Degenerate orbit transitions in a one-dimensional inelastic particle system

Area of Science:

• Physics
• Dynamical Systems
• Classical Mechanics

Background:

• Periodic orbits are fundamental in understanding dynamical systems.
• Inelastic collisions introduce complexities not present in elastic systems.
• Characterizing transitions between orbits is crucial for system analysis.

Purpose of the Study:

• To investigate continuous transitions between periodic orbits in a 1D inelastic particle system.
• To determine the generic codimension of these transitions.
• To identify and explain degenerate transitions and their classification.

Main Methods:

• Full mechanical description of the one-dimensional inelastic particle system.
• Analysis of continuous transitions occurring with single collision changes.
• Theoretical framework for classifying degenerate transitions based on collision sets.

Main Results:

• Generic continuous transitions are of co-dimension 2.
• An infinite set of degenerate transitions of co-dimension 1 were discovered.
• A simple criterion using discrete collision sets was developed to classify degenerate transitions.
• The nature of degeneracy was elucidated, with potential for higher degrees.

Conclusions:

• The study provides a comprehensive mechanical description of transitions in inelastic particle systems.
• It reveals a surprising abundance of degenerate transitions, offering new insights into system dynamics.
• The developed classification criterion simplifies the analysis of complex orbital behaviors.