Chaotic Diffusion on Periodic Orbits: The Perturbed Arnol'd Cat Map

Chaotic diffusion on periodic orbits (POs) is studied for the perturbed Arnol'd cat map on a cylinder, in a range of perturbation parameters corresponding to an extended structural-stability regime of the system on the torus. The diffusion coefficient is calculated using the following PO formulas: (a) The curvature expansion of the Ruelle zeta function. (b) The average of the PO winding-number squared, $w^{2}$, weighted by a stability factor. (c) The uniform (nonweighted) average of $w^{2}$. The results from formulas (a) and (b) agree very well with those obtained by standard methods, for all the perturbation parameters considered. Formula (c) gives reasonably accurate results for sufficiently small parameters corresponding also to cases of a considerably nonuniform hyperbolicity. This is due to {\em uniformity sum rules} satisfied by the PO Lyapunov eigenvalues at {\em fixed} $w$. These sum rules follow from general arguments and are supported by much numerical evidence.Comment: 6 Tables, 2 Figures (postscript); To appear in Physical Review

Stochasticity in the Josephson Map

"The Josephson map describes nonlinear dynamics of systems characterized by standard map with the uniform external bias superposed. The intricate structures of the phase space portrait of the Josephson map are examined on the basis of the tangent map associated with the Josephson map. Numerica1 observation of the stochastic dliffusion in the Josephson map is examined in comparison with the renormalized diffusion coefficient, calculated by the method of characteristic function. The global stochastisity of the Josephson map occurs at the values of far smaller stochastic parameter than the case of the standard map.

Separatrix Reconnection and Periodic Orbit Annihilation in the Harper Map

"Structure of the periodic accelerator orbits of the Harper map is investigated in detail from the view point of underlying scenario of chaos in the area preserving nontwist map. Since the twist function of the Harper map admits rigorous treatment for the entire range of phase variable, the results obtained in the present analysis describes generic novel phenomena, which are outside of the applicability of the Kolmogorov-Arnol\u27d-Moser theory.