Parameterization of slow-stable manifolds and their invariant vector bundles: Theory and numerical implementation

J. B. van den  Berg; J. D. Mireles  James

doi:10.3934/dcds.2016002

Article Contents

2016, Volume 36, Issue 9: 4637-4664. Doi: 10.3934/dcds.2016002

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Parameterization of slow-stable manifolds and their invariant vector bundles: Theory and numerical implementation

J. B. van den Berg^1, and
J. D. Mireles James^2,

1.
VU University Amsterdam, Department of Mathematics, de Boelelaan 1081, 1081 HV Amsterdam
2.
Florida Atlantic University, Department of Mathematical Sciences, 777 Glades Road, Boca Raton, FL 33431

Received: April 30, 2015

Revised: January 31, 2016

Published: May 2017

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Abstract

The present work deals with numerical methods for computing slow stable invariant manifolds as well as their invariant stable and unstable normal bundles. The slow manifolds studied here are sub-manifolds of the stable manifold of a hyperbolic equilibrium point. Our approach is based on studying certain partial differential equations equations whose solutions parameterize the invariant manifolds/bundles. Formal solutions of the partial differential equations are obtained via power series arguments, and truncating the formal series provides an explicit polynomial representation for the desired chart maps. The coefficients of the formal series are given by recursion relations which are amenable to computer calculations. The parameterizations conjugate the dynamics on the invariant manifolds and bundles to a prescribed linear dynamical systems. Hence in addition to providing accurate representation of the invariant manifolds and bundles our methods describe the dynamics on these objects as well. Example computations are given for vector fields which arise as Galerkin projections of a partial differential equation. As an application we illustrate the use of the parameterized slow manifolds and their linear bundles in the computation of heteroclinic orbits.

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Mathematics Subject Classification: 34C45, 37D10, 37C10, 37C29, 37M99.

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