% Encoding: UTF-8
@Article{AACI,
Title = {Recycling of strange sets: {I}. {Cycle} expansions},
Author = {R. Artuso and E. Aurell and P. Cvitanovi{\'{c}}},
Journal = {Nonlinearity},
Year = {1990},
Pages = {325--359},
Volume = {3}
}
@Article{AgCheb02,
Title = {On Determining the Eigenprojection and Components of a Matrix},
Author = {R. P. Agaev and P. Yu. Chebotarev},
Journal = {Automation and Remote Control},
Year = {2002},
Pages = {1537--1545},
Volume = {63}
}
@Article{AACII,
Title = {Recycling of strange sets: {II}. {Applications}},
Author = {R. Artuso and E. Aurell and P. Cvitanovi{\'{c}}},
Journal = {Nonlinearity},
Year = {1990},
Pages = {361},
Volume = {3}
}
@Article{AakHoeEhHe07,
Title = {Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes},
Author = {{\AA}kervik, E. and H{\oe}pffner, J. and Ehrenstein, U. and {Henningson}, D. S.},
Journal = {J. Fluid Mech.},
Year = {2007},
Pages = {305--314},
Volume = {579},
DOI = {10.1017/S0022112007005496}
}
@Article{Ab09,
Title = {Effective actions for statistical data assimilation},
Author = {Abarbanel, H. D. I.},
Journal = {Phys. Lett. A},
Year = {2009},
Pages = {4044--4048},
Volume = {373}
}
@Misc{Ab10,
Title = {Self-consistent stochastic model errors in data assimilation},
Author = {Abarbanel, H. D. I.},
Note = {\arXiv{1012.2031}},
Year = {2010}
}
@Article{AbBrKe91,
Title = {{Lyapunov} exponents in chaotic systems: their importance and their evaluation using observed data},
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Journal = {Int. J. Mod. Phys. B},
Year = {1991},
Pages = {1347--1375},
Volume = {5}
}
@Article{AbBrKe91a,
Title = {Variation of {Lyapunov} exponents on a strange attractor},
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Journal = {J. Nonlinear Sci.},
Year = {1991},
Pages = {175--199},
Volume = {1}
}
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Publisher = {Cambridge Univ. Press},
Year = {1992},
Address = {Cambridge}
}
@Book{AbrSte64,
Title = {Handbook of Mathematical Functions},
Editor = {Abramowitz, M. and Stegun, I. A.},
Publisher = {Dover},
Year = {1964},
Address = {New York}
}
@Article{ACFK09,
Title = {Dynamical state and parameter estimation},
Author = {H. D. I. Abarbanel and D. R. Creveling and R. Farsian and M. Kostuk},
Journal = {SIAM J. Appl. Math.},
Year = {2009},
Pages = {1341--1381},
Volume = {8}
}
@Article{ACHKW11,
Title = {Revealing the state space of turbulent pipe flow by symmetry reduction},
Author = {Willis, A. P. and Cvitanovi{\'c}, P. and Avila, M.},
Journal = {J. Fluid Mech.},
Year = {2013},
Note = {\arXiv{1203.3701}},
Pages = {514--540},
Volume = {721},
DOI = {10.1017/jfm.2013.75}
}
@Article{ACK89,
Title = {Phase transitions on strange irrational sets},
Author = {Artuso, R. and Cvitanovi\'{c}, P. and Kenny, B. G.},
Journal = {Phys. Rev. A},
Year = {1989},
Pages = {268--281},
Volume = {39},
DOI = {10.1103/PhysRevA.39.268}
}
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Title = {Characterization of unstable periodic orbits in chaotic attractors and repellers},
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Journal = {Phys. Rev. Lett.},
Year = {1989},
Pages = {819},
Volume = {63}
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@Article{AFT06,
Title = {{H}ermite function expansions versus {H}ermite polynomial expansions},
Author = {Abu-Falahah, I. and Torrea, J. L.},
Journal = {Glasgow Math. J.},
Year = {2006},
Pages = {203--215},
Volume = {48}
}
@Article{AG93,
Title = {Fast dynamos and determinants of singular intergral-opertaors},
Author = {E. Aurell and A. D. Gilbert},
Journal = {Geophys. Astrophys. Fluid Dyn.},
Year = {1993},
Pages = {5--32},
Volume = {73}
}
@Article{AGHks88,
Title = {Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry},
Author = {D. Armbruster and J. Guckenheimer and P. Holmes},
Journal = {Physica D},
Year = {1988},
Pages = {257--282},
Volume = {29},
Abstract = {Consider the normal form on the center manifold for the KSe and
discuss bifurcations with intrinsic O(2) symmetry.}
}
@Article{AGHks89,
Title = {{Kuramoto-Sivashinsky} Dynamics on the Center-Unstable manifold},
Author = {D. Armbuster and J. Guckenheimer and P. Holmes},
Journal = {SIAM J. Appl. Math.},
Year = {1989},
Pages = {676--691},
Volume = {49}
}
@Article{AhRo07,
Title = {Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators},
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Journal = {J. Fluid Mech.},
Year = {2007},
Note = {\arXiv{0902.1207}},
Pages = {447--478},
Volume = {645}
}
@Article{AKcgl02,
Title = {The World of the Complex {G}inzburg-{L}andau Equation},
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Pages = {99--143},
Volume = {74}
}
@Misc{akesson98discrete,
Title = {Discrete eigenfunction expansion of multi-dimensional {Brownian} motion and the {Ornstein-Uhlenbeck} process},
Author = {Akesson, F. and Lehoczky, J. P. },
Note = {preprint, Math. Sciences, Carnegie Mellon Univ.},
Year = {1998}
}
@Article{AlBa11,
Title = {Normal forms approach to diffusion near hyperbolic equilibria},
Author = {Almada Monter, S. A. and Bakhtin, Y.},
Journal = {Nonlinearity},
Year = {2011},
Number = {6},
Pages = {1883},
Volume = {24},
DOI = {10.1088/0951-7715/24/6/011}
}
@Article{AlEyRe05,
Title = {Accelerated {Monte Carlo} for optimal estimation of time series},
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Year = {2005},
Pages = {1331--1345},
Volume = {119}
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Journal = {Phys. Scr.},
Year = {1985},
Pages = {15},
Volume = {31},
Abstract = {The systematic appearance of limit cycles, homoclinic and
heteroclinic orbits in the Lorenze model has been mapped in the r-b
parameter plane at $\sigma = 10$. New structures in the model are thereby
revealed.}
}
@Article{Alfv50,
Title = {Discussion of the origin of the terrestrial and solar magnetic fields},
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Year = {1950},
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@Misc{AlHa11,
Title = {Quantum chaos and effective thermalization},
Author = {Altland, A. and Haake, F.},
Note = {\arXiv{1110.1270}},
Year = {2011}
}
@Article{AltTel08,
Title = {Poincar\'e recurrences and transient chaos in systems with leaks},
Author = {Altmann, E. G. and T\'el, T.},
Journal = {Phys. Rev. E},
Year = {2009},
Note = {\arXiv{0808.3785}},
Pages = {016204},
Volume = {79},
Abstract = {Simulating observational and experimental
situations, we consider a leak in the phase space of a chaotic
dynamical system. We obtain an expression for the escape rate
of the survival probability applying the theory of transient
chaos. This expression improves previous estimates based on the
properties of the closed system and explains dependencies on
the position and size of the leak and on the initial ensemble.
With a subtle choice of the initial ensemble, we obtain an
equivalence to the classical problem of Poincare recurrences in
closed systems, which is treated in the same framework.
Finally, we show how our results apply to weakly chaotic
systems and justify a split of the invariant saddle in
hyperbolic and non-hyperbolic components, related,
respectively, to the intermediate exponential and asymptotic
power-law decays of the survival probability.},
Numpages = {12}
}
@Incollection{ampatt,
Title = {Amplitude equations for pattern forming systems},
Author = {M. van Hecke and P. C. Hohenberg and W. van Saarloos},
Booktitle = {Fundamental Problems in Statistical Mechanics},
Publisher = {North-Holland},
Year = {1994},
Address = {Amsterdam},
Editor = {H. van Beijeren and M. H. Ernst},
Volume = {VIII}
}
@Misc{angelo-2003,
Title = {Ordered quantization and the {E}hrenfest time scale},
Author = {R. M. Angelo and L. Sanz and K. Furuya},
Note = {\arXiv{quant-ph/0302020}}
}
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Title = {The periodic orbits of an area preserving twist-map},
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Year = {1988},
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Volume = {115}
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Title = {{Lyapunov}, {Lanczos}, and inertia},
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Pages = {137--150},
Volume = {326}
}
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Author = {K. An and J. Yang},
Booktitle = {Trends in Nano- and Micro-Cavities},
Year = {2011},
Editor = {Kwon, O. D. and Lee, B. and An, K.},
Note = {\arXiv{0912.0164v3}},
Pages = {40--61},
Publisher = {Bentham Science}
}
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Title = {Sampling the posterior: {An} approach to non-{Gaussian} data assimilation},
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Year = {2007},
Pages = {50--64},
Volume = {230}
}
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Title = {Mathematical Methods of Classical Mechanics},
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Publisher = {Springer},
Year = {1978},
Address = {Berlin}
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Title = {Kolmogorov's hydrodynamic attractors},
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Year = {1991},
Pages = {19--22},
Volume = {434},
Abstract = {A review of Kolmogorov's efforts relating the Navier-Stokes equation
to the theory of dynamical system. Several interesting questions regarding the
connection are exposed.}
}
@Book{arnold92,
Title = {Ordinary Differential Equations},
Author = {V. I. Arnol'd},
Publisher = {Springer},
Year = {1992},
Address = {Berlin}
}
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Title = {Stochastic Differential Equations: Theory and Applications},
Author = {Arnold, L.},
Publisher = {Wiley},
Year = {1974},
Address = {New York},
ISBN = {9780486482361}
}
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Title = {Cycle expansions for intermittent maps},
Author = {R. Artuso and P. Cvitanovi\'{c} and G. Tanner},
Journal = {Proc. Theo. Phys. Supp.},
Year = {2003},
Pages = {1--21},
Volume = {150}
}
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Title = {A study of hairpin vortices in a laminar boundary layer},
Author = {Acarlar, M.S. and Smith, C.R.},
Journal = {J. Fluid Mech.},
Year = {1987},
Pages = {1-41 and 45-83},
Volume = {175}
}
@Article{Ascher95,
Title = {Implicit-explicit methods for time-dependent partial differential equations},
Author = {U. Ascher and S. Ruuth and B. Wetton},
Journal = {SIAM J. Numer. Anal.},
Year = {1995},
Pages = {797--823},
Volume = {32}
}
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Title = {The design of experiments for discriminating between two rival models},
Author = {Atkinson, A. C. and Fedorov, V. V.},
Journal = {Biometrika},
Year = {1975},
Pages = {57--70},
Volume = {62},
Abstract = {Sequential experimental designs are developed for
discriminating between two rival regression models, which
need not be linear in the parameters. Nonsequential
procedures are also derived and numerical examples given of
both classes of design. }
}
@Article{aub95ant,
Title = {Anti-integrability in dynamical and variational problems},
Author = {S. Aubry},
Journal = {Physica D},
Year = {1995},
Pages = {284--296},
Volume = {86}
}
@Article{Aubry88,
Title = {The dynamics of coherent structures in the wall region of turbulent boundary layer},
Author = {N. Aubry and P. Holmes and J.L. Lumley and E. Stone},
Journal = {J. Fluid Mech.},
Year = {1988},
Pages = {115--173},
Volume = {192}
}
@Incollection{autod89,
Title = {On automatic differentiation},
Author = {A. Griewank},
Booktitle = {Mathematical Programming: Recent Developments and Applications},
Publisher = {Kluwer},
Year = {1989},
Address = {Amsterdam},
Editor = {M. Iri and K. Tanabe},
Pages = {83--108}
}
@Article{ave99,
Title = {Average patterns of spatiotemporal chaos: {A} boundary effect},
Author = {V. M. Egu\'{i}luz and P. Alstr{\o}m and E. Hern\'{a}ndez-Garc\'{\i}a and O. Piro},
Journal = {Phys. Rev. E},
Year = {1999},
Pages = {2822},
Volume = {59}
}
@Misc{aver_dasbuch,
Author = {P. Cvitanovi\'{c}},
Note = {Chapter {\em ``Averaging''}, in \refref{DasBuch}},
Year = {2012}
}
@Book{AvrFiHow13,
Title = {Analytic Perturbation Theory and Its Applications},
Author = {K. E. Avrachenkov and J. A. Filar and P. G. Howlett},
Publisher = {SIAM},
Year = {2013},
Address = {Philadelphia},
DOI = {10.1137/1.9781611973143}
}
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Title = {Linear Operators in Spaces with an Indefinite Metric},
Author = {T. Ya. Azizov and I. S. Iohvidov},
Publisher = {Wiley},
Year = {1989},
Address = {New York}
}
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Title = {The evolution of disturbances in shear flows at high {R}eynolds numbers},
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Year = {1984},
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Volume = {70}
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Title = {Fast dynamo action in unsteady flows and maps in three dimensions},
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Year = {1987},
Pages = {1573},
Volume = {59}
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@Article{BaChi88,
Title = {Construction of fast dynamos using unsteady flows and maps in three dimensions},
Author = {Bayly, B. J. and Childress, S.},
Journal = {Geophys. Astrophys. Fluid Dyn.},
Year = {1988},
Pages = {211--240},
Volume = {44}
}
@Article{Backer1,
Title = {Regular-to-chaotic tunneling rates using a fictitious integrable system},
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Year = {2008},
Volume = {100}
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@Article{BaHuVa14,
Title = {Pseudo-orbits, stationary measures and metastability},
Author = {W. Bahsoun and H. Hu and S. Vaienti},
Journal = {Dyn. Sys.},
Year = {2014},
Pages = {322--336},
Volume = {29},
Abstract = { We characterize absolutely continuous stationary measures (acsms) of randomly perturbed dynamical systems in terms of pseudo-orbits linking the ergodic components of absolutely continuous invariant measures (acims) of the unperturbed system. We focus on those components, called least elements, which attract pseudo-orbits. Under the assumption that the transfer operators of both systems, the random and the unperturbed, satisfy a uniform Lasota--Yorke inequality on a suitable Banach space, we show that each least element is in a one-to-one correspondence with an ergodic acsm of the random system. },
DOI = {10.1080/14689367.2014.890172}
}
@Book{baker:makemson,
Title = {An Introduction to Astrodynamics},
Author = {Baker Jr., R. M. L. and M. W. Makemson},
Publisher = {Academic},
Year = {1960},
Address = {New York}
}
@InProceedings{Baladi_lr,
Title = {Linear response, or else},
Author = {Baladi, V.},
Booktitle = {Proceedings of the International Congress of Mathematicians},
Year = {2014},
Address = {Seoul, South Korea},
Editor = {Jang, S. Y. and Kim, Y. R. and Lee, D.-W. and Yie, I.},
Note = {\arXiv{1408.2937}},
Pages = {525--545},
Publisher = {Kyung Moon Sa Co. Ltd.}
}
@Article{BarChor98,
Title = {New perspectives in turbulence: scaling laws, asymptotics, and intermittency},
Author = { G.I. Barenblatt and A.J. Chorin },
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Year = {1998},
Pages = {265--291},
Volume = {40}
}
@Article{BarHuss11,
Title = {The {Moore-Penrose} pseudoinverse. {A} tutorial review of the theory},
Author = {Barata, J. C. A. and Hussein, M. S.},
Journal = {Braz. J. Phys.},
Year = {2012},
Note = {\arXiv{1110.6882}},
Pages = {146--165},
Volume = {42},
Abstract = {In the last decades the Moore-Penrose pseudoinverse
has found a wide range of applications in many areas of Science and
became a useful tool for physicists dealing, for instance, with optimization
problems, with data analysis, with the solution of linear integral equations,
etc. The existence of such applications alone should attract the interest
of students and researchers in the Moore-Penrose pseudoinverse and
in related subjects, like the singular values decomposition theorem for
matrices. In this note we present a tutorial review of the theory of the
Moore-Penrose pseudoinverse. We present the first definitions and
some motivations and, after obtaining some basic results, we center
our discussion on the Spectral Theorem and present an algorithmically
simple expression for the computation of the Moore-Penrose
pseudoinverse of a given matrix.
