Pruning of orbits in four-disk and hyperbola billiards

Area of Science:

• Dynamical systems theory
• Geometric mechanics
• Topology

Background:

• Understanding the behavior of complex dynamical systems is crucial.
• Symbolic dynamics offers a powerful framework for analyzing chaotic systems.
• Previous methods struggled to fully capture the topological properties of orbits in certain systems.

Purpose of the Study:

• To develop a novel symbolic plane for the four-disk system and hyperbola billiard.
• To preserve the orientation of the dynamical space within the symbolic representation.
• To clearly distinguish between allowed and forbidden orbits.

Main Methods:

• Construction of a new symbolic plane.
• Mapping of physical orbits from the dynamical space to the symbolic plane.
• Identification of a pruning front separating allowed and forbidden orbits.

Main Results:

• A new orientation-preserving symbolic plane was successfully constructed.
• Physical orbits were mapped to a topologically well-ordered plane.
• A monotone pruning front was identified, effectively separating allowed and forbidden orbits.

Conclusions:

• The developed symbolic plane provides a clear and ordered representation of complex dynamical systems.
• This method enhances the understanding of orbit behavior and predictability.
• The pruning front serves as a key feature for analyzing system dynamics.