Detecting the symmetry of attractors

This paper addresses the issue of how to determine numerically the symmetry of an attractor for dynamical systems. (The symmetr ies of attractors in phase space are related to patterns in the t ime-average of the solution.) Our approach to this quest ion proceeds in two parts. First, we prove a general theorem, based on group-theoret ic and differential topological ideas, which states that generically the symmetry of a ( thickened) attractor can be computed from the symmetr ies of a point in an auxiliary space. This theorem proceeds by integrating an equivariant mapping over the thickened attractor. Once this is done, the numerical computat ion of symmetr ies reduces to showing that a certain nonnegat ive number is zero. Numerically, demonst ra t ing that this number is zero can be difficult. Thus the second part of the algori thm is to consider how this number varies with parameters and noting that sudden jumps towards zero can be associated with increases in symmetry. The paper is divided into two parts. In the first we prove the general theorem and in the second we illustrate how the numerical techniques work on several examples including discrete dynamical systems with tetrahedral symmet ry in ~ and systems of three coupled cells. In high dimensions the integral ment ioned previously is difficult to compute . For such examples, we assume that an crgodic theorem is valid and that symmetr ies can be computcd using a t ime-average. We compare both of these methods on the low-dimensional examples as well as detect points of symmetry creation for a react ion-diffusion equat ion on an interval. This technique can also be used in principle to compute the symmetr ies of an attractor in an exper iment from a time-series.