We present a complete and self-contained
tutorial review, useful for those more devoted to applications, for
those more theoretically oriented and for those who already have
some working knowledge of the subject.}
}
@Article{Barkley94,
Title = {{Euclid}ean symmetry and the dynamics of rotating spiral waves},
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Year = {1994},
Pages = {164--167},
Volume = {72}
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@Article{Barraud77,
Title = {A numerical algorithm to solve {$A^T XA - X = Q$}},
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Year = {1977},
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Volume = {22}
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@Article{BasLee08,
Title = {Asymptotic properties of an estimator of the drift coefficients of multidimensional {Ornstein-Uhlenbeck} processes that are not necessarily stable},
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Journal = {Electronic J. Statistics},
Year = {2008},
Note = {\arXiv{0805.4535}},
Pages = {1309--1344},
Volume = {2}
}
@Article{BaSt72,
Title = {Algorithm 432: {Solution} of the matrix equation {AX + XB = C}},
Author = {R. H. Bartels and G. W. Stewart},
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Year = {1972},
Pages = {820--826},
Volume = {15}
}
@Article{BaVa13,
Title = {Escape rates formulae and metastablilty for randomly perturbed maps},
Author = {W. Bahsoun and S. Vaienti},
Journal = {Nonlinearity},
Year = {2013},
Pages = {1415},
Volume = {26},
Abstract = {We provide escape rates formulae for piecewise expanding interval maps with {\textquoteleft}random holes{\textquoteright}. Then we obtain rigorous approximations of invariant densities of randomly perturbed metastable interval maps. We show that our escape rates formulae can be used to approximate limits of invariant densities of randomly perturbed metastable systems.},
DOI = {10.1088/0951-7715/26/5/1415}
}
@Article{BaVo86,
Title = {Chaos on the pseudosphere},
Author = {N. Balasz and A. Voros},
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Year = {1986},
Pages = {109--240},
Volume = {143}
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@Article{BBB,
Title = {Orbital motion of spiral waves in excitable media},
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Year = {2010},
Pages = {58302},
Volume = {104}
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@Article{BBW09,
Title = {Noise management by molecular networks},
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Journal = {PloS Computational Biology},
Year = {2009},
Pages = {e1000506},
Volume = {5}
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@Article{BC89,
Title = {A mean flow first harmonic theory for hydrodynamic instabilities},
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Volume = {80}
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@Article{BCOR11,
Title = {Bifurcations in the {Lozi} map},
Author = {Botella-Soler, V. and Castelo, J. M. and Oteo, J. A. and Ros, J.},
Journal = {J. Phys. A},
Year = {2011},
Note = {\arXiv{1102.0034}},
Pages = {D5101},
Volume = {44}
}
@Book{BeGe06,
Title = {Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach},
Author = {Berglund, N. and Gentz, B.},
Publisher = {Springer},
Year = {2005},
Address = {Berlin},
ISBN = {9781846281860}
}
@Article{BeHa87,
Title = {Optimal projection equations for discrete-time fixed-order dynamic compensation of linear systems with multiplicative white noise},
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Journal = {Int. J. Control},
Year = {1987},
Pages = {65--73},
Volume = {46}
}
@Article{BeiramiZigzag,
Title = {Zigzag map: a variability-aware discrete-time chaotic-map truly random number generator},
Author = {Beirami, A. and Nejati, H. and Ali, W. H.},
Journal = {Electron. Lett.},
Year = {2012},
Pages = {1537--1538},
Volume = {48},
DOI = {10.1049/el.2012.2762}
}
@Book{Bell57,
Title = {{Dynamic Programming}},
Author = {Bellman, R.},
Publisher = {Princeton Univ. Press},
Year = {1957},
Address = {Princeton},
ISBN = {9780486317199}
}
@Book{Bell60,
Title = {{Introduction to Matrix Analysis}},
Author = {Bellman, R.},
Publisher = {SIAM},
Year = {1997},
Address = {Philadelphia},
Edition = {2},
ISBN = {9781611971170}
}
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Title = {On the indices of instability of unbounded operators},
Author = {V. S. Belonosov},
Journal = {Dokl. Akad. Nauk SSSR},
Year = {1983},
Pages = {11--14},
Volume = {273}
}
@Article{Belonosov84,
Title = {On instability indices of unbounded operators. {I}},
Author = {V. S. Belonosov},
Journal = {Some Applications of Functional Analysis to Problems in Mathematical Physics [in Russian], Novosibirsk},
Year = {1984},
Pages = {25--51},
Volume = {2}
}
@Article{Belonosov85,
Title = {On instability indices of unbounded operators. {I}},
Author = {V. S. Belonosov},
Journal = {Some Applications of Functional Analysis to Problems in Mathematical Physics [in Russian], Novosibirsk},
Year = {1985},
Pages = {5--33},
Volume = {2}
}
@Article{Bender87,
Title = {{Lyapunov}-like equations and reachability / observabiliy {Gramians} for descriptor systems},
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Journal = {IEEE Trans. Auto. Contr.},
Year = {1987},
Pages = { 343--348},
Volume = {32}
}
@Article{benks66,
Title = {Long waves in liquid films},
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@Article{Berkooz94,
Title = {Observations regarding ``{C}oherence and chaos in a model of turbulent boundary layer'' by {X}. {Z}hou and {L}. {S}irovich},
Author = {G. Berkooz and P. Holmes and J. L. Lumley and N. Aubry and E. Stone},
Journal = {Phys. Fluids},
Year = {1994},
Pages = {1574--1578},
Volume = {6}
}
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Title = {Semiclassical theory of spectral rigidity},
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Year = {1985},
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Volume = {400}
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Title = {Catastrophe optics: Morphologies of caustics and their diffraction patterns},
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Year = {1980},
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Volume = {18}
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@Misc{BeSa10,
Title = {A {Galerkin-{Newton}-ADI} method for solving large-scale algebraic {Riccati} equations},
Author = {P. Benner and J. Saak},
Year = {2010}
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@Book{Bevington,
Title = {Data Reduction and Error Analysis for the Physical Sciences},
Author = {P. R. Bevington and D. K. Robinson},
Publisher = {McGraw-Hill},
Year = {1992},
Address = {New York}
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Year = {2001},
Addendum = {See also \weblink{www.mathstat.concordia.ca/pg/bilbollt.html}.},
Pages = {365},
Volume = {12},
URL = {http://people.clarkson.edu/~bolltem/Papers/csf_2001.pdf}
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@Misc{BoGaMcK09,
Title = {Limit cycles, complex {Floquet} multipliers and intrinsic noise},
Author = {Boland, R. P. and Galla, T. and {McKane}, A. J.},
Note = {\arXiv{0903.5248}},
Year = {2009},
Abstract = {We study the effects of intrinsic noise on
chemical reaction systems, which in the deterministic limit
approach a limit cycle in an oscillatory manner. Previous
studies of systems with an oscillatory approach to a fixed
point have shown that the noise can transform the oscillatory
decay into sustained coherent oscillations with a large
amplitude. We show that a similar effect occurs when the
stable attractors are limit cycles. We compute the
correlation functions and spectral properties of the
fluctuations in suitably co-moving Frenet frames for several
model systems including driven and coupled Brusselators, and
the {Willamowski-Roessler} system. The effect occurs whenever
the Floquet multipliers of the limit cycle are complex, with
the amplitude of the oscillations increasing as the
instability boundary is approached.}
}
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title = {{Lyapunov} decoherence rate in classically chaotic systems},
year = {2010},
note = {\arXiv{1010.2115}},
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Volume = {11},
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@Book{BoyGor97,
Title = {Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension},
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Year = {1997},
Address = {Boston},
ISBN = {9781461220244}
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@Book{Brack,
Title = {Semiclassical Physics},
Author = {M. Brack and R. K. Bhaduri},
Publisher = {Addison-Wesley},
Year = {1997},
Address = {Reading, MA}
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Title = {Computational aspects of astrophysical {MHD} and turbulence},
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Publisher = {Taylor \& Francis},
Year = {2003},
Address = {London},
Editor = {A. Ferriz-Mas and M. N{\'{u}\~{n}ez}},
Note = {\arXiv{astro-ph/0109497}},
Pages = {269--344}
}
@Misc{Braun11,
Title = {Beyond the {Heisenberg} time: {Semiclassica} treatment of spectral correlations in chaotic systems with spin 1/2},
Author = {Braun, P.},
Note = {\arXiv{1110.4567}},
Year = {2011}
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Pages = {341--350},
Volume = {148}
}
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Title = {Applied Optimal Control},
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Publisher = {Hemisphere Publ.},
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@Article{broer97,
Title = {Algorithms for computing {N}ormally {H}yperbolic {I}nvariant {M}anifolds},
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@Misc{Brouwer07,
Title = {Semiclassical theory of the {E}hrenfest-time dependence of quantum transport},
Author = {{Brouwer}, P.~W.},
Note = {\arXiv{0706.1487}},
Year = {2007}
}
@Book{BrPetr,
Title = {The Theory of Open Quantum Systems},
Author = {H. P. Breuer and F. Petruccione},
Publisher = {Oxford Univ. Press},
Year = {2002},
Address = {Oxford}
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Year = {2003},
Pages = {152--167},
Volume = {174}
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Title = {Modulated amplitude waves and defect formation in the one-dimensional complex {{G}inzburg-{L}andau} equation},
Author = {L. Brusch and A. Torcini and M. van Hecke and M. G. Zimmermann and M. B\"{a}r},
Journal = {Physica D},
Year = {2001},
Pages = {127--148},
Volume = {160}
}
@Article{BTU,
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Year = {1993},
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@Article{BuChu11,
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Author = {A. Burchard and M. Chugunova},
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Year = {2011},
Note = {\arXiv{0905.1834}},
Pages = {367--388},
Volume = {43},
DOI = {10.1137/090759495}
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@Article{BuFengDong12,
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Year = {2012},
Pages = {631--640},
Volume = {38},
DOI = {10.1007/s12190-011-0501-4}
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@Article{BuKe03,
Title = {Estimating good discrete partitions from observed data: {Symbolic} false nearest neighbors},
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Journal = {Phys. Rev. Lett.},
Year = {2003},
Note = {\arXiv{nlin/0304054}},
Pages = {084102},
Volume = {91}
}
@InProceedings{BuKe03a,
Title = {Estimating good discrete partitions from observed data: symbolic false nearest neighbors},
Author = {Kennel, M. B. and Buhl, M.},
Booktitle = {Experimental Chaos: Seventh Experimental Chaos Conference},
Year = {2003},
Address = {Melville, NY},
Editor = {Ko\v{c}arev, L. and Carroll, T. L. and Gluckman, B. J. and Boccaletti, S. and Kurths, J.},
Pages = {380--380},
Publisher = {American Institute of Physics}
}
@Article{BuKe05,
Title = {Statistically relaxing to generating partitions for observed time-series data},
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Year = {2005},
Pages = {046213},
Volume = {71},
DOI = {10.1103/PhysRevE.71.046213}
}
@Article{BuMoMe12,
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Journal = {Chaos},
Year = {2012},
Note = {\arXiv{1206.3164}},
Pages = {047510},
Volume = {22},
DOI = {10.1063/1.4772195},
Numpages = {33}
}
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Volume = {451}
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@Misc{CaLiZh16,
Title = {Explore Stochastic Instabilities of Periodic Points by Transition Path Theory},
Author = {Y. Cao and L. Lin and X. Zhou},
Note = {\arXiv{1601.01424}}
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Pages = {040403},
Volume = {89},
Abstract = {By using numerical and semiclassical methods, we
evaluate the quantumbreaking, or Ehrenfest time for a
wave packet localized around classicalequilibrium
points of autonomous one-dimensional systems with
polynomialpotentials. We find that the Ehrenfest time
diverges logarithmically with theinverse of the Planck
constant whenever the equilibrium point is
exponentiallyunstable. For stable equilibrium points,
we have a power law divergence withexponent determined
by the degree of the potential near the equilibrium
point.}
}
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Title = {On the analytical and numerical approximation of invariant manifolds},
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Year = {1989},
Editor = {D. Benest and C. Froeschl\'{e}},
Pages = {285--329}
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@Misc{CatCaf07,
Title = {From Information Geometry to {N}ewtonian Dynamics},
Author = {A. Caticha and C. Cafaro},
Note = {\arXiv{0710.1071}},
Year = {2007}
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@Article{CB92,
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Year = {1992},
Pages = {511--527},
Volume = {234}
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@Article{CC92,
Title = {Periodic orbit quantization of the anisotropic {Kepler} problem},
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Journal = {Chaos},
Year = {1992},
Pages = {61},
Volume = {2}
}
@Article{CCKFNRS12,
Title = {Neural population dynamics during reaching},
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Journal = {Nature},
Year = {2012},
Number = {7405},
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Title = {A new property of a class of {J}acobi polynomials},
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Year = {2005},
Pages = {3551--3560},
Volume = {133 }
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@Article{CDKK10,
Title = {A fast explicit operator splitting method for passive scalar advection},
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@Article{CEEks93,
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Journal = {Physica D},
Year = {1993},
Pages = {321--326},
Volume = {67},
Abstract = {Proved the analyticity of the solution of the KSe on a periodic interval
near the real axis and the width of the region is estimated. Numerical calculations
show that this width should not depend on the size of the interval.}
}
@Article{CEEksgl93,
Title = {A global attracting set for the {Kuramoto-Sivashinsky} equation},
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Year = {1993},
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Volume = {152},
Abstract = {New bounds are given for the L^2 norm of solutions of the KSe (not
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}
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Title = {Cellular Automata and Complexity},
Author = {S. Wolfram},
Publisher = {Addison-Wesley},
Year = {1994},
Address = {Red-wood City}
}
@Book{CG96,
Title = {Stretch, twist, fold: the fast dynamo},
Author = {S. Childress and A. D. Gilbert},
Publisher = {Springer},
Year = {1995},
Address = {Berlin}
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Author = {C. Gol\'{e}},
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Year = {1992},
Pages = {441--448},
Volume = {210},
Abstract = {A auxiliary infinite gradient-like flow is constructed for
the area-preserving map such that orbits of the original system are rest points
of the new system. In this way, existence of orbits of any rotation number is
proved.}
}
@Article{ChaKer12,
Title = {Invariant recurrent solutions embedded in a turbulent two-dimensional {Kolmogorov} flow},
Author = {Chandler, G. J. and Kerswell, R. R.},
Journal = {J. Fluid Mech.},
Year = {2013},
Note = {\arXiv{1207.4682}},
Pages = {554--595},
Volume = {722},
DOI = {10.1017/jfm.2013.122}
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@Article{chand03tf,
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Journal = {Physica D},
Year = {2003},
Pages = {171},
Volume = {181},
Abstract = {Use wavelets to extract local frequencies and use the major
frequencies to identify the characteristics of phase space motion.}
}
@Article{Chandr43,
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@Article{chate94,
Title = {Spatiotemporal intermittency regimes of the one-dimensional complex {Ginzburg-Landau} equation},
Author = {H. Chat\'e},
Journal = {Nonlinearity},
Year = {1994},
Pages = {185--204},
Volume = {7},
Abstract = {In the Benjamin-Feir stable region, spatiotemporal intermittency
regimes are identified which consists of patches of 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 "laminar" state.}
}
@Article{chfield,
Title = {Chaotic Field Theory: a Sketch},
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Journal = {Physica A},
Year = {2000},
Note = {\arXiv{nlin/0001034}},
Pages = {61},
Volume = {288}
}
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Title = {Model reduction by the low-frequency approximation balancing method for unstable systems},
Author = {Chiu, T.},
Journal = {IEEE Trans. Automatic Control},
Year = {1996},
Pages = {995},
Volume = {AC-41}
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Title = {Optimal prediction with memory},
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Journal = {Physica D},
Year = {2002},
Pages = {239--257},
Volume = {166}
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@Book{Chossat94,
Title = {The {T}aylor-{C}ouette Problem},
Author = {P. Chossat and G. Iooss},
Publisher = {Springer},
Year = {1994}
}
@Article{Christiansen97,
Title = {Spatiotemporal chaos in terms of unstable recurrent patterns},
Author = {F. Christiansen and P. Cvitanovi\'{c} and V. Putkaradze},
Journal = {Nonlinearity},
Year = {1997},
Note = {\arXiv{chao-dyn/9606016}},
Pages = {55--70},
Volume = {10}
}
@Article{ChrLueOtt11,
Title = {Stretching and folding versus cutting and shuffling: {An} illustrated perspective on mixing and deformations of continua},
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Year = {2011},
Note = {\arXiv{1010.2256}},
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author = {P. Cvitanovi{\'c} and Y. Lan},
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editor = {N. Antoniou},
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pages = {313--325},
address = {Singapore},
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Pages = {P10026},
Volume = {2013},
Abstract = {Using path-integral methods, a formula is deduced for the
noise-induced escape rate from an attracting fixed point across an unstable
fixed point in one-dimensional maps. The calculation starts from the trace
formula for the eigenvalues of the Frobenius�Perron operator ruling the
time evolution of the probability density in noisy maps. The escape rate is
determined from the loop formed by two heteroclinic orbits connecting back
and forth the two fixed points of the one-dimensional map extended to a
two-dimensional symplectic map. The escape rate is obtained with the
