Scattering off two oscillating disks: Dilute chaos

P. K. Papachristou, F. K. Diakonos, V. Constantoudis, P. Schmelcher, and L. Benet
Phys. Rev. E 70, 056215 – Published 19 November 2004

Abstract

We investigate the role of the unstable periodic orbits and their manifolds in the dynamics of a time-dependent two-dimensional scattering system. As a prototype we use two oscillating disks on the plane with the oscillation axes forming an angle θ. The phase space of the system is five dimensional and it possesses a variety of families of unstable periodic orbits (UPOs) with intersecting manifolds. We perform numerical experiments to probe the structure of distinct scattering functions, in one and two dimensions, near the location of the UPOs. We find that the corresponding manifolds occur only in a very particular and localized way in the high-dimensional phase space. As a consequence the underlying fractal structure is ubiquitous only in higher-dimensional, e.g., two-dimensional, scattering functions. Both two-dimensional and one-dimensional scattering functions are dominated by seemingly infinite sequences of discontinuities characterized by small values of the magnitude of the projectile’s outgoing velocity. These peaks accumulate toward the phase-space locations of the UPOs, with a rate which monotonically depends on the corresponding instability exponent. They represent the intersections of the set of the initial conditions with invariant sets of larger dimensionality embedded in the phase space of the system, which are not directly related with the UPOs. We adopt the term “dilute chaos” to characterize these phenomenological aspects of the scattering dynamics.

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  • Received 6 July 2004

DOI:https://doi.org/10.1103/PhysRevE.70.056215

©2004 American Physical Society

Authors & Affiliations

P. K. Papachristou and F. K. Diakonos

  • Department of Physics, University of Athens, GR-15771, Athens, Greece

V. Constantoudis

  • Institute of Microelectronics (IMEL), NCSR “Demokritos,” P. O. Box 60228, Aghia Paraskevi, Attiki, Greece 15310 and Physics Department, National Technical University, Athens, Greece

P. Schmelcher

  • Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, INF 229, D-69120 Heidelberg, Germany and Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany

L. Benet

  • Centro de Ciencias Fisicas, UNAM, Apdo. Postal 48-3, 62251 Cuernavaca, Mexico

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Vol. 70, Iss. 5 — November 2004

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