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Dynamics in chaotic zones of area preserving maps: close to separatrix and global instability zones. (English) Zbl 1217.37051

This paper investigates the dynamics near the separatrices of area preserving maps, providing analytical and numerical results, with “realistic” simulations.
Both the “fish” case (in which only one branch of the stable manifold intersects transversally one branch of the unstable one) and the “figure 8” (like the pendulum) are taken into account, and some care is needed when infinitely many resonances overlap.
The width of the inner and outer chaotic zones and the amplitude of the stochastic layers are estimated rigorously, and the formation of tiny islands close to the separatrices and of large domains without invariant curves are studied in detail, with the introduction of a “biseparatrix” map model.

MSC:

37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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