E. Barany, M. Dellnitz and M. Golubitsky
Detecting the symmetry of attractors
Physica D. 67 (1993) 66-87.
This paper addresses the issue of how to determine numerically the symmetry of an attractor for dynamical systems. (The symmetries of attractors in phase space are related to patterns in the time-average of the solution.) Our approach to this question proceeds in two parts. First, we prove a general theorem, based on group-theoretic and differential topological ideas, which states that generically the symmetry of a (thickened) attractor can be computed from the symmetries of a point in an auxilliary space. This theorem proceeds by integrating an equivariant mapping over the thickened attractor.
Once this is done, the numerical computation of symmetries reduces to showing that a certain non-negative number is zero. Numerically, demonstrating that this number is zero can be difficult. Thus the second part of the algorithm 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 symmetry in R^3 and systems of three coupled cells. In high dimensions the integral mentioned previously is difficult to compute. For such examples, we assume that an ergodic theorem is valid and that symmetries can be computed using a time-average. We compare both of these methods on the low-dimensional examples as well as detect points of symmetry creation for a reaction-diffusion equation on an interval. This technique can also be used in principle to compute the symmetries of an attractor in an experiment from a time-series.