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.
|Publication status||Published - 2004|
Bibliographical noteAdditional 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