Finite-scale geometric invariants for chaotic and weakly chaotic dynamics

TL;DR

This paper introduces a finite scale geometric measure to characterize chaotic and weakly chaotic dynamics, providing a finite-time, resolution-dependent way to understand entropy and chaos in various dissipative systems.

Contribution

It proposes a novel finite scale geometric observable that captures chaotic behavior without relying on symbolic dynamics, applicable to both hyperbolic and intermittent systems.

Findings

Observable converges to Kolmogorov Sinai entropy in hyperbolic systems.

Decays to zero in hyperbolic systems without entropy production.

Remains well defined in open intermittent systems, indicating weak chaos signatures.

Abstract

We introduce a finite scale geometric observable that quantifies the growth rate of localized sets under time evolution in dissipative dynamical systems. Defined at finite time and resolution without reference to symbolic dynamics or Markov partitions this observable converges, in uniformly hyperbolic systems, to a resolution dependent plateau whose logarithmic scaling coefficient equals the Kolmogorov Sinai entropy. In merely hyperbolic systems, it decays to zero, reflecting the absence of entropy production, while remaining well defined at finite scales. Numerical results for the Henon map and Feigenbaum point illustrate these behaviors. Our findings yield a finite scale geometric characterization of chaotic dynamics, consistent with classical entropy theory where applicable. We further demonstrate that the observable remains well defined in open intermittent systems, where trajectories escape and classical asymptotic invariants fail, revealing finite scale signatures of transient weak chaos.