Detection of symmetry of attractors from observations I. Theory

Abstract

Barany et al. (1993) proposed a method for determining the symmetries of attractors of equivariant systems by averaging certain classes of equivariant maps. We use an idea in Barany et al. to re-cast definitions of symmetry detectives assuming that we only have access to (equivariant) observations from the system. Detecting from observations allows one to perform averaging in spaces that may have much lower dimension than the phase space. This paper generalises and develops their suggestion. Among the generalisations we consider is the use of nonpolynomial detectives, and we show using the notion of “prevalence” of Hunt et al. (1992) that our detectives and the detectives of Barany et al. (1993), Dellnitz et al. (1994), and Golubitsky and Nicol (1995) give the correct symmetry of attractors “almost certainly” in a measure-theoretic sense. We show that detectives can persistently give incorrect symmetries at isolated points in parametrised systems and discuss how to overcome this. We show how one can find the symmetry of an attractor from examination of a Poincaré section. In Part II of this article, Ashwin and Tomes apply these results to find symmetries of attractors in a physical system of four coupled electronic oscillators with S 4 symmetry.