Dynamics of a ring of three unidirectionally coupled Duffing oscillators with time-dependent damping

TL;DR

This paper investigates the complex dynamics of three unidirectionally coupled Duffing oscillators arranged in a ring, under different damping conditions, revealing routes to chaos, stability, and transient behaviors through bifurcation analysis.

Contribution

It introduces a detailed analysis of how time-dependent damping affects the dynamics of coupled Duffing oscillators, highlighting new transient and bifurcation phenomena.

Findings

Route from stable state to hyperchaos via bifurcations

System converges to stable equilibria with proportional damping

Transient toroidal hyperchaos observed with inverse damping

Abstract

We study dynamics of a ring of three unidirectionally coupled double-well Duffing oscillators for three different values of the damping coefficient: fixed dumping, proportional to time, and inversely proportional to time. The dynamics in all cases is analyzed through time series, Fourier and Hilbert transforms, Poincar\'e sections, as well as bifurcation diagrams and Lyapunov exponents with respect to the coupling strength. In the first case, we observe a well-known route from a stable steady state to hyperchaos through Hopf bifurcation and a series of torus bifurcations, as the coupling strength is increased. In the second case, the system is highly dissipative and converges into one of stable equilibria. Finally, in the third case, transient toroidal hyperchaos takes place.