 97150 Amadeu Delshams, Rafael RamirezRos
 Melnikov potential for exact symplectic maps
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Mar 26, 97

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Abstract. The splitting of separatrices of hyperbolic fixed points for exact
symplectic maps of $n$ degrees of freedom is considered.
The nondegenerate critical points of a realvalued function
(called the Melnikov potential) are associated to transverse
homoclinic orbits and an asymptotic expression for the
symplectic area between homoclinic orbits is given.
Moreover, if the unperturbed invariant manifolds are completely
doubled, it is shown that there exist, in general, at least $4$
primary homoclinic orbits ($4n$ in antisymmetric maps).
Both lower bounds are optimal.
Two examples are presented: a $2n$dimensional central standardlike
map and the Hamiltonian map associated to
a magnetized spherical pendulum.
Several topics are studied about these examples:
existence of splitting, explicit computations of Melnikov potentials,
transverse homoclinic orbits, exponentially small splitting, etc.
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