Using rotation number to detect sticky orbits in Hamiltonian systems

Area of Science:

• * Physics, specifically nonlinear dynamics and Hamiltonian systems.
• * Computational physics and numerical analysis.

Background:

• * Hamiltonian systems exhibit 'stickiness,' where orbits temporarily trap near islands in phase space.
• * Existing methods for detecting stickiness (e.g., recurrence analysis, Lyapunov exponents) are computationally intensive and require prior knowledge of island locations.
• * The standard map is a canonical model used to study the dynamics of various nonlinear systems and often displays stickiness.

Purpose of the Study:

• * To introduce a novel, efficient method for identifying sticky orbits in Hamiltonian systems.
• * To demonstrate that the divergence in rotation number calculations can serve as an indicator of stickiness.
• * To validate the method's effectiveness on the standard map across different control parameter values.

Main Methods:

• * Analysis of the small divergence of bursts in the rotation number calculation.
• * Application of the developed method to the standard map.
• * Utilizing a minimal number of map iterations for analysis.

Main Results:

• * The proposed method successfully identifies stickiness without prior knowledge of island positions.
• * The technique requires significantly fewer map iterations compared to traditional methods.
• * The method is effective for detecting stickiness across a range of control parameter values in the standard map.

Conclusions:

• * The divergence of rotation number bursts offers a robust and efficient tool for detecting stickiness in Hamiltonian systems.
• * This new procedure simplifies the identification of sticky orbits, making it more accessible for analyzing complex nonlinear dynamics.
• * The method's efficiency and independence from island position knowledge represent a significant advancement in dynamical systems analysis.