From the Hénon conservative map to the Chirikov standard map for large parameter values. (English) Zbl 1417.37191

Summary: In this paper we consider conservative quadratic Hénon maps and Chirikov’s standard map, and relate them in some sense.
First, we present a study of some dynamical properties of orientation-preserving and orientation-reversing quadratic Hénon maps concerning the stability region, the size of the chaotic zones, its evolution with respect to parameters and the splitting of the separatrices of fixed and periodic points plus its role in the preceding aspects.
Then the phase space of the standard map, for large values of the parameter \(k\), is studied. There are some stable orbits which appear periodically in \(k\) and are scaled somehow. Using this scaling, we show that the dynamics around these stable orbits is one of the above Hénon maps plus some small error, which tends to vanish as \(k\to\infty\). Elementary considerations about diffusion properties of the standard map are also presented.


37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37C05 Dynamical systems involving smooth mappings and diffeomorphisms


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