Denis Gouvêa Ladeira

Scaling properties of the Fermi-Ulam accelerator model

The chaotic low energy region (chaotic sea) of the Fermi-Ulam accelerator model is discussed within a scaling framework near the integrable to non-integrable transition. Scaling results for the average quantities (velocity, roughness, energy etc.) of the simplified version of the model are reviewed and it is shown that, for small oscillation amplitude of the moving wall, they can be described by scaling functions with the same characteristic exponents. New numerical results for the complete model are presented. The chaotic sea is also characterized by its Lyapunov exponents.

Dynamical properties of a dissipative hybrid Fermi-Ulam-bouncer model

Some consequences of dissipation are studied for a classical particle suffering inelastic collisions in the hybrid Fermi-Ulam bouncer model. The dynamics of the model is described by a two-dimensional nonlinear area-contracting map. In the limit of weak and moderate dissipation we report the occurrence of crisis and in the limit of high dissipation the model presents doubling bifurcation cascades. Moreover, we show a phenomena of annihilation by pairs of fixed points as the dissipation varies.

Scaling properties of a simplified bouncer model and of Chirikov's standard map

Scaling properties of Chirikovs standard map are investigated by studying the average value of I2, where I is the action variable, for initial conditions in (a) the stability island and (b) the chaotic component. Scaling behavior appears in three regimes, defined by the value of the control parameter K: (i) the integrable to non-integrable transition (K ≈ 0) and K < Kc (Kc ≈ 0.9716); (ii) the transition from limited to unlimited growth of I2, K Kc; (iii) the regime of strong nonlinearity, K Kc. Our scaling results are also applicable to Pustylnikovs bouncer model, since it is globally equivalent to the standard map. We also describe the scaling properties of a stochastic version of the standard map, which exhibits unlimited growth of I2 even for small values of K.

Scaling features of a breathing circular billiard

We investigate the chaotic lowest energy region of the simplified breathing circular billiard, a two-dimensional generalization of the Fermi model. When the oscillation amplitude of the breathing boundary is small and we are near the integrable to non-integrable transition, we obtain numerically that average quantities can be described by scaling functions. We also show that the map that describes this model is locally equivalent to Chirikovs standard map in the region of the phase space near the first invariant spanning curve.

On the localization of invariant tori in a family of generalized standard mappings and its applications to scaling in a chaotic sea

1 Departamento de F́ısica, UNESP Univ Estadual Paulista, Av.24A, 1515, 13506-900, Rio Claro, SP Brazil 2 Instituto de F́ısica da USP, Rua do Matão, Travessa R, 187 Cidade Universitária, 05314-970, São Paulo, SP Brazil 3 Universidade Federal de São João Del-Rei, Departamento de F́ısica e Matemática. Rodovia MG 443, Km 07, 36420-000, Ouro Branco, MG Brazil 4 Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy

Scaling investigation for the dynamics of charged particles in an electric field accelerator

Some dynamical properties of an ensemble of trajectories of individual (non-interacting) classical particles of mass m and charge q interacting with a time-dependent electric field and suffering the action of a constant magnetic field are studied. Depending on both the amplitude of oscillation of the electric field and the intensity of the magnetic field, the phase space of the model can either exhibit: (i) regular behavior or (ii) a mixed structure, with periodic islands of regular motion, chaotic seas characterized by positive Lyapunov exponents, and invariant Kolmogorov-Arnold-Moser curves preventing the particle to reach unbounded energy. We define an escape window in the chaotic sea and study the transport properties for chaotic orbits along the phase space by the use of scaling formalism. Our results show that the escape distribution and the survival probability obey homogeneous functions characterized by critical exponents and present universal behavior under appropriate scaling transformations. We show the survival probability decays exponentially for small iterations changing to a slower power law decay for large time, therefore, characterizing clearly the effects of stickiness of the islands and invariant tori. For the range of parameters used, our results show that the crossover from fast to slow decay obeys a power law and the behavior of survival orbits is scaling invariant.