BASIN EROSION IN THE TWIN-WELL DUFFING OSCILLATOR: TWO DISTINCT BIFURCATION SCENARIOS

Abstract

A study is made of the control-phase portraits encountered in the twin-well Duffing oscillator, concentrating on the loss of stability of attractors confined to a single well of the potential. This loss of stability may be understood by examining the bifurcations which precede a single-well attractor touching its basin boundary. Two distinct bifurcation scenarios are described in which the basin boundary develops a fractal structure. This fractal structure accompanies the development of a homoclinic tangency between the inset and outset of the saddle whose inset determines the separatrix action. In the first scenario described, this fractal structure, or chaotic saddle, grows due to a sequence of fractal–fractal basin implosions which are caused by the completion of Smale cycles or heteroclinic chains; the subharmonics involved occur in decreasing order, after the main homoclinic tangency. The second scenario focuses on the bifurcational events which necessarily prepare the creation of a chaotic saddle; the subharmonics involved appear in increasing order, before the main homoclinic event. Both basin erosion scenarios are consistent with a geometric model of the twin-well Duffing oscillator based on a three layer template; in the first scenario, the full three-layer template participates; while in the second scenario, only two layers, a standard horseshoe, play a role.