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Numerical computation of connecting orbits on a manifold

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Abstract

In this paper we propose a numerical method for approximating connecting orbits on a manifold and its bifurcation parameters. First we extend the standard nondegeneracy condition to the connecting orbits on a manifold. Then we construct a well-posed system such that the nondegenerate connecting orbit pair on a manifold is its regular solution. We use a difference method to discretize the ODE part in this well-posed system and we find that the numerical solutions still remain on the same manifold. We also set up a modified projection boundary condition to truncate connecting orbits on a manifold onto a finite interval. Then we prove the existence of truncated approximate connecting orbit pairs and derive error estimates. Finally, we carry out some numerical experiments to illustrate the theoretical estimates.

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Authors and Affiliations

  1. Department of Mathematics, Jilin University, Changchun, 130012, China

    Yuanyuan Liu & Yongkui Zou

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  1. Yuanyuan Liu
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  2. Yongkui Zou
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Correspondence to Yongkui Zou.

Additional information

This work is supported by NSFC of No.11071102 and 10801062.

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Liu, Y., Zou, Y. Numerical computation of connecting orbits on a manifold. Numer Algor 61, 429–464 (2012). https://doi.org/10.1007/s11075-012-9542-5

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  • DOI: https://doi.org/10.1007/s11075-012-9542-5

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