Phase descriptions of a multidimensional Ornstein-Uhlenbeck process

Area of Science:

• Physics
• Mathematics
• Dynamical Systems

Background:

• Stochastic oscillators are crucial in various scientific domains.
• Existing phase definitions for stochastic oscillators lack clear intuition.
• The asymptotic phase, based on eigenfunction expansion, is a previously proposed definition.

Purpose of the Study:

• To analyze and compare two distinct phase definitions for stochastic oscillators: the backward and forward phases.
• To provide explicit mathematical expressions for these phases.
• To enhance the understanding of their properties and behaviors.

Main Methods:

• Utilizing the analytically tractable two-dimensional Ornstein-Uhlenbeck process with complex eigenvalues.
• Deriving explicit expressions for both backward and forward phases.
• Analyzing the isochrons (lines of constant phase) and their angular density.

Main Results:

• Isochrons for both phases resemble spokes of a wheel, but their angular density differs.
• Backward phase isochrons depend solely on the deterministic vector field.
• Forward phase isochrons are influenced by both the deterministic field and the noise matrix.
• Backward phase progression is uniformly distributed in time; forward phase progression is not, except in symmetric cases.

Conclusions:

• The study provides explicit analytical results for backward and forward phase definitions in a specific stochastic process.
• Differences in isochron spacing and temporal progression highlight distinct characteristics of each phase.
• This work offers a clearer mathematical and intuitive framework for understanding phases in stochastic systems.