Numerical estimate of infinite invariant densities: application to Pesin-type identity

Abstract

Weakly chaotic maps with unstable fixed points are investigated in the regime where the invariant density is non-normalizable. We propose that the infinite invariant density of these maps can be estimated using as the long time limit of t^(1-alpha) rho(x, t), in agreement with earlier work of Thaler. Here rho(x, t) is the normalizable density of particles. This definition uniquely determines the infinite density and is a valuable tool for numerical estimations. We use this density to estimate the subexponential separation lambda_alpha of nearby trajectories. For a particular map introduced by Thaler we use an analytical expression for the infinite invariant density to calculate lambda_alpha exactly, which perfectly matches simulations without fitting. Misunderstanding which recently appeared in the literature is removed.

16 pages, 9 figures

References in corpus (5)

• Pesin-Type Identity for Weak Chaos
• Distributional Response to Biases in Deterministic Superdiffusion
• Aging generates regular motions in weakly chaotic systems
• Instability statistics and mixing rates
• Comment on "Lyapunov statistics and mixing rates for intermittent systems"

Cited by in corpus (5)

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• Number of first-passage times as a measurement of information for weakly chaotic systems
• Infinite Ergodicity that Preserves the Lebesgue Measure
• Does an intermittent dynamical system remain (weakly) chaotic after drilling in a hole?