Nonexistence of observable chaos and its robustness in strongly monotone dynamical systems

TL;DR

This paper proves that strongly monotone dynamical systems do not exhibit observable chaos in a measure-theoretic sense and that this property remains stable under small smooth perturbations.

Contribution

It establishes the nonexistence of observable chaos in strongly monotone systems and demonstrates the robustness of this property under $C^1$-perturbations.

Findings

Largest Lyapunov exponent is positive on a shy set.

Strongly monotone systems admit no observable chaos.

The absence of observable chaos is stable under $C^1$-perturbations.

Abstract

For strongly monotone dynamical systems on a Banach space, we show that the largest Lyapunov exponent $\lambda_{\max}>0$ holds on a shy set in the measure-theoretic sense. This exhibits that strongly monotone dynamical systems admit no observable chaos, the notion of which was formulated by L.S. Young. We further show that such phenomenon of no observable chaos is robust under the $C^1$-perturbation of the systems.