Abstract
We compare one-dimensional representations for the isotopy stable dynamics of homeomorphisms in two dimensions. We consider the skeleton graph representative, which captures periodic behaviour, and the homotopy graph representative which captures homo-/heteroclinic behaviour. The main result of this paper is to show that the dual to the skeleton graph representative is the homotopy graph representative of the inverse map. This gives a strong link between different methods for computing the dynamics.
Highlights
Homoclinic tangles were first observed by Poincaré (1890) in his treatise on celestial mechanics, from which he concluded that the dynamics was non-integrable and extremely complicated
We have considered the relationship between the approach to studying homoclinic dynamics by the use of skeleton graphs of trellises, and of homotopy lobe dynamics
We have shown that the two approaches are dual to each other, in the sense that the “bridge classes” of the homotopy lobe dynamics of the inverse map are dual to the “free edges” of the skeleton graph representative
Summary
Homoclinic tangles were first observed by Poincaré (1890) in his treatise on celestial mechanics, from which he concluded that the dynamics was non-integrable and extremely complicated. For low-dimensional (such as two-dimensional discrete-time) systems, it turns out that detailed information about the behaviour in terms of symbolic dynamics and topological entropy can be calculated. This information is useful in studying properties of fluid mixing (Boyland et al 2000, 2003; Finn et al 2006; Finn and Thiffeault 2011; Stremler et al 2011; Sattari et al 2016), ionisation of hydrogen (Mitchell et al 2004a, b; Mitchell and Delos 2007; Mitchell 2012a), opti-. The main contribution of the paper is to formalise the duality relationship between the homotopy graph and the skeleton graph of the inverse map.
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