Isostables for Stochastic Oscillators

Area of Science:

• Physics
• Nonlinear Dynamics
• Stochastic Processes

Background:

• The asymptotic phase of stochastic oscillators was previously defined using the backward Kolmogorov operator.
• A complete phase-amplitude description for noisy oscillators is essential for understanding their dynamics.

Purpose of the Study:

• To define the stochastic isostable coordinate for noisy oscillators.
• To extend the phase-amplitude description of stochastic oscillators.
• To provide a framework for stochastic limit cycle dynamics.

Main Methods:

• Defining the stochastic isostable coordinate as a specific eigenfunction of the backward Kolmogorov operator.
• Utilizing complex eigenvalues with least negative nontrivial real parts.

Main Results:

• The stochastic isostable coordinate was successfully defined.
• This definition completes the phase-amplitude description for stochastic oscillators.
• The results suggest a framework encompassing noise-induced oscillations.

Conclusions:

• The stochastic isostable coordinate provides a new tool for analyzing noisy oscillators.
• This work advances the understanding of stochastic limit cycle dynamics.
• The framework may explain noise-induced oscillations in various systems.