
    nature of fluid turbulence.},
  DOI                      = {10.1126/science.1100393}
}

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  Pages                    = {2739},
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  Abstract                 = {Change stability of maps to locate periodic orbits.}
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  Pages                    = {395--400},
  Volume                   = {51}
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  Year                     = {2004},
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  Title                    = {Universal properties of the transition from quasi-periodicity to chaos in dissipative systems},
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  Year                     = {2010},
  Pages                    = {104501},
  Volume                   = {104},
  DOI                      = {10.1103/PhysRevLett.104.104501}
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}

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  Journal                  = {Physica D},
  Year                     = {1982},
  Pages                    = {405--411},
  Volume                   = {5},
  DOI                      = {10.1016/0167-2789(82)90033-1}
}

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  year    = {2015},
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}

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@Misc{ShiAvHof2015,
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}

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  Pages                    = {793--796},
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  Year                     = {2006},
  Pages                    = {557},
  Volume                   = {198}
}

@Misc{ShWi16,
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  year   = {2017},
  note   = {In preparation.},
}

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  Year                     = {1987},
  Pages                    = {2051--2054},
  Volume                   = {58}
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@Article{ShWi89,
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  Year                     = {1989},
  Pages                    = {587--599},
  Volume                   = {198}
}

@Article{ShWi89a,
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  Year                     = {1989},
  Pages                    = {514--518},
  Volume                   = {57}
}

@Article{SiCvi10,
  Title                    = {Continuous symmetry reduction and return maps for high-dimensional flows},
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  Journal                  = {Physica D},
  Year                     = {2011},
  Pages                    = {187--198},
  Volume                   = {240}
}

@Article{Sieber2002,
  Title                    = {Numerical bifurcation analysis for multisection semiconductor lasers},
  Author                   = {J. Sieber},
  Journal                  = {SIAM J. Appl. Math.},
  Year                     = {2002},
  Pages                    = {248--270},
  Volume                   = {1}
}

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  Title                    = {Iteration of analytic functions},
  Author                   = {C. L. Siegel},
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  Year                     = {1942},
  Pages                    = {607--612},
  Volume                   = {43}
}

@Article{SieStei90,
  Title                    = {Classical and quantum mechanics of a strongly chaotic billiard system},
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  Year                     = {1990},
  Pages                    = {248--266},
  Volume                   = {44},
  DOI                      = {10.1016/0167-2789(90)90058-W}
}

@Article{SieStei91,
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  Author                   = {Sieber, M. and Steiner, F.},
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  Year                     = {1991},
  Pages                    = {1941--1944},
  Volume                   = {67}
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@PhdThesis{SiminosThesis,
  author =  {Siminos, E.},
  title =   {{Recurrent Spatio-temporal Structures in Presence of Continuous Symmetries}},
  school =  {School of Physics, Georgia Inst. of Technology},
  year =    {2009},
  address = {Atlanta},
  note =    {\\ \wwwcb{/projects/theses.html}}
}

@InProceedings{simo89,
  Title                    = {On the analytical and numerical approximation of invariant manifolds},
  Author                   = {C. Simo},
  Booktitle                = {Les M\'{e}thodes Modernes de la M\'{e}canique C\'{e}leste (Goutelas '89)},
  Year                     = {1989},
  Address                  = {Gif-sur-Yvette},
  Editor                   = {D. Benest and C. Froeschl\'{e}},
  Pages                    = {285--329},
  Publisher                = {Editions Frontieres}
}

@Article{sinai,
  Title                    = {Gibbs measures in ergodic theory},
  Author                   = {Ya. G. Sinai},
  Journal                  = {Russian Mathematical Surveys},
  Year                     = {1972},
  Pages                    = {21},
  Volume                   = {166}
}

@Book{sinai76,
  title =     {{Introduction to Ergodic Theory}},
  publisher = {Princeton Univ. Press},
  year =      {1976},
  author =    {Ya. G. Sinai},
  address =   {Princeton}
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@Article{SipGol11,
  Title                    = {Directed percolation describes lifetime and growth of turbulent puffs and slugs},
  Author                   = {Sipos, M. and Goldenfeld, N.},
  Journal                  = {Phys. Rev. E},
  Year                     = {2011},
  Pages                    = {035304},
  Volume                   = {84},
  DOI                      = {10.1103/PhysRevE.84.035304}
}

@Article{sirovcgl90,
  author  = {J. D. Rodriguez and L. Sirovich},
  title   = {Low-dimensional dynamics for the complex {Ginzburg-Landau} equation},
  journal = {Physica D},
  year    = {1990},
  volume  = {43},
  pages   = {77--86},
  doi     = {10.1016/0167-2789(90)90017-j},
}

@Article{Sirovich87,
  author =  {Sirovich, L.},
  title =   {Turbulence and the dynamics of coherent structures part {I:} coherent structures},
  journal = {Quart. Appl. Math.},
  year =    {1987},
  volume =  {45},
  pages =   {561--571},
  url =     {http://www.jstor.org/stable/43637457}
}

@Article{Sirovich94,
  Title                    = {Reply to ``{O}bservations regarding '{C}oherence and chaos in a model of turbulent boundary layer' by {X}. {Z}hou and {L}. {S}irovich''},
  Author                   = {L. Sirovich and X. Zhou},
  Journal                  = {Phys. Fluids},
  Year                     = {1994},
  Pages                    = {1579--1582},
  Volume                   = {6}
}

@Article{siv,
  Title                    = {Nonlinear analysis of hydrodynamical instability in laminar flames - {I}. {D}erivation of basic equations},
  Author                   = {G. I. Sivashinsky},
  Journal                  = {Acta Astronaut.},
  Year                     = {1977},
  Pages                    = {1177},
  Volume                   = {4},
  Abstract                 = {A integro-differential equation (KSe) is derived directly from the flame propagation equations near the bifurcation point.}
}

