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Some dynamical and chaotic properties are studied for a classical particle bouncing between two rigid walls, of which one is fixed and the other moves in time, in the presence of an external field. The system is a hybrid, behaving not as a purely Fermi-Ulam model, nor as a bouncer, but as a combination of the two. We consider two different kinds of motion of the moving wall: (i) periodic; and (ii) random. The dynamics of the model is studied via a two-dimensional nonlinear area-preserving map. We confirm that, for periodic oscillations, our model recovers the well-known results of the Fermi-Ulam model in the limit of zero external field. For intense external fields, we establish the range of control parameters values within which invariant spanning curves are observed below the chaotic sea in the low energy domain. We characterise this chaotic low energy region in terms of Lyapunov exponents. We also show that the velocity of the particle, and hence also its kinetic energy, grow according to a power law when the wall moves randomly, yielding clear evidence of Fermi acceleration.

The chaotic low energy region of the Fermi-Ulam simplified accelerator model is characterized by the use of scaling analysis. It is shown that the average velocity and the roughness (variance of the average velocity) obey scaling functions with the same characteristic exponents. The formalism is widely applicable, including to billiards and to other chaotic systems.

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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 and experiences a transition from integrable ( = 0) to nonintegrable ( = 0). For small values of , the phase space shows a mixed structure where periodic islands, chaotic seas, and invariant tori coexist. As the parameter increases and reaches a critical value 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 , the survival probability decays exponentially when it turns into a slower decay as the control parameter is reduced. The slower decay is related to trapping dynamics, slowing the Fermi Acceleration, i.e., unbounded growth of the velocity.

We study dynamical properties of an ensemble of noninteracting particles in a time-dependent elliptical-like billiard. It was recently shown [Phys. Rev. Lett. 100, 014103 (2008)] that for the nondissipative dynamics, the particle experiences unlimited energy growth. Here we show that inelastic collisions suppress Fermi acceleration in a driven elliptical-like billiard. This suppression is yet another indication that Fermi acceleration is not a structurally stable phenomenon. DOI: 10.1103/PhysRevLett.104.224101 PACS numbers: 05.45.Pq, 05.45.Tp Fermi acceleration (FA) is a phenomenon that occurs when a classical particle acquires unlimited energy upon collisions with a heavy and moving wall. The original idea is due to Fermi [1] who assumed that the enormous energy of the cosmic particles comes from interactions with moving magnetic clouds. After that many different 1D Fermi accelerator models were studied [2][3][4][5]. Basically they are composed of a classical particle which experiences collisions with a moving wall. A source of returning for a next collision can be a fixed wall [3,4], a gravitational field [5], or both [6]. A simple generalization to 2D is to consider the dynamics of the particle inside a billiard domain that, depending on the shape of the boundary, demonstrates regular [7], mixed [8], or fully chaotic dynamics [9]. Applications of billiards to physical problems include superconducting [10] and confinement of electrons in semiconductors by electric potentials [11,12], ultracold atoms trapped in a laser potential [13][14][15][16], mesoscopic quantum dots [17], reflection of light from mirrors [18], waveguides [19,20], and microwave billiards [21,22].If the boundary is time dependent, the LoskutovRyabov-Akinshin (LRA) conjecture [23] claims that chaotic dynamics for a billiard with the static boundary is a sufficient condition to produce FA if a time perturbation of the boundary is introduced. This conjecture was confirmed in many models [24][25][26]. Recently, however, [27] a specific perturbation in the boundary of an integrable elliptical billiard led to the observation of a tunable FA. The result discussed in [27] was a break of two paradigms: (i) it was expected [28] that the elliptical billiard, which is integrable for static boundary and therefore demonstrates the most regular dynamics, does not exhibit FA; and (ii) since the static version of the elliptical billiard does not have chaotic dynamics, then the LRA conjecture [23] should be extended.In this Letter we show that the mechanism which produces FA in the time-dependent elliptical-like billiard can be broken by nonelastic collisions. Since the destruction is observed for very small dissipation, one can conjecture that FA is not a structurally stable phenomenon. We consider the dynamics of an ensemble of noninteracting particles in a time-dependent elliptical-like billiard. Our results show that initial conditions chosen along the separatrix curve of the billiard with a static boundary lead the particle to exhibit FA. Thus the LRA...

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.

We consider the phenomenon of Fermi acceleration for a classical particle inside an area with a closed boundary of oval shape. The boundary is considered to be periodically time varying and collisions of the particle with the boundary are assumed to be elastic. It is shown that the breathing geometry causes the particle to experience Fermi acceleration with a growing exponent rather smaller as compared to the no breathing case. Some dynamical properties of the particle's velocity are discussed in the framework of scaling analysis. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3227740͔The behavior of the average velocity for the time varying oval-shaped billiard with the breathing geometry is considered. A four dimensional mapping that describes the dynamics of the model is carefully constructed. We show that the average velocity of the particle is described by a scaling function with critical exponents. The exponents obtained do not match the exponents found for the bouncer model (the simplest one-dimensional model exhibiting unlimited energy growth), thus putting the oval billiard in a different class of universality of the bouncer model.