Transitions to chaos in the Ginzburg-Landau equation

Abstract

The amplitude evolution of instability waves in many dissipative systems is described close to criticality, by the Ginzburg-Landau partial differential equation. A numerical study of the long-time behavior of amplitude-modulated waves governed by this equation allows the identification of two distinct routes of the Ruelle-Takens-Newhouse type as the modulation wavenumber is decreased. The first route involves a sequence of bifurcations from a limit cycle to a two-torus to a three-torus and to a turbulent régime, the last stage being preceded by frequency locking. The turbulent régime is itself followed by a new two-torus. In the second route, this two-torus exhibits a single subharmonic bifurcation which immediately results in transition to chaos. A description of the various possible dynamical states is tentatively given in the plane of the two control parameters cd and cn. © 1983.