Dynamics of a two-dimensional map on nested circles and rings

Abstract

• We consider a two-dimensional map whose bounded dynamics can be restricted to an invariant circle, cyclic invariant circles, invariant annual regions or disks. • We prove that on such invariant sets the trajectories are always either periodic of the same period, or quasiperiodic and dense. • We show that the invariant sets may be transversely attracting or repelling, and undergo the typical cascade of period doubling bifurcations. • We show that homoclinic bifurcations lead to chaotic rings, annual regions filled with dense repelling cyclical circles and aperiodic trajectories. We consider a discrete dynamical system, a two-dimensional real map which represents a one-dimensional complex map. Depending on the parameters, its bounded dynamics can be restricted to an invariant circle, cyclic invariant circles, invariant annular regions or disks. We show that on such invariant sets the trajectories are always either periodic of the same period, or quasiperiodic and dense. Moreover, the invariant sets may be transversely attracting or repelling, and undergo the typical cascade of period doubling bifurcations. Homoclinic bifurcations can also occur, leading to chaotic rings, annular regions filled with dense repelling cyclical circles and aperiodic trajectories.