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

TL;DR

This work analyzes chaotic dynamics in the perturbed Duffing oscillator as the external forcing amplitude $\gamma$ varies, using Runge--Kutta integration and topological horseshoe theory to study first, second, and third return maps. It shows that increasing forcing can cause the Smale horseshoe to degenerate into a pseudo-horseshoe while the chaotic invariant set persists, reducing the lower bound on topological entropy. It also identifies a critical forcing threshold where the chaotic invariant set loses attractivity, as revealed by Lyapunov exponents becoming negative and the attraction basin shrinking, though the horseshoe structure can remain. Together, these results illuminate how external forcing shapes transitions between chaotic and regular behavior in the Duffing system and quantify the associated entropy and stability changes.

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.

Figures (32)

• Figure 1: Smale horseshoe of the first return map when $\gamma=0.4$.
• Figure 2: Schematic diagram of the Smale horseshoe in Figure \ref{['fig1:a']}.
• Figure 3: Smale horseshoe of the first return map when $\gamma=0.5$.
• Figure 4: Smale horseshoe of the first return map when $\gamma=0.6$.
• Figure 5: Smale horseshoe of the first return map when $\gamma=0.7$.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Theorem 2
• Definition 5
• Theorem 3
• proof
• Corollary 1