English L Q, Mertens David, Abdoulkary Saidou, Fritz C B, Skowronski K, Kevrekidis P G
Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA.
Department of Physics, Eckerd College, St. Petersburg, Florida 33711, USA.
Phys Rev E. 2016 Dec;94(6-1):062212. doi: 10.1103/PhysRevE.94.062212. Epub 2016 Dec 20.
We derive the Kuramoto-Sakaguchi model from the basic circuit equations governing two coupled Wien-bridge oscillators. A Wien-bridge oscillator is a particular realization of a tunable autonomous oscillator that makes use of frequency filtering (via an RC bandpass filter) and positive feedback (via an operational amplifier). In the past few years, such oscillators have started to be utilized in synchronization studies. We first show that the Wien-bridge circuit equations can be cast in the form of a coupled pair of van der Pol equations. Subsequently, by applying the method of multiple time scales, we derive the differential equations that govern the slow evolution of the oscillator phases and amplitudes. These equations are directly reminiscent of the Kuramoto-Sakaguchi-type models for the study of synchronization. We analyze the resulting system in terms of the existence and stability of various coupled oscillator solutions and explain on that basis how their synchronization emerges. The phase-amplitude equations are also compared numerically to the original circuit equations and good agreement is found. Finally, we report on experimental measurements of two coupled Wien-bridge oscillators and relate the results to the theoretical predictions.
我们从描述两个耦合维恩桥振荡器的基本电路方程推导出仓本-坂口模型。维恩桥振荡器是一种利用频率滤波（通过一个RC带通滤波器）和正反馈（通过一个运算放大器）的可调谐自治振荡器的具体实现。在过去几年里，这类振荡器已开始用于同步研究。我们首先表明，维恩桥电路方程可以转化为一对耦合的范德波尔方程的形式。随后，通过应用多时间尺度方法，我们推导出了描述振荡器相位和幅度缓慢演化的微分方程。这些方程直接让人联想到用于同步研究的仓本-坂口型模型。我们根据各种耦合振荡器解的存在性和稳定性对所得系统进行分析，并在此基础上解释它们的同步是如何出现的。还将相位-幅度方程与原始电路方程进行了数值比较，发现二者吻合良好。最后，我们报告了两个耦合维恩桥振荡器的实验测量结果，并将结果与理论预测相关联。
