Chaos Beyond Linearized Stability Analysis: Folding of the Phase Space and Distribution of Lyapunov Exponents

Abstract

We consider a mechanism for area preserving Hamiltonian systems which leads to the enhanced probability, $P(\lambda ,t)$, to find small values of the finite time Lyapunov exponent, $\lambda$. 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 $F(\lambda )={\lim }_{t\to \infty }{t}^{-1}\ln P(\lambda ,t)$ at the origin has a finite value $F(0)=-\tilde{\lambda }$ and a slope $F^{\prime }(0)\le 1$. This means that all negative moments of the distribution $<e^{-m\lambda t}>$ are saturated by rare events with $\lambda \to 0$. Extensive numerical simulations confirm our findings.

10 pages, 5 figures

References in corpus (5)

• Statistics of finite-time Lyapunov exponents in a random time-dependent potential
• Quantum Disorder and Quantum Chaos in Andreev Billiards
• Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics
• Adiabatic quantization of Andreev levels
• Spectrum of the Andreev Billiard and Giant Fluctuations of the Ehrenfest Time

Cited by in corpus (1)

• Fractal catastrophes