用基波平衡原理分析非线性振荡与混沌

Analysing nonlinear oscillation and chaos using fundamental-wave balance principle

用"基波平衡原理"求得注入网络的基波电流Is1。它的流向代表网络在脱离激励源以后,为维持自激振荡关于实功与虚功的盈亏情况,是判断网络稳定性和振荡性状的有力依据。当实功与虚功同时取得平衡时,能求得基波解的振荡频率ωs和幅值Um,网络必然存在有对应的周期解。结论的普遍性可推广到三阶非线性微分方程。并阐明微分方程存在有多个周期解是产生混沌振荡的重要原因。其正确性可以用SIMULINK仿真验证。

The fundamental-wave current Is1 can be found by the theory of fundamental harmonic balance. Its flow direction represents profit and loss of active and reactive power in order to maintain self-oscillation, when the exciting source applied to the network is cut off. And it is powerfull basis of judging stability and self-excited oscillation shape and properties. Oscillation frequencya ωs and amplitude Um of fundamentalwave solution can be found, when active and reactive power of the network obtain balance synchronously. The respective periodic solution exist inevitably. The universality of the conclusion can be spread to third-order nonlinear differential equation. It is expounded that a differential equation possessing multiple periodic solutions is the important reason of generating chaotic oscillation. Its correctness can be verified by the Simulink.