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

We study dynamics of a ring of three unidirectionally coupled double-well Duffing oscillators for three different values of the damping coefficient: fixed, proportional to time, and inversely proportional to time. The system dynamics in all cases are analyzed using time series, Fourier and Hilbert transforms, Poincaré sections, bifurcation diagrams, and Lyapunov exponents for various coupling strengths and damping coefficients. 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 the stable equilibria. Finally, in the third case, transient toroidal hyperchaos takes place.