@Article{sivmich80,
  Title                    = {On irregular wavy flow of a liquid down a vertical plane},
  Author                   = {G. I. Sivashinsky and D. M. Michelson},
  Journal                  = {Progr. Theor. Phys. Suppl.},
  Year                     = {1980},
  Pages                    = {2112--2114},
  Volume                   = {66},
  DOI                      = {On irregular wavy flow of a liquid down a vertical plane}
}

@Article{skeleton,
  author  = {P. Cvitanovi{\'c}},
  title   = {Periodic orbits as the skeleton of classical and quantum chaos},
  journal = {Physica D},
  year    = {1991},
  volume  = {51},
  pages   = {138--151},
  doi     = {10.1016/0167-2789(91)90227-z},
}

@Phdthesis{Skuf05,
  Title                    = {Understanding the chaotic saddle with focus on a 9-variable model of planar {Couette} flow},
  Author                   = {J. D. Skufca},
  School                   = {U. Maryland},
  Year                     = {2005},
  URL                      = {https://drum.umd.edu/dspace/handle/1903/2897}
}

@Article{SM,
  Title                    = {The study of hairpin vortices in a laminar boundary layer},
  Author                   = {C. R. Smith and S. P. Metzler},
  Journal                  = {J. Fluid Mech.},
  Year                     = {1983},
  Pages                    = {27},
  Volume                   = {129}
}

@Article{SM83,
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@Book{Smale00,
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  Year                     = {2000},
  Address                  = {Singapore}
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@Book{Smale80,
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@Book{Sou70,
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  Publisher                = {Dunod},
  Year                     = {1970},
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@Book{Sou97,
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  Year                     = {1991},
  Pages                    = {211--216},
  Volume                   = {434},
  Abstract                 = {A brief survey of modelling equations of fully developed
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  Pages                    = {2316},
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@Techreport{SpiekerBlog,
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  Year                     = {2009},
  Note                     = {{\tt Channelflow.org/dokuwiki/doku.php/gtspring2009:spieker\_blog}}
}

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@Book{StGo09,
  Title                    = {Mathematics for Physics: A Guided Tour for Graduate Students},
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  Year                     = {2009},
  Address                  = {Cambridge}
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  Title                    = {Velocity profile in a two-layer {Kolmogorov}-like flow},
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    limit orbit can be extended in a unique way to the small parameter
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    demonstrate the idea.},
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@Conference{STPGS15,
  Title                    = {Experimental Observation of Exact Coherent Structures in a Weakly Turbulent Quasi-Two-Dimensional Flow},
  Author                   = {B. Suri and J. Tithof and R. K. Pallantla and R. Grigoriev and M. Schatz},
  Booktitle                = {68\textsuperscript{th} Annual Meeting of the APS Division of Fluid Dynamics},
  Year                     = {2015},
  Series                   = {Bull. Am. Phys. Soc},
  Volume                   = {60},
  URL                      = {http://meetings.aps.org/link/BAPS.2015.DFD.D29.2}
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@Book{strogb,
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  Year                     = {2002},
  Number                   = {02},
  Pages                    = {267--276},
  Volume                   = {23},
  DOI                      = {10.1142/S0252959902000250},
  Eprint                   = {http://www.worldscientific.com/doi/pdf/10.1142/S0252959902000250}
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@InProceedings{sunil_ahuja_template-based_2006,
  Title                    = {Template-based stabilization of relative equilibria},
  Author                   = {S. Ahuja and {I.G.} Kevrekidis and {C.W.} Rowleym},
  Booktitle                = {2006 American Control Conference, 14-16 June 2006},
  Year                     = {2006},
  Address                  = {Piscataway, NJ},
  Pages                    = {3606--3611},
  Publisher                = {{IEEE}},
  DOI                      = {10.1109/ACC.2006.1657278}
}

@Article{SunKanZha11,
  Title                    = {A method of recovering the initial vectors of globally coupled map lattices based on symbolic dynamics},
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  Journal                  = {Chin. Phys. B},
  Year                     = {2011},
  Pages                    = {120507},
  Volume                   = {20},
  DOI                      = {10.1088/1674-1056/20/12/120507}
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@Article{TakOon84,
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@Book{Taub61,
  Title                    = {{John von Neumann}: Collected Works. {Volume II}: {Operators}, Ergodic Theory and Almost Periodic Functions in a Group},
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@Book{TE05,
  Title                    = {Spectra and pseudospectra: the behavior of nonnormal matrices and operators},
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  Publisher                = {Princeton Univ. Press},
  Year                     = {2005},
  Address                  = {Princeton}
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@Article{TeHaHe10,
  Title                    = {Spatial optimal growth in three-dimensional boundary layers},
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  Year                     = {2010},
  Pages                    = {5--37},
  Volume                   = {646},
  DOI                      = {10.1017/S0022112009993260}
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@Article{temam91edd,
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  year    = {2000},
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  pages   = {2319--2323},
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}

@Book{Tennekes72,
  Title                    = {A First Course in Turbulence},
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  Publisher                = {MIT Press},
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@Article{teramura2014,
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}

