周期激励Ueda电路中Hopf分岔与混沌的参数区域

Hopf Bifurcations and Chaotic Parameter Ranges in Periodically Excited Ueda′s Circuit

周期激励Ｕｅｄａ电路是一种具有奇异混沌行为的强非线性电路。本文基于Ｈｏｐｆ分岔条件给出了一个确定混沌参数区域的方法。首先使用谐波平衡法获得基谐波幅度随时间变化的方程；然后由该方程平衡点的稳定性得出平衡点的Ｈｏｐｆ分岔出现条件；最后通过数值方法，计算出在Ｈｏｐｆ分岔曲线周围系统的Ｌａｙａｐｕｎｏｖ分量，确定Ｕｅｄａ电路出现奇异混沌吸引子的参数区域。结果表明，Ｕｅｄａ电路的Ｈｏｐｆ分岔曲线附近确实存在着奇异混沌吸收子，这为预测Ｕｅｄａ电路奇异混沌吸收子的出现提供了一种有效的方法。文末还讨论了混沌与Ｈｏｐｆ分岔之间的关系。

The periodically excited Ueda′s circuit is one of strongly nonlinear circuits having chaotic behaviors. This paper presents a method for predicting parameter regions of the circuit within which the chaotic phenomena appear. It first utilizes the harmonic ba lance technique to obtain the dynamic equations governing the amplitude of the fundamental harmonic, then derive the conditions of Hopf bifurcations by analyzing the stability of the equation and finally determine chaotic parameter regions by numerically calculating the Lyapunov exponents of the Ueda′s circuit when the parameters vary in the vicinity of the Hopf bifurcation curves. The results show that the Ueda′s circuit indeed has chaotic attraotors near the curve. This provides a reliable method for predicting the occurrence of strange chaotic attractors in the Ueda′s circuit. Finally, the relations between the Hopf bifurcation and the occurrence of chaos are discussed.