Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

Grassberger, Peter; Badii, R.; Politi, Antonio

doi:10.1007/bf01015324

Cited by 419 publications

(225 citation statements)

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“…For finite time t the value of λ depends on initial conditions and it is called a finite time Lyapunov exponent. The probability distribution of finite time Lyapunov exponents P (λ, t) [2,3,4], especially its behavior at small λ, is important for many applications, where the measured quantity is sensitive to the existence of trajectories staying close for anomalously long time. Examples range from the problems of ocean acoustics [5] and branching of 2d electron flow [6] to Loschmidt echo [7] and mesoscopic superconductivity [8].…”

Section: Introductionmentioning

confidence: 99%

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Chaos beyond linearized stability analysis: Folding of the phase space and distribution of Lyapunov exponents

Silvestrov

Пономарев

2007

Physics Letters A

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We consider a mechanism for area preserving Hamiltonian systems which leads to the enhanced probability, P (λ, t), to find small values of the finite time Lyapunov exponent, λ. In our investigation of chaotic dynamical systems we go beyond the linearized stability analysis of nearby divergent trajectories and consider folding of the phase space in the course of chaotic evolution. We show that the spectrum of the Lyapunov exponents F (λ) = lim t→∞ t −1 ln P (λ, t) at the origin has a finite value F (0) = −λ and a slope F ′ (0) ≤ 1. This means that all negative moments of the distribution e −mλt are saturated by rare events with λ → 0. Extensive numerical simulations confirm our findings. An exponential divergency of nearby trajectories in phase space is commonly considered as a paradigm of classical chaos [1,2]. The time evolution of the distance between two trajectories is determined by the stability matrix M(t), whose largest eigenvalue grows exponentially like e λt . For finite time t the value of λ depends on initial conditions and it is called a finite time Lyapunov exponent. The probability distribution of finite time Lyapunov exponents P (λ, t) [2,3,4], especially its behavior at small λ, is important for many applications, where the measured quantity is sensitive to the existence of trajectories staying close for anomalously long time. Examples range from the problems of ocean acoustics [5] and branching of 2d electron flow [6] to Loschmidt echo [7] and mesoscopic superconductivity [8].

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Section: Introductionmentioning

confidence: 99%

“…For large times the probability distribution of finite-time Lyapunov exponents P (λ, t) [2,3,4] has a generic form (λ ≥ 0)…”

mentioning

confidence: 99%

Chaos beyond linearized stability analysis: Folding of the phase space and distribution of Lyapunov exponents

Silvestrov

Пономарев

2007

Physics Letters A

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“…Generalnie jednak wykładnicze rozprzestrzenianie się błędów uniemożliwia prognozy. Dynamika stopy rozbieżności jest reprezentowana przez efektywne wykładniki Lapunowa ( [14]), lokalne wykładniki Lapunowa (zob. [39]) oraz prognozowalne wzorce (zob.…”

Section: Przegląd Literaturyunclassified

Fractal features of structures of gas flow in selected gas stations of I order

Gurgul¹,

Suder²

2012

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“…Finite time Lyapunov exponents (FTL) have been introduced to quantify dynamical instabilities over a finite interval of time [22,23,24,25]. They depend on time and on the initial conditions of the dynamical system.…”

Section: Quantitative Measures To Characterize Transient Dynamicsmentioning

confidence: 99%

Thresholds in transient dynamics of signal transduction pathways

Rateitschak¹

2010

Journal of Theoretical Biology

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Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

Cited by 419 publications

References 25 publications

Chaos beyond linearized stability analysis: Folding of the phase space and distribution of Lyapunov exponents

Chaos beyond linearized stability analysis: Folding of the phase space and distribution of Lyapunov exponents

Fractal features of structures of gas flow in selected gas stations of I order

Thresholds in transient dynamics of signal transduction pathways

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