The Hopf bifurcation and spiral wave solution of the complex Ginzburg–Landau equation

When the dispersive and diffusive effects are negligible, the complex Ginzburg–Landau equation degenerates, in form, as the Landau equation, in which occurs the Hopf bifurcation. For the traveling wave solutions of the complex Ginzburg–Landau equation, which is reduced to its corresponding ordinary differential form by use of traveling wave frame, the spiral and plane waves correspond to the orbit near the focus and the limit cycle, respectively. The shock wave is a heteroclinic orbit between the focus and limit cycle.