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

TL;DR

This paper investigates how phase space structures in a bouncer model influence Fermi acceleration, revealing that trapping near stable regions slows down energy growth as the system transitions from integrable to chaotic.

Contribution

It introduces a detailed analysis of trapping effects and phase space transport in a bouncer model, highlighting how invariant tori destruction affects Fermi acceleration.

Findings

Trapping near stable regions slows velocity growth.

Survival probability decay shifts from exponential to slower as epsilon decreases.

Invariant tori destruction leads to unbounded energy growth.

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 non integrable ($\epsilon\ne 0$). 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