"Green" noise in quasistationary stochastic systems

Area of Science:

• Physics
• Nonlinear Dynamics
• Stochastic Processes

Background:

• Traditional analysis of stochastic systems often assumes white noise.
• Systems driven by "green" noise (zero spectral density at zero frequency) present unique challenges.
• Understanding non-stationary behavior under colored noise is crucial for many applications.

Purpose of the Study:

• To develop and apply a modified Krylov-Bogoliubov averaging method for stochastic systems driven by green noise.
• To investigate the emergence of effective potentials and metastable states in such systems.
• To analyze the impact of green noise spectral properties on system stability and lifetime.

Main Methods:

• Application of the Krylov-Bogoliubov averaging method adapted for non-stationary systems.
• Utilizing the ergodicity of nonlinear random functions within the averaging framework.
• Derivation of general equations for approximations of the theory.
• Numerical simulations to validate theoretical predictions.

Main Results:

• The first approximation reveals an effective potential governing the averaged motion of the system.
• Metastable states are identified in a phase-locked loop example.
• The lifetime of metastable states significantly increases as the green noise spectrum sharpens in the low-frequency region.
• Theoretical predictions show high accuracy when compared with numerical simulation results.

Conclusions:

• The proposed averaging method effectively describes stochastic systems driven by green noise.
• Green noise can induce high stability and long-lived metastable states.
• Further research into unsolved problems related to green noise is warranted.