Noise-enhanced stickiness in the Harper map

Area of Science:

• Dynamical systems
• Statistical mechanics
• Chaos theory

Background:

• The Poincaré recurrence statistic (PRS) describes the probability of a system returning to its initial state.
• In deterministic systems, PRS decay is often power-law due to 'stickiness' near invariant structures.
• Understanding recurrence is crucial for predicting system behavior over time.

Purpose of the Study:

• To investigate the effect of noise on the Poincaré recurrence statistic in dynamical systems.
• To analyze how noise perturbs trajectories and influences recurrence times.
• To elucidate the mechanisms behind altered PRS decay in the presence of noise.

Main Methods:

• Analysis of the Poincaré recurrence statistic (PRS) under noise perturbations.
• Study of the Harper map as a representative dynamical system.
• Comparison of trapping, visit, and recurrence times.
• Modeling with a finite-state Markov chain.

Main Results:

• Noise allows trajectories to access previously inaccessible regions, enhancing trapping.
• Noisy PRS exhibits an extended intermediate-time tail before asymptotic exponential decay.
• Recurrence times are significantly influenced by noise-induced entry into islands.
• Markov models confirm noise's role in causing slower PRS decay.

Conclusions:

• Noise fundamentally alters the recurrence properties of dynamical systems.
• The enhanced trapping effect due to noise leads to observable changes in the PRS.
• Noise-induced transitions between regions are key to understanding modified recurrence statistics.