Structure of motion near saddle points and chaotic transport in Hamiltonian systems

Generic symmetry and transport properties of near separatrix motion in 1 1/2-degree-of-freedom Hamiltonian systems are studied. First the rescaling invariance of motion near saddle points, with respect to the transformation epsilon-->lambda epsilon, chi-->chi+pi of the amplitude epsilon and phase chi, of the time-periodic perturbation, is recalled. The rescaling parameter lambda depends only on the frequency of the perturbation, and the behavior of an unperturbed Hamiltonian near a saddle point. Additional rescaling symmetry of the motion with respect to transformation epsilon-->lambda(1/2)epsilon, chi-->chi+/-pi/2 is found for some Hamiltonian systems possessing symmetry in the phase space. It is shown that these rescaling invariance properties of motion lead to strong periodic (or quasiperiodic) dependencies of all statistical characteristics of the chaotic motion near the separatrix on log,oa with the period log(10)lambda. These properties are examined for different models of chaotic motion by direct numerical integrations of equations of motion, as by well as using a computationally efficient method of the separatrix mapping.