RESUMO
In this work we investigate how the behavior of the Shannon entropy can be used to measure the diffusion exponent of a set of initial conditions in two systems: (i) standard map and (ii) the oval billiard. We are interested in the diffusion near the main island in the phase space, where stickiness is observed. We calculate the diffusion exponent for many values of the nonlinear parameter of the standard map where the size and shape of the main island change as the parameter varies. We show that the changes of behavior in the diffusion exponent are related with the changes in the area of the main island and show that when the area of the main island is abruptly reduced, due to the destruction of invariant tori and, consequently, creation of hyperbolic and elliptic fixed points, the diffusion exponent grows.
RESUMO
Some phase space transport properties for a conservative bouncer model are studied. The dynamics of the model is described by using a two-dimensional measure preserving mapping for the variables' velocity and time. The system is characterized by a control parameter ε and experiences a transition from integrable (ε=0) to nonintegrable (ε≠0). For small values of ε, the phase space shows a mixed structure where periodic islands, chaotic seas, and invariant tori coexist. As the parameter ε increases and reaches a critical value εc, all invariant tori are destroyed and the chaotic sea spreads over the phase space, leading the particle to diffuse in velocity and experience Fermi acceleration (unlimited energy growth). During the dynamics the particle can be temporarily trapped near periodic and stable regions. We use the finite time Lyapunov exponent to visualize this effect. The survival probability was used to obtain some of the transport properties in the phase space. For large ε, the survival probability decays exponentially when it turns into a slower decay as the control parameter ε is reduced. The slower decay is related to trapping dynamics, slowing the Fermi Acceleration, i.e., unbounded growth of the velocity.