Chaotic transients and generalized crises of a Duffing-van der Pol oscillator with two external periodic excitations

Abstract

Global property of a Duffing-van der Pol oscillator with two external periodic excitations is investigated by generalized cell mapping digraph method. As the bifurcation parameter varies, a chaotic transient appears in a regular boundary crisis. Two kinds of transient boundary crises are discovered to reveal some reasons for the discontinuous changes for domains of attraction and boundaries. A chaotic saddle collides with the stable manifold of a periodic saddle at the fractal boundary of domains when the crisis occurs, if the chaotic saddle lies in the basin of attraction, the basin of attraction decreases suddenly while the boundary increases after the crisis; if the chaotic saddle is at a boundary, the two boundaries merge into one because of the crisis. In addition, two chaotic saddles can be merged into a new one, when they touch each other in a transient merging crisis. Finally the chaotic transient disappears in an interior crisis. The characteristics of these generalized crises are quite important for the study of chaotic transients.