一阶微分电路构成的混沌

Abstract

混沌是有界非线性函数最一般的普遍形式，它普遍的存在于自然界的各个学科领域。混沌是各种各样有界非线性振荡的统一术语。本文证明三个频率不同谐波源混频构成的一阶微分电路，也可以产生混沌。它充分说明混沌函数存在的广泛性。该微分方程能够用谐波分析法和功率平衡定理求出它的主谐波解。并用仿真软件画出相图，验证求解结果的正确性。在上世纪混沌理论刚刚出现时，很多文献认为是奇异吸引子，这种认识显然是片面的。事实上，人们也可以做出一个完全相反的结论，相点的运动轨道既不发散趋于无穷，也不收敛为稳定极限环。相点自由任意的游荡在相空间不是随机的，是正常一般的运动形式。轨线不重复的混沌相图是无处不在的普遍现象，轨线不断重复的等周期振荡才是特殊的个别现象。 Chaos is the most universally common form of bounded nonlinear function. It commonly exists in the various subject areas of nature. Chaos is the universal term of variously bounded nonlinear aperiodic oscillation. This paper proves that the first order differential circuit which is constituted by mixing of three harmonic sources with different frequency also can produce chaos. It sufficiently explains that the extensiveness of chaotic functions exists in nature. The main harmonic components in the differential equations can be solved by using the harmonic balance principle and power balance theorem. Their correctness of solving results can be verified by phase portrait plotted by simulation. In last century, the era when chaos theory was first published, chaos was considered as a singular attractor in a lot of literatures. The recognition is obviously unilateral and wrong. In fact, people can also make a completely opposite conclusion, the motional trajectory of phase point will neither diverge to be infinite nor converge to stable limit cycle. The phase point freely and arbitrarily wandering in phase space is ordinary and universal motional form, but it is non-random. Chaotic phase portraits on which trajectories are not repeated are pervasive phe-nomenon. The constant periodic oscillation which continuously repeats original orbit is an indi-vidual and special motion form.