Nonequilibrium phase transitions and bifurcation of limit cycles and multiperiodic flows in continuous media

Abstract

This paper deals with higher order instabilities which may occur in various synergetic systems. We extend the method given in our previous paper in several ways. We include continuous (and inhomogeneous) media described by nonlinear partial differential equations. While in our previous paper we assumed that the bifurcating trajectories remain close to the corresponding old one we now relax this assumption. It is now only assumed that the newly developing manifolds remain close to the originally attracting manifold. Furthermore we may allow for stochastic forces, which are important at phase transition points, or for weak external driving fields. Our approach avoids the difficulty of small divisors known from other approaches treating bifurcation of limit cycles. Our paper shows that in many cases the enormous number of degrees of freedom of a system can exactly be reduced to few relevant degrees of freedom (order parameters) close to situations where bifurcation occurs. The resulting order parameter equations may describe various kinds of motion including chaos.