Stochastic Phase Reduction for a General Class of Noisy Limit Cycle Oscillators

We formulate a phase-reduction method for a general class of noisy limit cycle oscillators and find that the phase equation is parametrized by the ratio between time scales of the noise correlation and amplitude relaxation of the limit cycle. The equation naturally includes previously proposed and mutually exclusive phase equations as special cases. The validity of the theory is numerically confirmed. Using the method, we reveal how noise and its correlation time affect limit cycle oscillations.

Figure 1

Steady distribution function of Stuart-Landau oscillators driven by Ornstein-Uhlenbeck processes when $\mathit{G}=(1,0)$, $\sigma =0.3$, $\Omega =1$, and $c=0.1$. Symbols are numerical results and solid lines are theoretical predictions, Eq. (13). Dotted and dashed lines are Eq. (13) with $k=0$ and $k=\infty$, respectively. (a) $({\tau }_{\eta },{\tau }_{\rho })=(0.2,0.1)$ (triangles), (0.1, 0.1) (circles), and (0.1, 0.2) (squares). (b) $({\tau }_{\eta },{\tau }_{\rho })=(0.2,0.2)$ (triangles), (0.1, 0.1) (circles), and (0.05, 0.05) (squares).

Figure 2

Mean frequency $\Omega$ of Stuart-Landau oscillators driven by Ornstein-Uhlenbeck processes when $\mathit{G}=(x,0)$, $\sigma =0.3$, $\omega =1$, and $c=1$. Solid lines are theoretical predictions, Eq. (14). (a) ${\tau }_{\rho }=0.01$. (b) ${\tau }_{\eta }=0.01$.