Correlation times in stochastic equations with delayed feedback and multiplicative noise

Area of Science:

• Nonlinear dynamics
• Stochastic processes
• Complex systems

Background:

• Stochastic differential equations (SDEs) are crucial for modeling complex systems.
• Pitchfork bifurcations represent critical transitions in dynamical systems.
• Delayed feedback can significantly alter system dynamics.

Purpose of the Study:

• To investigate the characteristic correlation time in an SDE model featuring a pitchfork bifurcation and delayed feedback.
• To assess the validity of the statistical independence assumption between system states separated by a delay time (τ).

Main Methods:

• Numerical integration of the governing stochastic differential equation.
• Analytical determination of correlation time in the limit of small delay times (small τ).

Main Results:

• The correlation time (T) was found to diverge as T~a(-1) approaching the bifurcation threshold (a=0).
• Divergence of correlation time signals a sharp bifurcation threshold.
• The analysis revealed a failure of the statistical independence assumption near the bifurcation threshold.
• The small-τ expansion accurately predicted the bifurcation threshold location but showed deviations in correlation time magnitude.

Conclusions:

• The correlation time serves as a robust indicator of sharp bifurcation thresholds in systems with delayed feedback.
• The assumption of statistical independence is invalid near bifurcation points in these models.
• Delayed feedback introduces complexities that require careful consideration in dynamical system analysis.