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year = {2017},
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note = {{\tt {Channelflow.org}}},
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homogeneous fluids the new upper bounds for the imaginary part ci of
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improve on previous results, giving values within 10% of the known
exact solution in several cases.},
doi = {10.1017/s0022112072000400},
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doi = {10.1137/050623401},
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@Phdthesis{Crofts07thesis,
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Publisher = {Center for Nonlinear Science},
URL = {http://ChaosBook.org/tutorials}
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Note = {{\tt ChaosBook.org/projects}}
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School = {School of Physics, Georgia Inst. of Technology},
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
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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
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doi = {10.1007/bf00916425},
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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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abstract = {Some properties of the KSe are discussed. A unified
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in the 1:n resonances of a fixed point.},
doi = {10.1088/0951-7715/5/4/004},
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DOI = {10.1016/0022-0396(92)90143-B}
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Year = {2004},
Address = {Atlanta},
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Abstract = { In almost every scientific field, an experiment involves
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its uncertainty; this is known as Parameter Determination. An example
would be the determination of the mass of the top quark, from data
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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}
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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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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 = {046214},
doi = {10.1103/PhysRevE.84.046214},
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pages = {017205},
doi = {10.1103/physreve.66.017205},
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Year = {1994},
Note = {\arXiv{hep-th/9301111}},
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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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Year = {1961},
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author = {Taylor, G. I.},
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Title = {The spectrum of turbulence},
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Title = {The diversity of steady state solutions of the complex {Ginzburg-Landau} equation},
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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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Title = {Spatial optimal growth in three-dimensional boundary layers},
Author = {Tempelmann, D. and Hanifi, A. and Henningson, D. S.},
Journal = {J. Fluid Mech.},
Year = {2010},
Pages = {5--37},
Volume = {646},
DOI = {10.1017/S0022112009993260}
}
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author = {R. Temam},
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@Article{tenenbaum2000,
author = {Tenenbaum, J. B. and Silva, V. de and Langford, J. C.},
title = {A global geometric framework for nonlinear dimensionality reduction},
journal = {Science},
year = {2000},
volume = {290},
pages = {2319--2323},
doi = {10.1126/science.290.5500.2319},
}
@Book{Tennekes72,
Title = {A First Course in Turbulence},
Author = {H. Tennekes and J.L. Lumley},
Publisher = {MIT Press},
Year = {1972},
Addr = {Cambridge, MA}
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@Article{teramura2014,
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title = {Damping filter method for obtaining spatially localized solutions},
journal = {Phys. Rev. E},
year = {2014},
volume = {89},
pages = {052910},
doi = {10.1103/physreve.89.052910},
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@Article{TFGphase97,
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year = {1997},
volume = {55},
pages = {5073},
abstract = {The phase gradient is used as an order parameter.
Different states are identified. In the PT region, a modified KSe
rules the phase dynamics of the CGLe.},
}
@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},
Author = {Thiffeault, J.-L.},
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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author = {Thiffeault, J.-L.},
title = {Derivatives and constraints in chaotic flows: asymptotic behaviour and a numerical method},
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year = {2002},
volume = {172},
pages = {139--161},
note = {\arXiv{nlin/0101012}},
doi = {10.1016/s0167-2789(02)00588-2},
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journal = {College Math. J.},
year = {1998},
volume = {29},
pages = {408--411},
doi = {10.2307/2687256},
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@Article{TI03,
author = {S. Toh and T. Itano},
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}
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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.},
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}
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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},
}
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author = {Toronov, V. Y. and Derbov, V. L.},
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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},
}
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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},
}
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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},
}
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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},
}
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pages = {289--321},
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Publisher = {Cambridge Univ. Press},
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Address = {Cambridge}
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Author = {L. N. Trefethen},
Publisher = {SIAM},
Year = {2000},
Address = {Philadelphia}
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@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}
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Journal = {Science},
Year = {1993},
Pages = {578--584},
Volume = {261}
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@Article{TuckBar03,
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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,
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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}
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year = {1994},
volume = {72},
pages = {172--175},
doi = {10.1103/physrevlett.72.172},
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Year = {2011},
Pages = {114501},
Volume = {107},
DOI = {10.1103/PhysRevLett.107.114501}
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@Article{VaKaAt11,
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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}
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@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,
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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},
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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}
}
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note = {To appear},
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note = {To appear},
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note = {To appear},
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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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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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author = {P. Cvitanovi{\'c}},
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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},
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editor = {Chang, D. E. and Holm, D. D. and Patrick, G. and Ratiu, T.},
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