Stochastic Poincaré maps for a slow-fast system with white noises: Approximation and visualization

Area of Science:

• Dynamical Systems and Chaos Theory
• Stochastic Processes
• Computational Mathematics

Background:

• Slow-fast systems exhibit complex dynamics often influenced by noise.
• Poincaré maps are crucial for analyzing periodic and chaotic behavior in dynamical systems.
• Stochastic perturbations introduce randomness, complicating traditional analysis.

Purpose of the Study:

• To develop methods for approximating and visualizing stochastic Poincaré maps.
• To analyze the behavior of these maps in slow-fast systems with white noise.
• To investigate the convergence of stochastic maps to deterministic ones.

Main Methods:

• Utilizing Taylor expansions for approximation.
• Employing a specialized moving orthogonal system for visualization.
• Deriving analytical expressions for stochastic Poincaré maps.

Main Results:

• An expression for stochastic Poincaré maps in slow-fast systems was established.
• Visualization techniques revealed the shapes and behavior of these maps.
• The approximation was shown to converge to deterministic Poincaré maps as noise and perturbation parameters tend to zero.

Conclusions:

• The developed approximation and visualization methods are effective for stochastic Poincaré maps.
• The convergence property provides a link between stochastic and deterministic system analysis.
• The study offers insights into the dynamics of noisy slow-fast systems.