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November  2019, 39(11): 6541-6564. doi: 10.3934/dcds.2019284

## Moduli of stability for heteroclinic cycles of periodic solutions

 1 Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal 2 Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

Received  January 2019 Revised  June 2019 Published  August 2019

We consider $C^2$ vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic behavior and from the asymptotic properties of its orbits we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. As expected, this set contains the periods of the orbits involved in the cycle, a combination of their angular speeds, the rates of expansion and contraction in linearizing neighborhoods of them, besides information regarding the transition maps and the transition times between these neighborhoods. We conclude with an application of this result to a class of cycles obtained by the lifting of an example of R. Bowen.

Citation: Maria Carvalho, Alexander Lohse, Alexandre A. P. Rodrigues. Moduli of stability for heteroclinic cycles of periodic solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6541-6564. doi: 10.3934/dcds.2019284
##### References:

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##### References:
Local data near a periodic solution ${\mathcal C}$
Linear components of the global maps
Scheme for the global transition
Phase diagram of (23)
Bowen's example (24) with $\varepsilon>0$
Illustration of the properties that are conveyed from (25) to its lifting (26)
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