Transient motion and chaotic dynamics in a pair of van der Pol oscillators

The transient chaos and the stable chaotic dynamics of coupled autonomous van der Pol (VdP) oscillators with cubic term are investigated. Transient chaos is a common phenomenon in externally driven van der Pol oscillators. Nevertheless, in coupled autonomous VdP oscillators the occurrence of transient chaos, even stable chaos, is a rare scenario. To the best of our knowledge, transient chaos has not often been observed in coupled autonomous van der Pol systems. We demonstrate that the nonlinear restoring forces in a pair of van der Pol oscillators can induce a transient chaotic route to deterministic chaos. The symmetric coupling has been considered and provided by perturbing the amplitude of one oscillator by a fraction another oscillator's amplitude. The coupled systems undergo a crisis when the coupling parameter passes through a certain threshold. The crisis occurs when the chaotic attractor behaves as a chaotic repeller for a transient time and transient chaos emerges. After transient motion, the system's dynamics is attracted either to a periodic motion or stationary state. The Lyapunov spectrum, bifurcation diagram, phase space trajectories and Poincare section were used to study the chaotic motion. The effects of nonlinear restoring force have been investigated through bifurcation diagram.