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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
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.
Received: 28.02.2022 Accepted: 08.04.2022
Citation:
V. Mendoza, “The Dynamical Core of a Homoclinic Orbit”, Regul. Chaotic Dyn., 27:4 (2022), 477–491
Linking options:
https://www.mathnet.ru/eng/rcd1176 https://www.mathnet.ru/eng/rcd/v27/i4/p477
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