A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms

Abstract

In this paper we develop several numerical algorithms for the computation of invariant manifolds in quasi-periodically forced systems. The invariant manifolds we consider are invariant tori and the asymptotic invariant manifolds (whiskers) to these tori.The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.

Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems

“…Recently developed parallel algorithms may remedy this problem [74], although the computation of the normal behavior by the approach of [40] may still remain intractable for larger systems. Appealing alternatives may be the fractional iteration method [48,49] (see Section 3.1) and the reducibility method proposed in [44][45][46]. The latter seems particularly powerful in that it allows to compute the dynamics on the torus, as well as an approximation for the torus itself.…”

“…Quasi-periodic bifurcations of invariant circles are here computed by Fourier-based numerical continuation and analysis of normal behavior. We follow [38][39][40], see [41][42][43][44][45][46][47][48][49] for alternative methods. Let f : R n → R n be a diffeomorphism and let T 1 = R/2πZ denote the unit circle.…”

“…This is due to the non-orientability of the corresponding linear bundles (eigendirections). Hence we use the double covering trick described in [44][45][46]. This shows that the two complex eigenvalues λ ⊥ 1,2 actually correspond to the real and distinct eigenvalues λ 2 of the Fourier coefficients of the x-component of the parametrization of the circle on top as a function of k. Bottom row: eigenvalues of (3.11) in C. Magnifications in the insets illustrate the existence of three classes of related eigenvalues of (3.11), corresponding to three separate circles in C. Eigenvalues which our algorithm identifies as sufficiently precise are plotted with either thick dots (green), triangles (red), or squares (blue) according to which class they are assigned to.…”

Validated numerics for equilibria of analytic vector fields; Invariant manifolds and connecting orbits

“…The user of the method is free to choose the model system, though a poor choice may well lead to no solution. This freedom makes the method very flexible, and it applies to problems as diverse as invariant circles and their stable/unstable manifolds [77,78,79], breakdown/collisions of invariant bundles associated with quasi periodic dynamics [80,81], stable/unstable manifolds of periodic orbits of differential equations and diffeomorphisms [76,82,83,84,85], study slow stable manifolds [74,76] and their invariant vector bundles [86], and invariant tori for differential equations [87,81]. The parameterization methods has also been used to develop KAM strategies not requiring action angle variables [88,89,90,91], as well as to study invariant objects for PDEs [92,93,56] and DDEs [94,95,96].…”

Two Periodic Models for the Earth-Moon System

“…The center manifold can be computed by means of normal form techniques [31][32][33], with the parametrization method [34][35][36][37] and also numerically [29]. The dynamics restricted to the center manifold can be described by a Hamiltonian with two degrees of freedom.…”