@Article{ThiBoo99,
  author   = {Thiffeault, {J.-L.} and Boozer, A. H.},
  title    = {Geometrical constraints on finite-time {Lyapunov} exponents in two and three dimensions},
  journal  = {Chaos},
  year     = {2001},
  volume   = {11},
  pages    = {16--28},
  note     = {\arXiv{physics/0009017}},
  abstract = {Constraints are found on the spatial variation of finite-time Lyapunov
    exponents of two and three-dimensional systems of ordinary differential
    equations. In a chaotic system, finite-time Lyapunov exponents describe
    the average rate of separation, along characteristic directions,
    of neighboring trajectories. The solution of the equations is a coordinate
    transformation that takes initial conditions (the Lagrangian coordinates)
    to the state of the system at a later time (the Eulerian coordinates).
    This coordinate transformation naturally defines a metric tensor,
    from which the Lyapunov exponents and characteristic directions are
    obtained. By requiring that the Riemann curvature tensor vanish for
    the metric tensor (a basic result of differential geometry in a flat
    space), differential constraints relating the finite-time Lyapunov
    exponents to the characteristic directions are derived. These constraints
    are realized with exponential accuracy in time. A consequence of
    the relations is that the finite-time Lyapunov exponents are locally
    small in regions where the curvature of the stable manifold is large,
    which has implications for the efficiency of chaotic mixing in the
    advection-diffusion equation. The constraints also modify previous
    estimates of the asymptotic growth rates of quantities in the dynamo
    problem, such as the magnitude of the induced current.},
  doi      = {10.1063/1.1342079},
}

@Article{Thiffeault11,
  Title                    = {Using multiscale norms to quantify mixing and transport},
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  Journal                  = {Nonlinearity},
  Year                     = {2012},
  Note                     = {\arXiv{1105.1101}},
  Pages                    = {1--44},
  Volume                   = {25},
  DOI                      = {10.1088/0951-7715/25/2/R1}
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@Article{Thiffeault2001,
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  note    = {\arXiv{nlin/0101012}},
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  title   = {A periodic-like solution in channel flow},
  journal = {J. Fluid Mech.},
  year    = {2003},
  volume  = {481},
  pages   = {67--76},
  doi     = {10.1017/s0022112003003768},
}

@Book{Tinkham,
  Title                    = {Group Theory and Quantum Mechanics},
  Author                   = {Tinkham, M.},
  Publisher                = {Dover},
  Year                     = {2003},
  Address                  = {New York}
}

@Unpublished{tithof16,
  author = {J. Tithof and B. Suri and R. K. Pallantla and R. O. Grigoriev and M. F. Schatz},
  title  = {Bifurcations in a quasi-two-dimensional {Kolmogorov}-like flow},
  note   = {Submitted to J. Fluid Mech.},
  year   = {2016},
  url    = {https://arXiv.org/abs/1601.00243},
}

@PhdThesis{Tithof2016,
  author = {J. Tithof},
  title =  {Dimensionality reduction of spatiotemporal chaos in a quasi-two-dimensional flow},
  school = {Georgia Institute of Technology},
  year =   {2016}
}

@Article{TKSSG14,
  author =  {Tuckerman, L. S. and Kreilos, T. and Schrobsdorff, H. and Schneider, T. M. and Gibson, J. F.},
  title =   {Turbulent-laminar patterns in plane Poiseuille flow},
  journal = {Phys. Fluids},
  year =    {2014},
  volume =  {26},
  pages =   {114103},
  doi =     {10.1063/1.4900874}
}

@Article{ToDe94,
  author   = {Toronov, V. Y. and Derbov, V. L.},
  title    = {Geometric phases in lasers and liquid flows},
  journal  = {Phys. Rev. E},
  year     = {1994},
  volume   = {49},
  pages    = {1392--1399},
  abstract = {Pancharatnam's geometric phase is introduced for such nonlinear dissipative
    systems as lasers and liquid flows. Two types of geometric; phases
    are shown to arise in these systems: the phase induced by the inner
    dynamics of the system and the one caused by the cyclic and adiabatic
    variation of the system parameters. A possible generalization of
    the geometric-effects theory in other dissipative systems is discussed.},
  doi      = {10.1103/physreva.49.1392},
}

@Article{ToDe94a,
  author   = {Toronov, V. Y. and Derbov, V. L.},
  title    = {Geometric-phase effects in laser dynamics},
  journal  = {Phys. Rev. A},
  year     = {1994},
  volume   = {50},
  pages    = {878--881},
  abstract = {We show that such phenomena of laser dynamics as mean-phase-slope
    jumps and temporal phase jumps at resonance between the cavity and
    spectral line frequencies are intrinsically connected with the topology
    of attractors in the space of rays and can be interpreted as the
    manifestations of the geometric-phase properties of the evolution
    operator.},
  doi      = {10.1103/physreva.50.878},
  keywords = {complex Lorenz, geometric phase, Laser, symmetry},
}

@Article{ToDe97,
  author   = {Toronov, V. Y. and Derbov, V. L.},
  title    = {Geometric phases in a ring laser},
  journal  = {Quantum Electronics},
  year     = {1997},
  volume   = {27},
  pages    = {644--648},
  abstract = {An investigation is made of the geometric phases in a ring laser with
    counterpropagating waves. It is shown that the frequency splitting
    of the counterpropagating waves that appears in the case of radiation
    pulsations or is induced by displacement of an external mirror can
    be regarded as a manifestation of a geometric phase.},
  doi      = {10.1070/qe1997v027n07abeh001007},
}

@Article{ToDe97a,
  author  = {Toronov, V. Y. and Derbov, V. L.},
  title   = {Boundedness of attractors in the complex {Lorenz} model},
  journal = {Phys. Rev. E},
  year    = {1997},
  volume  = {55},
  pages   = {3689--3692},
  doi     = {10.1103/physreve.55.3689},
}

@Article{ToDe98,
  author  = {Toronov, V. Y. and Derbov, V. L.},
  title   = {Topological properties of laser phase},
  journal = {J. Optical Soc. of America B},
  year    = {1998},
  volume  = {15},
  pages   = {1282--1290},
  doi     = {10.1364/josab.15.001282},
}

@Article{tompaid96,
  author   = {S. Tompaidis},
  title    = {Numerical Study of Invariant Sets of a Quasi-periodic Perturbation of a Symplectic Map},
  journal  = {Experimental Mathematics},
  year     = {1996},
  volume   = {5},
  pages    = {211--230},
  abstract = {{N}ewton method is used to construct periodic orbits of
    longer and longer period to approach a invariant torus with specific
    rotation vector. Behavior after the break of a torus is described.},
  doi      = {10.1080/10586458.1996.10504589},
}

