Dynamical and statistical properties of a rotating oval billiard Diogo Ricardo da Costa a,b,⇑ , Diego F.M. Oliveira c , Edson D. Leonel c,d a Instituto de Física, Univ São Paulo, Rua do Matão, Cidade Universitária, 05314-970 São Paulo, SP, Brazil b School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom c UNESP, Univ Estadual Paulista, Departamento de Física, Av. 24A 1515, Bela Vista, 13506-900 Rio Claro, SP, Brazil d The Abdus Salam – ICTP, Strada Costiera, 11, 34151 Trieste, Italy article info Article history: Received 26 March 2013 Received in revised form 6 August 2013 Accepted 6 October 2013 Available online 22 October 2013 Keywords: Rotating oval billiard Dynamical properties Chaos abstract Some dynamical and statistical properties of a time-dependent rotating oval billiard are studied. We considered cases with (i) positive and (ii) negative curvature for the boundary. For (i) we show the system does not present unlimited energy growth. For case (ii) how- ever the average velocity for an ensemble of noninteracting particles grows as a power law with acceleration exponent well defined. Finally, we show for both cases that after introducing time-dependent perturbation, the mixed structure of the phase space observed for static case is recovered by making a suitable transformation in the angular position of the particle. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction As an attempt to explain the origin of cosmic rays acceleration, Enrico Fermi [1], in his pioneering work, proposed that charged particles could be accelerated by collisions/interactions with time-dependent magnetic structures. Since then, many models have been proposed in order to understand/explain Fermi’s idea [2–13] (and references in therein). One of the most studied version of the problem is the one-dimensional Fermi–Ulam model [14–17]. Such a system con- sists of a classical particle (representing the cosmic particle) confined and bouncing between two rigid walls, one of them is assumed to be fixed (working as a returning mechanism) and the other one moves periodically in time (denoting the time- dependent magnetic structure). It has been proved [18] that in such a system the unlimited energy growth, also known as Fermi acceleration, is not observed due to the existence of a set of invariant spanning curves in the phase space, therefore the model fails to produce unlimited energy growth. On the other hand, if the fixed wall is replaced by a constant gravitational field, for a specific combination of initial conditions and control parameters the unlimited energy growth can be observed [19,20]. This happens due to the loss of correlation between two collisions since as the velocity increases, the time between collisions also increases. A natural extension of one-dimension systems are two-dimensional billiards model. In such systems one or many (non- interacting) particles are confined in a closed domain experiencing collisions with the boundary [21–24]. Basically they can be classified in three different classes namely, (i) integrable, (ii) ergodic and (iii) mixed. One of the main questions about such systems is whether or not the unlimited energy growth of the particle is observed. In this sense, a conjecture by Loskutov– Ryabov–Akinshin (LRA) has been proposed [25]. Basically the conjecture states that, if the system has a chaotic component in the phase space for the static version of the problem, after the introduction of a time-dependent perturbation on the bound- ary, the phenomenon of Fermi acceleration must be observed. Such a conjecture has been confirmed in many billiards including, oval [26,27], stadium [28], Lorentz gas [29–31] and many others. Recently it has been shown that even elliptic 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.10.007 ⇑ Corresponding author at: Instituto de Física, Univ São Paulo, Rua do Matão, Cidade Universitária, 05314-970 São Paulo, SP, Brazil. Tel.: +55 1982781422. E-mail address: drcosta@usp.br (D.R. da Costa). Commun Nonlinear Sci Numer Simulat 19 (2014) 1926–1934 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns