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Transition tori near an elliptic-fixed point

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  • Let $F:(M,\omega) \mapsto (M,\omega)$ be a smooth symplectic diffeomorphism with a fixed point a and a heteroclinic orbit in the sense of been in the intersection of the central stable and the central unstable manifolds of the fixed point. It is studied the case when the tangent space of a point in the heteroclinic orbit is the direct sum of three subspaces. The first one is the characteristic bundle of the central stable manifold of $\mathbf a$ the second one is the characteristic bundle of the central unstable manifold of $\mathbf a$, and the third one is tangent to the intersection of the central stable and unstable manifolds.
    In this situation, the homoclinic map $\Lambda$ is a smooth and symplectic diffeomorphism of open subsets of the central manifold of $\mathbf a$.
    Moreover, if an invariant circle intersects the domain of definition of $\Lambda$ and its image intersects other circle, there are orbits that wander from one circle to the other. This phenomenon is similar to the Arnold diffusion.
    The Melnikov Method gives sufficient conditions for the existence of homoclinic maps, and non identity homoclinic maps in a perturbation of a Hamiltonian system.
    Mathematics Subject Classification: 58F05,70H05.

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