expression of the prefactor to the Arrhenius�van�t�Hoff exponential
factor.},
DOI = {10.1088/1742-5468/2013/10/P10026}
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Year = {2010},
Pages = {2010--1},
URL = {gauss.uc3m.es/web/personal_web/fteran}
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Address = {Redwood City}
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Volume = {90}
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@Article{diag_Fred,
Title = {Spectrum of stochastic evolution operators: {Local} matrix representation approach},
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Journal = {Phys. Rev. E},
Year = {1999},
Note = {\arXiv{chao-dyn/9904027}},
Pages = {3936},
Volume = {60}
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Title = {{Lyapunov} exponential stability of linear hyperbolic systems of balance laws},
Author = {A. Diagne and G. Bastin and J-M Coron},
Journal = {Automatica},
Year = {2011},
Note = {accepted for publication}
}
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@Book{ditt01cq,
Title = {Classical and Quantum Dynamics: From Classical Paths to Path Integrals},
Author = {W. Dittrich and M. Reuter},
Publisher = {Springer},
Year = {2001},
Address = {New York}
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@Article{DLBconv92,
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Year = {1992},
Pages = {287--308},
Volume = {55},
Abstract = {Describe the occurrence of the spatio-temporal intermittency in a 1-d
convective system that shows time-independent patterns based on the amplitude equation
approach}
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@Article{doercgl87,
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@Article{doercgl88,
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Year = {1988},
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Volume = {1},
Abstract = {The stability of plane waves of the CGLe is analyzed in a
periodic domain. The mass and energy integrals are estimated to give an
upper bound of the magnitude of the field variable. After that, cone
condition is established for the CGLe which indicates the existence of
the inertial manifold and proves the existence of finite dimensionality of
the attractor in this infinite-dimensional system. The Lyapunov dimension
is estimated. Note the defition of Lyapunov exponents which is a little
different from usual.}
}
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Volume = {155}
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URL = {http://www.numdam.org/item?id=BSMF_1957__85__431_0}
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@Book{DR81,
Title = {Hydrodynamic Stability},
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Publisher = {Cambridge Univ. Press},
Year = {1981},
Address = {Cambridge, UK}
}
@Article{Drazin58,
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Year = {1996},
Pages = {204--210},
Volume = {211},
Keywords = {Frobenius-Perron, piecewise-linear, map}
}
@Article{DrTe11,
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Pages = {056203},
Volume = {83}
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@Article{du87,
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Year = {1987},
Pages = {1731},
Volume = {58},
Abstract = {A quantitative theory has been developed which shows that each
classical closed orbit that begins and ends near the nucleus contributes
an oscillatory term to the average oscillator strength.}
}
@Book{Durb_Koop,
Title = {Time series analysis by state space methods},
Author = {J. Durbin and S. J. Koopman},
Publisher = {Oxford Univ. Press},
Year = {2012},
Address = {Oxford}
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@Article{DV2000,
Title = {Phase space analysis of a dynamical model for the subcritical transition to turbulence in plane {C}ouette flow},
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Year = {2000},
Pages = {377--381},
Volume = {14}
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Year = {1993},
Pages = {1646},
Volume = {48}
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@Article{DyMiSm94,
Title = {Observable and hidden singular features of large fluctuations in nonequilibrium systems},
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Year = {1994},
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Year = {1987},
Pages = {87},
Volume = {A413}
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@Article{Eben02,
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Year = {2002},
Pages = {557--563},
Volume = {35},
Abstract = {A class of stochastic integer sequences are introduced with
every element being a sum of two previous elements. The interplay between
randomness and memory leads to a wide variety of behaviors rangin from
stretched exponential to log-normal to algebraic growth. The set of all
possible sequence values has an intricate structure.}
}
@Article{eck81,
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@Article{eckcgl,
Title = {The {E}ckhaus instability for traveling waves},
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Year = {1992},
Pages = {269--286},
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Year = {1985},
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Volume = {57}
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@Article{ecklp88,
Title = {Liapunov spectra for infinite chains of nonlinear oscillators},
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Year = {1988},
Pages = {853--878},
Volume = {50}
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@Book{eckman80map,
Title = {Iterated Maps on the Interval as Dynamical Systems},
Author = {J.-P. Eckman and P. Collet},
Publisher = {Birkh\"{a}user},
Year = {1980},
Address = {Boston}
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@Article{EFKKLM08,
Title = {Vacuum stress-energy density and its gravitational implications},
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Year = {2008},
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pages = {281--321},
address = {Boston},
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@Phdthesis{GibsonPhD,
Title = {Dynamical-systems models of wall-bounded, shear-flow turbulence},
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School = {Cornell Univ.},
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@Book{gil_lf,
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Address = {New York}
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Note = {\arXiv{1107.4501}},
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@Book{Gitterman05,
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Note = {\arXiv{1208.1691}},
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Volume = {92},
Abstract = {A brief review of experiments on film flows, surface waves, and thermal convection.
In 1-d, the transition from cellular states to spatiotemporal chaos is described. In 2-d,
periodic and quasiperiodic patterns including defect-mediating ones are discussed.
Statistical emphasis on the transport and mixing phenomena in fluids is put and some open
problems are stated. }
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@Article{golubsym84,
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Title = {A {Hessenberg-Schur} method for the problem {AX + XB = C}},
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Title = {Stationary distributions of a noisy logistic process},
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Note = {\arXiv{cond-mat/0502389}},
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@Article{GoSuOr87,
Title = {Stability and {Lyapunov} stability of dynamical systems: A differential approach and a numerical method},
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@Article{GoTeNaEr10,
Title = {Dynamics of limit-cycle oscillators subject to general noise},
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@Inbook{GottYan,
Title = {Quantum Mechanics: Fundamentals},
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Publisher = {Springer},
Year = {2003},
Address = {New York},
Edition = {2},
ISBN = {9780387216232}
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Title = {Matrix Computations},
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Address = {Baltimore, MD}
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@Misc{GoZa10,
Title = {Noise reduces disorder in chaotic dynamics},
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defficiency: 1. It converges to a limit cycle sometimes. 2. Two symbol
sequence converges to the same cycle.}
}
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Title = {An Introduction to Continuum Mechanics},
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Publisher = {Academic},
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Publisher = {Springer},
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Address = {Berlin}
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@Book{haake01,
title = {{Quantum Signatures of Chaos}},
publisher = {Springer},
year = {2010},
author = {Haake, F.},
address = {Berlin},
edition = {3},
doi = {10.1007/978-3-662-04506-0},
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@Book{HaCh08,
Title = {Nonlinear Dynamical Systems and Control: A {Lyapunov}-Based Approach},
Author = {Haddad, W. M. and Chellaboina, V.},
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@Misc{HagJoy99,
Title = {Exponentially accurate semiclassical dynamics: propagation, localization, {E}hrenfest times, scattering and more general states},
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Note = {\arXiv{math-ph/9911036}},
Year = {1999},
Abstract = {We prove six theorems concerning exponentially
accurate semiclassical quantum mechanics. Two of these
theorems are known results, but have new proofs. Under
appropriate hypotheses, they conclude that the exact
and approximate dynamics of an initially localized wave
packet agree up to exponentially small errors in
$\hbar$ for finite times and for Ehrenfest times. Two
other theorems state that for such times the wave
packets are localized near a classical orbit up to
exponentially small errors. The fifth theorem deals
with infinite times and states an exponentially
accurate scattering result. The sixth theorem provides
extensions of the other five by allowing more general
initial conditions.}
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Year = {1982},
Pages = {303--325},
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@Phdthesis{hansen,
Title = {Symbolic Dynamics in Chaotic systems},
Author = {K. T. Hansen},
School = {Univ. of Oslo},
Year = {1993},
Address = {Box 1048, Blindern, N-0316, Oslo, Norway}
}
@Article{hansen92,
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Year = {1992},
Pages = {100--104},
Volume = {165},
Abstract = {Discuss the change of the symbol sequence of one particular
periodic orbit in the {H\'{e}non} map.}
}
@Book{hao90b,
Title = {Chaos II},
Author = {B.-L. Hao},
Publisher = {World Scientific},
Year = {1990},
Address = {Singapore}
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@Book{hao98,
Title = {Applied Symbolic Dynamics and Chaos},
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Publisher = {World Scientific},
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Address = {Singapore}
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@Book{hartmanb,
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Year = {1964},
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@InProceedings{Heller89,
Title = {Wavepacket dynamics and quantum chaology},
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Booktitle = {Les Houches Lecture Notes, Chaos and Quantum Physics},
Year = {1989},
Address = {Amsterdam },
Editor = {M.-J. Giannoni and A. Voros and J. Zinn-Justin},
Pages = {547},
Publisher = {North Holland}
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Title = {Semiclassical time evolution without root searches - {Comments and Perspective - Reply}},
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@Article{HenCviLip14,
Title = {Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system},
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Journal = {Phys. Rev. E},
Year = {2015},
Note = {\arXiv{1507.00462}},
Pages = {062922},
Volume = {92},
DOI = {10.1103/PhysRevE.92.062922}
}
@Article{HenLipCvi15,
author = {J. M. Heninger and D. Lippolis and P. Cvitanovi\'c},
title = {Perturbation theory for the {Fokker-Planck} operator in chaos},
journal = {Commun. Nonlinear Sci. Numer. Simul.},
year = {2018},
volume = {55},
pages = {16--28},
doi = {10.1016/j.cnsns.2017.06.025},
}
@Book{henonrtb2,
Title = {Generating Families in the Restricted Three-Body Problem II. Quantitative Study of Bifurcations},
Author = {M. H\'{e}non},
Publisher = {Springer},
Year = {2001},
Address = {New York}
}
@Unpublished{HerGot10,
Title = {The large core limit of spiral waves in excitable media},
Author = {S. Hermann and G. A. Gottwald},
Note = {\arXiv{1003.5830}, SIAM J. Appl. Dyn. Syst., to appear},
Year = {2010}
}
@Article{heron87,
Title = {Dimension of the attractors associated to the {G}inzburg-{L}andau partial differential equation},
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Year = {1987},
Pages = {282--304},
Volume = {28}
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@Article{HeusMull,
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@Article{HGC08,
Title = {Equilibrium and traveling-wave solutions of plane {Couette} flow},
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Journal = {J. Fluid Mech.},
Year = {2009},
Note = {\arXiv{0808.3375}},
Pages = {243--266},
Volume = {638}
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Title = {The correlation spectrum for hyperbolic analytic maps},
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Year = {1992},
Pages = {1237},
Volume = {5}
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Year = {1988},
Pages = {381--396},
Volume = {14}
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@Book{hille,
Title = {Ordinary Differential Equations in the Complex Domain},
Author = {E. Hille},
Publisher = {Dover},
Year = {1997},
Address = {New York}
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@Article{HJM03,
Title = {Scaling of the Turbulence Transition Threshold in a Pipe},
Author = {Hof, B. and Juel, A. and Mullin, T.},
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Pages = {244502},
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@Article{HNks86,
Title = {The {Kuramoto-Sivashinsky} equation: a bridge between PDE's and dynamical systems},
Author = {J. M. Hyman and B. Nicolaenko},
Journal = {Physica D},
Year = {1986},
Pages = {113--126},
Volume = {18},
Abstract = {Numerical work on the transition to chaos of the KSe. The attracting
solution manifold undergoes a complex bifurcation sequence including fixed points, limit
cycles, travelling waves, homoclinic orbits, invariant tori.}
}
@Article{HNZks86,
Title = {Order and complexity in the {Kuramoto-Sivashinsky} model of weakly turbulent interfaces},
Author = {J. M. Hyman and B. Nicolaenko and S. Zaleski},
Journal = {Physica D},
Year = {1986},
Pages = {265--292},
Volume = {23},
Abstract = {The properties of the KSe is discussed in even larger domains. Still
the dynamical behavior is dominated by the skeleton of low dimensional vector fields.
More complex dynamics happens in large domain where intermittency is accurately
identified only by high-precision numerical method. The presence of strange fixed points
overrun the cellular fixed points in large scales which generates complex spatial
structure to the solutions. The Large scale statistical behavior
of the system is computed.}
}
@Misc{ho07,
Title = {Deformed multi-variable {Fokker-Planck} equations},
Author = {C.-L. Ho and R. Sasaki},
Year = {2007},
Eprint = {cond-mat/0703291}
}
@Book{HoJo90,
Title = {Matrix Analysis},
Author = {Horn, R. A. and Johnson, C. R.},
Publisher = {Cambridge Univ. Press},
Year = {1990},
Address = {Cambridge}
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@Book{Holmes96,
Title = {Turbulence, Coherent Structures, Dynamical Systems and Symmetry},
Author = {P. Holmes and J.L. Lumley and G. Berkooz},
Publisher = {Cambridge Univ. Press},
Year = {1996},
Address = {Cambridge}
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Year = {2009},
Note = {\arXiv{0808.1513}},
Pages = {056202},
Volume = {79},
Abstract = {Based on information theory, we present a
method to determine an optimal Markov approximation for
modelling and prediction from time series data. The method
finds a balance between minimal modelling errors by taking
as much as possible memory into account and minimal
statistical errors by working in embedding spaces of rather
small dimension. A key ingredient is an estimate of the
statistical error of entropy estimates.}
}
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Year = {1937},
Address = {Berlin}
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@Misc{GoPaDu16,
author = {A. Gouasmi and E. Parish and K. Duraisamy},
title = {Characterizing memory effects in coarse-grained nonlinear systems using the {Mori-Zwanzig} formalism},
year = {2016},
url = {https://arxiv.org/abs/1611.06277},
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Title = {Ordinary Differential Equations},
Author = {E. L. Ince},
Publisher = {Dover},
Year = {1956},
Address = {New York}
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@Book{infdymnon,
title = {{Infinite-Dimensional Dynamical Systems in Mechanics and Physics}},
publisher = {Springer},
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Annote = {a second-order partial differential equation, found by Laplace in 1810, is the Fokker-Planck equation for a one-dimensional Ornstein-Uhlenbeck process. It is argued that Laplace's reasoning, although not rigorous, can be entirely justified by using the modern theory of weak convergence of stochastic processes. The solutions to the differential equation found by Laplace and others, using expansions in terms of Hermite polynomials, are discussed. },
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Abstract = {A UPO is fo und in the chaotic region of one shell turbulence
model (GOY). The scaling exponents of the structure function are calculated
to verify that the statistics are well approximated by the UPO.}
}
@Book{Kato80,
Title = {{Perturbation Theory for Linear Operators}},
Author = {Kato, T.},
Publisher = {Springer},
Year = {1980},
Address = {Berlin},
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@Book{Katok95,
Title = {Introduction to the Modern Theory of Dynamical Systems},
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@Article{KawKida01,
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Abstract = {Two time-periodic solutions are found in a 3-d constrained
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mainly follows the periodic orbit with strong variations. The gentle one
is related to the bursting behavior of the system. Heteroclinic orbits
between these two periodic orbits are found.}
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@Article{Keane98,
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@Book{Keizer87,
Title = {Statistical Thermodynamics of Nonequilibrium Processes},
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Year = {1987},
Address = {Berlin}
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@Book{kellbv,
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Address = {New York}
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Author = {G. Keller and P. J. Howard and Klages, R.},
Journal = {Nonlinearity},
Year = {2008},
Pages = {1719--1743},
Volume = {21},
Abstract = {We consider families of dynamics that can be described in
terms of Perron--Frobenius operators with exponential mixing properties.
