Moduli of stability for heteroclinic cycles of periodic solutions

Maria Carvalho; Alexander Lohse; Alexandre A. P. Rodrigues

doi:10.3934/dcds.2019284

Article Contents

2019, Volume 39, Issue 11: 6541-6564. Doi: 10.3934/dcds.2019284

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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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 34C28, 34C37; Secondary: 37C29, 37D05, 37G35.

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Figure 1. Local data near a periodic solution $ {\mathcal C} $

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Figure 2. Linear components of the global maps

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Figure 3. Scheme for the global transition

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Figure 4. Phase diagram of (23)

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Figure 5. Bowen's example (24) with $ \varepsilon>0 $

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Figure 6. Illustration of the properties that are conveyed from (25) to its lifting (26)

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