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Related Experiment Videos

Scattering off two oscillating disks: dilute chaos.

P K Papachristou1, F K Diakonos, V Constantoudis

  • 1Department of Physics, University of Athens, GR-15771, Athens, Greece.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
PubMed
Summary

Unstable periodic orbits (UPOs) and their manifolds significantly influence scattering dynamics in time-dependent systems. Numerical experiments reveal localized manifolds and fractal structures in higher-dimensional scattering functions, leading to "dilute chaos".

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Area of Science:

  • Physics
  • Dynamical Systems
  • Chaos Theory

Background:

  • Scattering systems exhibit complex dynamics influenced by unstable periodic orbits (UPOs).
  • The role of UPOs and their stable/unstable manifolds in high-dimensional phase spaces remains an active area of research.
  • Understanding these structures is crucial for characterizing chaotic behavior in time-dependent systems.

Purpose of the Study:

  • To investigate the influence of unstable periodic orbits (UPOs) and their manifolds on the dynamics of a time-dependent two-dimensional scattering system.
  • To analyze the structure of scattering functions near UPOs and understand the emergence of fractal patterns.
  • To introduce and characterize the phenomenon of "dilute chaos" in scattering dynamics.

Main Methods:

  • Numerical experiments were conducted on a prototype system of two oscillating disks with intersecting UPO manifolds.

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  • Scattering functions in one and two dimensions were analyzed near UPO locations.
  • Phase space analysis was employed to probe the localization of manifolds and the nature of discontinuities.

    • Main Results:

    • Unstable periodic orbits (UPOs) and their manifolds are found to be highly localized in the high-dimensional phase space.
    • Fractal structures are ubiquitous in higher-dimensional (e.g., 2D) scattering functions.
    • Scattering functions exhibit sequences of discontinuities accumulating towards UPO locations, related to invariant sets, not directly UPOs.
    • The term "dilute chaos" is proposed to describe these observed dynamics.

    Conclusions:

    • The localization of manifolds significantly impacts the observability of fractal structures in scattering functions.
    • "Dilute chaos" provides a new perspective on the complex dynamics arising from the interaction of invariant sets in scattering systems.
    • The study highlights the intricate relationship between UPOs, their manifolds, and the emergent fractal and discontinuous features in chaotic scattering.