Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration

Abstract

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 epsilon and experiences a transition from integrable (epsilon = 0) to nonintegrable (epsilon not equal 0). For small values of epsilon, the phase space shows a mixed structure where periodic islands, chaotic seas, and invariant tori coexist. As the parameter epsilon increases and reaches a critical value epsilon(c), 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 epsilon, the survival probability decays exponentially when it turns into a slower decay as the control parameter epsilon is reduced. The slower decay is related to trapping dynamics, slowing the Fermi Acceleration, i.e., unbounded growth of the velocity.