Symmetry aspects of instabilities in driven systems

Abstract

The authors investigate bifurcations initiated by either soft-mode or hard-mode instabilities in autonomous dynamical systems, with strong emphasis on symmetry considerations. The analysis is performed within the Landau theory of phase transitions extended to driven systems. The bifurcation behaviour is described in terms of an n-component order parameter which transforms either as an irreducible representation of the symmetry group of the system (symmetry-induced bifurcations) or as a reducible representation consisting of an irreducible representation occurring twice (coupling induced bifurcations). Symmetry criteria are derived for bifurcations to stationary and to time-periodic structures. The types of bifurcation may be separated into different classes characterised by the distinct matrix groups which are induced in the order parameter space. They state a number of general stability criteria, and give a complete classification of bifurcations occurring for order parameter dimensions n=1, 2 and 3.