Exploring conservative islands using correlated and uncorrelated noise

Area of Science:

• Dynamical Systems
• Statistical Physics

Background:

• Conservative dynamical systems often exhibit mixed phase spaces with coexisting regular and chaotic regions.
• Understanding the influence of noise on the dynamics within these systems is crucial for predicting long-term behavior.

Purpose of the Study:

• To investigate the role of noise in the penetration of regular islands within conservative dynamical systems.
• To analyze how different noise distributions affect dynamical behaviors and phase space exploration.

Main Methods:

• Utilized the standard map with specific nonlinearity parameters to create a mixed phase space.
• Employed noise with three distributions: uniform, Gaussian, and power-law correlated.
• Analyzed system dynamics using recurrence time statistics (RTS), Lyapunov exponents, and phase space occupation rates.

Main Results:

• The standard deviation of noise distributions is the primary factor driving island penetration.
• Island penetration leads to power-law decays in RTS due to enhanced trapping.
• Power-law correlated noise induces algebraic RTS decay, even without sticky motion.
• High noise intensities result in ergodic-like behavior, but Lyapunov exponents retain signatures of regular islands.

Conclusions:

• Noise significantly impacts the accessibility of regular structures in mixed phase spaces.
• Recurrence time statistics effectively reveal the effects of noise-induced trapping and island penetration.
• The nature of noise, particularly its correlation properties, plays a key role in determining the decay rates of dynamical processes.