# Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism

## Abstract

Some dynamical properties for an oval billiard with a scatterer in its interior are studied. The dynamics consists of a classical particle colliding between an inner circle and an external boundary given by an oval, elliptical, or circle shapes, exploring for the first time some natural generalizations. The billiard is indeed a generalization of the annular billiard, which is of strong interest for understanding marginally unstable periodic orbits and their role in the boundary between regular and chaotic regions in both classical and quantum (including experimental) systems. For the oval billiard, which has a mixed phase space, the presence of an obstacle is an interesting addition. We demonstrate, with details, how to obtain the equations of the mapping, and the changes in the phase space are discussed. We study the linear stability of some fixed points and show both analytically and numerically the occurrence of direct and inverse parabolic bifurcations. Lyapunov exponents and generalized bifurcation diagrams are obtained. Moreover, histograms of the number of successive iterations for orbits that stay in a cusp are studied. These histograms are shown to be scaling invariant when changing the radius of the scatterer, and they have a power law slope around −3. Themore »

- Authors:

- Instituto de Física da USP, Rua do Matão, Travessa R, 187, Cidade Universitária, CEP 05314-970 São Paulo, SP (Brazil)
- (United Kingdom)
- (Brazil)
- School of Mathematics, University of Bristol, Bristol (United Kingdom)
- UNESP-Univ Estadual Paulista, Câmpus de São João da Boa Vista, São João da Boa Vista, SP (Brazil)
- Departamento de Física, UNESP-Univ Estadual Paulista, Av. 24A, 1515, 13506-900 Rio Claro, SP (Brazil)

- Publication Date:

- OSTI Identifier:
- 22402538

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 25; Journal Issue: 3; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BIFURCATION; CHAOS THEORY; CUSPED GEOMETRIES; DIAGRAMS; DYNAMICS; EQUATIONS; LYAPUNOV METHOD; MAPPING; PARTICLES; PHASE SPACE; SCALING

### Citation Formats

```
Costa, Diogo Ricardo da, School of Mathematics, University of Bristol, Bristol, Departamento de Física, UNESP-Univ Estadual Paulista, Av. 24A, 1515, 13506-900 Rio Claro, SP, Dettmann, Carl P., Oliveira, Juliano A. de, and Leonel, Edson D.
```*Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism*. United States: N. p., 2015.
Web. doi:10.1063/1.4915474.

```
Costa, Diogo Ricardo da, School of Mathematics, University of Bristol, Bristol, Departamento de Física, UNESP-Univ Estadual Paulista, Av. 24A, 1515, 13506-900 Rio Claro, SP, Dettmann, Carl P., Oliveira, Juliano A. de, & Leonel, Edson D.
```*Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism*. United States. doi:10.1063/1.4915474.

```
Costa, Diogo Ricardo da, School of Mathematics, University of Bristol, Bristol, Departamento de Física, UNESP-Univ Estadual Paulista, Av. 24A, 1515, 13506-900 Rio Claro, SP, Dettmann, Carl P., Oliveira, Juliano A. de, and Leonel, Edson D. Sun .
"Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism". United States.
doi:10.1063/1.4915474.
```

```
@article{osti_22402538,
```

title = {Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism},

author = {Costa, Diogo Ricardo da and School of Mathematics, University of Bristol, Bristol and Departamento de Física, UNESP-Univ Estadual Paulista, Av. 24A, 1515, 13506-900 Rio Claro, SP and Dettmann, Carl P. and Oliveira, Juliano A. de and Leonel, Edson D.},

abstractNote = {Some dynamical properties for an oval billiard with a scatterer in its interior are studied. The dynamics consists of a classical particle colliding between an inner circle and an external boundary given by an oval, elliptical, or circle shapes, exploring for the first time some natural generalizations. The billiard is indeed a generalization of the annular billiard, which is of strong interest for understanding marginally unstable periodic orbits and their role in the boundary between regular and chaotic regions in both classical and quantum (including experimental) systems. For the oval billiard, which has a mixed phase space, the presence of an obstacle is an interesting addition. We demonstrate, with details, how to obtain the equations of the mapping, and the changes in the phase space are discussed. We study the linear stability of some fixed points and show both analytically and numerically the occurrence of direct and inverse parabolic bifurcations. Lyapunov exponents and generalized bifurcation diagrams are obtained. Moreover, histograms of the number of successive iterations for orbits that stay in a cusp are studied. These histograms are shown to be scaling invariant when changing the radius of the scatterer, and they have a power law slope around −3. The results here can be generalized to other kinds of external boundaries.},

doi = {10.1063/1.4915474},

journal = {Chaos (Woodbury, N. Y.)},

number = 3,

volume = 25,

place = {United States},

year = {Sun Mar 15 00:00:00 EDT 2015},

month = {Sun Mar 15 00:00:00 EDT 2015}

}