Determination of the basin of attraction of a periodic orbit in two dimensionsusing meshless collocation

Peter Giesl; James  McMichen

doi:10.3934/jcd.2016010

Article Contents

2016, Volume 3, Issue 2: 191-210. Doi: 10.3934/jcd.2016010

This issue Previous Article Asymptotic invariance and the discretisation of nonautonomous forward attracting sets Next Article

Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation

Peter Giesl^1, and
James McMichen^1,

1.
Department of Mathematics, University of Sussex, Falmer BN1 9QH

Received: December 2016

Revised: February 2017

Published: June 2016

Abstract / Introduction Related Papers Cited by

Abstract

A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of a periodic orbit without requiring information about its position or stability. Moreover, it is robust to small perturbations of the system.
In two-dimensional systems, a contraction metric can be characterised by a scalar-valued function. In [9], the function was constructed as solution of a first-order linear Partial Differential Equation (PDE), and numerically constructed using meshless collocation. However, information about the periodic orbit was required, which needed to be approximated.
In this paper, we overcome this requirement by studying a second-order PDE, which does not require any information about the periodic orbit. We show that the second-order PDE has a solution, which defines a contraction metric. We use meshless collocation to approximate the solution and prove error estimates. In particular, we show that the approximation itself is a contraction metric, if the collocation points are dense enough. The method is applied to two examples.

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Mathematics Subject Classification: Primary: 37C27, 65N35; Secondary: 37C75, 65P40.

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