Linear and nonlinear stability of periodic orbits in annular billiards

Abstract

An annular billiard is a dynamical system in which a particle moves freely in a disk except for elastic collisions with the boundary, and also a circular scatterer in the interior of the disk. We investigate stability properties of some periodic orbits in annular billiards in which the scatterer is touching or close to the boundary. We analytically show that there exist linearly stable periodic orbits of arbitrary period for scatterers with decreasing radii that are located near the boundary of the disk. As the position of the scatterer moves away from a symmetry line of a periodic orbit, the stability of periodic orbits changes from elliptic to hyperbolic, corresponding to a saddle-center bifurcation. When the scatterer is tangent to the boundary, the periodic orbit is parabolic. We prove that slightly changing the reflection angle of the orbit in the tangential situation leads to the existence of KAM islands. Thus we show that there exists a decreasing to zero sequence of open intervals of scatterer radii, along which the billiard table is not ergodic.A billiard is a dynamical system where a point particle moves with constant velocity inside a domain and experiences elastic collisions with the boundary of the domain. The shape of the boundary determines the dynamics of the billiard. Billiards in a disk on a plane are completely integrable, while annular billiard tables consisting of a particle confined between two nonconcentric disks generically display mixed phase space due to a family of regular orbits that never touch the scatterer. Billiard models find applications in a variety of problems in statistical 1 , classical and quantum 2 physics. In this paper, we consider annular billiard tables formed of a small circular scatterer placed in the interior of a unit circle; this is a popular geometry for microwave billiard experiments 3 . Circular boundaries allow us to analytically examine linear and nonlinear stability of some periodic orbits. Depending on the parameters of the problem, we find that there exist linearly stable orbits of arbitrarily large period. We show the existence of a saddle-center bifurcation as the parameters vary, corresponding to a change of stability from linearly elliptic to saddle type. Placing the scatterer tangentially to the external circle creates a cusp that is a source of singularities in the billiard. We use KAM theory to establish that in the cusp case, the periodic orbits are nonlinearly stable.

Hyperbolicity and Abundance of Elliptical Islands in Annular Billiards

“…The annular billiard has also many other periodic normal trajectories, as the orbits presented in Section 4, which play a very important role in the dynamics and in our analysis, as in [11,19]. The stability of some of these orbits was established in [15]. It is clear that the stability of periodic orbits and so the dynamics depend on the parameters.…”

“…In this work we focus on this last situation and look for hyperbolic behavior. More recently, Dettmann and Fain [15] have exhibited families of stable normal periodic orbits in the annular billiard when the obstacle is small and near the boundary, concluding that the system can not be ergodic for open sets of values of parameters close to this limit. The result is obtained through an explicit construction of suitable orbits and a direct computation of their non linear stability.…”

Hyperbolicity and abundance of elliptical islands in annular billiards

“…In this work, we focus on this last situation and look for hyperbolic behavior. More recently, Dettmann and Fain [15] exhibited families of stable normal periodic orbits in the annular billiard when the obstacle is small and near the boundary, concluding that the system cannot be ergodic for open sets of values of parameters close to this limit. The result is obtained through an explicit construction of suitable orbits and a direct computation of their nonlinear stability.…”

Time Recurrence Analysis of a Near Singular Billiard

“…In this work, we investigate the classical dynamics of a billiard with a very small scatterer through its recurrence properties and estimations of the largest Lyapunov exponent (LLE). The chosen model is the annular billiard, which has been the subject of many analytical and numerical studies [28,[30][31][32][33][34][35]. It consists of a particle confined in the region between two circumscribed circumferences of radii R and r, with r < R. When the circumferences are concentric, the energy and angular momentum are both conserved and therefore the system is integrable, since the plane billiard has two degrees of freedom.…”