M. Dellnitz, M. Golubitsky and I. Melbourne
Mechanisms of symmetry creation
In: Bifurcation and Symmetry. (E. Allgower, K. Boehmer and M. Golubitsky, eds.) ISNM 104 Birkhausser, Basel, 1992, 99-109.
Numerical simulation in Chossat and Golubitsky indicates that symmetry increasing bifurcations of chaotic attractors occur with great frequency in the dynamics of symmetric mappings. The pictures in Chossat and Golubitsky, Field and Golubitsky, and King and Stewart also demonstrate that an unexpected kind of pattern formation occurs in symmetric chaotic dynamics where an order based on symmetry is forced on the randomness that is related to chaotic dynamics. Inspection of various numerical simulations in the literature also show that symmetry increasing bifurcations have been observed in ODEs (the Lorenz equation) and in Galerkin approximations of PDEs (the Ginzburg-Landau equation in).
In this paper we make a more detailed numerical analysis of how symmetry creation can occur and how it may be related to pattern formation as the term is used in Physics and Engineering. In applications, the fundamental question concerns how the symmetry of an attractor in phase space manifests itself in physical space. The important point to be noted here is that the symmetry of the attractor in phase space is only `on average'. One must either iterate the mapping a relatively large number of times, or analogously integrate the differential equation for a relatively long time in order to see that symmetry. It follows that if the symmetry of the attractor is to be seen in physical variables (as a pattern) it must be seen through some averaged quantity.
In Section 2 we indicate by example how averaged quantities can undergo symmetry creation. We consider both the Brusselator and the Ginzburg-Landau equation. Although these equations each have only a single reflectional symmetry, the types of symmetry increasing they exhibit are quite different. In Section 3 we illustrate these differences by considering the discrete dynamics of odd maps on the line. In this section we also consider how the parameter values where symmetry creation occurs may be computed by methods other than direct simulation. These techniques are based on theoretical results in Melbourne, Dellnitz and Golubitsky.
Generally speaking we think of symmetry creation in systems with finite symmetry as occurring through the `collision' of symmetry related conjugate attractors. As we show in Section 3 this is not always the basis for symmetry creation --- but collisions do occur frequently and it is a useful way of thinking. In systems with continuous symmetry there is another method by which symmetry creation can occur --- drifting along group orbits. We illustrate this phenomenon in Section 4 by using an example of a mapping on R^4 having O(2) symmetry. We discuss briefly why this type of symmetry creation may be the method by which turbulent wavy vortices turn into turbulent Taylor vortices in the Couette-Taylor experiment.