Geodesic
Separatrix maps in~Hamiltonian systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 2, pp. 219-322

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The separatrix map is constructed for some classes of problems in Hamiltonian dynamics. The formulae obtained are used to study two-dimensional symplectic maps close to integrable maps: elliptic periodic trajectories passing through separatrix lobes are constructed, and some estimates for the width of the stochastic layer are given. For Hamiltonian systems with two and a half degrees of freedom it is proved that the Arnol'd diffusion in the a priori unstable case is generic, and in the Mather problem trajectories are constructed for which the mean energy growth is linear in time. 
@article{RM_2007_62_2_a0,
     author = {G. N. Piftankin and D. V. Treschev},
     title = {Separatrix maps {in~Hamiltonian} systems},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {219--322},
     publisher = {mathdoc},
     volume = {62},
     number = {2},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2007_62_2_a0/}
}
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G. N. Piftankin; D. V. Treschev. Separatrix maps in~Hamiltonian systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 2, pp. 219-322. http://geodesic.mathdoc.fr/item/RM_2007_62_2_a0/