Geodesic
Separatrix splitting from the point of view of symplectic geometry
Matematičeskie zametki, Tome 61 (1997) no. 6, pp. 890-906

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Generally, the invariant Lagrangian manifolds (stable and unstable separatrices) asymptotic with respect to a hyperbolic torus of a Hamiltonian system do not coincide. This phenomenon is called separatrix splitting. In this paper, a symplectic invariant qualitatively describing separatrix splitting for hyperbolic tori of maximum (smaller by one than the number of degrees of freedom) dimension is constructed. The construction resembles that of the homoclinic invariant found by Lazutkin for two-dimensional symplectic maps and of Bolotin's invariant for splitting of asymptotic manifolds of a fixed point of a symplectic diffeomorphism. 
@article{MZM_1997_61_6_a9,
     author = {D. V. Treschev},
     title = {Separatrix splitting from the point of view of symplectic geometry},
     journal = {Matemati\v{c}eskie zametki},
     pages = {890--906},
     publisher = {mathdoc},
     volume = {61},
     number = {6},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_61_6_a9/}
}
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D. V. Treschev. Separatrix splitting from the point of view of symplectic geometry. Matematičeskie zametki, Tome 61 (1997) no. 6, pp. 890-906. http://geodesic.mathdoc.fr/item/MZM_1997_61_6_a9/