Continuation of Quasi-periodic Invariant Tori

Many systems in science and engineering can be modeled as coupled or forced nonlinear oscillators, which may possess quasi-periodic or phase-locked invariant tori. Since there exist routes to chaos involving the break-down of invariant tori, these phenomena attract considerable attention. This paper presents a new algorithm for the computation and continuation of quasi-periodic invariant tori of ODEs that is based on a natural parametrization of such tori. Since this parametrization is uniquely defined, the proposed method requires neither the computation of a base of a transversal bundle nor remeshing during continuation. It is independent of the stability type of the torus, and examples of attracting and saddle-type tori are given. The algorithm is robust in the sense that it can compute approximations to weakly resonant tori. The performance of the method is demonstrated with examples.