Lyapunov Modes in Extended Systems

Yang, Hongliu; Radons, Günter

doi:10.1002/9783527658701.ch12

Cited by 4 publications

(4 citation statements)

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“…It has been suggested that the Lyapunov vectors associated with the large positive Lyapunov exponents may not be as significant in the long-time dynamics as the Lyapunov vectors associated with the Lyapunov exponents near zero (c.f. [34]). The dynamics of the vectors associated with the large exponents represent fast dynamics where the dynamics of the vectors associated with Lyapunov exponents near zero represent much longerlived dynamics.…”

Section: Introductionmentioning

confidence: 99%

Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection

Paul

2016

Phys. Rev. E

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We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.

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Section: Introductionmentioning

confidence: 99%

Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection

Paul

2016

Phys. Rev. E

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“…Systems of many hard particles have also been instrumental in the discovery of a fascinating connection between microscopic and macroscopic dynamical effects, that of Lyapunov modes, although later found in systems with soft potentials; for a review see [74]. There are many other connections between dynamics in general and statistical mechanics [29,54].…”

Section: Physical Applicationsmentioning

confidence: 99%

Recent Advances in Open Billiards with Some Open Problems

Dettmann

2011

World Scientific Series on Nonlinear Science Series B

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Much recent interest has focused on "open" dynamical systems, in which a classical map or flow is considered only until the trajectory reaches a "hole", at which the dynamics is no longer considered. Here we consider questions pertaining to the survival probability as a function of time, given an initial measure on phase space. We focus on the case of billiard dynamics, namely that of a point particle moving with constant velocity except for mirror-like reflections at the boundary, and give a number of recent results, physical applications and open problems.

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“…The theory and procedures of Lyapunov exponent calculation have been properly developed [1][2][3][4] and now Lyapunov exponents are used widely in very different fields of science, including (but not limited to) physics [5], astronomy [6], medicine [7], economy [8], etc. Due to their great efficiency Lyapunov exponents are applied to a large number of complex systems, including spatially extended ones [9][10][11][12][13][14][15]). …”

Section: Introductionmentioning

confidence: 99%