Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent

TL;DR

This paper studies intermittent maps with zero Lyapunov exponents, deriving a generalized entropy measure linked to trajectory separation and complexity, supported by analytical and numerical results.

Contribution

It introduces a generalized Pesin's identity connecting a new trajectory separation measure with Krengel's entropy for systems with zero Lyapunov exponents.

Findings

Derived the infinite invariant density for Pomeau-Manneville map.

Established the equality of $ ext{alpha} imes ext{mean}( ext{lambda}_ ext{alpha})$ with Krengel's entropy.

Validated theoretical predictions with numerical simulations.

Abstract

One dimensional intermittent maps with stretched exponential separation of nearby trajectories are considered. When time goes infinity the standard Lyapunov exponent is zero. We investigate the distribution of $\lambda_{\alpha}= \sum_{i=0}^{t-1} \ln \left| M'(x_i) \right|/t^{\alpha}$, where $\alpha$ is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of $\lambda_{\alpha}$ is determined by the infinite invariant density. Using semi analytical arguments we calculate the infinite invariant density for the Pomeau-Manneville map, and with it obtain excellent agreement between numerical simulation and theory. We show that $\alpha \left< \lambda_{\alpha}\right>$ is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that $\left< \lambda_{\alpha}\right>$ and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.