Bistable chaos. II. Bifurcation analysis

Abstract

Results of a bifurcation analysis are given for the model of a Van der Pol–Duffing autonomous electronic oscillator. The oscillator is described by three ordinary differential equations and consists of a RC oscillator resistively coupled to an LC oscillator. The steady-state problem is described by the unfolding of the quartic potential F=1/4${\mathit{X}}^4$-1/2α${\mathit{X}}^2$+μX, giving rise to the elementary cusp catastrophe. We show how the bifurcation diagram evolves with μ and recover a ‘‘cross-shaped diagram’’ reminiscent of the one obtained by Boissonade and De Kepper for the Belousov-Zhabotinskii chemical system [J. Phys. Chem. 84, 501 (1980)]. We also show that nonzero values of μ result in coexisting attractors with different dynamics. Specifically, we show a limit cycle attractor in one potential well coexisting with a chaotic attractor in the other well.