For piecewise C2 expanding interval maps we rigorously prove
continuity properties of the drift and of the diffusion
coefficient D under parameter variation. Our main result is that D
is Lipschitz continuous up to quadratic logarithmic corrections.
For a special class of piecewise linear maps we provide more precise
estimates at specific parameter values. Our analytical findings are
quantified numerically for the latter class of maps by using exact
expansions for the transport coefficients that can be evaluated
numerically. We numerically observe strong local variations of all
continuity properties.},
DOI = {10.1088/0951-7715/21/8/003}
}
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Title = {Necessary and sufficient conditions for balancing unstable systems},
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Journal = {IEEE Trans. Automatic Control},
Year = {1987},
Pages = {157},
Volume = {AC-32}
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@Article{Kerswell05,
Title = {Recent progress in understanding the transition to turbulence in a pipe},
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Journal = {Nonlinearity},
Year = {2005},
Pages = {17--44},
Volume = {18},
Abstract = {The problem of understanding the nature of fluid
flow through a circular straight pipe remains one of the
oldest problems in fluid mechanics. So far no explanation
has been substantiated to rationalize the transition
process by which the steady unidirectional laminar flow
state gives way to a temporally and spatially disordered
three-dimensional (turbulent) solution as the flow rate
increases. Recently, new travelling wave solutions have
been discovered which are saddle points in phase space.
These plausibly represent the lowest level in a hierarchy
of spatio-temporal periodic flow solutions which may be
used to construct a cycle expansion theory of turbulent
pipe flows. We summarize this success against the backdrop
of past work and discuss its implications for future
research. }
}
@Article{ketzWeyl,
Title = {Hierarchical fractal {Weyl} laws for chaotic resonance states in open mixed systems},
Author = {K\"orber, M. J. and Michler, M. and B\"acker, A. and Ketzmerick, R.},
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Year = {2013},
Pages = {114102},
Volume = {111},
DOI = {10.1103/PhysRevLett.111.114102}
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@Article{kev01ks,
Title = {The {Oseberg} transition: {Visualization} of global bifurcations for the {Kuramoto-Sivashinsky} equation},
Author = {M. E. Johnson and M. S. Jolly and I. G. Kevrekidis},
Journal = {Int. J. Bifur. Chaos},
Year = {2001},
Pages = {1--18},
Volume = {11}
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@Inbook{Kiefer85,
chapter = {Design of Experiments},
pages = {718},
title = {{Jack Carl Kiefer Collected Papers}},
publisher = {Springer},
year = {1985},
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volume = {III},
address = {Berlin},
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Pages = {1083--1107},
Volume = {8}
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@Article{Kim87,
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Journal = {J. Fluid Mech.},
Year = {1987},
Pages = {133--166},
Volume = {177}
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@Article{Kitagawa77,
Title = {An algorithm for solving the matrix equation {X = F X F' + S}},
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Pages = {745--753},
Volume = {25}
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@Incollection{Kitahara75,
Title = {{The Hamilton-Jacobi} equation approach to fluctuation phenomena},
Author = {Kitahara, K.},
Booktitle = {Advances in Chemical Physics: Membranes, Dissipative Structures and Evolution},
Publisher = {John Wiley \&\ Sons},
Year = {1975},
Address = {Hoboken, NJ},
Editor = {G. Nicolis and R. Lefever},
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@Book{Kitch98,
Title = {Symbolic Dynamics. One-sided, Two-sided, and Countable {Markov} shifts},
Editor = {B. Kitchens},
Publisher = {Springer},
Year = {1998},
Address = {Berlin}
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@InProceedings{KK05,
Title = {Unstable periodic motion in plane {C}ouette system: The skeleton of turbulence},
Author = { G. Kawahara and S. Kida and M. Nagata},
Booktitle = {One Hundred Years of Boundary Layer Research},
Year = {2005},
Publisher = {Kluwer}
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@InProceedings{Kleiser80,
Title = {Treatment of incompressibility and boundary conditions in {3-D} numerical spectral simulations of plane channel flows},
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Booktitle = {Proc. 3\textsuperscript{rd} GAMM Conf. Numerical Methods in Fluid Mechanics},
Year = {1980},
Address = {Viewweg, Braunschweig},
Editor = {E. Hirschel},
Organization = {GAMM},
Pages = {165--173}
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@Article{KLH94,
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Year = {1994},
Pages = {175--198},
Volume = {270}
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@Article{KNSks90,
Title = {Back in the saddle again: a computer assisted study of the {Kuramoto-Sivashinsky} equation},
Author = {I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel},
Journal = {Siam J. Appl. Math.},
Year = {1990},
Pages = {760--790},
Volume = {50},
Abstract = {The initial bifurcations of the KSe are examined analytically and
numerically. The heteroclinic connections between symmetric solutions proved to play
an important role in the dynamics.}
}
@Article{KoGrPo13,
Title = {Classifying and measuring geometry of a quantum ground state manifold},
Author = {Kolodrubetz, M. and Gritsev, V. and Polkovnikov, A.},
Journal = {Phys. Rev. B},
Year = {2013},
Pages = {064304},
Volume = {88},
DOI = {10.1103/PhysRevB.88.064304}
}
@Misc{KoKo11,
Title = {{Navier-Stokes} equation by stochastic variational method},
Author = {T. Koide and T. Kodama},
Note = {\arXiv:1105.6256}
}
@Article{kolm91,
Title = {The local structure of turbulence in incompressible viscous fluid for very large {Reynolds} numbers},
Author = {A. N. Kolmogorov},
Journal = {Proc. R. Soc. Lond. A},
Year = {1991},
Pages = {9--13},
Volume = {434},
Abstract = {Study the distribution of velocity difference of neighboring points
based on the locally homogeneous and locally isotropic hypothesized poperties of
turbulence motion. Two more similarity hypothesis suggested the existence of a
universal distribution for different viscosity and energy dissipation rate. The
large distance asymptotic behavior of the distribution is also deduced.}
}
@Article{Kolov94,
Title = {A remark on the problem of quantum-classical correspondence in the case of chaotic dynamics},
Author = {A. R. Kolovsky},
Journal = {Europhys. Lett.},
Year = {1994},
Pages = {79},
Volume = {27}
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@Article{Kolov96,
Title = {Quantum coherence, evolution of the {Wigner} function, and transition from quantum to classical dynamics for a chaotic system},
Author = {A. R. Kolovsky},
Journal = {Chaos},
Year = {1996},
Pages = {534--542},
Volume = {6},
DOI = {10.1063/1.166201}
}
@Article{kook89,
Title = {Periodic orbits for reversible symplectic mappings},
Author = {H-T. Kook and J. D. Meiss},
Journal = {Physica D},
Year = {1989},
Pages = {65--86},
Volume = {35},
Abstract = {By constructing a 2N-dimensional symplectic map for a
Langrangian system, the orbit structure of the phase space is discussed for
a time-reversible system. Periodic orbits are classfied according to the
rotation number, symmetry, and morse index. Unstable orbits repel other
periodic orbits to form resonances where a orbit can be trapped for long with
the same rotation number. The chain of resonances with the same
commensurability forms channel which provides the communication bridge
between different resonances. The connection of the orbit structure with the
continued fraction representation of real numbers is discussed.}
}
@Article{kooknewt,
Title = {Application of {N}ewton's method to {L}agrangian mappings},
Author = {H-T Kook and J. D. Meiss},
Journal = {Physica D},
Year = {1989},
Pages = {317--326},
Volume = {36},
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.}
}
@Misc{koopmania,
Author = {P. Cvitanovi\'{c}},
Note = {Appendix {\em ``Implementing evolution''}, in \refref{DasBuch}},
Year = {2013}
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@Article{KoSch90,
Title = {Noise reduction--finding the simplest dynamical system consistent with the data},
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Pages = {183--196},
Volume = {41}
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@Article{KoSch93,
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Year = {1993},
Pages = {1752--1763},
Volume = {48}
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@Article{kovalev,
Title = {Sur le probl\`eme de la rotation d'un corps solide d'un point fixe},
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Journal = {J. Phys. A: Math. Gen.},
Year = {2004},
Pages = {2365--2370},
Volume = {37},
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.}
}
@Book{KrReBe10,
Title = {A Kinetic View of Statistical Physics},
Author = {Krapivsky, P. L. and Redner, S. and Ben-Naim, E.},
Publisher = {Cambridge Univ. Press},
Year = {2010},
Address = {Cambridge, UK}
}
@Article{KRSR67,
Title = {The structure of turbulent boundary layers},
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Journal = {J. Fluid Mech.},
Year = {1967},
Pages = {741--773},
Volume = {30}
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@Article{kruskal62,
Title = {Asymptotic theory of {H}amiltonian and other systems with all solutons nearly periodic},
Author = {M. Kruskal},
Journal = {J. Math. Phys.},
Year = {1962},
Pages = {806},
Volume = {3},
Abstract = {Asymptotic expansion to all orders. A systematic way to
construct the adiabatic invariants.}
}
@Article{kruskal65,
Title = {Interaction of "solitons" in a collisionless plasma and the recurrence of initial states},
Author = {N. J. Zabusky and M. D. Kruskal},
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Year = {1965},
Pages = {240},
Volume = {15}
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@Article{ks04com,
Title = {Fourth-order time stepping for stiff {PDE}s},
Author = {A. -K. Kassam and L. N. Trefethen},
Journal = {SIAM J. Sci. Comp.},
Year = {2004},
Abstract = {A contour integral method is presented to evaluate accurately
the matrix functions with removable singularities.}
}
@Article{kschang86,
Title = {Travelling waves on fluid interfaces: {N}ormal form analysis of the {K}uramoto-{S}ivashisky equation},
Author = {H.-C. Chang},
Journal = {Phys. Fluids},
Year = {1986},
Pages = {3142},
Volume = {29},
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.}
}
@Article{ksgreene88,
Title = {The steady states of the {Kuramoto-Sivashinsky} equation},
Author = {J. M. Greene and J. -S. Kim},
Journal = {Physica D},
Year = {1988},
Pages = {99--120},
Volume = {33},
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 {K}uramoto-{S}ivashinsky equation},
Author = {R. Grimshaw and A. P. Hooper},
Journal = {Physica D},
Year = {1991},
Pages = {231--238},
Volume = {50},
Abstract = {If the effect of short-wave stability is small, there are no
regular shocks.}
}
@Article{ksham95,
Title = {Hamiltonian structure and integrability of the stationary {Kuramoto-Sivashinsky} equation},
Author = {S. Bouquet},
Journal = {J. Math. Phys.},
Year = {1995},
Pages = {1242},
Volume = {36},
Abstract = {The stationary KSe can be transformed into a time-depent 1-d
Hamiltonian system. There exist a large class of initial conditions under which
the system will explode in a finite time like 120/(x-x_0)^3.}
}
@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},
Abstract = {Numerically calculated connections of the KSe}
}
@Article{kskent92,
Title = {Travelling-waves of the {Kuramoto-Sivashinsky} equation: period-multiplying bifurcations},
Author = {P. Kent and J. Elgin},
Journal = {Nonlinearity},
Year = {1992},
Pages = {899--919},
Volume = {5},
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.}
}
@Article{kstroy89,
Title = {The existence of steady solutions of the {K}uramoto-{S}ivashinsky 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.}
}
@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},
Abstract = {Using asymptotic analysis (beyond all orders)in the weak
shock limit, the oscillatory
shocks and the solitary saves are constructed. It shows that the oscillatory
shocks can be only be anti-symmetric and numurical results support the
anlytical calculation.}
}
@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,
Title = {Persistent propagation of concentration waves in dissipative media far from thermal equilibrium},
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}
@Book{kura84tur,
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@Phdthesis{Lan:Thesis,
Title = {Dynamical systems approach to $1-d$ spatiotemporal chaos -- {A} cyclist's view},
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School = {School of Physics, Geogia Institute of Technology},
Year = {2004},
Address = {Atlanta}
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Title = {Stationary modulated-amplitude waves in the {1D} complex {G}inzburg-{L}andau equation},
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Title = {One-dimensional iterative maps},
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Booktitle = {Chaos},
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ISBN = {9780691610535}
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Title = {Two-dimensional iterative maps},
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Booktitle = {Chaos},
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ISBN = {9780691610535}
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Abstract = {This paper establishes several almost sure asymptotic
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@Article{LecDenSol90,
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Title = {Ergodic theory and visualization {I}: {Visualization} of ergodic partition and invariant sets},
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Note = {\arXiv{0805.4221}},
Year = {2008},
Abstract = {We present a computational study of the
invariant sets visualization method based on ergodic
partition theory. The algorithms
for computation of the time averages of many L1 functions
are developed and employed producing the approximation of
the phase space ergodic partitioning. The method is
exposed in the context of discrete-time dynamical systems
producing a graphical representation of the phase space
in terms of the invariant set structure that gives a
substantial insight into the global and local properties
of the dynamics. We use the Chirikov standard map in
order to show the implementation of the method, followed
by applications to other multi-dimensional maps. We examine
the visualization of periodic sets using harmonic time
averages.}
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Year = {2009},
Note = {\arXiv{0901.2986}},
Pages = {550--564},
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@Misc{Liao11,
Title = {Chaos: a bridge from microscopic uncertainty to macroscopic randomness},
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Note = {\arXiv{1108.4472}},
Year = {2011}
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@Book{Lichtenberg92,
Title = {Regular and Chaotic Dynamics},
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@Book{liede,
Title = {Applications of Lie Groups to Differential Equations},
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Publisher = {Spring-Verlag},
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Year = {2004},
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Volume = {71},
Abstract = {a linear part of a co-dimension two-bifurcation system that
is on a three-dimensional central manifold and subjected to parametric
excitation by an ergodic real noise. an n-dimensional Ornstein-Uhlenbeck
vector process which is the output of a linear filter system, while both
the detailed balance condition and the strong mixing condition are
removed. Through a perturbation method and the spectrum representations of
the Fokker Planck operator and its adjoint operator of the linear filter
system, the explicit asymptotic expressions of the maximal Lyapunov
exponent for three case studies, in which different forms of the
coefficient matrix included in the noise excitation term are assumed, are
obtained.}
}
@Unpublished{LipCvi07,
Title = {Optimal resolution of the state space of a chaotic flow in presence of noise},
Author = {P. Cvitanovi\'c and D. Lippolis},
Note = {In preparation},
Year = {2013}
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Title = {How well can one resolve the state space of a chaotic map?},
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Year = {2010},
Note = {\arXiv{0902.4269}},
Pages = {014101},
Volume = {104},
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}
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author = {D. Lippolis},
title = {Mapping densities in a noisy state space},
booktitle = {{2013 International Symposium on Nonlinear Theory and its Applications, NOLTA2013}},
year = {2013},
volume = {2},
pages = {318--321},
address = {Japan},
publisher = {IEICE},
doi = {10.15248/proc.2.318},
}
@Phdthesis{LippolisThesis,
Title = {How well can one resolve the state space of a chaotic map?},
Author = {Lippolis, D.},
School = {School of Physics, Georgia Inst. of Technology},
Year = {2010},
Address = {Atlanta},
URL = {http://ChaosBook.org/projects/theses.html}
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Year = {2005}
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Address = {Providence, RI}
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Abstract = {A 28-variable model of the atmosphere is constructed by
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double-Fourier series. The model includes the nonlinear interactions
among disturbances of three different wave lengths. Nonperiodic
time-dependent solutions are determined by numerical integration.By
comparing separate solutions with slightly different initial
conditions, the growth rate of small initial errors is studied. The
time required for errors comparable to observational errors in the
atmosphere to grow to intolerable errors is strongly dependent upon
the current circulation pattern, and varies from a few days to a few
weeks.Some statistical predictability of certain quantities seems to
be present even after errors in the complete circulation pattern are
no longer small. The feasibility of performing similar studies with
much larger atmospheric models is considered.},
DOI = {10.1111/j.2153-3490.1965.tb01424.x}
}
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Abstract = {Some early ideas concerning the general circulation of
the atmosphere are reviewed. A model of the general circulation,
consisting of three ordinary differential equations, is introduced.
