Noise-induced distortion of the mean limit cycle of nonlinear oscillators

We study the change in the size and shape of the mean limit cycle of a stochastically driven nonlinear oscillator as a function of noise amplitude. Such dynamics occur in a variety of nonequilibrium systems, including the spontaneous oscillations of hair cells of the inner ear. The noise-induced distortion of the limit cycle generically leads to its rounding through the elimination of sharp (high-curvature) features through a process we call corner cutting. We provide a criterion that may be used to identify limit cycle regions most susceptible to such noise-induced distortions. By using this criterion, one may obtain more meaningful parametric fits of nonlinear dynamical models from noisy experimental data, such as those coming from spontaneously oscillating hair cells.

Figure 1

Stochastic trajectories of the hair bundle model. A representative stochastic trajectory (green) superimposed upon the deterministic (red) and mean (dashed black) limit cycles. The noise amplitude is by room temperature and the fluctuation-dissipation theorem $2k_{\text{B}}T\lambda$, with viscosity $\lambda$. This figure as reproduced from Ref. [26] exemplifies the effect of noise that we study here.

Figure 2

Hopf scalar potential. The deterministic limit cycle (red curve) lies in the minimum potential region of the Goldstone sombrero potential described by Eq. (5). The color map runs from dark blue (low potential) to light yellow (high potential). The vector potential (not shown) is a constant azimuthal vector field which drives the limit cycle dynamics in a counterclockwise circular limit cycle of radius $R_0$. See text for details.

Figure 3

Numerical simulation of the stochastic Hopf oscillator. Calculations were performed using Eq. (1). (a) The finite-temperature (light blue) trajectories and the mean (red) limit cycle. (b) A typical time series (black) of the stochastic dynamics of $X\left(t\right)$ and $Y\left(t\right)$.

Figure 4

Scalar potential map for generalized Hopf. Three-dimensional plot of the scalar potential in Eq. (7), for $n=4$ with the valleys seen in dark blue and hills in between them. The color map spans across dark blue (low potential) to light green (high potential). The deterministic limit cycle (yellow) for small vector potential skirts around the hills and pinches at the valleys.

Figure 5

Stochastic trajectories with variation in temperature and $\omega$. Quarter lobes of the trajectories obtained by solving Eq. (7) using $\omega$ values of $\left\{0,101,200\right\}$ and $\langle {\eta }_z^2\rangle$ values of $\left\{0,80,320,600\right\}$.

Figure 6

Examples of corner cutting. (a) The deterministic oscillator tracks the local minimum potential regions. (b) One lobe of the potential landscape. (c) Oscillator at $\langle {\eta }^2\rangle =10$. (d) Oscillator at $\langle {\eta }^2\rangle =30.$ A (black) arrow points to an example of a corner cutting trajectory. These have been simulated using parameter values $\mu =80,b=1,\alpha =2000,n=4$, resulting in ${\omega }^{★}=100$ [see Eq. (9)] and $\omega =101.$

Figure 7

Direction of $f_v$. One of the lobes of the mean limit cycle (gray) of the $\langle {\eta }^2\rangle =30$ stochastic system, with the (blue) regions (A, B) corresponding to arc lengths amid which the corner-cutting trajectories deviate from the particle's average behavior. The direction of ${\mathbf{f}}_V$ is illustrated by (orange) arrows. This lobe corresponds to the marked lobe in the (upper right) inset.

Figure 8

Confining potential. (A–D) Energy landscapes in the $\widehat{n}$ direction to the zero-temperature limit cycle, corresponding to the A–B arc length in Fig. 7. The (red) cross is indicative of the noiseless particle position, with negative values pointing towards (0,0). These positions correspond to the (black) cuts along the (yellow) limit cycle atop.

Figure 9

Nonuniform distortion of the limit cycle. The distance of the mean limit cycle at $\langle {\eta }^2\rangle =2000$ from the underlying noiseless curve shown as a color map. These values are normalized to the average value of the zero-temperature cycle, $\sqrt{\mu /b}$.

Figure 10

Curvature of the confining potential. The plot depicts $\kappa$ values along the zero-temperature limit cycle. We depict the inner curve, since the asymmetry of the problem renders it more susceptible to corner cutting. Additional plots exhibiting Eq. (10) illustrate the time of decay for the total number of trajectories that have escaped the mean path of a system with noise variance $\langle {\eta }^2\rangle =30$. This points to a theoretical method of determining regions in the oscillatory system that are prone to distortion in the presence of noise and hence less reliable when fitting parameters to experimental data.

Figure 11

Return time distribution. The histogram represents the distribution of the stochastic trajectories that lie $3T$ above the minimum potential. We consider trajectories that leave the mean limit cycle in a region of small potential curvature ($\kappa =0.6$) as shown in Fig. 8 B. The overlaying plot is the theoretical prediction of Eq. (10).