Dynamical systems with \(\varepsilon \) small random perturbations appear in both continuous mechanical motions and discrete stochastic chemical kinetics. The present work provides a detailed analysis of the central limit theorem (CLT), with a time-inhomogeneous Gaussian process, near a deterministic limit cycle in \(\mathbb {R}^n\). Based on respectively the theory of random perturbations of dynamical systems and the WKB approximation that codes the large deviations principle (LDP), results are developed in parallel from both standpoints of stochastic trajectories and transition probability density and their relations are elucidated. We show rigorously the correspondence between the local Gaussian fluctuations and the curvature of the large deviation rate function near its infimum, connecting the CLT and the LDP of diffusion processes. We study uniform asymptotic behavior of stochastic limit cycles through the interchange of limits of time \(t\rightarrow \infty \) and \(\varepsilon \rightarrow 0\). Three further characterizations of stochastic limit cycle oscillators are obtained: (i) An approximation of the probability flux near the cycle; (ii) Two special features of the vector field for the cyclic motion; (iii) A local entropy balance equation along the cycle with clear physical meanings. Lastly and different from the standard treatment, the origin of the \(\varepsilon \) in the theory is justified by a novel scaling hypothesis via constructing a sequence of stochastic differential equations.

具有\（\ varepsilon \）小随机扰动的动力学系统会同时出现在连续的机械运动和离散的随机化学动力学中。本工作通过时间非均匀的高斯过程，在\（\ mathbb {R} ^ n \）的确定性极限周期附近，对中心极限定理（CLT）进行了详细的分析。。分别基于动力系统的随机扰动理论和编码大偏差原理（LDP）的WKB逼近，从随机轨迹和转移概率密度两个角度并行开发了结果，并阐明了它们之间的关系。我们严格显示了局部高斯涨落与大偏差率函数曲率接近其最大值之间的对应关系，将扩散过程的CLT和LDP连接起来。我们通过时间极限（\ t \ rightarrow \ infty \）和\（\ varepsilon \ rightarrow 0 \）的互换研究随机极限周期的一致渐近行为。获得了随机极限周期振荡器的三个进一步的特征：（i）周期附近的概率通量的近似值；ii循环运动矢量场的两个特殊特征；（iii）沿循环的局部熵平衡方程，具有明确的物理意义。最后，与标准处理不同，该理论中\（\ varepsilon \）的起源通过构造一系列随机微分方程的一种新颖的缩放假设得以证明。