For different intensities of the axially symmetric and asymmetric
thermal forcing, the equations may possess one or two stable
steady-state solutions, one or two stable periodic solutions, or
irregular (aperiodic) solutions. Qualitative reasoning which has been
applied to the real atmosphere may sometimes be applied to the model,
and checked for soundness by comparing the conclusions with numerical
solutions. The implications of irregularity for the atmosphere and
for atmospheric science are discussed.},
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Year = {2013},
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Volume = {7},
DOI = {10.1002/lpor.201300042}
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Year = {1998},
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Volume = {19}
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@Misc{LSTvW14,
Title = {Large-scale fluctuations of {Lyapunov} exponents in diffusive systems},
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Note = {\arXiv{1406.6370}},
Year = {2014}
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Volume = {91}
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@Article{LundJohan91,
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Publisher = {Springer-Verlag},
Year = {1993},
Address = {Berlin, Germany}
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Volume = {19},
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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.}
}
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DOI = {10.1080/00207179208934253}
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@Book{lyschm,
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Author = {N. Sidorov and B. Loginov and A. Sinitsyn and M. Falaleev},
Publisher = {Kluwer},
Year = {2002},
Address = {Dordrecht},
Series = {mathematics and its applications},
Volume = {550}
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Volume = {18}
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@Article{MajBran12,
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Year = {2012},
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Volume = {32},
DOI = {10.3934/dcds.2012.32.3133}
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@Misc{malinin07,
Title = {Classical nonlinear response of a chaotic system: Langevin dynamics and spectral decomposition},
Author = {S. V. Malinin and V. Y. Chernyak},
Note = {\arXiv{nlin/0703014}},
Year = {2007}
}
@Book{man90b,
Title = {Dissipative structures and weak turbulence},
Author = {P. Manneville},
Publisher = {Academic},
Year = {1990},
Address = {Boston}
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Author = {B. B. Mandelbrot},
Publisher = {WH Freeman Co.},
Year = {1982},
Address = {New York}
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Year = {2004},
Pages = {169--181},
Volume = {18}
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Title = {Topological characterization of spatial-temporal chaos},
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Journal = {submitted},
Year = {2003},
Pages = {1},
Volume = {1},
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.}
}
@Book{marsdenbb,
Title = {The {H}opf Bifurcation and its Applications},
Author = {J. E. Marsden and M. McCracken},
Publisher = {Springer},
Year = {1976}
}
@Article{mart87lfa,
Title = {Local frequency analysis of chaotic motion in multidimensional systems: {Energy} transport and bottlenecks in planar {OCS}},
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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 "noble ratio" consideration.}
}
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Volume = {4}
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@Article{mawcgl,
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Year = {2000},
Pages = {86},
Volume = {85},
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.}
}
@Article{May76,
Title = {Simple mathematical models with very complicated dynamics},
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Abtract = {instability of plane waves}
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Year = {2003},
Pages = {134},
Volume = {174},
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.}
}
@Article{mcord86,
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Volume = {114}
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@Article{MEF04,
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Journal = {Chaos},
Year = {2004},
Pages = {S11},
Volume = {14}
}
@Article{Mein95s,
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Year = {1995},
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Volume = {25}
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@Article{MFE04,
Title = {A low-dimensional model for turbulent shear flows},
Author = {Moehlis, J. and Faisst, H. and Eckhardt, B.},
Journal = {New J. of Physics},
Year = {2004},
Pages = {56+17},
Volume = {6}
}
@Article{MFE04b,
Title = {Periodic orbits and chaotic sets in a low-dimensional model for shear flows},
Author = {Moehlis, J. and Faisst, H. and Eckhardt, B.},
Journal = {SIAM J. Appl. Dyn. Syst.},
Year = {2004},
Pages = {352--376},
Volume = {4}
}
@Article{mfind,
Title = {Efficient algorithm for detecting unstable periodic orbits in chaotic systems},
Author = {R. L. Davidchack and Y.-C. Lai},
Journal = {Phys. Rev. E},
Year = {1999},
Pages = {6172},
Volume = {60}
}
@Article{MHMwind96,
Title = {Winding number instability in the phase-turbulence regime of the complex {G}inzburg-{L}andau equation},
Author = {R. Montagne and E. Hern\'{a}ndez-Garc\'{i}a and M. S. Miguel},
Journal = {Phys. Rev. Lett.},
Year = {1996},
Pages = {267},
Volume = {77},
Abstract = {There exist a band of winding numbers associated with stable states in
the phase turbulent regime. The states with large winding numbers decay to the ones
with smaller winding numbers. Defect chaos happens when the range of stable winding
numbers vanishes.}
}
@Article{michsiv77,
Title = {Nonlinear analysis of hydrodynamic instability in laminar flames - II. Numerical experiments},
Author = {D. M. Michelson and G. I. Sivashinsky},
Journal = {Acta Astr.},
Year = {1977},
Pages = {1207--1221},
Volume = {4},
Abstract = {Consider the effect of hydrodynamics instability and diffusion thermal
instability through the numerical calculation of the KSe.}
}
@Misc{MiFeFl08,
Title = {Preservation of hyperbolic equilibrium points and synchronization in dynamical systems},
Author = {C. Miranda-Reyes and G. Fern and J. J. Flores-Godoy},
Year = {2008}
}
@Article{Milb_sqa,
Title = {Quantum and classical Liouville dynamics in the anharmonic oscillator},
Author = {G. J. Milburn},
Journal = {Phys. Rev. A},
Year = {1986},
Number = {1},
Pages = {674--685},
Volume = {33}
}
@Article{Milb_sqa2,
Title = {Dissipative quantum and classical Liouville dynamics in the anharmonic oscillator},
Author = {G. J. Milburn and C. A. Holmes},
Journal = {Phys. Rev. Lett.},
Year = {1986},
Number = {21},
Pages = {2237--2240},
Volume = {56}
}
@Book{MiraGum80,
Title = {Recurrances and Discrete Dynamical Systems},
Author = {I. Gumowski and C. Mira},
Publisher = {Springer},
Year = {1980},
Address = {Berlin}
}
@Article{mis01ks,
Title = {Rigorous Numerics for Partial Differential Equations: the {Kuramoto-Sivashinsky} equation},
Author = {P. Zgliczynski and K. Mischaikow},
Journal = {Foundations of Comp. Math.},
Year = {2001},
Pages = {255--288},
Volume = {1}
}
@Article{misdis,
Title = {Construction of symbolic dynamics from experimental time series},
Author = {K. Mischaikow and M. Mrozek and J. Reiss and A. Szymczak},
Journal = {Phys. Rev. Lett.},
Year = {1999},
Pages = {1144},
Volume = {82}
}
@Article{Misiurewicz80,
Title = {Strange attractors for the {Lozi} mappings},
Author = {Misiurewicz, Michal},
Journal = {Ann. New York Acad. Sci.},
Year = {1980},
Pages = {348--358},
Volume = {357}
}
@Article{mislor1,
Title = {Chaos in the {L}orenz equations: A computer assisted proof Part II: Details},
Author = {K. Mischaikow and M. Mrozek},
Journal = {Math. Comp.},
Year = {1998},
Pages = {1023--1046},
Volume = {67}
}
@Article{Mks86,
Title = {Steady solutions of the {Kuramoto-Sivashinsky} equation},
Author = {D. Michelson},
Journal = {Physica D},
Year = {1986},
Pages = {89--111},
Volume = {19},
Abtract = {The variety of steady solution of the KSe is discussed. For large c, there is only one odd front-like bounded solution. In decreasing c, odd solutions with more zeros are born until finally a periodic solution is born. Associated with the periodic solution, infinite many tori will appear in the elliptic case and Cantor-type set of chaotic solutions, with infinite many homoclinic odd solutions.},
DOI = {10.1016/0167-2789(86)90055-2}
}
@Misc{MoMiAo14,
Title = {Optimal colored noise for estimating phase response curves},
Author = {{Morinaga}, K. and {Miyata}, R. and {Aonishi}, T.},
Note = {\arXiv{1409.1637}},
Year = {2014}
}
@Article{Moore81,
Title = {Principal component analysis in linear systems: {Controllability}, observability, and model reduction},
Author = {B. Moore},
Journal = {IEEE Trans. Automat. Contr.},
Year = {1981},
Pages = {17--32},
Volume = {26}
}
@InProceedings{Morkovin69,
Title = {On the many faces of transition},
Author = {Morkovin, M.V},
Booktitle = {Viscous Drag Reduction},
Year = {1969},
Editor = {Wells, C.S.},
Pages = {1--31},
Publisher = {Plenum, New York}
}
@Article{Morrison02,
Title = {Reynolds number dependence of streamwise velocity spectra in turbulent pipe flow},
Author = {J.F. Morrison and W. Jiang and B.J. McKeon and A.J. Smits},
Journal = {Phys. Rev. Lett.},
Year = {2002},
Pages = {214501},
Volume = {88}
}
@Article{Morrison04,
Title = {Scaling of the streamwise velocity component in turbulent pipe flow},
Author = {J.F. Morrison and B.J. McKeon and W. Jiang and A.J. Smits},
Journal = {J. Fluid Mech.},
Year = {2004},
Pages = {99--131},
Volume = {508}
}
@Book{Morse73,
Title = {Variational Analysis: Critical Extremals and Sturmian Extensions},
Author = {M. Morse},
Publisher = {Wiley},
Year = {1973},
Address = {New York}
}
@Article{moser96,
Title = {A rapidly converging iteration method and nonlinear partial differential equations -{I}},
Author = {J. Moser},
Journal = {Ann. Scuola Norm. Super. Pisa},
Year = {1966},
Pages = {265--315},
Volume = {20},
URL = {http://archive.numdam.org/ARCHIVE/ASNSP/ASNSP_1966_3_20_2/ASNSP_1966_3_20_2_265_0/ASNSP_1966_3_20_2_265_0.pdf}
}
@Article{moser96a,
Title = {A rapidly convergent iteration method and non-linear differential equations - {II}},
Author = {J. Moser},
Journal = {Ann. Scuola Norm. Super. Pisa},
Year = {1966},
Pages = {499--535},
Volume = {20},
URL = {http://archive.numdam.org/ARCHIVE/ASNSP/ASNSP_1966_3_20_3/ASNSP_1966_3_20_3_499_0/ASNSP_1966_3_20_3_499_0.pdf}
}
@Book{mosersr,
Title = {Stable and Random Motions in Dynamical Systems},
Author = {J. Moser},
Publisher = {Princeton Univ. Press},
Year = {1973},
Address = {Princeton}
}
@Article{MR94,
Title = {Periodic orbit expansions for the {Lorentz} gas},
Author = {G. P. Morriss and L. Rondoni},
Journal = {J. Stat. Phys.},
Year = {1994},
Pages = {553--584},
Volume = {75},
DOI = {10.1007/BF02186872}
}
@Article{MS66,
Title = {A thermally excited nonlinear oscillator},
Author = {D. W. Moore and E. A. Spiegel},
Journal = {Astrophys. J.},
Year = {1966},
Pages = {871},
Volume = {143}
}
@Article{MTYYA14,
Title = {Potential function in a continuous dissipative chaotic system: {Decomposition} scheme and role of strange attractor},
Author = {Ma, Y. and Tan, Q. and Yuan, R. and Yuan, B. and Ao, P.},
Journal = {Int. J. Bifur. Chaos},
Year = {2014},
Pages = {1450015},
Volume = {24},
DOI = {10.1142/S0218127414500151}
}
@Article{MvH98,
Title = {Building blocks of spatiotemporal intermittency},
Author = {M. van Hecke},
Journal = {Phys. Rev. Lett.},
Year = {1998},
Pages = {1896},
Volume = {80}
}
@Article{MvHhole01,
Title = {Ordered and Self-disordered dynamics of holes and defects in the one-dimensional complex {G}inzburg-{L}andau equation},
Author = {M. van Hecke and M. Howard},
Journal = {Phys. Rev. Lett.},
Year = {2001},
Pages = {2018},
Volume = {86},
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.}
}
@Article{N90,
Title = {Three-dimensional finite-amplitude solutions in plane {Couette} flow: bifurcation from infinity},
Author = {Nagata, M.},
Journal = {J. Fluid Mech.},
Year = {1990},
Pages = {519--527},
Volume = {217},
DOI = {10.1017/S0022112090000829}
}
@Article{name,
Title = {Nonuniform stationary measure of the invariant unstable foliation in {Hamiltonian} and fluid mixing systems},
Author = {M. Giona and A. Adrover},
Journal = {Phys. Rev. Lett.},
Year = {1998},
Pages = {3864},
Volume = {81},
Abstract = {The article demonstrates the existence of a nonuniform stationary measure
associated with the space-filling properties of the unstable invariant foliation in 2D
differentiable area-preserving systems and derives a sequence of analytical approximations
for it. In fluid convection systems, it is just the convected material line.}
}
@Article{NaSaWa13,
Title = {{Hamilton-Jacobi} method for molecular distribution function in a chemical oscillator},
Author = {Nakanishi, H. and Sakaue, T. and Wakou, J.},
Journal = {J. Chem. Phys.},
Year = {2013},
Pages = {214105},
Volume = {139},
DOI = {10.1063/1.4834636}
}
@Misc{natmsr,
Author = {P. Cvitanovi\'{c} and R. Artuso and L. Rondoni and E. A. Spiegel},
Note = {Chapter {\em Transporting densities}, in \refref{DasBuch}},
Year = {2012}
}
@Article{NeSa11,
Title = {Thermodynamic formula for the cumulant generating function of time-averaged current},
Author = {Nemoto, T. and Sasa, S.-i.},
Journal = {Phys. Rev. E},
Year = {2011},
Pages = {061113},
Volume = {84},
DOI = {10.1103/PhysRevE.84.061113},
Numpages = {18}
}
@Article{NeuSch01,
Title = {Estimation of parameters and eigenmodes of multivariate autoregressive models},
Author = {A. Neumaier and T. Schneider},
Journal = {A.C.M. Trans. Math. Softw.},
Year = {2001},
Pages = {27--57},
Volume = {27},
Abstract = {Dynamical characteristics of a complex system can
often be inferred from analyses of a stochastic time
series model fitted to observations of the system.
