Sensitivity, entropy, and escape rates at the onset of chaos

TL;DR

This paper analytically connects sensitivity, entropy, and escape rates in nonlinear dynamical systems at chaos and its onset, unifying known cases of strong and weak chaos.

Contribution

It provides a unified analytical framework linking key dynamical properties at the transition to chaos, covering both positive and zero Lyapunov exponents.

Findings

Unified relations at chaos and chaos onset.

Reconciliation of entropy and escape rate behaviors.

Extension of Pesin-like identities to systems with escape.

Abstract

We analytically link three properties of nonlinear dynamical systems, namely sensitivity to initial conditions, entropy production, and escape rate, in $z$-logistic maps for both positive and zero Lyapunov exponents. We unify these relations at chaos, where the Lyapunov exponent is positive, and at its onset, where it vanishes. Our result unifies, in particular, two already known cases, namely (i) the standard entropy rate in the presence of escape, valid for exponential functionality rates with strong chaos, and (ii) the Pesin-like identity with no escape, valid for the power-law behavior present at points such as the Feigenbaum one.