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Note = {\arXiv{nlin.CD/0706.1940}}
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Publisher = {Center for Nonlinear Science},
URL = {http://ChaosBook.org/tutorials}
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Author = {J. F. Gibson},
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Year = {2002}
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Note = {{\tt ChaosBook.org/projects}}
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Year = {2008},
Address = {Atlanta},
Note = {{\tt ChaosBook.org/projects/theses.html}}
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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
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the {K-L} decomposition systematically destroys dissipation and leads
to blow up solutions.},
doi = {10.1007/bf00916425},
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year = {2000},
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address = {New York},
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@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}
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Year = {2013},
Note = {\arXiv{1209.0593}},
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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.},
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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.},
}
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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},
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Abstract = {For c=1, the existence of two periodic orbits and two
heteroclinic orbits are proved and suggestions for
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DOI = {10.1016/0022-0396(89)90134-4}
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@Article{kstroy92,
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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}
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Title = {Dynamical systems approach to $1-d$ spatiotemporal chaos -- {A} cyclist's view},
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School = {School of Physics, Georgia Inst. of Technology},
Year = {2004},
Address = {Atlanta},
Note = {{\tt ChaosBook.org/projects/theses.html}}
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year = {2016}
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Note = {In preparation},
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Abstract = {Transition to turbulence in pipe flow is one of the most
fundamental and longest-standing problems in fluid dynamics.
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rates, but in practice pipe flow becomes turbulent even at moderate
speeds. This transition drastically affects the transport efficiency
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unstable traveling waves in computational studies of the
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confirming the proposed transition scenario and suggesting that the
dynamics associated with these unstable states may indeed capture the
nature of fluid turbulence.},
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volume = {11},
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note = {\arXiv{physics/0009017}},
abstract = {Constraints are found on the spatial variation of finite-time Lyapunov
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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},
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note = {\arXiv{nlin/0101012}},
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@Book{Tinkham,
Title = {Group Theory and Quantum Mechanics},
Author = {Tinkham, M.},
Publisher = {Dover},
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Address = {New York}
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@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.},
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url = {https://arXiv.org/abs/1601.00243},
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@PhdThesis{Tithof2016,
author = {J. Tithof},
title = {Dimensionality reduction of spatiotemporal chaos in a quasi-two-dimensional flow},
school = {Georgia Institute of Technology},
year = {2016}
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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},
}
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volume = {50},
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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},
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pulsations or is induced by displacement of an external mirror can
be regarded as a manifestation of a geometric phase.},
doi = {10.1070/qe1997v027n07abeh001007},
}
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journal = {Phys. Rev. E},
year = {1997},
volume = {55},
pages = {3689--3692},
doi = {10.1103/physreve.55.3689},
}
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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},
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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},
}
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Note = {\arXiv{1404.7774}},
Year = {2014}
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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,
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title = {Mean flow and modeling of turbulent-laminar patterns in plane {Couette} flow},
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year = {2007},
volume = {117},
pages = {224--226},
doi = {10.1007/978-3-540-72604-3_71},
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address = {New York},
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Publisher = {World Scientific},
Year = {1985},
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@Article{VaKaAt11,
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@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,
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Year = {2009},
Pages = {215--233},
Volume = {627},
DOI = {10.1017/S0022112009006041}
}
@Article{ViSa09,
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Year = {2010},
Pages = {294--314},
Volume = {52},
DOI = {10.1137/090753474}
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@Article{VK11,
author = {L. van Veen and G. Kawahara},
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journal = {Phys. Rev. Lett.},
year = {2011},
volume = {107},
pages = {114501},
doi = {10.1103/physrevlett.107.114501},
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author = {Vladimirov, A. G. and Toronov, V. Y. and Derbov, V. L.},
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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},
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@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}
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Volume = {110},
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@Book{Vorst03,
Title = {Iterative Krylov Methods for Large Linear Systems},
Author = {H. A. Vorst},
Publisher = {Cambridge Univ. Press},
Year = {2003},
Address = {Cambridge}
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@Article{voth02,
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Pages = {254501},
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@Article{voth07,
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year = {2007},
volume = {78},
pages = {02374},
doi = {10.1063/1.2536719},
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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,
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author = {F. Waleffe},
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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},
}
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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}
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@Article{W95a,
author = {F. Waleffe},
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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,},
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editor = {P. Cvitanovi{\'c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
address = {Copenhagen},
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