Hidden renormalization group for the near-separatrix Hamiltonian dynamics

Some properties of near-separatrix motion general for a wide class of Hamiltonian systems with 112 and 2 degrees of freedom are examined. Separatrix map is reviewed, and its invariance with respect to a renormalization of the perturbation parameter is demonstrated. The corresponding invariance of the shifted separatrix map leads to the similarity of the phase-space topology near the saddle point for systems with considerably different perturbation strengths. Motion of the harmonically perturbed pendulum exemplifies the above properties. Similar invariance property is formulated for the two degrees of freedom system describing motion of two particles in a fourth-order polynomial potential. It is shown, that the renormalization procedure can be modified to incorporate the case of large coupling. Numerical integration illustrates the similarity of the phase space structure before and after the renormalization. Possible applications of the results are discussed. Condition for the existence of the invariance under consideration is formulated, it is shown that the standard map is not renormalized in the described above sense.