Symmetric Chaos

Abstract

Chaos is the occurrence of highly irregular behaviour in a deterministic dynamical system. It is well known that both ordered and chaotic states can be obtained in such systems by varying parameters. In contrast we here consider circumstances in which aspects of both order and chaos may be present in the same behaviour and at the same parameter values. Namely, in a dynamical system with symmetry, chaotic attractors may themselves possess a degree of symmetry. The non-chaotic dynamics of symmetric systems has attracted increasing attention of late, and the time appears to be ripe to extend the investigation into the chaotic regime. We describe some experimental situations in which this type of behaviour appears to be implicated, including patterned turbulence in fluids and nonlinear oscillations in electronic circuits. We describe sample results from various numerical experiments, to illustrate some of the phenomena that can occur, and describe some simple but fundamental mathematical results that go some way towards explaining them. The field has many open questions, some of which are indicated. © 1992, ACADEMIC PRESS, INC.