Phase Transition in Dynamical Systems: Defining Classes of Universality for Two-Dimensional Hamiltonian Mappings via Critical Exponents

摘要

A phase transition from integrability to nonintegrability in two-dimensional Hamiltonian mappings is described and characterized in terms of scaling arguments. The mappings considered produce a mixed structure in the phase space in the sense that, depending on the combination of thecontrol parameters and initial conditions, KAM islands which are surrounded by chaotic seas thatare limited by invariant tori are observed. Some dynamical properties for the largest component ofthe chaotic sea are obtained and described in terms of the control parameters. The average valueand the deviation of the average value for chaotic components of a dynamical variable are describedin terms of scaling laws, therefore critical exponents characterizing a scaling function that describesa phase transition are obtained and then classes of universality are characterized. The three modelsconsidered are: The Fermi-Ulam accelerator model, a periodically corrugate waveguide, and variant ofthe standard nontwist map.