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Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps. (English) Zbl 1181.37077

Authors’ abstract: We consider a one-parameter family of area preserving maps in a neighbourhood of an elliptic fixed point. As the parameter evolves, hyperbolic and elliptic periodic orbits are created. The exceptional resonances of order less than 5 have to be considered separately. The invariant manifolds of the hyperbolic periodic points bound islands containing the elliptic periodic points. Generically, these manifolds split. It turns out that the inner and outer splittings are different under suitable conditions. We provide accurate formulae describing the splittings of these manifolds as a function of the parameter and the relative values of these magnitudes as a function of geometric properties. The numerical agreement is illustrated using mainly Henon map as example.

MSC:

37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37G05 Normal forms for dynamical systems
37G10 Bifurcations of singular points in dynamical systems
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37M99 Approximation methods and numerical treatment of dynamical systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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