Continuation of Quasi-periodic Invariant Tori

Abstract

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 breakdown of invariant tori, these phenomena attract considerable attention. This paper presents a new algorithm for the computation and continuation of quasiperiodic 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.

Computing Arnol′d tongue scenarios

“…Hence, the rotation number assumes rational values on an open and dense set along this path. This leads to a failure of the algorithm because the Denjoy parametrisation exists only for invariant circles with irrational rotation number; see [36] for example computations illustrating this effect.…”

“…The coupled Van der Pol system has been studied previously in [17,34,36]. We re-investigate it in more depth in §4.3.…”

“…We use a method based on the Denjoy parametrisation for two reasons: this parametrisation is unique up to a phase shift, which leads to a particularly simple algorithm; see also [36]. Furthermore, it contains the rotation number explicitly.…”

“…In both settings the (β, α) parameter plane features Arnol d tongues with tips at the line α = 0. The computations of Arnol d tongues and quasi-periodic arcs are two-parameter continuations by nature [2,3,17,36]. In all the considerations that follow, we aim at using pseudo arc-length continuation.…”

“…In §3 we combine ideas in [10,22,36] with a two-point boundary value problem setting and continuation techniques to compute quasi-periodic arcs that sit in between the Arnol d tongues. The performance of our methods is demonstrated with three examples in §4: an embedded Arnol d family in §4.1, a generic caricature example in §4.2 and the system of coupled Van der Pol oscillators in §4.3.…”