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Regular and Chaotic Dynamics, 2022, Volume 27, Issue 4, Pages 477–491
DOI: https://doi.org/10.1134/S1560354722040062
(Mi rcd1176)
 

Alexey Borisov Memorial Volume

The Dynamical Core of a Homoclinic Orbit

V. Mendoza

Instituto de Matemática e Computacão, Universidade Federal de Itajubá, Av. BPS 1303, Bairro Pinheirinho, CEP 37500-903 Itajubá, Brazil
References:
Abstract: The complexity of a dynamical system exhibiting a homoclinic orbit is given by its dynamical core which, due to Cantwell, Conlon and Fenley, is a set uniquely determined in the isotopy class, up to a topological conjugacy, of the end-periodic map relative to that orbit. In this work we prove that a sufficient condition to determine the dynamical core of a homoclinic orbit of a Smale diffeomorphism on the 2-disk is the non-existence of bigons relative to this orbit. Moreover, we propose a pruning method for eliminating bigons that can be used to find a Smale map without bigons and hence for finding the dynamical core.
Keywords: Homoclinic orbits, dynamical core, Smale horseshoe, pruning theory.
Funding agency Grant number
Fundação de Amparo à Pesquisa do Estado de São Paulo 2010/20159-6
This work was supported by the FAPESP grant 2010/20159-6.
Received: 28.02.2022
Accepted: 08.04.2022
Bibliographic databases:
Document Type: Article
Language: English
Citation: V. Mendoza, “The Dynamical Core of a Homoclinic Orbit”, Regul. Chaotic Dyn., 27:4 (2022), 477–491
Citation in format AMSBIB
\Bibitem{Men22}
\by V.~Mendoza
\paper The Dynamical Core of a Homoclinic Orbit
\jour Regul. Chaotic Dyn.
\yr 2022
\vol 27
\issue 4
\pages 477--491
\mathnet{http://mi.mathnet.ru/rcd1176}
\crossref{https://doi.org/10.1134/S1560354722040062}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4462434}
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