Phase space structure and chaotic scattering in near-integrable systems

Area of Science:

• Classical Mechanics
• Dynamical Systems Theory
• Chaos Theory

Background:

• Investigates bifurcation phenomena and phase space structure changes.
• Focuses on the transition from regular to chaotic scattering in classical systems with unbounded dynamics.
• Examines integrable systems with degenerated unstable periodic orbits at infinity.

Purpose of the Study:

• To analyze the transition from regular to chaotic scattering in classical systems.
• To apply the McGehee transformation and Melnikov method to perturbed systems.
• To model stretching dynamics in ABC molecules using coupled Morse oscillators.

Main Methods:

• Utilized the McGehee transformation to remove degeneracy in periodic orbits.
• Applied the Melnikov method to predict homoclinic crossings of stable and unstable manifolds.
• Calculated subharmonic and homoclinic Melnikov functions for theoretical analysis.

Main Results:

• Proved the existence of chaotic scattering and periodic orbits (elliptic and hyperbolic).
• Quantified the width of the main stochastic layer and resonances.
• Predicted initial conditions leading to singularities in the scattering function.
• Calculated the perturbation parameter for channel transitions in coupled Morse oscillators.

Conclusions:

• The Melnikov method effectively predicts chaotic scattering and phase space structures.
• The study provides a framework for understanding complex dynamics in perturbed systems.
• Numerical experiments supplement theoretical findings, validating the models used.