@Article{torc96,
  author   = {A. Torcini},
  title    = {Order parameter for the transition from phase to amplitude turbulence},
  journal  = {Phys. Rev. Lett.},
  year     = {1996},
  volume   = {77},
  pages    = {1047},
  abstract = {the maximal consersed phase gradient is introduced as an
    order parameter to characterize the transition from phase to defect
    turbulence in the cgle. it has a finite value in the pt regime and
    decreases to zero when the transition to defect turbulence is
    approached. a modified kse is able to reproduce the main feature of
    the stable waves and to explain their origin.},
  doi      = {10.1103/physrevlett.77.1047},
}

@Article{toricgl,
  author  = {P. Tak{\'a}{\v{c}}},
  title   = {Invariant 2-tori in the time-dependent {Ginzburg-Landau} equation},
  journal = {Nonlinearity},
  year    = {1992},
  volume  = {5},
  pages   = {289--321},
  doi     = {10.1088/0951-7715/5/2/002},
}

@Article{Townsend1961,
  author =  {A. A. Townsend},
  title =   {Equilibrium layers and wall turbulence},
  journal = {J. Fluid Mech.},
  year =    {1961},
  volume =  {11},
  pages =   {97--120},
  doi =     {10.1017/S0022112061000883}
}

@Book{Townsend76,
  Title                    = {The Structure of Turbulent Shear Flows, 2\textsuperscript{nd} Edn.},
  Author                   = {A. A. Townsend},
  Publisher                = {Cambridge Univ. Press},
  Year                     = {1976},
  Address                  = {Cambridge}
}

@Article{transgol,
  Title                    = {Transport by capillary waves, {Part II}: {Scalar} dispersion and the structure of the concentration field},
  Author                   = {R. Ramshankar and J. P. Gollub},
  Journal                  = {Phys. Fluids A},
  Year                     = {1991},
  Pages                    = {1344},
  Volume                   = {3}
}

@Article{Tre97,
  author  = {L. Trefethen},
  title   = {Pseudospectra of linear operators},
  journal = {SIAM Rev.},
  year    = {1997},
  volume  = {39},
  pages   = {383--406},
  doi     = {10.1137/s0036144595295284},
}

@Book{Tref00,
  Title                    = {Spectral methods in {M}atlab},
  Author                   = {L. N. Trefethen},
  Publisher                = {SIAM},
  Year                     = {2000},
  Address                  = {Philadelphia}
}

@Book{Trefethen97,
  Title                    = {Numerical Linear Algebra},
  Author                   = {L. N. Trefethen and D. Bau},
  Publisher                = {SIAM},
  Year                     = {1997},
  Addr                     = {Philadelphia}
}

@Unpublished{TrThNi14,
  Title                    = {Knotted strange attractors and matrix {Lorenz} systems},
  Author                   = {{Tranchida}, J. and {Thibaudeau}, P. and {Nicolis}, S.},
  Note                     = {\arXiv{1404.7774}},
  Year                     = {2014}
}

@Article{TTRD93,
  Title                    = {Hydrodynamic stability without eigenvalues},
  Author                   = {N. Trefethen and A. E. Trefethen and S. C. Reddy and T. A. Driscoll},
  Journal                  = {Science},
  Year                     = {1993},
  Pages                    = {578--584},
  Volume                   = {261}
}

@Article{TuckBar03,
  author   = {L. S. Tuckerman and D. Barkley},
  title    = {Symmetry breaking and turbulence in perturbed plane {Couette} flow},
  journal  = {Theor. Comp. Fluid Dyn.},
  year     = {2002},
  volume   = {16},
  pages    = {91--97},
  note     = {\arXiv{physics/0312051}},
  abstract = {Perturbed plane Couette flow containing a thin
spanwise-oriented ribbon undergoes a subcritical bifurcation at
Re = 230 to a steady 3D state containing streamwise vortices.
This bifurcation is followed by several others giving rise to a
fascinating series of stable and unstable steady states of
different symmetries and wavelengths. First, the
backwards-bifurcating branch reverses direction and becomes
stable near Re = 200. Then, the spanwise reflection symmetry is
broken, leading to two asymmetric branches which are themselves
destabilized at Re = 420. Above this Reynolds number, time
evolution leads first to a metastable state whose spanwise
wavelength is halved and then to complicated time-dependent
behavior. These features are in agreement with experiments.},
  doi      = {10.1007/s00162-002-0064-y},
}

@Article{TuckBar07,
  author  = {L. S. Tuckerman and D. Barkley},
  title   = {Mean flow and modeling of turbulent-laminar patterns in plane {Couette} flow},
  journal = {Springer Proc. Physics},
  year    = {2007},
  volume  = {117},
  pages   = {224--226},
  doi     = {10.1007/978-3-540-72604-3_71},
}

@Article{tucker2002ros,
  author    = {Tucker, W.},
  title     = {A rigorous {ODE} solver and {Smale's 14th} problem},
  journal   = {Found. Comp. Math},
  year      = {2002},
  volume    = {2},
  pages     = {53--117},
  address   = {New York},
  doi       = {10.1007/s002080010018},
  publisher = {Springer},
}

@Book{Tung85,
  Title                    = {Group Theory in Physics},
  Author                   = {Tung, W.-K.},
  Publisher                = {World Scientific},
  Year                     = {1985},
  Address                  = {Singapore}
}

@Article{umb94pat,
  author  = {F. Melo and P. Umbanhowar and H. L. Swinney},
  title   = {Transition to parametric wave patterns in a vertically oscillated granular layer},
  journal = {Phys. Rev. Lett.},
  year    = {1994},
  volume  = {72},
  pages   = {172--175},
  doi     = {10.1103/physrevlett.72.172},
}

