Finite-time recurrence analysis of chaotic trajectories in Hamiltonian systems

Area of Science:

• Non-linear dynamics
• Chaos theory
• Statistical mechanics

Background:

• Understanding chaotic trajectories in non-linear Hamiltonian systems is crucial for predicting their long-term behavior.
• Traditional methods often struggle to characterize complex dynamical behaviors like phase space trapping.
• Recurrence analysis offers a potential tool for uncovering hidden dynamical patterns.

Purpose of the Study:

• To introduce and validate a finite-time recurrence analysis method for characterizing chaotic trajectories.
• To demonstrate how recurrence rates can identify dynamically distinct regions within phase space.
• To visually illustrate and differentiate regions of 'stickiness' in non-linear systems.

Main Methods:

• Generating an ensemble of initial conditions for chaotic trajectories.
• Evolving trajectories over a finite time and computing their recurrence rates.
• Analyzing the relationship between recurrence rates and dynamical trapping (stickiness).
• Applying the method to three distinct non-linear maps.

Main Results:

• Finite-time recurrence analysis effectively distinguishes between average and anomalous trajectory behaviors.
• Orbits with high recurrence rates consistently exhibit dynamical trapping or 'stickiness'.
• The method successfully identified and characterized sticky regions in the analyzed non-linear maps.
• Distinct dynamical behaviors were visually illustrated through recurrence rate patterns.

Conclusions:

• Finite-time recurrence analysis provides valuable prior knowledge of dynamical behavior in non-linear Hamiltonian systems.
• This approach offers a robust method for identifying and characterizing phase space regions with distinct dynamical properties, particularly sticky regions.
• The proposed method serves as a powerful visualization tool for complex chaotic systems.