Characterization of a continuous phase transition in a chaotic system

Some characteristics of scaling invariance are discussed for a transition from integrability to non-integrability in a class of dynamical systems described by a two-dimensional, nonlinear and area-preserving mapping. The dynamical variables are action I and angle θ and that the angle diverges in the limit of vanishingly action. The transition is controlled by a parameter closely related to the order parameter. A scaling invariance observed for the average squared action along the chaotic sea gives evidence that the transition observed from integrability to non-integrability is equivalent to a second-order, also called continuous, phase transition since when the order parameter approaches zero at the same time the response of the order parameter to the conjugate field (susceptibility) diverges. This investigation allows application to a wide class of systems and transitions, including transition from limited to unlimited chaotic diffusion in dissipative systems and also in a transition from limited to unlimited Fermi acceleration in time-dependent billiard systems.