Invariant tori of the weakly dissipative version of the Ginzburg—Landau equation

We consider a periodic boundary value-problem for a weakly dissipative variant of the complex Ginzburg– Landau equation in the case where the period (wavelength) is small. The possibility of the existence of finite-dimensional invariant tori is proved. For solutions that belong to such tori, asymptotic formulas are obtained. We prove that all invariant tori, except for tori of dimension one (i.e., limit cycles), are unstable. We used various methods of the theory of dynamical systems with an infinite-dimensional space of initial conditions, for example, the method of integral (invariant) manifolds, the method of normal forms, and methods of perturbation theory.