Modulation effects on the dynamics of a ring of three Duffing oscillators coupled unidirectionally with time-dependent damping

We study the dynamics of a ring of three unidirectionally coupled modulated double-well Duffing oscillators for three different values of the damping coefficient: fixed damping, proportional to time, and inversely proportional to time. The dynamics in all cases are analyzed through time series, Fourier transforms, Poincaré sections, as well as bifurcation diagrams and Lyapunov exponents with respect to the coupling strength and modulation parameters (amplitude Γ and frequency θ). In the first case, we observe how a well-known route from a stable steady state to hyperchaos through Hopf bifurcation and a series of torus bifurcations change when modulation parameters (amplitude and frequency) and the coupling strength are increased. In the second case, the system is highly dissipative and converges into one of the small limit cycles generated from the attraction basins. Finally, transient toroidal hyperchaos coexists with quasi-periodic states in the third case.