Time-Dependent and Chaotic Behaviour in Systems with O(3)-Symmetry

It is by now well known that chaotic temporal behaviour may be generated by completely deterministic evolution laws. The main theoretical interest in chaotic phenomena has been devoted to the examination of the stochastic properties of low dimensional dynamical systems, although experimental confirmations of certain theoretical predictions have been mainly pursued in spatially extended systems, e.g. in lasers, hydrodynamic flows, and chemical reactions. In such systems low dimensional temporal chaotic behaviour evidently has to be strongly interrelated with coherent spatial patterns. The success of synergetics in the explanation of spontaneous pattern formation in selforganizing systems allows a straightforward treatment of time-dependent and chaotic behaviour in spatially extended systems of finite size which is caused by the nonlinear interaction of several order parameters, each connected with a coherent spatial pattern. In these cases a complete examination of the interrelations between temporal disorder and spatial coherence becomes possible. Spatial patterns are naturally connected to the symmetries of a system. Therefore, it can be expected that quite different systems, which, however, possess the same spatial symmetries, may show comparable behaviour. To demonstrate this we shall consider instabilities in systems with 0(3)-symmetry. For the special case of mode interaction between two groups of modes belonging to two different irreducible representations of the 0(3)-group with l=1 and l=2 we shall derive the order parameter equations.

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