Slow time-periodic solutions of the Ginzburg-Landau equation

In this paper we study the behaviour of solutions of the form if(z, t) = q~(z) e- i~wt (e << 1) of the rescaled Ginzburg-Landau equation, ~k, = [ 1 - (1 + iB)l~kl2]~k + (1 + iA)~kzz, for A = ca, B = eb, w plays the role of free parameter. This leads to a perturbation analysis on a complex Duffing equation (similar to the analysis of Holmes [Physica D 23 (1986) 84]). We show that the spatial quasiperiodic solutions (of the unperturbed, e = 0, case) disappear due to the perturbation and prove the existence of degenerated periodic solutions which oscillate through the origin. We also establish the existence of several types of heteroclinic orbits connecting counterrotating periodic patterns.