@Article{VaKa11,
  Title                    = {Homoclinic tangle on the edge of shear turbulence},
  Author                   = {L. van Veen and G. Kawahara},
  Journal                  = {Phys. Rev. Lett.},
  Year                     = {2011},
  Pages                    = {114501},
  Volume                   = {107},
  DOI                      = {10.1103/PhysRevLett.107.114501}
}

@Article{VaKaAt11,
  Title                    = {On matrix-free computation of {2D} unstable manifolds},
  Author                   = {L. van Veen and G. Kawahara and M. Atsushi},
  Journal                  = {SIAM J. Sci. Comput.},
  Year                     = {2011},
  Pages                    = {25--44},
  Volume                   = {33},
  DOI                      = {10.1137/100789804}
}

@Phdthesis{vanDoorne04,
  Title                    = {Stereoscopic {PIV} on transition in pipe flow},
  Author                   = {C. W. H. van Doorne},
  School                   = {Delft},
  Year                     = {2004},
  URL                      = {www.ahd.tudelft.nl}
}

@Article{VC08,
  Title                    = {Stable manifolds and the transition to turbulence in pipe flow},
  Author                   = {Viswanath, D. and Cvitanovi{\'c}, P.},
  Journal                  = {J. Fluid Mech.},
  Year                     = {2009},
  Pages                    = {215--233},
  Volume                   = {627},
  DOI                      = {10.1017/S0022112009006041}
}

@Article{ViSa09,
  Title                    = {Complex singularities and the {Lorenz} attractor},
  Author                   = {Viswanath, D. and \c{S}ahuto\u{g}lu, S.},
  Journal                  = {SIAM Rev.},
  Year                     = {2010},
  Pages                    = {294--314},
  Volume                   = {52},
  DOI                      = {10.1137/090753474}
}

@Article{VK11,
  author  = {L. van Veen and G. Kawahara},
  title   = {Homoclinic tangle on the edge of shear turbulence},
  journal = {Phys. Rev. Lett.},
  year    = {2011},
  volume  = {107},
  pages   = {114501},
  doi     = {10.1103/physrevlett.107.114501},
}

@Article{VlToDe98,
  author  = {Vladimirov, A. G. and Toronov, V. Y. and Derbov, V. L.},
  title   = {The complex {Lorenz} model: {Geometric} structure, homoclinic bifurcation and one-dimensional map},
  journal = {Int. J. Bifur. Chaos},
  year    = {1998},
  volume  = {8},
  pages   = {723--729},
  doi     = {10.1142/s0218127498000516},
}

@Article{VlToDe98a,
  author  = {Vladimirov, A. G. and Toronov, V. Y. and Derbov, V. L.},
  title   = {Properties of the phase space and bifurcations in the complex {Lorenz} model},
  journal = {Technical Physics},
  year    = {1998},
  volume  = {43},
  pages   = {877--884},
  doi     = {10.1134/1.1259094},
}

@Article{vN32,
  author  = {J. von Neumann},
  title   = {{Zur Operatorenmethode In Der Klassischen Mechanik}},
  journal = {Ann. Math.},
  year    = {1932},
  volume  = {33},
  pages   = {587--642},
  note    = {See additions \cite{vonNeum32}. Reprinted in \cite{Taub61}.},
  doi     = {10.2307/1968537},
}

@Article{vonNeum32,
  Title                    = {{Zus{\"a}tze zur Arbeit ``Zur Operatorenmethode in der klassischen Mechanik''}. ({German}) [{Additions} to the work ``{On} operator methods in classical mechanics'']},
  Author                   = {J. von Neumann},
  Journal                  = {Ann. Math.},
  Year                     = {1932},
  Pages                    = {789--791},
  Volume                   = {33}
}

@Article{Voros87,
  Title                    = {Spectral functions, special functions and the {Selberg} zeta function},
  Author                   = {Voros, A.},
  Journal                  = {Commun. Math. Phys.},
  Year                     = {1987},
  Pages                    = {439--465},
  Volume                   = {110},
  DOI                      = {10.1007/BF01212422}
}

@Article{Voros92,
  Title                    = {Spectral zeta functions},
  Author                   = {Voros, A.},
  Journal                  = {Adv. Stud. Pure Math},
  Year                     = {1992},
  Pages                    = {327--358},
  Volume                   = {21}
}

@Book{Vorst03,
  Title                    = {Iterative Krylov Methods for Large Linear Systems},
  Author                   = {H. A. Vorst},
  Publisher                = {Cambridge Univ. Press},
  Year                     = {2003},
  Address                  = {Cambridge}
}

@Article{voth02,
  Title                    = {Experimental measurements of streching fields in fluid mixing},
  Author                   = {G. A. Voth and G. Haller and J. P. Gollub},
  Journal                  = {Phys. Rev. Lett.},
  Year                     = {2002},
  Pages                    = {254501},
  Volume                   = {88},
}

@Article{voth07,
  author  = {K.-Y. Chan and D. Stich and G. A. Voth},
  title   = {Real-time image compression for high-speed particle tracking},
  journal = {Rev. Sci. Inst.},
  year    = {2007},
  volume  = {78},
  pages   = {02374},
  doi     = {10.1063/1.2536719},
}

@Article{vp98,
  author   = {A. Varga and S. Pieters},
  title    = {Gradient-Based Approach to Solve Optimal Periodic Output Feedback Control Problems},
  journal  = {Automatica},
  year     = {1998},
  volume   = {34},
  pages    = {477--481},
  abstract = {A numerical method based on the gradient minimization
    and Periodic Schur Decomposition (PSD) is devised to solve the
    linear-quadratic (LQ) optimization problem the linear periodic
    discrete-time control systems. It can be used as a discretized
    version of the similar problem in the continuous regime.},
  doi      = {10.1016/s0005-1098(97)00214-8},
}

