Nonlinear behaviors in an extended medium : complex hysteresis in Ginzburg-Landau system

Abstract

We analyze dynamics of the complex Ginzburg-Landau equation by sequential reduction. First the complex Ginzburg-Landau equation is reduced into a high-dimensional ordinary differential equation by proper mode truncation, and analyzed with varying control parameters. The parameter representing energy change determines the stability of single-lobed tori, and thus it crucially affects the destabilizing route of periodicity. Various features occur in the bifurcation sequences including complex hysteresis. A simple hysteresis refers to subcritical transitions between two different dynamical phases at two different values of a control parameter in a simple dynamical system. In contrast to such simple hysteresis, we report here a more complex form where more than two different dynamical phases are participating in the process. One-dimensional return maps are used to investigate the basic characteristics of this system.