Scattering off two oscillating disks: dilute chaos.

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.
• 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.