@Article{VSE98,
  author  = {J. Vollmer and T. M. Schneider and B. Eckhardt},
  title   = {Basin boundary, edge of chaos, and edge state in a two-dimensional model},
  journal = {New J. Phys.},
  year    = {2009},
  volume  = {11},
  pages   = {013040},
  doi     = {10.1088/1367-2630/11/1/013040},
}

@Article{W01,
  author  = {F. Waleffe},
  title   = {Exact coherent structures in channel flow},
  journal = {J. Fluid Mech.},
  year    = {2001},
  volume  = {435},
  pages   = {93--102},
  doi     = {10.1017/s0022112001004189},
}

@InProceedings{W02,
  Title                    = {Exact coherent structures and their instabilities: {Toward} a dynamical-system theory of shear turbulence},
  Author                   = {F. Waleffe},
  Booktitle                = {Proceedings of the International Symposium on ``Dynamics and Statistics of Coherent Structures in Turbulence: Roles of Elementary Vortices''},
  Year                     = {2002},
  Editor                   = {Shigeo Kida},
  Pages                    = {115--128},
  Publisher                = {National Center of Sciences, Tokyo, Japan}
}

@Article{W03,
  author  = {F. Waleffe},
  title   = {Homotopy of exact coherent structures in plane shear flows},
  journal = {Phys. Fluids},
  year    = {2003},
  volume  = {15},
  pages   = {1517--1543},
  doi     = {10.1063/1.1566753},
}

@InProceedings{W90b,
  Title                    = {On the origin of the streak spacing in turbulent shear flow},
  Author                   = {F. Waleffe},
  Booktitle                = {Annual Research Briefs},
  Year                     = {1990},
  Pages                    = {159--168},
  Publisher                = {Center for Turbulence Research, Stanford University}
}

@Article{W95a,
  author  = {F. Waleffe},
  title   = {Hydrodynamic stability and turbulence: {Beyond} transients to a {Self-Sustaining Process}},
  journal = {Stud. Applied Math.},
  year    = {1995},
  volume  = {95},
  pages   = {319--343},
  doi     = {10.1002/sapm1995953319},
}

@Article{W95b,
  author  = {F. Waleffe},
  title   = {Transition in shear flows: {Nonlinear} normality versus non-normal linearity},
  journal = {Phys. Fluids},
  year    = {1995},
  volume  = {7},
  pages   = {3060--3066},
  doi     = {10.1063/1.868682},
}

@Article{W97,
  Title                    = {On a {Self-Sustaining Process} in shear flows},
  Author                   = {F. Waleffe},
  Journal                  = {Phys. Fluids},
  Year                     = {1997},
  Pages                    = {883--900},
  Volume                   = {9},
  DOI                      = {10.1063/1.869185}
}

@Article{W98,
  author  = {F. Waleffe},
  title   = {Three-dimensional coherent states in plane shear flows},
  journal = {Phys. Rev. Lett.},
  year    = {1998},
  volume  = {81},
  pages   = {4140--4143},
  doi     = {10.1103/physrevlett.81.4140},
}

@Misc{Waleffe2005,
  author = {F. Waleffe},
  title =  {On transition scenarios in channel flows},
  year =   {1990},
  note =   {preprint},
  url =    {http://www.math.wisc.edu/~waleffe/M812/obtran.pdf}
}

@Unpublished{Waleffe90,
  Title                    = {Proposal for a self-sustaining mechanism in shear flows},
  Author                   = {F. Waleffe},
  Note                     = {Center for Turbulence Research, Stanford University/NASA Ames, unpublished preprint (1990)},
  Year                     = {1990},
  URL                      = {http://www.math.wisc.edu/~waleffe/ECS/sspctr90.pdf}
}

@Incollection{WalPant97,
  Title                    = {How streamwise Rolls and Streaks Self-Sustain in a Shear Flow},
  Author                   = {F. Waleffe and J. Kim},
  Booktitle                = {Self-Sustaining Mechanisms of Wall Bounded Turbulence},
  Publisher                = {Computational Mechanics Publications},
  Year                     = {1997},
  Address                  = {Southampton},
  Editor                   = {R. L. Panton},
  Pages                    = {385--422},
  URL                      = {www.math.wisc.edu/~waleffe/ECS/WK97.pdf}
}

@Phdthesis{Way08,
  Title                    = {Dynamics in the {Hopf} bundle, the geometric phase and implications for dynamical systems},
  Author                   = {Way, R.},
  School                   = {Dept. of Mathematics, U. Surrey},
  Year                     = {2008}
}

@Article{WBHZ14,
  Title                    = {Unstable flow structures in the {Blasius} boundary layer},
  Author                   = {H. Wedin and A. Bottaro and A. Hanifi and G. Zampogna},
  Journal                  = {Eur. Phys. J. E},
  Year                     = {2014},
  Pages                    = {34},
  Volume                   = {37},
  DOI                      = {10.1140/epje/i2014-14034-1}
}

@InProceedings{WeBoHaZa13,
  Title                    = {Three-dimensional nonlinear states in the {Blasius} boundary layer},
  Author                   = {Wedin, H. and Bottaro, A. and Hanifi, A. and Zampogna, G.},
  Booktitle                = {AIMETA Conference, 17-20 September 2013},
  Year                     = {2013},
  Address                  = {Torino, Italy},
  Volume                   = {XXI},
  URL                      = {http://www.aimeta.dicam.unibo.it/sites/www.aimeta.dicam.unibo.it/files/static/www.aimetatorino2013.it/cdrom/cdrom_pdf_fullpaper/000550004201.pdf}
}

@Article{WegSol02,
  Title                    = {On some mathematics for visualizing high dimensional data},
  Author                   = {Wegman, E. J. and Solka, J. L.},
  Journal                  = {Sankhy{\=a}: Indian J. Statistics, Ser. A},
  Year                     = {2002},
  Pages                    = {429--452}
}

