Attractors on an N-torus: Quasiperiodicity versus chaos

Abstract

The occurrence of quasiperiodic motions in nonconservative dynamical systems is of great fundamental importance. However, current understanding concerning the question of how prevalent such motions should be is incomplete. With this in mind, the types of attractors which can exist for flows on an N-torus are studied numerically for N = 3 and 4. Specifically, nonlinear perturbations are applied to maps representing N-frequency quasiperiodic attractors. These perturbations can cause the original N-frequency quasiperiodic attractors to bifurcate to other types of attractors. Our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows: N-frequency quasiperiodic attractors are the most common, followed by (N - 1)-frequency quasiperiodic attractors,…, followed by periodic attractors. However, as the nonlinearity is further increased, N-frequency quasiperiodicity becomes less common, ceasing to occur when the map becomes noninvertible. Chaotic attractors are very rare for N = 3 for small to moderate nonlinearity, but are somewhat more common for N = 4. Examination of the types of chaotic attractors that occur for N = 3 reveals a rich variety of structure and dynamics. In particular, we see that there are chaotic attractors which apparently fill the entire N-torus (i.e., limit sets of orbits on these attractors are the entire torus); furthermore, these are the most common types of chaotic attractors at moderate nonlinearities.