New Period-Doubling and Equiperiod Bifurcations of the Reversible Area-Preserving Map

Abstract

Two types of period-doubling and equiperiod bifurcations of the reversible areapreserving map are studied. Ordinary period-doubling bifurcation means that the eigenvalue of the mother elliptic periodic orbit (u) is −1, u becomes a saddle periodic orbit with reflection, and an elliptic daughter periodic orbit (v) appears, where the period of v is twice that of u. The other period-doubling bifurcation named the reverse period-doubling bifurcation means that the eigenvalue of the mother saddle periodic orbit with reflection (u′) is −1, u′ becomes an elliptic orbit, and a daughter periodic orbit (v′) appears, where the period of v′ is twice that of u′. The daughter periodic orbit is a saddle with reflection. We prove that both the daughters v and v′ exist in the reversible Smale horseshoe. The forcing relation of the ordinary and reverse period-bifurcations is obtained. Similarly, the ordinary equiperiod and reverse equiperiod bifurcations are also discussed.