EXACT SOLUTIONS OF CYCLICALLY SYMMETRIC OSCILLATOR EQUATIONS WITH NON‐LINEAR COUPLING. PART III: GLOBALLY STABLE SINGLE‐FREQUENCY LIMIT CYCLES

Several authors have found very simple exact limit cycle solutions in certain sets of N cyclically symmetric coupled non-linear oscillator equations. In part I of this paper it was shown that the search for sets of equations with similar solutions is made much simpler if the equations are expressed in terms of the normal modes of the unperturbed (linear) system. In various applications it is desirable that the equations have a limit cycle solution with a single frequency and that this solution be globally stable. It is found here that the transformation of the equations to the unperturbed normal modes can greatly simplify the construction of such systems.

几位作者在某些N个循环对称耦合非线性振子方程组中发现了非常简单的精确极限环解。在本文的第一部分表明，如果方程是根据未受扰动（线性）系统的简正模来表示的，那么寻找具有类似解的方程组就会变得简单得多。在各种应用中，希望方程具有单一频率的极限环解，并且该解是全局稳定的。在此发现，将方程变换为未受扰动的简正模可以极大地简化此类系统的构建。