Abstract
Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent λ was calculated using the FTLE method, which for conservative systems, λ > 0 indicates chaotic behavior and λ = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater’s theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of λ, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.
Highlights
The problem of a one-point particle, or an ensemble of non-interacting point particles, moving with constant velocity inside a bounded region, subject to elastic collisions with the boundaries, is generically known as a billiard. Such dynamical systems are described by nonlinear mappings and have great interest in several branches of physics [1,2,3,4,5,6]
Even billiards with simple geometry exhibit rich dynamical behaviors, typical of Hamiltonian systems, and depending on the geometry of the boundaries and the control parameters, the associated phase space can be: I—regular, consisting of quasi-periodic or periodic orbits lying on Kolmogorov–Arnold–Moser (KAM) tori; II—chaotic, with orbits that densely fill the whole phase space; or III—mixed, in which regular motion coexists with chaotic motion
This is the case with most non-integrable Hamiltonian systems, wherein the characterization of orbits as regular or chaotic is of great interest [7]
Summary
The problem of a one-point particle, or an ensemble of non-interacting point particles, moving with constant velocity inside a bounded region, subject to elastic collisions with the boundaries, is generically known as a billiard. Even billiards with simple geometry exhibit rich dynamical behaviors, typical of Hamiltonian systems, and depending on the geometry of the boundaries and the control parameters, the associated phase space can be: I—regular, consisting of quasi-periodic or periodic orbits lying on Kolmogorov–Arnold–Moser (KAM) tori; II—chaotic, with orbits that densely fill the whole phase space; or III—mixed, in which regular motion coexists with chaotic motion. This is the case with most non-integrable Hamiltonian systems, wherein the characterization of orbits as regular or chaotic is of great interest [7]. This phenomenon is known as stickiness and it influences the transport properties of the system [6,8,9,10]
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