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Ordered intricacy of Shilnikov saddle-focus homoclinics in symmetric systems. (English) Zbl 1481.37053

The Shilnikov saddle-focus bifurcation is one of the keys to understand the origin and the structure of deterministic chaos in dynamical systems. Furthermore, it has several applications in physics, neuroscience, and economics. This article is meant to disclose the structure of bifurcation unfoldings, including multiple shapes of bifurcation curves in a parameter plane of typical \(Z_2\)-symmetric systems.
The authors develop a new symbolic approach based on the Poincaré’s technique of inverse mappings that allows them to illustrate the universality and richness of the sets of homoclinic and heteroclinic bifurcations of Shilnikov saddle foci by using two representative examples of systems of ordinary differential equations: a smooth adaptation of the Chua circuit and a three-dimensional normal form.

MSC:

37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37G05 Normal forms for dynamical systems
37G10 Bifurcations of singular points in dynamical systems
37G40 Dynamical aspects of symmetries, equivariant bifurcation theory
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References:

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