Many systems in science and engineering can be modelled 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 ordinary differential equations that is based on a natural parametrisation of such tori. Since this parametrisation is uniquely defined, the proposed method requires neither the computation of a base of a transversal bundle, nor re-meshing 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.
Additional information: Later published by Society for Industrial and Applied Mathematics (2005), SIAM Journal on Applied Dynamical Systems, 4(3), pp. 459-488, ISSN 1536-0040
Sponsorship: The work of Frank Schilder is supported by EPSRC grant GR/R72020/01.
- finite-difference method, invariant tori, invariance condition, continuation