Strange attractors that are not chaotic

Abstract

It is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic. Here we use the word strange to refer to the geometry or shape of the attracting set, while the word chaotic refers to the dynamics of orbits on the attractor (in particular, the exponential divergence of nearby trajectories). We first give examples for which it can be demonstrated that there is a strange nonchaotic attractor. These examples apply to a class of maps which model nonlinear oscillators (continuous time) which are externally driven at two incommensurate frequencies. It is then shown that such attractore are persistent under perturbations which preserve the original system type (i.e., there are two incommensurate external driving frequencies). This suggests that, for systems of the typw which we have considered, nonchaotic strange attractors may be expected to occur for a finite interval of parameter values. On the other hand, when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit. Thus we conjecture that, in general, continuous time systems (“flows”) which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space.