The phenomenon of Fermi acceleration is addressed for the problem of a classical and dissipative bouncer model, using a scaling description. The dynamics of the model, in both the complete and simplified versions, is obtained by use of a two-dimensional nonlinear mapping. The dissipation is introduced using a restitution coefficient on the periodically moving wall. Using scaling arguments, we describe the behavior of the average chaotic velocities on the model both as a function of the number of collisions with the moving wall and as a function of the time. We consider variations of the two control parameters; therefore critical exponents are obtained. We show that the formalism can be used to describe the occurrence of a transition from limited to unlimited energy growth as the restitution coefficient approaches unity. The formalism can be used to characterize the same transition in two-dimensional time-varying billiard problems.

The chaotic low-energy region of a simplified Fermi-Ulam accelerator model is investigated numerically to determine the average energy and number of collisions as functions of time. We find that these properties exhibit scaling when the oscillation amplitude of the moving wall is small. Following a transient regime, the average energy increases in time, reaches a maximum and then shows a surprising slow decay.

Scaling properties of Chirikov's standard map are investigated by studying the average value of I 2 , 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 < K c (K c ≈ 0.9716); (ii) the transition from limited to unlimited growth of I 2 , K K c ; (iii) the regime of strong nonlinearity, K K c . Our scaling results are also applicable to Pustylnikov's 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 I 2 even for small values of K.

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.

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 twodimensional 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. Specifically, we characterize events of the crisis as the damping coefficients are varied. Moreover, in the limit of high dissipation the system exhibits a sequence of doubling bifurcation cascade. We also report the annihilation of pairs of fixed points as the damping coefficients are varied.

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 Chirikov's standard map in the region of the phase space near the first invariant spanning curve.

The behavior of the average velocity for a classical particle in the one-dimensional Fermi accelerator model under sawtooth external force is considered. For elastic collisions, it is known that the average velocity of the particle grows unlimitedly because of the discontinuities of the derivative of the moving wall's position with respect to time. However, and contrary to what was expected to be observed, the introduction of a friction force generated from a slip of a body against a rough surface leads to a boundary separating different regions of the phase space that yields the particle to either experience unlimited energy growth or suppression of Fermi acceleration. The Fermi acceleration is described by using scaling arguments. The formalism presented can be extended to two-dimensional time-dependent billiards as well as to higher-order mappings.

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