Abstract:
In this paper we consider an attracting heteroclinic cycle made by a 1-dimensional and a 2-dimensional separatrices between two hyperbolic saddles having complex eigenvalues. The basin of the global attractor exhibits historic behavior and, from the asymptotic properties of these nonconverging time averages, we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. These invariants are determined by the quotient of the real parts of the eigenvalues of the equilibria, a linear combination of their imaginary components and also the transition maps between two cross sections on the separatrices.
Keywords:
Bykov attractor, historic behavior, conjugacy, complete set of invariants.
MC and AR were partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. AR also acknowledges financial support from Program INVESTIGADOR FCT (IF/00107/2015). This work has greatly benefited from AR’s visit to Nizhny Novgorod University, supported by the grant RNF 14-41-00044.
\Bibitem{CarRod18}
\by Maria Carvalho, Alexandre P. Rodrigues
\paper Complete Set of Invariants for a Bykov Attractor
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 3
\pages 227--247
\mathnet{http://mi.mathnet.ru/rcd320}
\crossref{https://doi.org/10.1134/S1560354718030012}
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https://www.mathnet.ru/eng/rcd320
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This publication is cited in the following 5 articles:
Haijun Wang, Guiyao Ke, Guili Dong, Qifang Su, Jun Pan, “Singularly Degenerate Heteroclinic Cycles with Nearby Apple-Shape Attractors”, Int. J. Bifurcation Chaos, 33:01 (2023)
Rodrigues A.A.P., “Unfolding a Bykov Attractor: From An Attracting Torus to Strange Attractors”, J. Dyn. Differ. Equ., 34:2 (2022), 1643–1677
R. Barrio, M. Carvalho, L. Castro, A. A. P. Rodrigues, “Experimentally accessible orbits near a bykov cycle”, Int. J. Bifurcation Chaos, 30:10 (2020), 2030030
M. Carvalho, A. Lohse, A. A. P. Rodrigues, “Moduli of stability for heteroclinic cycles of periodic solutions”, Discret. Contin. Dyn. Syst., 39:11 (2019), 6541–6564
Hashimoto Sh., Kiriki Sh., Soma T., “Moduli of 3-Dimensional Diffeomorphisms With Saddle-Foci”, Discret. Contin. Dyn. Syst., 38:10 (2018), 5021–5037
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