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A Numerical Scheme for Computing Stable and Unstable Manifolds in Nonautonomous Flows

Sanjeeva Balasuriya

Sanjeeva Balasuriya

School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia

E-mail Address: sanjeevabalasuriya@yahoo.com

Supported by the Australian Research Council through Future Fellowship Grant FT130100484.

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https://doi.org/10.1142/S021812741630041XCited by:3 (Source: Crossref)

Abstract

There are many methods for computing stable and unstable manifolds in autonomous flows. When the flow is nonautonomous, however, difficulties arise since the hyperbolic trajectory to which these manifolds are anchored, and the local manifold emanation directions, are changing with time. This article utilizes recent results which approximate the time-variation of both these quantities to design a numerical algorithm which can obtain high resolution in global nonautonomous stable and unstable manifolds. In particular, good numerical approximation is possible locally near the anchor trajectory. Nonautonomous manifolds are computed for two examples: a Rossby wave situation which is highly chaotic, and a nonautonomus (time-aperiodic) Duffing oscillator model in which the manifold emanation directions are rapidly changing. The numerical method is validated and analyzed in these cases using finite-time Lyapunov exponent fields and exactly known nonautonomous manifolds.

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