Oscillations in geophysical systems, for example, are
sometimes characterized by principal oscillation
patterns, eigenmodes of estimated autoregressive (AR)
models of first order. This paper describes the
estimation of eigenmodes of AR models of arbitrary
order. AR processes of any order can be decomposed into
eigenmodes with characteristic oscillation periods,
damping times, and excitations. Estimated eigenmodes
and confidence intervals for the eigenmodes and their
oscillation periods and damping times can be computed
from estimated model parameters. As a computationally
efficient method of estimating the parameters of AR
models from high-dimensional data, a stepwise least
squares algorithm is proposed. This algorithm computes
model coefficients and evaluates criteria for the
selection of the model order stepwise for AR models of
successively decreasing order. Numerical simulations
indicate that, with the least squares algorithm, the AR
model coefficients and the eigenmodes derived from the
coefficients are estimated reliably and that the
approximate 95\% confidence intervals for the
coefficients and eigenmodes are rough approximations of
the confidence intervals inferred from the
simulations.}
}
@Misc{NeuSch97,
Title = {Multivariate autoregressive and {Ornstein-Uhlenbeck} processes: {Estimates} for order, parameters, spectral information, and confidence regions},
Author = {A. Neumaier and T. Schneider},
Note = {Submitted to A.C.M. Trans. Math. Softw., unpublished}
}
@Article{newcgl1,
Title = {Instabilities of {G}inzburg-{L}andau equation: periodic solutions},
Author = {P. K. Newton and L. Sirovich},
Journal = {Quart. Appl. Math.},
Year = {1986},
Pages = {49--58},
Volume = {XLIV}
}
@Article{newcgl2,
Title = {Instabilities of {Ginzburg-{L}andau} equation: secondary bifurcation},
Author = {P. K. Newton and L. Sirovich},
Journal = {Quart. Appl. Math.},
Year = {1986},
Pages = {367--374},
Volume = {XLIV}
}
@Article{newenv74,
Title = {Envelope equations},
Author = {A. C. Newell},
Journal = {Lect. Appl. Math.},
Year = {1974},
Pages = {157--163},
Volume = {15}
}
@Article{nhouse78,
Title = {Occurrence of strange {A}xiom {A} attractors near quasi-periodic flows on $T^m(m=3$ or more)},
Author = {S. E. Newhouse and D. Ruelle and F. Takens},
Journal = {Commun. Math. Phys.},
Year = {1978},
Pages = {35},
Volume = {64}
}
@Misc{Nico90,
Title = {Chaotic dynamics and markovian coarse-graining in nonlinear dynamical systems},
Author = {G. Nicolis},
Note = {in \textit{Noise and Chaos in Nonlinear Dynamical Systems}, edited by F. Moss, L. A. Lugiato, and W. Schleich (Cambridge Univ. Pres)},
Year = {1990}
}
@Article{Nico95,
Title = {Chaotic dynamics applied to information processing},
Author = {J. S. Nicolis},
Journal = {Rep. Prog. Phys.},
Year = {1986},
Pages = {1109--1196},
Volume = {49}
}
@Article{nlin0001013,
Title = {{Frobenius-Perron} resonances for maps with a mixed phase space},
Author = {{Weber}, J. and {Haake}, F. and {{\v S}eba}, P.},
Journal = {Phys. Rev. Lett.},
Year = {2000},
Note = {\arXiv{nlin/0001013}},
Pages = {3620--3623},
Volume = {85}
}
@Article{nlin0102039,
Title = {Periodic orbit quantization of a {Hamiltonian} map on the sphere},
Author = {A. J. Scott and G. J. Milburn},
Journal = {J. Phys. A},
Year = {2001},
Note = {\arXiv{nlin/0102039}},
Pages = {7541},
Volume = {34},
Abstract = {In a previous paper we introduced examples of
Hamiltonian mappings with phase space structures
resembling circle packings. It was shown that a vast
number of periodic orbits can be found using special
properties. We now use this information to explore the
semiclassical quantization of one of these maps.}
}
@Article{nlin0105047,
Title = {Resonances of the {Frobenius-Perron} operator for a {Hamiltonian} map with a mixed phase space},
Author = {{Weber}, J. and {Haake}, F. and {Braun}, P.~A. and {Manderfeld}, C. and {Seba}, P.},
Journal = {J. Phys. A},
Year = {2001},
Note = {\arXiv{nlin/0105047}},
Pages = {7195--7211},
Volume = {34}
}
@Article{nlin0107020,
Title = {Classical versus quantum time evolution of (quasi-) probability densities at limited phase-space resolution},
Author = {{Manderfeld}, C. and {Weber}, J. and {Haake}, F.},
Journal = {J. Phys. A},
Year = {2001},
Note = {\arXiv{nlin/0107020}},
Pages = {9893--9905},
Volume = {34}
}
@Article{nlin0312062,
Title = {Spectral properties and classical decays in quantum open systems},
Author = {Garc{\'{\i}}a-Mata, I. and {Saraceno}, M.},
Journal = {Phys. Rev. E},
Year = {2004},
Note = {\arXiv{nlin/0312062}},
Pages = {056211},
Volume = {69}
}
@Article{NLSmal91,
Title = {Modulational instabilities and soliton solutions of a generalized nonlinear {S}chr\"{o}dinger equation},
Author = {B. A. Malomed and L. Stenflo},
Journal = {J. Phys. A},
Year = {1991},
Pages = {1149--1153},
Volume = {24},
Abtract = {A Hamiltonian equation is proposed for the generalized nonlinear Schr\"{o}dinger equation. Soliton solutions are found and the stability of the plane waves is investigated.}
}
@Article{NLSper77,
Title = {Nonlinear Schr\"{o}dinger equation including growth and damping},
Author = {N. R. Pereira and L. Stenflo},
Journal = {Phys. Fluids},
Year = {1977},
Pages = {1733--1734},
Volume = {20},
Abstract = {A nonlinear Schr\"{o}dinger equation is found with growth and damping
terms in the weakly nonlinear regime. An exact solitary solution is found for the
equation.}
}
@Article{NLSstn03,
Title = {Oscillons at a plasma surface},
Author = {L. Stenflo and M. Y. Yu},
Journal = {Phys. Plasmas},
Year = {2003},
Pages = {912--913},
Volume = {10},
Abstract = {Under the external driving force, an oscillon solutions is found.}
}
@Article{NockelStone,
Title = {Ray and wave chaos in asymmetric resonant optical cavities},
Author = {Nockel, J. U. and Stone, A. D.},
Journal = {Nature},
Year = {1997},
Number = {6611},
Pages = {45--47},
Volume = {385},
Abstract = {OPTICAL resonators are essential components of lasers and
other optical devices, A resonator is characterized by a set of modes,
each with a resonant frequency omega and resonance width delta omega =
1/tau, where tau is the lifetime of a photon in the mode, Cylindrical
or spherical dielectric resonators have extremely long-lived
resonances(1) due to 'whispering gallery' modes in which light
circulates around the perimeter trapped by total internal reflection,
These resonators emit light isotropically, Recently a new category of
asymmetric resonant cavities has been proposed in which substantial
deformation of the cavity from cylindrical or spherical symmetry leads
to partially chaotic ray dynamics, This has been predicted(2-4) to
give rise to a universal, frequency-independent broadening of
the,whispering-gallery resonances, and to highly anisotropic emission,
Here we present solutions of the wave equation for asymmetric resonant
cavities which confirm these predictions but also reveal interesting
frequency-dependent effects characteristic of quantum chaos, For small
deformations the lifetime is controlled by evanescent leakage, the
optical analogue of quantum tunnelling(5); here the lifetime is
significantly shortened by a process known as 'chaos-assisted
tunnelling'(6-7). In contrast, for large deformations (similar to 10%)
some resonances are found to have longer lifetimes than predicted by
the ray chaos model due to the phenomenon of 'dynamical
localization'(8).},
DOI = {Doi 10.1038/385045a0}
}
@InProceedings{NoeckChang,
Title = {{2D} microcavities: {Theory} and experiemtns},
Author = {N\"ockel, J. U. and Chang, R. K.},
Booktitle = {Cavity-Enhanced Spectroscopies},
Year = {2002},
Editor = {Van Zee, R. D. and Looney, J. P.},
Pages = {185--226},
Publisher = {Academic},
ISBN = {9780124759879}
}
@Article{noisy_Fred,
Title = {Trace formulas for stochastic evolution operators: {Weak} noise perturbation theory},
Author = {P. Cvitanovi\'{c} and C. P. Dettmann and R. Mainieri and G. Vattay },
Journal = {J. Stat. Phys.},
Year = {1998},
Note = {\arXiv{chao-dyn/9807034}},
Pages = {981},
Volume = {93}
}
@Article{nongal89,
Title = {Nonlinear {G}\"{a}lerkin methods},
Author = {M. Martine and R. Temam},
Journal = {SIAM J. Numer. Anal.},
Year = {1989},
Pages = {1139--1157},
Volume = {26},
Abstract = {nonlinear {G}\"{a}lerkin methods}
}
@Book{nr,
Title = {Numerical Recipes in {C}},
Author = {W. H. Press and S. A. Teukolsky and W. T. Vetterling and B. P. Flannery},
Publisher = {Cambridge Univ. Press},
Year = {1992},
Address = {Cambridge, England}
}
@Book{ns,
Title = {Numerical Partial Differential Equations},
Author = {J. W. Thomas},
Publisher = {Spring-Verlag},
Year = {1995},
Address = {New York}
}
@Article{NSTks85,
Title = {Some global dynamical properties of the {Kuramoto-Sivashinsky} equations: nonlinear stability and attractors},
Author = {B. Nicolaenko and B. Scheurer and R. Temam},
Journal = {Physica D},
Year = {1985},
Pages = {155--183},
Volume = {16},
Abstract = {The KSe shows coherent spatial patterns but temporal chaos. The
anti-diffusion term destablizes the solutin while stability is achived by the collective
action of all modes through the nonlinear convection term. An estimate of the L^2 norm of
the solution in the antisymmetric periodic function space is given. The number of
determining Fourier modes is calculated together with fractal dimension of the strange
attractor.}
}
@Article{OcToHe92,
Title = {Accuracy of semiclassical dynamics in the presence of chaos},
Author = {P. W. Oconnor and S. Tomsovic and Heller, E. J. },
Journal = {J. Stat. Phys.},
Year = {1992},
Pages = {131--152},
Volume = {68}
}
@Article{OHSZKCKY04,
Title = {A local ensemble {Kalman} filter for atmospheric data assimilation},
Author = {Ott, E. and Hunt, B. and Szunyogh, I. and Zimin, A. and Kostelich, E. and Corrazza, M. and Kalnay, E. and Yorke, J.},
Journal = {Tellus A},
Year = {2004},
Pages = {415--428},
Volume = {56}
}
@Article{OkEckThiDoe08,
Title = {Mixing effectiveness depends on the source--sink structure: simulation results},
Author = {Okabe, T. and Eckhardt, B. and Thiffeault, J.-L. and Doering, C. R.},
Journal = {J. Stat. Phys},
Year = {2008},
Pages = {P07018},
Volume = {2008},
DOI = {10.1088/1742-5468/2008/07/P07018}
}
@Book{Oksendal03,
Title = {Stochastic Differential Equations: An Introduction with Applications},
Author = {{\O}ksendal, B.K.},
Publisher = {Springer},
Year = {1998},
Address = {New York},
ISBN = {9783540637202}
}
@Article{OlGo92,
Title = {Isotropic-nematic transition in shear flow: {State} selection, coexistence, phase transitions, and critical behavior},
Author = {Olmsted, P. D. and Goldbart, P. M.},
Journal = {Phys. Rev. A},
Year = {1992},
Pages = {4966--4993},
Volume = {46},
DOI = {10.1103/PhysRevA.46.4966}
}
@Article{Olmsted94,
Title = {Rotational invariance and {Goldstone modes} in nematic elastomers and gels},
Author = {Olmsted, P. D.},
Journal = {J. Phys. II France},
Year = {1994},
Pages = {2215--2230},
Volume = {4},
DOI = {10.1051/jp2:1994257}
}
@Article{OnKa79,
Title = {Nonequilibrium steady state of critical fluids under shear flow: {A} renormalization group approach},
Author = {A. Onuki and K. Kawasaki},
Journal = {Ann. Phys.},
Year = {1979},
Pages = {456--528},
Volume = {121},
DOI = {10.1016/0003-4916(79)90105-2}
}
@Article{Onsager53,
Title = {Fluctuations and irreversible processes},
Author = {L. Onsager and S. Machlup},
Journal = {Phys. Rev.},
Year = {1953},
Pages = {1505--1512},
Volume = {91},
DOI = {10.1103/PhysRev.91.1505}
}
@Article{Onsager53II,
Title = {Fluctuations and irreversible processess. {II. Systems} with kinetic energy},
Author = {Machlup, S. and Onsager, L.},
Journal = {Phys. Rev.},
Year = {1953},
Pages = {1512--1515},
Volume = {91},
DOI = {10.1103/PhysRev.91.1512}
}
@Article{Onuki87,
Title = {Homogeneous block copolymer systems under shear flow},
Author = {A. Onuki},
Journal = {J. Chem. Phys.},
Year = {1987},
Pages = {3692--3697},
Volume = {87},
DOI = {10.1063/1.452967}
}
@Article{optp1,
Title = {Optimal periodic orbits of chaotic systems},
Author = {B. R. Hunt and E. Ott},
Journal = {Phys. Rev. Lett.},
Year = {1996},
Pages = {2254},
Volume = {76}
}
@Article{optp2,
Title = {Optimal periodic orbits of continuous time chaotic systems},
Author = {T.-H. Yang and B. R. Hunt and E. Ott},
Journal = {Phys. Rev. E},
Year = {2000},
Pages = {1950},
Volume = {62}
}
@Article{Ordonez96,
Title = {Spectral decomposition of tent maps using symmetry considerations},
Author = {G. E. Ordonez and D. J. Driebe},
Journal = {J. Stat. Phys.},
Year = {1996},
Pages = {269--276},
Volume = {84},
Keywords = {Frobenius-Perron, coupled systems, tent map, symmetry}
}
@Article{Orland11,
Title = {Generating transition paths for {Langevin} bridges},
Author = {H. Orland},
Journal = {J. Chem. Phys.},
Year = {2011},
Pages = {174114},
Volume = {134},
DOI = {10.1063/1.3586036}
}
@Article{OrlMaj15,
Title = {Effective Langevin equations for constrained stochastic processes},
Author = {S. N. Majumdar and H. Orland},
Journal = {J. Stat. Mech.},
Year = {2015},
Pages = {P06039},
DOI = {10.1088/1742-5468/2015/06/P06039}
}
@Article{OrnUhl30,
Title = {On the theory of the {Brownian} motion},
Author = {Uhlenbeck, G. E. and Ornstein, L. S.},
Journal = {Phys. Rev.},
Year = {1930},
Pages = {823--841},
Volume = {36},
DOI = {10.1103/PhysRev.36.823}
}
@Article{Orszag1,
Title = {Boundary conditions for incompressible flows},
Author = {S. A. Orszag and M. Israeli and M. Deville},
Journal = {J. Sci. Comput.},
Year = {1986},
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Abstract = {{G}\"{a}lerkin method}
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Year = {1996},
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Year = {2000},
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Year = {2009},
Note = {\arXiv{0912.1581}},
Pages = {19},
Volume = {136},
Abstract = {Transferring information from observations of a dynamical system to estimate
the fixed parameters and unobserved states of a system model can be formulated
as the evaluation of a discrete time path integral in model state space. The
observations serve as a guiding potential working with the dynamical rules of
the model to direct system orbits in state space. The path integral
representation permits direct numerical evaluation of the conditional mean path
through the state space as well as conditional moments about this mean.}
}
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Year = {1883},
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Volume = {174}
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Pages = {123--161},
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Volume = {156}
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Volume = {3}
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Volume = {231}
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Title = {The {Fokker-Planck} Equation},
Author = {H. Risken},
Publisher = {Springer},
Year = {1996},
Address = {New York},
ISBN = {978-3-540-61530-9}
}
@Misc{RMChaosHist,
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Note = {Appendix {\em ``A brief history of chaos''}, in \refref{DasBuch}},
Year = {2012}
}
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Author = {Robinson, S. K.},
Journal = {Ann. Rev. Fluid Mech.},
Year = {1991},
Pages = {601--639},
Volume = {23}
}
@Article{Romanov73,
Title = {Stability of plane-parallel {C}ouette flow},
Author = {Romanov, V.A.},
Journal = {Functional Anal.\ \& its Applic.},
Year = {1973},
Pages = {137--146},
Volume = {7}
}
@Article{Rome69,
Title = {A direct solution to the linear variance equation of a time-invariant linear system},
Author = {Rome, H.},
Journal = {IEEE Transactions on Automatic Control},
Year = {1969},
Pages = {592--593},
Volume = {14}
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@Article{RoMeBaSchHe09,
Title = {Spectral analysis of nonlinear flows},
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Year = {2009},
Pages = {115},
Volume = {641}
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Year = {1995},
Pages = {4661},
Volume = {52},
DOI = {10.1103/PhysRevE.52.4661}
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@Article{ross,
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Year = {1976},
Pages = {397},
Volume = {57}
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@Article{roth99,
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Author = {D. Rothstein and E. Henry and J. P. Gollub},
Journal = {Nature},
Year = {1999},
Pages = {770--772},
Volume = {401},
Abstract = {Experimentally generate a 2-d periodic(or weakly turbulent)
velocity field using glycerol-water on magnets driven by electric current.
