Hopf bifurcation and bifurcation from constant oscillations to a torus path for delayed complex Ginzburg-Landau equations

Abstract

We consider the complex Ginzburg-Landau equation with feedback control given by some delayed linear terms (possibly dependent of the past spatial average of the solution). We prove several bifurcation results by using the delay as parameter. We start proving a Hopf bifurcation result for the equation without diffusion (the so-called Stuart-Landau equation) when the amplitude of the delayed term is suitably chosen. The diffusion case is considered first in the case of the whole space and later on a bounded domain with periodicity conditions. In the first case a linear stability analysis is made with the help of computational arguments (showing evidence of the fulfillment of the delicate transversality condition). In the last section the bifurcation takes place starting from an uniform oscillation and originates a path over a torus. This is obtained by the application of an abstract result over suitable functional spaces.