Nonequilibrium phase transitions and bifurcation of limit cycles and multiperiodic flows

Abstract

In many nonequilibrium physical systems such as lasers, fluids, chemical reactions, semiconductors etc. new structures or patterns occur when external parameters are changed. Furthermore with further changing parameters a whole hierarchy of new pattern formations may occur where the first step starting from a homogeneous state is well understood. The present paper investigates what happens at the next steps where a system performs complicated motions in the form of a limit cycle or, still more generally, in the form of a multiperiodic flow. In this latter case the time dependence is composed of periodic motions of different frequencies. We study the case in which by change of an external parameter the stable motion (trajectory) becomes unstable and is replaced by a set of new trajectories (bifurcation). It is assumed that the bifurcating trajectories remain in the neighbourhood of the former stable trajectory. Our procedure allows for a number of applications to lasers, fluid dynamics, chemical waves etc. as will be shown in subsequent papers.