Persistent spatial patterns are observed, which decays exponentially.}
}
@Article{rowley_reduction_2003,
Title = {Reduction and reconstruction for self-similar dynamical systems},
Author = {Rowley, C. W. and Kevrekidis, I. G. and Marsden, J. E. and Lust, K.},
Journal = {Nonlinearity},
Year = {2003},
Pages = {1257--1275},
Volume = {16}
}
@Article{Rowley05,
Title = {Model reduction for fluids using balanced proper orthogonal decomposition},
Author = {C. W. Rowley},
Journal = {Int. J. Bifur. Chaos},
Year = {2005},
Pages = {997--1013},
Volume = {15}
}
@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},
Address = {Philadelphia},
Publisher = {SIAM}
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@Article{RSBH98,
Title = {On stability of streamwise streaks and transition thresholds in plane channel flows},
Author = {Reddy, S. C. and Schmid, P. J. and Baggett, J. S. and Henningson, D. S. },
Journal = {J. Fluid Mech.},
Year = {1998},
Pages = {269--303},
Volume = {365}
}
@Book{rtb,
Title = {Theory of Orbits},
Author = {V. Szebehely},
Publisher = {Academic},
Year = {1967},
Address = {New York}
}
@Article{rue04ne,
Title = {Conversations on nonequilibrium physics with an extraterrestrial},
Author = {D. Ruelle},
Journal = {Physics Today},
Year = {2004},
Pages = {48--53},
Volume = {57},
Abstract = {New ideas of nonequilibrium physics are introduced. Emphasize
the SRB measure approach and energy fluctuation theorem.}
}
@Article{ruell68,
author = {D. Ruelle},
title = {Statistical mechanics of a one-dimensional lattice gas},
journal = {Commun. Math. Phys.},
year = {1968},
volume = {9},
pages = {267--278},
doi = {10.1007/BF01654281},
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@Article{ruelext,
Title = {Large volume limit of the distribution of characteristic exponents in turbulence},
Author = {D. Ruelle},
Journal = {Commun. Math. Phys.},
Year = {1982},
Pages = {287--302},
Volume = {87}
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@Article{ruell71,
Title = {On the nature of turbulence},
Author = {D. Ruelle and F. Takens},
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Year = {1971},
Pages = {167},
Volume = {20}
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@Book{ruelle,
Title = {Statistical Mechanics, Thermodynamic Formalism},
Author = {D. Ruelle},
Publisher = {Addison-Wesley},
Year = {1978},
Address = {Reading, MA}
}
@Article{Ruelle09,
Title = {A review of linear response theory for general differentiable dynamical systems},
Author = {D. Ruelle},
Journal = {Nonlinearity},
Year = {2009},
Note = {\arXiv{0901.0484}},
Pages = {855--870},
Volume = {22}
}
@Article{Ruelle76,
Title = {Zeta functions for expanding maps and {Anosov} flows},
Author = {D. Ruelle},
Journal = {Invent. Math.},
Year = {1976},
Pages = {231},
Volume = {34},
DOI = {10.1007/BF01403069}
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@Article{Ruelle76b,
Title = {A measure associated with {Axiom-A} attractors},
Author = {Ruelle, D.},
Journal = {Amer. J. Math.},
Year = {1976},
Pages = {619--654},
Volume = {98},
Abstract = {The future orbits of a diffeomorphism near an Axiom-A
attractor are investigated. It is found that their asymptotic
behavior is in general described by a fixed probability measure �
carried by the attactor. The measure has an exponential cluster
property, and satisfies a variational principle.}
}
@Article{ruelle79,
Title = {Ergodic theory of differentiable dynamical systems},
Author = {D. Ruelle},
Journal = {Publ. Math. IHES},
Year = {1979},
Pages = {275},
Volume = {50}
}
@Article{Ruelle80,
Title = {Measures describing a turbulent flow},
Author = {Ruelle, D.},
Journal = {Ann. N. Y. Acad. Sci.},
Year = {1980},
Pages = {1--9},
Volume = {357},
DOI = {10.1111/j.1749-6632.1980.tb29669.x}
}
@Article{ruelle86,
Title = {Resonances of chaotic dynamical systems},
Author = {D. Ruelle},
Journal = {Phys. Rev. Lett.},
Year = {1986},
Pages = {405},
Volume = {56},
Abstract = {We present analytic properties of the power spectrum for
a class of chaotic dynamical systems (Axiom-A systems). The power spectrum
is meromorphic in a strip; the position of the poles (resonances) depends
on the system considered, but only their residues depend on
the observable monitored. In relation with these results we also discuss
the exponential or nonexponential decay of correlation functions at infinity.
In conclusion, it appears desirable to analyze the decay of correlation functions
and the possible analyticity of power spectra for physical time evolutions,
and for computer generated simple dynamical systems (non-Axiom-A in general).}
}
@Article{Ruelle97,
Title = {Differentiation of {SRB} states},
Author = {D. Ruelle},
Journal = {Commun. Math. Phys.},
Year = {1997},
Pages = {227},
Volume = {187}
}
@Article{Ruelle98,
Title = {Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics},
Author = {D. Ruelle},
Journal = {J. Stat. Phys.},
Year = {1999},
Note = {\arXiv{chao-dyn/9812032}},
Pages = {393--468},
Volume = {95}
}
@Article{Ruelle98a,
Title = {Nonequilibrium statistical mechanics near equilibrium: {Computing} higher order terms},
Author = {D. Ruelle},
Journal = {Nonlinearity},
Year = {1998},
Note = {\arXiv{arXiv.org:0901.0484}},
Pages = {5--18},
Volume = {11}
}
@Misc{SaGoGh11,
Title = {Two novel costs for determining the tuning parameters of the {Kalman} filter},
Author = {{Saha}, M. and {Goswami}, B. and {Ghosh}, R.},
Note = {\arXiv{1110.3895}},
Year = {2011}
}
@Article{SaitoMatsu,
Title = {Linear response theory formulated from the chaotic dynamics},
Author = {N. Saito and Y. Matsunaga},
Journal = {J. Phys. Soc. Japan},
Year = {1989},
Pages = {3089--3105},
Volume = {58}
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@Article{SaiYam98,
Title = {Time averaged properties along unstable periodic orbits and chaotic orbits in two map systems},
Author = {Y. Saiki and M. Yamada},
Journal = {Nonlin. Proc. Geophys.},
Year = {2008},
Pages = {675--680},
Volume = {15},
DOI = {10.5194/npg-15-675-2008}
}
@Article{saka90br,
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Navier-Stokes equations and relate its nonlinear
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dynamically orthogonal equations to perform efficient order-reduction and
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action of the quadratic terms. In particular, we illustrate the
connection of the low-dimensionality of the attractor with the
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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.
},
Eprint = {http://www.sciencemag.org/cgi/reprint/305/5690/1594.pdf},
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Pages = {640--643},
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@Article{tbohr,
Title = {The diversity of Steady state solutions of the complex {G}inzburg-{L}andau equation},
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@Book{TelGr06,
Title = {Chaotic Dynamics: An Introduction Based on Classical Mechanics},
Author = {T\'el, T. and Gruiz, M.},
Publisher = {Cambridge Univ. Pres},
Year = {2006},
Address = {Cambridge}
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@Article{temam91edd,
Title = {Approximation of attractors, large eddy simulations and multiscale methods},
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Journal = {Proc. R. Soc. Lond. A},
Year = {1991},
Pages = {23--39},
Volume = {434},
Abstract = {Estimates on the dimension of the attractors and the decay rates of
Fourier mode of NSe are given. By approximation of the inertial manifold, the
interaction between the large and small eddies are derived. Such laws open the way to
numerically efficient algorithms via multiscale mthods.}
}
@Article{TFGphase97,
Title = {Studies of Phase Turbulence in the One Dimensional Complex {G}inzburg-{L}andau Equation},
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Journal = {Phys. Rev. E},
Year = {1997},
Pages = {5073},
Volume = {55},
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{Therapos89,
Title = {Balancing transformation for unstable nonminimal linear systems},
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Journal = {IEEE Trans. Automatic Control},
Year = {1989},
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Volume = {34}
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@Article{Thomas2014,
Title = {Self-sustaining turbulence in a restricted nonlinear model of plane {Couette} flow},
Author = {Thomas, V.L. and Lieu, B.K. and Jovanovi\'c, M.R. and Farrell, B.F. and Ioannou, P.J. and Gayme, D.F.},
Journal = {Phys. Fluids},
Year = {2014},
Volume = {26},
DOI = {10.1063/1.4898159}
}
@Article{TiCo01,
Title = {Adjoints and low-rank covariance representation},
Author = {M. K. Tippett and S. E. Cohn},
Journal = {Nonlinear Proc. Geoph.},
Year = {2001},
Pages = {331--340},
Volume = {8},
URL = {https://hal.archives-ouvertes.fr/hal-00302042}
}
@Article{ToHe93,
Title = {Long-time semiclassical dynamics of chaos --- the stadium billiard},
Author = {S. Tomsovic and Heller, E. J. },
Journal = {Phys. Rev. E},
Year = {1993},
Pages = {282 -- 299},
Volume = {47}
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@Article{ToHe93a,
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Journal = {Phys. Rev. Lett.},
Year = {1993},
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@Article{ToHe93b,
Title = {Postmodern quantum mechanics},
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Journal = {Physics Today},
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Year = {1974},
Pages = {1731--1749},
Volume = {51},
Abstract = {The notion of irreversible circulation of fluctuation around
the steady state, or cyclic balance, is proposed. In addition to the
shape of the probability distribution function this concept is necessary
in describing the behaviour of the thermodynamically coupled degrees of
freedom at off-equilibrium situation. When the circulation entails
instability, a macroscopic orbital revolution, i.e. limit cycle, may
appear, which is characteristic of a far from equilibrium situation.
