Bifurcations to invariant 2-tori for the complex Ginzburg-Landau equation

摘要

We present a numerical study of the large-time asymptotic behavior of solutions to the one-dimensional complex Ginzburg-Landau equation with periodic boundary conditions. Our parameters belong to the Benjamin-Feir unstable region. Our solutions start near a pure-mode rotating wave that is stable under sideband perturbations for the Reynolds number R ranging over an interval (Rsub, Rsup). We find sub- and supercritical bifurcations from this stable rotating wave to a stable 2-torus as the parameter R is decreased or increased past the critical value Rsub or Rsup. As R > Rsup further increases, we observe additional stable 2-tori together with a variety of other dynamical phenomena. We compare our numerical simulations to both rigorous mathematical results and experimental observations for binary fluid mixtures.