Bifurcation and chaos in the double-well Duffing–van der Pol oscillator: Numerical and analytical studies

Abstract

The behavior of a driven double-well Duffing--van der Pol oscillator for a specific parametric choice $(|\ensuremath{\alpha}|=\ensuremath{\beta})$ is studied. The existence of different attractors in the system parameters $f$-$\ensuremath{\omega}$ domain is examined and a detailed account of various steady states for fixed damping is presented. The transition from quasiperiodic to periodic motion through chaotic oscillations is reported. The intervening chaotic regime is further shown to possess islands of phase-locked states and periodic windows (including period-doubling regions), boundary crisis, three classes of intermittencies, and transient chaos. We also observe the existence of local-global bifurcation of intermittent catastrophe type and global bifurcation of blue-sky catastrophe type during the transition from quasiperiodic to periodic solutions. Using a perturbative periodic solution, an investigation of the various forms of instabilities allows one to predict Neimark instability in the $f$-$\ensuremath{\omega}$ plane and eventually results in the approximate predictive criteria for the chaotic region.