On the hysteresis and dynamics of a ring of three unidirectionally coupled bistable Duffing oscillators

We study a ring of three Duffing oscillators coupled unidirectionally, focusing on the effects of the coupling strength, time-dependent damping and attractor following using time series, phase portrait, Fourier transformations, bifurcation diagrams, potential planes, Lyapunov exponents and hysteresis paths for two cases of damping: constant \((\alpha=0.4)\) and linearly increasing \((\alpha(t)=\frac{t}{\tau})\). For constant damping term the dynamics shows quasiperiodic 2D torus, limit cycles, heteroclinic orbits, fixed points, chaos, and hyperchaos. These behavior shows the existence of hysteresis and an important change in the potential energy. If damping term is dependent of the time \((\alpha = \frac{t}{\tau} )\), the dynamics of the system shows bottleneck behavior influenced by higher frequencies which persists for a small range of values of \((\sigma)\). In resume, the potential energy in this set of three Duffing oscillators depends strongly of three factors: first the attractor tracking (past memories), second the coupling strength and third the damping term.