Stochastic switching in slow-fast systems: a large-fluctuation approach

Area of Science:

• Stochastic Systems Analysis
• Nonlinear Dynamics
• Computational Physics

Background:

• Rare events in singularly perturbed stochastic systems are challenging to predict.
• Existing methods like stochastic normal form approaches have limitations.
• Understanding large fluctuation-induced rare events is crucial for system stability.

Purpose of the Study:

• To develop a novel perturbation method for predicting rare event rates in singularly perturbed stochastic systems.
• To model rare event occurrences probabilistically using a probability density function approach.
• To analyze the dynamics and switching behavior in systems like the stochastic damped Duffing oscillator.

Main Methods:

• Utilized a probability density function approach and a WKB ansatz.
• Formulated a two-point boundary value problem to model state variable and noise force interactions.
• Leveraged vastly different time scales for dimension reduction and prediction on the slow manifold.
• Employed center manifold theory for system dynamics reduction.

Main Results:

• Developed a method to compute an exponent determining rare event probability.
• Successfully analyzed a stochastic damped Duffing oscillator with multiple equilibrium points.
• Predicted switching times between states using optimal paths in an expanded phase space.
• Demonstrated excellent agreement between predicted and simulated exponential scaling of switching rates.

Conclusions:

• The developed perturbation method accurately predicts rare event rates and switching dynamics.
• The reduced system dynamics align with the original system dynamics in an exponentially scaling sense.
• This approach offers a robust framework for analyzing rare events in complex stochastic systems.