@Article{Weil49,
  author  = {Weil, A.},
  title   = {Numbers of solutions of equations in finite fields},
  journal = {Bull. Amer. Math. Soc},
  year    = {1949},
  volume  = {55},
  pages   = {497--508},
  doi     = {10.1007/978-1-4757-1705-1_50},
}

@Article{weiss82,
  author  = {J. Weiss and M. Tabor and G. Carnevale},
  title   = {The {P}ainlev{\'e} property for partial differential equations},
  journal = {J. Math. Phys.},
  year    = {1983},
  volume  = {24},
  pages   = {522},
  doi     = {10.1063/1.525721},
}

@Article{weiss83,
  Title                    = {The {P}ainlev{\'e} for partial differential equations. {II}: {B\"a}cklund transformation, {Lax} pairs, and the {Schwarzian} derivative},
  Author                   = {J. Weiss},
  Journal                  = {J. Math. Phys.},
  Year                     = {1983},
  Pages                    = {1405},
  Volume                   = {24}
}

@Article{weiss86,
  author  = {J. Weiss},
  title   = {{B\"a}cklund transformation and the {P}ainlev{\'e} property},
  journal = {J. Math. Phys.},
  year    = {1986},
  volume  = {27},
  pages   = {1293},
  doi     = {10.1063/1.527134},
}

@Misc{WeissFeig,
  Title                    = {Feigenbaum constant},
  Author                   = {Weisstein, E. W.},
  Note                     = {mathworld.wolfram.com/FeigenbaumConstant.html},
  Year                     = {2012}
}

@Book{Weyl39,
  Title                    = {The Classical Groups: Their Invariants and Representations},
  Author                   = {Weyl, H.},
  Publisher                = {Princeton Univ. Press},
  Year                     = {1939},
  Address                  = {Princeton, NJ},
  ISBN                     = {9780691057569}
}

@Article{WFSBC15,
  author  = {Budanur, N. B. and Short, K. Y. and Farazmand, M. and Willis, A. P. and Cvitanovi{\'c}, P.},
  title   = {Relative periodic orbits form the backbone of turbulent pipe flow},
  journal = {J. Fluid Mech.},
  year    = {2017},
  volume  = {833},
  pages   = {274--301},
  doi     = {10.1017/jfm.2017.699},
}

@Article{WGW07,
  author  = {J. Wang and J. F. Gibson and F. Waleffe},
  title   = {Lower branch coherent states in shear flows: {Transition} and control},
  journal = {Phys. Rev. Lett.},
  year    = {2007},
  volume  = {98},
  pages   = {204501},
  note    = {\arXiv{physics/0703072}},
  doi     = {10.1103/physrevlett.98.204501},
}

@Article{WHC10,
  author  = {A. P. Willis and Y. Hwang and C. Cossu},
  title   = {Optimally amplified large-scale streaks and drag reduction in turbulent pipe flow},
  journal = {Phys. Rev. E},
  year    = {2010},
  volume  = {82},
  pages   = {036321},
  doi     = {10.1103/physreve.82.036321},
}

@Unpublished{WiAvCv12,
  Title                    = {Symmetries of turbulent flows: {How} to slice them},
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@Article{OZSVL17,
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@Article{ChTuBa17,
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@Article{Lasagna17,
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@Article{SheGra17,
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@Misc{TuGiWi17,
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  title  = {Stokes preconditioning for rapid calculation of steady states and traveling waves},
  year   = {2017},
  note   = {In preparation},
}

@Article{BudHof17a,
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  title   = {Complexity of the laminar-turbulent boundary in pipe flow},
  journal = {Physical Review Fluids},
  year    = {2018},
  volume  = {3},
  doi     = {10.1103/PhysRevFluids.3.054401},
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@Article{HaMeZaGa17,
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  title   = {Geometric decomposition of the conformation tensor in viscoelastic turbulence},
  journal = {J. Fluid Mech.},
  year    = {2017},
  note    = {To appear},
}

@Article{HamGay17,
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  title   = {A model-based scale by scale analysis of convective velocities in turbulent channels},
  journal = {J. Fluid Mech.},
  year    = {2017},
  note    = {To appear},
}

@Article{GuWiHoiAv17,
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  title   = {Transport Properties of the Azimuthal Magnetorotational Instability},
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  year    = {2017},
  volume  = {849},
  pages   = {92},
  doi     = {10.3847/1538-4357/aa917d},
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@Article{GuHoWiAv17,
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  volume  = {119},
  pages   = {164501},
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@Article{ZhScScXi17,
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  title   = {Distinct transition in flow statistics and vortex dynamics between low- and high-extent turbulent drag reduction in polymer fluids},
  journal = {J. Non-Newt. Fluid Mech.},
  year    = {2018},
  doi     = {10.1016/j.jnnfm.2018.03.017},
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@Article{Shankar_2017,
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  title   = {Topological Sound and Flocking on Curved Surfaces},
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  year    = {2017},
  volume  = {7},
  doi     = {10.1103/physrevx.7.031039},
}

@Article{DelMarVen17,
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  title   = {Topological origin of equatorial waves},
  journal = {Science},
  year    = {2017},
  pages   = {eaan8819},
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@Article{LecKer17,
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  title   = {Connection between nonlinear energy optimization and instantons},
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  note    = {To appear},
}

