Critical exponents that describe a transition from integrability to nonintegrability in a two-dimensional, nonlinear and area-preserving map are obtained via localization of the first invariant spanning curve (invariant tori) in the phase space. In a general class of systems, the position of the first invariant tori is estimated by reducing the mapping of the system to the standard mapping where a transition takes place from local to global chaos. The phase space of the mapping shows a large chaotic sea surrounding periodic islands and limited by a set of invariant tori whose position of the first of them depends on the control parameters. The formalism leads us to obtain analytically critical exponents that describe the behaviour of the average variable (action) along the chaotic sea. The result is compared to several models in the literature confirming the approach is of large interest. The formalism used is general and the procedure can be extended to many other different systems.

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 ␥ in one of the dynamical variables, the critical exponents that characterize the scaling functions are obtained for many different values of ␥. Therefore classes of universality are defined.

Decay to asymptotic steady state in one-dimensional logistic-like mappings is characterized by considering a phenomenological description supported by numerical simulations and confirmed by a theoretical description. As the control parameter is varied bifurcations in the fixed points appear. We verified at the bifurcation point in both; the transcritical, pitchfork and period-doubling bifurcations, that the decay for the stationary point is characterized via a homogeneous function with three critical exponents depending on the nonlinearity of the mapping. Near the bifurcation the decay to the fixed point is exponential with a relaxation time given by a power law whose slope is independent of the nonlinearity. The formalism is general and can be extended to other dissipative mappings.

We consider a family of two-dimensional nonlinear area-preserving mappings that generalize the Chirikov standard map and model a variety of periodically forced systems. The action variable diffuses in increments whose phase is controlled by a negative power of the action and hence effectively uncorrelated for small actions, leading to a chaotic sea in phase space. For larger values of the action the phase space is mixed and contains a family of elliptic islands centered on periodic orbits and invariant Kolmogorov-Arnold-Moser (KAM) curves. The transport of particles along the phase space is considered by starting an ensemble of particles with a very low action and letting them evolve in the phase until they reach a certain height h. For chaotic orbits below the periodic islands, the survival probability for the particles to reach h is characterized by an exponential function, well modeled by the solution of the diffusion equation. On the other hand, when h reaches the position of periodic islands, the diffusion slows markedly. We show that the diffusion coefficient is scaling invariant with respect to the control parameter of the mapping when h reaches the position of the lowest KAM island.

A dynamical phase transition from integrability to non-integrability for a family of 2-D Hamiltonian mappings whose angle, θ , diverges in the limit of vanishingly action, I, is characterised. The mappings are described by two parameters: (i) , controlling the transition from integrable ( = 0) to non-integrable ( = 0); and (ii) γ , denoting the power of the action in the equation which defines the angle. We prove the average action is scaling invariant with respect to either or n and obtain a scaling law for the three critical exponents.