Bifurcations of Mode-Locked Periodic Orbits in Three-Dimensional Maps

Abstract

In this paper, we report the bifurcations of mode-locked periodic orbits occurring in maps of three or higher dimensions. The “torus” is represented by a closed loop in discrete time, which contains stable and unstable cycles of the same periodicity, and the unstable manifolds of the saddle. We investigate two types of “doubling” of such loops: (a) two disjoint loops are created and the iterates toggle between them, and (b) the length of the closed invariant curve is doubled. Our work supports the conjecture of Gardini and Sushko, which says that the type of bifurcation depends on the sign of the third eigenvalue. We also report the situation arising out of Neimark–Sacker bifurcation of the stable and saddle cycles, which creates cyclic closed invariant curves. We show interesting types of saddle-node connection structures, which emerge for parameter values where the stable fixed point has bifurcated but the saddle has not, and vice versa.