@Article{Bonolis05,
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@Article{KYdSOVML17,
  author  = {Krug, D. and Yang, X. I. A. and de Silva, C. M. and Ostilla-Mónico, R. and Verzicco, R. and Maru{\v{s}}i{\'c}, I. and Lohse, D.},
  title   = {Statistics of turbulence in the energy-containing range of {Taylor–Couette} compared to canonical wall-bounded flows},
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  year    = {2017},
  volume  = {830},
  pages   = {797–-819},
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@Article{ChKlWhMo17,
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@Article{Jimenez18,
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  year    = {2018},
  volume  = {842},
  pages   = {P1},
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@Article{JimenezJFM01,
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@Article{JimenezSimens2001,
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@Article{LiTom17b,
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@Article{LiTom17,
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  booktitle = {{Chaos: Classical and Quantum}},
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  year      = {2017},
  editor    = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
  address   = {Copenhagen},
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@Article{BevBas00,
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  title     = {{Chaotic Modelling and Simulation: Analysis of Chaotic Models, Attractors and Forms}},
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  pages    = {115002},
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  abstract = {We consider two stochastic variants of a Fibonacci sequence generated by x n + 1 = x n  ±  x n − 1 and x n + 1 = | x n  ±  x n − 1 |, respectively. Both the sequences begin with x 0 = x 1 = 1 and the ± signs in each iteration are independently chosen with probability p for +. We focus on the generalized Lyapunov exponent τ( q ) of { x n }, which gives the growth (when q > 0) or decay (when q < 0) rate of the ensemble average of | x n | q when n → ∞. By using a technique that reduces the non-increasing segments in the sequences, we map the random sequences to a product of random 2 × 2 nonnegative matrices, which is closely related to the Farey fraction spin chain model. We show that there exists a critical value q * below which the ensemble average of | x n | q is dominated by the non-increasing or linearly increasing samples, and hence τ( q ) is a constant. Both q * and τ( q ) (and its derivatives) can be precisely calculated with the transfer operator method. Moreover, when q is a positive integer, τ( q ) is exactly determined by a system of polynomial equations. The analysis can be readily generalized to the random sequence generated by x n + 1 = 2cos (π/ k ) x n ± x n − 1 (or x n + 1 = |2cos (π/ k ) x n ± x n − 1 |) for any integer k ≥ 3.},
}

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  title     = {{Hamiltonian Dynamical Systems: A Reprint Selection}},
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  year     = {2018},
  abstract = {Abstract A resolvent analysis of exact coherent states (ECS) of the Navier-Stokes equations (NSE) in a low Reynolds number channel is performed. The resolvent framework recasts the NSE into an input/output form in which the nonlinear term is treated as an internal forcing that drives the linear dynamics of the system. The framework has previously shown promise with regards to producing low-dimensional representations of ECS; here,},
  doi      = {10.1088/1873-7005/aab1ab},
}

@Article{SchDab16,
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  year    = {2017},
  volume  = {27},
  pages   = {053104},
  doi     = {10.1063/1.4982720},
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  pages   = {093102},
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}

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  number  = {2207},
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@Article{DetMor96,
  author  = {Dettmann, C. P. and Morriss, G. P.},
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@Article{Madelung27,
  author  = {Madelung, E.},
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@Incollection{CBMarkov,
  author    = {P. Cvitanovi{\'c}},
  title     = {Walkabout: {Transition} graphs},
  booktitle = {{Chaos: Classical and Quantum}},
  publisher = {Niels Bohr Inst.},
  year      = {2017},
  editor    = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
  address   = {Copenhagen},
  url       = {http://ChaosBook.org/paper.shtml#Marko},
}

@Book{GodRoy13,
  title     = {{Algebraic Graph Theory}},
  publisher = {Springer},
  year      = {2013},
  author    = {Godsil, C. and Royle, G. F.},
  address   = {New York},
  isbn      = {9781461301639},
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@InCollection{Jacobs15,
  author    = {H. O. Jacobs},
  booktitle = {{Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden}},
  publisher = {Springer},
  title     = {The role of {SE(3)} reduction for swimming in {Stokes} and {Navier-Stokes} fluids},
  year      = {2015},
  address   = {New York},
  editor    = {Chang, D. E. and Holm, D. D. and Patrick, G. and Ratiu, T.},
  doi       = {10.1007/978-1-4939-2441-7},
}

@Article{SYCMR15,
  author  = {Saiki, Y. and Yamada, M. and Chian, A. C.-L. and Miranda, R. A. and Rempel, E. L.},
  title   = {Reconstruction of chaotic saddles by classification of unstable periodic orbits: {Kuramoto-Sivashinsky} equation},
  journal = {Chaos},
  year    = {2015},
  volume  = {25},
  pages   = {103123},
  doi     = {10.1063/1.4933267},
}

@Misc{FarSer18,
  author = {Farazmand, M. and Serra, M.},
  title  = {Variational {Lagrangian} formulation of the {Euler} equations for incompressible flow: {A} simple derivation},
  year   = {2018},
  url    = {https://arxiv.org/abs/1807.02726},
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@Article{Arnold66,
  author  = {V. I. Arnol'd},
  title   = {Sur la g{\'{e}}om{\'{e}}trie diff{\'{e}}rentielle des groupes de {Lie} de dimension infinie et ses applications {\`{a}} l'hydrodynamique des fluides parfaits},
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@Article{HKTH16,
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  pages   = {063107},
  doi     = {10.1103/PhysRevE.93.063107},
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@Article{KKSGMM07,
  author    = {K. Krishan and H. Kurtuldu and M. F. Schatz and M. Gameiro and K. Mischaikow and S. Madruga},
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  journal   = {Phys. Fluids},
  year      = {2007},
  volume    = {19},
  pages     = {117105},
  doi       = {10.1063/1.2800365},
  publisher = {{AIP} Publishing},
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@Article{KLTSXPSM16,
  author  = {M. Kram{\'{a}}r and R. Levanger and J. Tithof and B. Suri and M. Xu and M. Paul and M. F. Schatz and K. Mischaikow},
  title   = {Analysis of {Kolmogorov} flow and {Rayleigh-B{\'{e}}nard} convection using persistent homology},
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  year    = {2016},
  volume  = {334},
  pages   = {82--98},
  doi     = {10.1016/j.physd.2016.02.003},
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