Sytem of Dynamic and Differential Physics Kentu Notes-156.pdf

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302 A. Delshams et al. In the following, we will apply this idea to g ε being several of the families ap- pearing in the problem. We will keep the convention of keeping the same letter for the objects corresponding to a family. We will use caligraphic for the vector field and capitals for the Hamiltonian. In [DLS06a], it is shown that there are remarkably simple formulas for ˜ S ε , the generator of the the map ˜ s ε – the expression of s ε in coordinates. ˜ S ε = lim N ± + N 1 j = 0 F ε f j ε ( Γ ε ε ) 1 s 1 ε k ε F ε f j ε s 1 ε k ε + N + j = 1 F ε f j ε ( Γ ε ε + ) 1 k ε F ε f j ε k ε = lim N ± + N 1 j = 0 F ε f j ε ( Γ ε ε ) 1 k ε s 1 ε F ε k ε r j ε s 1 ε + N + j = 1 F ε f j ε ( Γ ε ε + ) 1 k ε F ε k ε r j ε (10) Similarly, for Hamiltonian flows, we have S ε = lim T ± 0 T dH ε d ε Φ u , ε ( Γ ε ε ) 1 ( s ε ) 1 k ε dH ε d ε Φ u , ε ( s ε ) 1 k ε + T + 0 dH ε d ε Φ u , ε ( Γ ε ε + ) 1 k ε dH ε d ε Φ u , ε k ε (11) It is not difficult to see that the sums or the integrals converge uniformly. The formulas (10) and (11) give the hamiltonian of the deformation as the integral of the generator of the perturbation over the homoclinic orbit minus the generator of the perturbation evaluated on the asymptotic orbits.

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