Tuning limit cycles with a noise: Survival and collapse

Area of Science:

• Nonlinear Dynamics
• Stochastic Processes
• Physics

Background:

• Limit cycle oscillators are fundamental in various scientific fields.
• Understanding noise effects on oscillatory systems is crucial for accurate modeling.

Purpose of the Study:

• To derive and solve the slow dynamics equation for limit cycle oscillators driven by Gaussian white noise.
• To investigate the impact of noise strength on limit cycle behavior and identify the collapse threshold.

Main Methods:

• Averaging over fast motion based on timescale separation.
• Analytic solution of the averaged slow dynamics equation.
• Perturbation series analysis to determine the noise threshold.

Main Results:

• Noise strength decreases the limit cycle loop area.
• A critical noise level causes limit cycle collapse.
• Analytic solutions were obtained for Van der Pol and Duffing-Van der Pol oscillators.

Conclusions:

• The study provides a theoretical framework for noise-induced collapse in oscillators.
• Noise characteristics significantly modify oscillator dynamics and stability.
• The findings are validated through numerical simulations.