Amadeu Delshams, Rafael Ramirez-Ros
Melnikov potential for exact symplectic maps
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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 non-degenerate critical points of a real-valued 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 standard-like
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.