Two-particle circular billiards versus randomly perturbed one-particle circular billiards

Rankovic, Sandra; Porter, Mason A.
March 2013
Chaos;Mar2013, Vol. 23 Issue 1, p013123
Academic Journal
We study a two-particle circular billiard containing two finite-size circular particles that collide elastically with the billiard boundary and with each other. Such a two-particle circular billiard provides a clean example of an 'intermittent' system. This billiard system behaves chaotically, but the time scale on which chaos manifests can become arbitrarily long as the sizes of the confined particles become smaller. The finite-time dynamics of this system depends on the relative frequencies of (chaotic) particle-particle collisions versus (integrable) particle-boundary collisions, and investigating these dynamics is computationally intensive because of the long time scales involved. To help improve understanding of such two-particle dynamics, we compare the results of diagnostics used to measure chaotic dynamics for a two-particle circular billiard with those computed for two types of one-particle circular billiards in which a confined particle undergoes random perturbations. Importantly, such one-particle approximations are much less computationally demanding than the original two-particle system, and we expect them to yield reasonable estimates of the extent of chaotic behavior in the two-particle system when the sizes of confined particles are small. Our computations of recurrence-rate coefficients, finite-time Lyapunov exponents, and autocorrelation coefficients support this hypothesis and suggest that studying randomly perturbed one-particle billiards has the potential to yield insights into the aggregate properties of two-particle billiards, which are difficult to investigate directly without enormous computation times (especially when the sizes of the confined particles are small).


Related Articles

  • May chaos always be suppressed by parametric perturbations? Schwalger, Tilo; Dzhanoev, Arsen; Loskutov, Alexander // Chaos;Jun2006, Vol. 16 Issue 2, p023109 

    The problem of chaos suppression by parametric perturbations is considered. Despite the widespread opinion that chaotic behavior may be stabilized by perturbations of any system parameter, we construct a counterexample showing that this is not necessarily the case. In general, chaos suppression...

  • Small-scale structure of nonlinearly interacting species advected by chaotic flows. Herna´ndez-Garcı´a, Emilio; Lo´pez, Cristo´bal; Neufeld, Zolta´n // Chaos;Jun2002, Vol. 12 Issue 2, p470 

    We study the spatial patterns formed by interacting biological populations or reacting chemicals under the influence of chaotic flows. Multiple species and nonlinear interactions are explicitly considered, as well as cases of smooth and nonsmooth forcing sources. The small-scale structure can be...

  • Finite-Time Evolution of Small Perturbations Superposed on a Chaotic Solution: Experiment with an Idealized Barotropic Model. Yamane, Shozo; Yoden, Shigeo // Journal of the Atmospheric Sciences;5/1/2001, Vol. 58 Issue 9, p1066 

    Provides information on a study which examined the fundamental principles of finite-time evolution of small perturbations in chaotic systems using an idealized barotropic model on a rotating sphere. Details on Lyapunov exponents and Lyapunov vectors; Properties of the root-main-square...

  • Competitive autocatalytic reactions in chaotic flows with diffusion: Prediction using finite-time Lyapunov exponents. Schlick, Conor P.; Umbanhowar, Paul B.; Ottino, Julio M.; Lueptow, Richard M. // Chaos;Mar2014, Vol. 24 Issue 1, p1 

    We investigate chaotic advection and diffusion in autocatalytic reactions for time-periodic sine flow computationally using a mapping method with operator splitting. We specifically consider three different autocatalytic reaction schemes: a single autocatalytic reaction, competitive...

  • Scattering from a classically chaotic repellor. Gaspard, Pierre; Rice, Stuart A. // Journal of Chemical Physics;2/15/1989, Vol. 90 Issue 4, p2225 

    Studies the scattering of a point particle from circular discs in a plane. Fractal and chaotic metastable state of the system; Particle escape rate; Lyapunov exponent of the repellor; Kolmogorov-Sinai entropy per unit time.

  • Enhancement of Lyapunov exponents in one-dimensional, randomly phased waves. Stoltz, Peter H.; Cary, John R. // Physics of Plasmas;Jun94, Vol. 1 Issue 6, p1817 

    Reports on the finding that the Lyapunov exponent asymptotes to a value of 0.4 times the resonance broadening frequency for a particle moving in the electric field of randomly phase waves. Peak of the Lyapunov exponent at intermediate values of the overlap parameter; Evidence that the standard...

  • Chaotic itinerancy based on attractors of one-dimensional maps. Sauer, Timothy // Chaos;Sep2003, Vol. 13 Issue 3, p947 

    Describes a general methodology for constructing systems that have a slowly converging Lyapunov exponent near zero based on one-dimensional maps with chaotic attractors. Mechanism of chaotic itinerancy; Aspects of chaotic itinerancy.

  • On the dynamics of ocean ambient noise: Two decades later. Siddagangaiah, Shashidhar; Yaan Li; Xijing Guo; Kunde Yang // Chaos;2015, Vol. 25 Issue 10, p103117-1 

    Two decades ago, it was shown that ambient noise exhibits low dimensional chaotic behavior. Recent new techniques in nonlinear science can effectively detect the underlying dynamics in noisy time series. In this paper, the presence of low dimensional deterministic dynamics in ambient noise is...

  • ANTI-SYNCHRONIZATION OF 4-DIMENSIONAL HYPERCHAOTIC LI AND HYPERCHAOTIC LÃœ SYSTEMS VIA ACTIVE CONTROL. Vaidyanathan, Sundarapandian // International Journal of Control Theory & Computer Modeling;Nov2012, Vol. 2 Issue 6, p25 

    In this paper, we derive new results for the anti-synchronization of identical and non-identical hyperchaotic Li systems (Li, Tang and Chen, 2005) and hyperchaotic Lü systems (Bao and Liu, 2008). Active control method has been deployed for achieving the four-dimensional hyperchaotic systems...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics