Generalized separatrix mapping in a system with more than two degrees of freedom

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Statistical Mechanics

Background:

• The separatrix map is crucial for understanding chaotic layers and Arnold diffusion.
• Previous studies assumed a 3D map simplifies to a 2D version in certain limits.
• This simplification allows direct application of 2D separatrix map results.

Purpose of the Study:

• Investigate a 3D symplectic map derived from the Arnold Hamiltonian.
• Analyze the dynamics of this 3D map, particularly in low-order resonance frequency domains.
• Compare numerical and analytical results with the standard 2D separatrix map.

Main Methods:

• Developed a three-dimensional (3D) symplectic map from the Arnold Hamiltonian.
• Employed numerical and analytical estimation techniques.
• Computed maximum Lyapunov exponent and metric entropy.

Main Results:

• The 3D map's dynamics deviate from expected results in low-order resonance frequency domains.
• Calculated maximum Lyapunov exponent and metric entropy show discrepancies.
• Findings challenge the applicability of 2D separatrix map results in these specific regions.

Conclusions:

• The generalization of the separatrix map to 3D exhibits complex dynamics not captured by 2D models.
• Low-order resonances significantly alter Arnold diffusion and chaotic layer behavior.
• Further research is needed to refine models for higher-dimensional chaotic systems.