A universal description of stochastic oscillators.

Area of Science:

• Physics, chemistry, and biology
• Nonlinear dynamics
• Stochastic processes

Background:

• Many natural systems exhibit oscillations with significant random components, arising from diverse mechanisms like noise-perturbed limit cycles or excitable systems.
• Despite varied origins, these stochastic oscillations often share similar observable behaviors.
• Existing methods struggle to unify the analysis of these diverse oscillatory systems.

Purpose of the Study:

• To introduce a novel nonlinear transformation for unifying the mathematical description of stochastic oscillators.
• To simplify the analysis of spontaneous activity, response to perturbations, and correlation statistics in weakly coupled oscillators.
• To provide a universal framework for comparing and characterizing different types of random oscillations.

Main Methods:

• Developed a complex-valued function, denoted as ψ(x), derived from stochastic oscillators.
• Identified ψ(x) as the eigenfunction of the Kolmogorov backward operator with eigenvalue λ₁ = μ₁ + iω₁.
• Utilized this transformation to derive exact formulas for power spectra, susceptibility, and cross-spectra.

Main Results:

• The complex-valued function's power spectrum precisely follows a Lorentz profile with peak frequency ω₁ and half-width μ₁.
• The system's susceptibility to weak external forcing is characterized by a simple one-pole filter centered at ω₁.
• Cross-spectra of coupled oscillators are readily expressed using spontaneous power spectra and susceptibilities of individual systems.

Conclusions:

• The introduced nonlinear transformation offers a unified and simplified mathematical framework for stochastic oscillators.
• This method allows for direct comparison of qualitatively different stochastic oscillators and provides clear coherence characteristics.
• The framework facilitates the description and analysis of weakly coupled oscillatory systems across scientific disciplines.