We consider a general class of limit cycle oscillators driven by an additive Gaussian white noise. Based on the separation of timescales, we construct the equation of motion for slow dynamics after appropriate averaging over the fast motion. The equation for slow motion whose coefficients are modified by noise characteristics is solved to obtain the analytic solution in the long time limit. We show that with increase of noise strength, the loop area of the limit cycle decreases until a critical value is reached, beyond which the limit cycle collapses. We determine the noise threshold from the condition for removal of secular divergence of the perturbation series and work out two explicit examples of Van der Pol and Duffing-Van der Pol oscillators for corroboration between the theory and numerics.

中文翻译：

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用噪声调整极限环：生存和崩溃

我们考虑由加性高斯白噪声驱动的一类通用极限环振荡器。基于时间尺度的分离，我们在对快速运动进行适当平均后构建了慢速动力学的运动方程。求解慢运动方程，其系数根据噪声特性进行修正，得到长时间极限内的解析解。我们表明，随着噪声强度的增加，极限环的环路面积减小，直到达到临界值，超过该值极限环就会崩溃。我们根据消除扰动序列的长期发散的条件确定噪声阈值，并计算出范德波尔和杜芬-范德波尔振荡器的两个明确的例子，以验证理论和数值。