On the Computation of Reducible Invariant Tori on a Parallel Computer

Abstract

Abstract. We present an algorithm for the computation of reducible invariant tori of discrete dynamical systems that is suitable for tori of dimensions larger than 1. It is based on a quadratically convergent scheme that approximates, at the same time, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. The Floquet matrix describes the linearization of the dynamics around the torus and, hence, its linear stability. The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution. For these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies. The paper includes some examples (flows) to show the efficiency of the method in a parallel computer. In these flows we compute invariant tori of dimensions up to 5, by taking suitable sections.

Numerical computation of rotation numbers of quasi-periodic planar curves

“…where the coefficients c (m) j are also given by equation (19). The operator Θ d q,p,p−d corresponds to the Richardson extrapolation of order p − d of equation (21).…”

“…To extrapolate in this expression using the coefficients (19) we require the denominators (L + p − l) · · · (L + p − 1) in (27) not to depend on p.…”

“…For example, the methods in [5,7,19] have been applied efficiently in a wide set of contexts. However, they require to compute a representation -by means of a trigonometric polynomial-of the curve which solves the invariance equation of the problem, so it is required to solve large systems of equations -as large as the used number of Fourier modes, say M. One possibility to face this difficulty is to solve these full linear systems, with a cost O(M 3 ) in time and O(M 2 ) in memory, by means of efficient parallel algorithms as it is proposed in [19]. Another recent approach presented in [7], based on the analytic and geometric ideas developed in [6], allows to reduce the computational effort of the problem to a cost of order O(M log M) in time and O(M) in memory.…”

Quasi-Periodic Frequency Analysis Using Averaging-Extrapolation Methods

“…As considered in [20], in the computations presented we take into account the gravitational effect of Saturn, Uranus, Neptune, and Earth. To numerically integrate the vector fields, we use a high-order Taylor method (the order depending on the desired precision).…”

“…Specifically, we consider the effect of Saturn, Uranus, Neptune, and Earth, the first perturbation being the most significant. This model was studied in [20], where the four-dimensional invariant torus that replaces the point L 5 was computed. Note that we expect the invariant tori of the autonomous spatial system to enlarge their dimension by adding the frequencies of the planets (see [23,38]).…”