论文标题
随机扰动下非线性系统的随机极限周期振荡
Stochastic limit-cycle oscillations of a nonlinear system under random perturbations
论文作者
论文摘要
具有$ε$的动力系统在连续机械运动和离散的随机化学动力学中都出现。目前的工作提供了对中央限制定理（CLT）的详细分析，并具有时间固定的高斯进程，接近$ \ mathbb {r}^n $的确定性限制周期。基于动态系统和WKB近似的随机扰动理论，从随机轨迹的角度和过渡概率密度及其关系及其关系均已阐明，结果是并行开发的。我们严格地显示了局部高斯波动与大偏差率功能的曲率之间的对应关系，连接了CLT和扩散过程的大偏差原理。我们研究了随机极限循环的统一行为，通过时间$ t \ to \ infty $和$ε\ to 0 $的时间互换。随机极限循环振荡器的三个进一步表征将获得：（i）循环附近的概率通量的近似； （ii）循环运动的矢量场的两个特殊特征； （iii）沿周期的局部熵平衡方程，具有明显的物理含义。最后，与标准处理不同，理论中$ε$的起源是通过构建一系列随机微分方程的序列来证明了新型缩放假设的合理性。
Dynamical systems with $ε$ 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 the theory of random perturbations of dynamical systems and the WKB approximation respectively, 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 large deviation principle of diffusion processes. We study uniform asymptotic behavior of stochastic limit cycles through the interchange of limits of time $t\to\infty$ and $ε\to 0$. 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 $ε$ in the theory is justified by a novel scaling hypothesis via constructing a sequence of stochastic differential equations.
