Finding critical exponents for two-dimensional Hamiltonian maps

Abstract

The transition from integrability to nonintegrability for a set of two-dimensional Hamiltonian mappings exhibiting mixed phase space is considered. The phase space of such mappings show a large chaotic sea surrounding Kolmogorov-Arnold-Moser islands and limited by a set of invariant tori. The description of the phase transition is made by the use of scaling functions for average quantities of the mapping averaged along the chaotic sea. The critical exponents are obtained via extensive numerical simulations. Given the mappings considered are parametrized by an exponent γ in one of the dynamical variables, the critical exponents that characterize the scaling functions are obtained for many different values of γ. Therefore classes of universality are defined. © 2010 The American Physical Society.