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A Lipschitz Version of the \(\lambda \)-Lemma and a Characterization of Homoclinic and Heteroclinic Orbits

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Abstract

In this paper we consider finite dimensional dynamical systems generated by a Lipschitz function. We prove a version of the Whitney’s Extension Theorem on compact manifolds to obtain a version of the well-known \(\lambda \)-lemma for Lipschitz functions. The notions of Lipschitz transversality and hyperbolicity are investigated in the finite dimensional framework with a norm between \(C^1\)-norm and \(C^0\)-norm. As an application, we study homoclinic and heteroclinic orbits obtaining, as a consequence, a stability result for Lipschitz Morse–Smale functions.

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Acknowledgements

We express our thanks to the referees for their helpful and interesting comments and suggestions which allowed us to improve the presentation of our paper.

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  1. Departamento de Matemática e Estatástica, Universidade Estadual de Ponta Grossa, 84030-900, Ponta Grossa, PR, Brazil

    Leonardo Pires & Giuliano G. La Guardia

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  1. Leonardo Pires
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  2. Giuliano G. La Guardia
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Correspondence to Leonardo Pires.

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This research has been partially supported by the Brazilian Agencies CAPES and CNPq. The authors declare that there is not conflict of interests in the publication of this paper. The authors declare the data availability.

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Pires, L., La Guardia, G.G. A Lipschitz Version of the \(\lambda \)-Lemma and a Characterization of Homoclinic and Heteroclinic Orbits. Qual. Theory Dyn. Syst. 20, 82 (2021). https://doi.org/10.1007/s12346-021-00521-6

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