Spatial homogenization by perturbation on the complex Ginzburg–Landau equation

Ginzburg-Landau equation has two types of behavior: one is spatio-temporal chaos lying inside the limit cycle on the two dimensional space, the other is a spatially homogeneous periodic solution on the limit cycle. If we perturb the solution behaving spatio-temporal chaos to the outside of a limit cycle, it is numerically observed that the perturbed solution converges to a spatially homogeneous periodic oscillation. This is the transition from chaos to regular motions based on a spatial homogenization by the perturbation. By constructing the invariant sets and using the asymptotic stability of the limit cycle, we prove analytically that the solution starting from an initial condition far from the limit cycle converges to the limit cycle oscillation.