Artigo
Circular, elliptic and oval billiards in a gravitational field
Registro en:
Communications In Nonlinear Science And Numerical Simulation. Amsterdam: Elsevier Science Bv, v. 22, n. 1-3, p. 731-746, 2015.
1007-5704
10.1016/j.cnsns.2014.08.030
WOS:000345700500056
6130644232718610
Autor
Costa, Diogo Ricardo da [UNESP]
Dettmann, Carl P.
Leonel, Edson D. [UNESP]
Resumen
We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the energy regimes is made. The linear stability of fixed points is studied, yielding exact analytical expressions for parameter values at which a period-doubling bifurcation occurs. The dynamics is apparently ergodic at certain energies in all three models, in contrast to the regularity of the circular and elliptic billiard dynamics in the field-free case. This finding is confirmed using a sensitive test involving Lyapunov weighted dynamics. In the last part of the paper a time dependence is introduced in the billiard boundary, where it is shown that for the circular billiard the average velocity saturates for zero gravitational force but in the presence of gravitational it increases with a very slow growth rate, which may be explained using Arnold diffusion. For the oval billiard, where chaos is present in the static case, the particle has an unlimited velocity growth with an exponent of approximately 1/6. (C) 2014 Elsevier B.V. All rights reserved. Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação para o Desenvolvimento da UNESP (FUNDUNESP) Center for Scientific Computing (NCC/GridUNESP) of the Sao Paulo State University (UNESP) Univ Sao Paulo, Inst Fis, BR-05508090 Sao Paulo, Brazil Univ Bristol, Sch Math, Bristol, Avon, England UNESP, Dept Fis, BR-13506900 Rio Claro, SP, Brazil Abdus Salaam Int Ctr Theoret Phys, Abdus Salam, I-34151 Trieste, Italy Universidade Estadual Paulista, Departamento de Física, BR-13506900 Rio Claro, SP, Brazil FAPESP: 2013/22764-2 FAPESP: 2012/18962-0 FAPESP: 2010/52709-5 FAPESP: 2012/23688-5