Examples are mentioned, including chemical oscillations and undamped
spiking in lasers.},
DOI = {10.1143/PTP.51.1731}
}
@Article{tompaid96,
Title = {Numerical study of invariant sets of a quasi-periodic perturbation of a symplectic map},
Author = {S. Tompaidis},
Journal = {Experimental Mathematics},
Year = {1996},
Pages = {211--230},
Volume = {5},
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.}
}
@Article{torc96,
Title = {Order Parameter for the Transition from Phase to Amplitude Turbulence},
Author = {A. Torcini},
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Year = {1996},
Pages = {1047},
Volume = {77},
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.}
}
@Article{toricgl,
Title = {Invariant 2-tori in the time-dependent {G}inzburg-{L}andau equation},
Author = {P. Tak\'{a}\v{c}},
Journal = {Nonlinearity},
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Volume = {5}
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@Book{Townsend76,
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@Book{Trefethen97,
Title = {Numerical Linear Algebra},
Author = {L. N. Trefethen and D. Bau},
Publisher = {SIAM},
Year = {1997},
Addr = {Philadelphia}
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@Article{TrePan98,
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Year = {1998},
Pages = {390},
Volume = {55},
Abstract = {A periodic orbit analysis in the Lorenz system
and the study of the properties of the associated tangent
linear equations are performed. A set of vectors are found
that satisfy the Oseledec (1968) theorem and reduce to
Floquet eigenvectors in the particular case of a periodic
orbit. These vectors, called Lyapunov vectors, can be
considered the generalization to aperiodic orbits of the
normal modes of the instability problem and are not
necessarily mutually orthogonal. The relation between
singular vectors and Lyapunov vectors is clarified. The
mechanism responsible for super-Lyapunov growth is shown to
be related to the nonorthogonality of Lyapunov vectors. The
leading Lyapunov vectors, as defined here, as well as the
asymptotic final singular vectors, are tangent to the
attractor, while the leading initial singular vectors, in
general, point away from it. Perturbations that are on the
attractor can be found in the subspace of the leading
Lyapunov vectors.}
}
@Book{Truesdell91,
Title = {A First Course in Rational Continuum Mechanics: {General} concepts},
Author = {Truesdell, C.},
Publisher = {Academic},
Year = {1991},
Address = {New York},
Volume = {1}
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@Article{TTRD93,
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Journal = {Science},
Year = {1993},
Pages = {578--584},
Volume = {261}
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@Article{TuckBar03,
Title = {Symmetry breaking and turbulence in perturbed plane {Couette} flow},
Author = {L. S. Tuckerman and D. Barkley},
Journal = {Theor. Comp. Fluid Dyn.},
Year = {2002},
Note = {\arXiv{physics/0312051}},
Pages = {91--97},
Volume = {16},
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.}
}
@Article{Uhlig73,
Title = {Simultaneous block diagonalization of two real symmetric matrices},
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Journal = {Linear Algebra Appl.},
Year = {1973},
Pages = {281--289},
Volume = {7},
Abstract = {The simultaneous diagonalization of two real symmetric
(r.s.) matrices has long been of interest. This subject is
generalized here to the following problem raised by Taussky-Tod: What
is the first simultaneous block diagonal structure of a nonsingular
pair of r.s. matrices S and T? If the number of blocks is maximal,
then one can speak of the finest simultaneous block diagonalization
of S and T.},
DOI = {10.1016/S0024-3795(73)80001-1}
}
@Book{Ulam60,
title = {{A Collection of Mathematical Problems}},
publisher = {Interscience},
year = {1960},
author = {Ulam, S. M.},
address = {New York},
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Year = {1994},
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@Book{VanZeeLooney2002,
Title = {Cavity-enhanced Spectroscopies},
Author = {Van Zee, R.D. and Looney, J.P.},
Publisher = {Academic Press},
Year = {2002},
ISBN = {9780124759879},
Series = {Experimental methods in the physical sciences},
URL = {https://books.google.com/books?id=NQU8vTYJt1UC}
}
@InProceedings{Varga01,
Title = {On balancing and order reduction of unstable periodic systems},
Author = {Varga, A.},
Booktitle = {Periodic Control Systems 2001},
Year = {2001},
Address = {New York},
Editor = {S. Bittanti and P. Colaneri},
Organization = {IFAC Workshop},
Pages = {177--182},
Publisher = {Elsevier},
ISBN = {978-0-08-043682-1},
URL = {http://elib.dlr.de/22687/1/varga_como01p2.pdf}
}
@InProceedings{Varga05,
Title = {A periodic systems toolbox for {MATLAB}},
Author = {Varga, A},
Booktitle = {Proc. of IFAC Workshop on Periodic Control Systems},
Year = {2005},
Pages = {474--474},
Volume = {16}
}
@InProceedings{Varga11,
Title = {An overview of recent developments in computational methods for periodic system},
Author = {Varga, A.},
Booktitle = {Proc. of IFAC Workshop on Periodic Control Systems},
Year = {2011}
}
@Article{Varga97,
Title = {Periodic {Lyapunov} equations: {Some} applications and new algorithms},
Author = {Varga, A.},
Journal = {Int. J. Control},
Year = {1997},
Pages = {69--88},
Volume = {67},
DOI = {10.1080/002071797224360}
}
@InProceedings{VarVanD01,
Title = {Computational methods for periodic systems-an overview},
Author = {Varga, A. and Van Dooren, P.},
Booktitle = {Periodic Control Systems 2001},
Year = {2001},
Address = {New York},
Editor = {S. Bittanti and P. Colaneri},
Organization = {IFAC Workshop},
Pages = {171--176},
Publisher = {Elsevier},
ISBN = {978-0-08-043682-1},
URL = {http://elib.dlr.de/11743/1/varga_como01p1.pdf}
}
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Volume = {5},
Abstract = {The paper derives a general form of the term
structure of interest rates. The following assumptions are
made: (A.1) The instantaneous (spot) interest rate follows a
diffusion process; (A.2) the price of a discount bond depends
only on the spot rate over its term; and (A.3) the market is
efficient. Under these assumptions, it is shown by means of an
arbitrage argument that the expected rate of return on any bond
in excess of the spot rate is proportional to its standard
deviation. This property is then used to derive a partial
differential equation for bond prices. The solution to that
equation is given in the form of a stochastic integral
representation. An interpretation of the bond pricing formula
is provided. The model is illustrated on a specific case.}
}
@Misc{Vattay,
Author = {G. Vattay and P. Cvitanovi\'{c}},
Note = {Chapter {\em ``Noise''}, in \refref{DasBuch}},
Year = {2012}
}
@Article{VeQi10,
Title = {On the {Poincar{\'e}-Hill} cycle map of rotational random walk: locating the stochastic limit cycle in a reversible {Schnakenberg} model},
Author = {Vellela, M. and Qian, H.},
Journal = {Proc. Roy. Soc. Lond. Ser A},
Year = {2010},
Pages = {771--788},
Volume = {466},
Abstract = {Recent studies on stochastic oscillations mostly focus on
the power spectral analysis. However, the power spectrum yields
information only on the frequency of oscillation and cannot differentiate
between a stable limit cycle and a stable focus. The cycle flux,
introduced by Hill (Hill 1989 Free energy transduction and biochemical
cycle kinetics), is a quantitative measure of the net movement over a
closed path, but it is impractical to compute for all possible cycles in
systems with a large state space. Through simple examples, we introduce
concepts used to quantify stochastic oscillation, such as the cycle flux,
the Hill{\textendash}Qian stochastic circulation and rotation number. We
introduce a novel device, the Poincar{\'e}{\textendash}Hill cycle map
(PHCM), which combines the concept of Hill{\textquoteright}s cycle flux
with the Poincar{\'e} map from nonlinear dynamics. Applying the PHCM to a
reversible extension of an oscillatory chemical system, the Schnakenberg
model, reveals stable oscillations outside the Hopf bifurcation region in
which the deterministic system contains a limit cycle. Bistable behaviour
is found on the small volume scale with high probabilities around both
the fixed point and the limit cycle. Convergence to the deterministic
system is found in the thermodynamic~limit. },
DOI = {10.1098/rspa.2009.0346}
}
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Title = {Semiclassical construction of resonances with hyperbolic structure: the scar function},
Author = {Vergini, E. G. and G. Carlo},
Journal = {J. Phys. A},
Year = {2000},
Pages = {4525--4552},
Volume = {34}
}
@Article{VentKar14,
Title = {Convolutionless {Nakajima-Zwanzig} equations for stochastic analysis in nonlinear dynamical systems},
Author = {D. Venturi and G. E. Karniadakis},
Journal = {Proc. R. Soc. A},
Year = {2014},
Pages = {20130754},
Volume = {470},
DOI = {10.1098/rspa.2013.0754}
}
@Article{VerCar00,
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Author = {Vergini, E. G. and G. Carlo},
Journal = {J. Phys. A},
Year = {2000},
Note = {\arXiv{chao-dyn/9904039}},
Pages = {4717--4724},
Volume = {33}
}
@Article{Verg00,
Title = {Semiclassical theory of short periodic orbits in quantum chaos},
Author = {Vergini, E. G.},
Journal = {J. Phys. A},
Year = {2000},
Note = {\arXiv{chao-dyn/9904038}},
Pages = {4709--4716},
Volume = {33}
}
@Article{VeSchRi08,
Title = {The short periodic orbit approach for {Quantum Cat Maps}},
Author = {Vergini, E. G. and D. M. Schneider and A. M. F. Rivas},
Journal = {J. Phys. A},
Year = {2008},
Pages = {405102},
Volume = {41}
}
@Unpublished{ViSeLi11,
Title = {Metric entropy and the optimal prediction of chaotic signals},
Author = {D. Viswanath and X. Liang and K. Serkh},
Note = {\arXiv{nlin/1102.3202}},
Year = {2011}
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@Article{VK79,
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@Book{vKampen92,
Title = {Stochastic Processes in Physics and Chemistry},
Author = {N. G. van Kampen},
Publisher = {North-Holland},
Year = {1992},
Address = {Amsterdam}
}
@Article{vN32,
Title = {{Zur Operatorenmethode in der klassischen Mechanik}. ({German}) [{On} the operator methods of classical mechanics]},
Author = {J. von Neumann},
Journal = {Ann. Math.},
Year = {1932},
Note = {See additions \cite{vonNeum32}. Reprinted in \cite{Taub61}.},
Pages = {587--642},
Volume = {33}
}
@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{voth02,
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Journal = {Phys. Rev. Lett.},
Year = {2002},
Pages = {254501},
Volume = {88},
Abstract = {Based on the experimentally generated velocity field, fixed points on
poincar\'{e} section are identified as well as their stable and unstable manifolds.
These information can be related to the gradient of the stretching field and the
distribution of passive scalar quantities. It seems that on these manifolds the
stretching field gains its maximum.}
}
@Article{vp98,
Title = {Gradient-based approach to solve optimal periodic output feedback control problems},
Author = {A. Varga and S. Pieters},
Journal = {Automatica},
Year = {1998},
Pages = {477--481},
Volume = {34},
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.}
}
@Article{W01,
Title = {Exact Coherent Structures in Channel Flow},
Author = {Waleffe, F. },
Journal = {J. Fluid Mech.},
Year = {2001},
Pages = {93--102},
Volume = {435}
}
@InProceedings{W02,
Title = {Exact coherent structures and their instabilities: {Toward} a dynamical-system theory of shear turbulence},
Author = {Waleffe, F. },
Booktitle = {Proceedings of the International Symposium on ``Dynamics and Statistics of Coherent Structures in Turbulence: Roles of Elementary Vortices''},
Year = {2002},
Editor = {Kida, Shigeo},
Pages = {115--128},
Publisher = {National Center of Sciences, Tokyo, Japan}
}
@Article{W03,
Title = {Homotopy of exact coherent structures in plane shear flows},
Author = {Waleffe, F.},
Journal = {Phys. Fluids},
Year = {2003},
Pages = {1517--1543},
Volume = {15}
}
@Article{W90,
Title = {Proposal for a self-sustaining process in shear flows},
Author = {Waleffe, F.},
Journal = {Working paper, available at \verb+www.math.wisc.edu/~waleffe/ECS/sspctr90.pdf+},
Year = {1990}
}
@InProceedings{W90b,
Title = {On the origin of the streak spacing in turbulent shear flow},
Author = {Waleffe, F.},
Booktitle = {Annual Research Briefs},
Year = {1990},
Pages = {159--168},
Publisher = {Center for Turbulence Research, Stanford Univ.}
}
@Article{W95a,
Title = {Hydrodynamic Stability and Turbulence: Beyond transients to a self-sustaining process},
Author = {Waleffe, F.},
Journal = {Stud. Appl. Math.},
Year = {1995},
Pages = {319--343},
Volume = {95}
}
@Article{W95b,
Title = {Transition in shear flows: Nonlinear normality versus non-normal linearity},
Author = {Waleffe, F.},
Journal = {Phys. Fluids},
Year = {1995},
Pages = {3060--3066},
Volume = {7}
}
@Article{W98,
Title = {Three-Dimensional Coherent States in Plane Shear Flows},
Author = {Waleffe, F.},
Journal = {Phys. Rev. Lett.},
Year = {1998},
Pages = {4140--4148},
Volume = {81}
}
@Misc{WaKo12,
author = {G. M. Wadsworth and D. T. Kovari},
title = {Optimal resolution of state space in chaotic hyperbolic 2-dimensional maps},
year = {2012},
note = {ChaosBook.org project},
url = {http://ChaosBook.org/projects/Lippolis/WaKo12.pdf},
}
@Unpublished{Waleffe90,
Title = {Proposal for a self-sustaining mechanism in shear flows},
Author = {F. Waleffe},
Note = {Center for Turbulence Research, Stanford Univ./NASA Ames, unpublished preprint (1990)}
}
@Article{Waleffe97,
Title = {On a self-sustaining process in shear flows},
Author = {Waleffe, F.},
Journal = {Phys. Fluids},
Year = {1997},
Pages = {883--900},
Volume = {9}
}
@Article{WaLiLi11,
Title = {Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of {Lorenz } equations},
Author = {Wang, P. F. and Li, J. P. and Li, Q.},
Journal = {Numerical Algorithms},
Year = {2011},
DOI = {10.1007/s11075-011-9481-6}
}
@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},
Note = {www.math.wisc.edu/~waleffe/ECS/WK97.pdf},
Pages = {385--422}
}
@Article{Wan11,
Title = {An adaptive high-order minimum action method},
Author = {X. Wan},
Journal = {J. Comput. Physics},
Year = {2011},
Pages = { - },
DOI = {10.1016/j.jcp.2011.08.006}
}
@Article{Wang13,
Title = {Forward and adjoint sensitivity computation of chaotic dynamical systems},
Author = {Wang, Q.},
Journal = {J. Comput. Phys.},
Year = {2013},
Addendum = {http://arXiv.org/abs/1202.5229},
Pages = {1--13},
Volume = {235},
DOI = {10.1016/j.jcp.2012.09.007}
}
@Article{WanZhoE10,
Title = {Study of the noise-induced transition and the exploration of the phase space for the {Kuramoto-Sivashinsky} equation using the minimum action method},
Author = {X. Wan and X. Zhou and W. E},
Journal = {Nonlinearity},
Year = {2010},
Pages = {475},
Volume = {23}
}
@Book{Wax,
Title = {Selected Papers on Noise and Stochastic Processes},
Author = {E. N. Wax},
Publisher = {Dover},
Year = {1954},
Address = {New York},
ISBN = {9780486495354}
}
@Article{WeezCoupLas,
Title = {Coherence between two coupled lasers from a dynamics perspective},
Author = {Ray, W. and Rogers, J. L. and Wiesenfeld, K.},
Journal = {Opt. Expr.},
Year = {2009},
Pages = {9357--68},
Volume = {17},
DOI = {10.1364/OE.17.009357}
}
@Article{WeezNonlInst,
Title = {Noisy precursors of nonlinear instabilities},
Author = {K. Wiesenfeld},
Journal = {J. Stat. Phys.},
Year = {1985},
Pages = {1071--1097},
Volume = {38}
}
@Article{weiss82,
Title = {The {P}ainlev\'{e} property for partial differential equations},
Author = {J. Weiss and M. Tabor and G. Carnevale},
Journal = {J. Math. Phys.},
Year = {1983},
Pages = {522},
Volume = {24}
}
@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,
Title = {{B}\"{a}cklund transformation and the {P}ainlev\'{e} property},
Author = {J. Weiss},
Journal = {J. Math. Phys.},
Year = {1986},
Pages = {1293},
Volume = {27}
}
@Misc{WeissFeig,
Title = {Feigenbaum constant},
Author = {Weisstein, E. W.},
Note = {mathworld.wolfram.com/FeigenbaumConstant.html},
Year = {2012}
}
@Article{Weyl,
Title = {The asymptotic distribution law for the eigenvalues of linear partial differential equations (with applications to the theory of black body radiation)},
Author = {H. Weyl},
Journal = {Math. Ann.},
Year = {1912},
Pages = {441--479},
Volume = {71}
}
@Article{WierMain,
Title = {Fractal {Weyl} law for chaotic microcavities: {Fresnel}'s laws imply multifractal scattering},
Author = {Wiersig, J. and Main, J.},
Journal = {Phys. Rev. E},
Year = {2008},
Pages = {036205},
Volume = {77},
DOI = {10.1103/PhysRevE.77.036205}
}
@Book{wiggb,
Title = {Global Bifurcations and Chaos},
Author = {S. Wiggins},
Publisher = {Springer},
Year = {1988},
Address = {New York}
}
@Book{wiggins,
Title = {Normally Hyperbolic Invariant Manifolds in Dynamical Systems},
Author = {S. Wiggins },
Publisher = {Springer},
Year = {1994},
Address = {New York}
}
@Book{wiggins2,
Title = {Chaotic Transport in Dynamical Systems},
Author = {S. Wiggins},
Publisher = {Springer},
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Title = {{Hermit}e polynomials},
Author = {Wikipedia},
Note = {{en.wikipedia.org/wiki/Hermite\_polynomials}}
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Standard techniques for computing Lyapunov vectors produce
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for constructing the n most rapidly growing Lyapunov
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Year = {2014},
Note = {\arXiv{1012.2721}},
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@Inproceedings{BajCha17,
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booktitle = {Proc. 11. UK Conference on Boundary Integral Methods UKBIM11},
year = {2017},
editor = {D. J. Chappell},
isbn = {9780993111297},
pages = {25--32},
address = {Nottingham UK},
publisher = {Nottingham Trent Univ.},
url = {http://irep.ntu.ac.uk/id/eprint/31465/},
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author = {Goldobin, Denis S.},
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journal = {European Physical Journal Special Topics},
year = {2014},
volume = {223},
pages = {1699--1709},
doi = {10.1140/epjst/e2014-02179-9},
abstract = {We evoke the idea of representation of the chaotic attractor by the set of unstable periodic orbits and disclose a novel noise-induced ordering phenomenon. For long unstable periodic orbits forming the strange attractor the weights (or natural measure) is generally highly inhomogeneous over the set, either diminishing or enhancing the contribution of these orbits into system dynamics. We show analytically and numerically a weak noise to reduce this inhomogeneity and, additionally to obvious perturbing impact, make a regularizing influence on the chaotic dynamics. This universal effect is rooted into the nature of deterministic chaos.},
day = {01},
issn = {1951-6401},
}
@Comment{jabref-meta: databaseType:bibtex;}