Variation of entropy in the Duffing system with the amplitude of the external force

TL;DR

This study explores how the entropy and chaotic dynamics of the Duffing system change with varying external force amplitude, revealing a transition from chaos to pseudo-horseshoe states and changes in attractivity.

Contribution

It introduces the analysis of entropy variation and the degeneracy of topological horseshoes in the Duffing system as the external force amplitude varies.

Findings

Topological horseshoes degenerate into pseudo-horseshoes at high force amplitudes.

The lower bound of topological entropy decreases with increasing force amplitude.

The chaotic invariant set's attractivity diminishes as the amplitude surpasses a critical value.

Abstract

In this paper, we revisit the well-known perturbed Duffing system and investigate its chaotic dynamics by means of numerical Runge--Kutta method based on topological horseshoe theory. Precisely, we investigate chaos through the topological horseshoes associated with the first, second, and third return maps, obtained by varying the amplitude of an external force term while keeping all other parameters fixed. Our new finding demonstrates that, when the force amplitude exceeds a certain value, the topological (Smale) horseshoe degenerates into a pseudo-horseshoe, while chaotic invariant set persists. This phenomenon indicates that the lower bound of the topological entropy decreases as the force amplitude increases, thereby enriching the dynamics in the perturbed Duffing system. Furthermore, we identify a critical value of the force amplitude governing the attractivity of the chaotic invariant set. For amplitudes slightly below this value, the basin of attraction of the chaotic invariant set progressively shrinks as the amplitude increases. In contrast, for larger amplitudes, both Lyapunov exponents become negative while the topological horseshoe persists, suggesting that the chaotic invariant set loses attractivity as the amplitude grows.