Finite-scale geometric invariants for chaotic and weakly chaotic dynamics

TL;DR

The paper introduces a finite-scale geometric invariant $I(\epsilon,t)$ that measures the time- and resolution-dependent growth of localized unstable sets without relying on symbolic partitions. In uniformly hyperbolic systems, $I(\epsilon,t)$ converges to a scaling $\frac{h_{KS}}{\log(1/\epsilon)}$, linking finite-scale growth to entropy; in non-hyperbolic regimes it vanishes, and in open/crisis regimes it reflects escape rates via $-\kappa/\log(1/\epsilon)$. It further shows that in open intermittent dynamics the invariant remains well-defined through survival-conditioned averages, capturing subexponential instability characteristic of weak chaos. Numerical tests on the hyperbolic Henon map, the Feigenbaum point, and the Pomeau–Manneville map corroborate these behaviors, illustrating the utility of finite-scale geometric observables as a versatile tool beyond classical asymptotic invariants.

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.

Figures (2)

• Figure 1: (a) Rescaled invariant $\log(1/\varepsilon)\,I(\varepsilon,t)$ for the Hénon map at $a=1.4$, $b=0.3$, showing convergence to the Kolmogorov--Sinai entropy $h_{\mathrm{KS}}$ (dashed line). (b) Same quantity for the logistic map at the Feigenbaum accumulation point $\mu=\mu_\infty$, where the invariant converges to zero for all resolutions. The same definition and normalization are used in both cases.
• Figure 2: Finite-scale geometric invariant for an open intermittent map. Shown is the normalized quantity $n\log(1/\varepsilon)\, I(\varepsilon,n)$ as a function of time $n$ for the open Pomeau--Manneville map $x_{n+1} = x_n + a x_n^{z}$ with $z=1.5$. Trajectories are removed upon reaching $x\ge1$, and averages are conditioned on survival up to time $n$. Curves corresponding to different resolutions $\varepsilon$ collapse onto a common bounded behavior, indicating subexponential instability characteristic of transient weak chaos. The dashed line marks the mean plateau value obtained from the smallest resolution.