Effects of Moderate Noise on a Limit Cycle Oscillator: Counterrotation and Bistability

The effects of noise on the dynamics of nonlinear systems is known to lead to many counterintuitive behaviors. Using simple planar limit cycle oscillators, we show that the addition of moderate noise leads to qualitatively different dynamics. In particular, the system can appear bistable, rotate in the opposite direction of the deterministic limit cycle, or cease oscillating altogether. Utilizing standard techniques from stochastic calculus and recently developed stochastic phase reduction methods, we elucidate the mechanisms underlying the different dynamics and verify our analysis with the use of numerical simulations. Last, we show that similar bistable behavior is found when moderate noise is applied to the FitzHugh-Nagumo model, which is more commonly used in biological applications.

Figure 1

Radial isochrone clock model [Eq. (1) with $c=0$], with streamlines indicating the direction of the vector field.

Figure 2

Deterministic phase plane for $c\ne 0$. Light blue curves are short (approximately half of one period) numerically simulated stochastic trajectories of (3) starting at $(x,y)=(0,-1)$ (red dot). Thin black curves are streamlines of the deterministic vector field. (a),(b) Counterrotating (using $Q_1$) with (a) $c=4$ and (b) $c=-4$. (c),(d) Using $Q_2$ with (c) uniformly rotating $c=4$ and (d) antirotating with $c=-15$. The lower panels are close-up sketches of a stochastic trajectory near the indicated part of the limit cycle.

Figure 3

The counterrotating case [see Figs. 2 and 2], with $G_x=G_y$. Parameter values are $\gamma =15$, $\omega =1$, and $\sigma =0.6$.

Figure 4

The uniformly rotating and antirotating cases [see Figs. 2 and 2], with $G_x=G_y$. Parameter values are $\gamma =15$, $\omega =1$, and $\sigma =0.63$.

Figure 5

Broken symmetry for the counterrotating case [see Fig. 2]. $c=4$, $\gamma =15$, $\sigma =0.6$, $G_x=0$, and $G_y=1$. $x(t)$ is a simulation trajectory showing bistablelike behavior. The stationary density is sharply peaked at the zeros of the drift $V_{\phi }$. Note that the zeros of $D_{\phi }$ are not exactly at the zeros of $V_{\phi }$.

Figure 6

Bistability in the FHN model. (a) Phase plane with a stationary histogram, for $\sigma =0.13$. (The inset shows the marginal histogram of the asymptotic phase.) The thick white line shows the deterministic limit cycle. (b) Sample stochastic trajectories. Parameters are $\tau =100$, $a=0.8$, and $m=1.2$.