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Normal Forms for Manifolds of Normally Hyperbolic Singularities and Asymptotic Properties of Nearby Transitions

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Abstract

This paper contains theory on two related topics relevant to manifolds of normally hyperbolic singularities. First, theorems on the formal and \( C^k \) normal forms for these objects are proved. Then, the theorems are applied to give asymptotic properties of the transition map between sections transverse to the centre-stable and centre-unstable manifolds of some normally hyperbolic manifolds. A method is given for explicitly computing these so called Dulac maps. The Dulac map is revealed to have similar asymptotic structures as in the case of a saddle singularity in the plane.

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The data that support the findings of this study are available within the article

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Acknowledgements

The author would like to thank Holger Dullin for all the discussions and constructive criticisms of which have made this paper possible. Thanks must also be given to Robert Roussarie for the many comments that greatly improved an earlier version of this manuscript, and to the reviewers for their careful reading

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  1. Department of Applied Mathematics, University of Colorado, Boulder, CO, USA

    Nathan Duignan

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  1. Nathan Duignan
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Correspondence to Nathan Duignan.

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Duignan, N. Normal Forms for Manifolds of Normally Hyperbolic Singularities and Asymptotic Properties of Nearby Transitions. Qual. Theory Dyn. Syst. 20, 26 (2021). https://doi.org/10.1007/s12346-021-00458-w

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