Bistable chaos. II. Bifurcation analysis

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=1/4X⁴-1/2{α}X²+{μ}X, 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.