广义BVP电路系统的振荡行为及其非光滑分岔机理

探讨了具有分段线性特性的广义BVP电路系统随参数变化的复杂动力学演化过程. 其非光滑分界面将相空间划分成不同的区域, 分析了各区域中平衡点的稳定性, 得到其相应的简单分岔和Hopf分岔的临界条件. 给出了不同分界面处广义Jacobian矩阵特征值随辅助参数变化的分布情况, 讨论了分界面处系统可能存在的分岔行为, 指出当广义特征值穿越虚轴时可能引起Hopf分岔, 导致系统由周期振荡转变为概周期振荡, 而当出现零特征值时则导致系统的振荡在不同平衡点之间转换. 针对系统的两种典型振荡行为, 结合数值模拟验证了理论分析的结果.

Oscillations and non-smooth bifurcations in a generalized BVP circuit system

The complicated dynamical evolution of a generalized BVP circuit system with piecewise linear characteristics is explored. The phase space is divided into different types of regions by the nonsmooth boundaries. In each region, the stabilities of the equilibrium points are investigated, from which the critical conditions related to simple bifurcations as well as Hopf bifurcations are obtained. By employing the analysis of the distribution of the eigenvalues of the generalized Jacobian matrix, the bifurcation behaviors related to the nonsmooth boundaries are explored in detail. It is pointed out that when pure imaginary eigenvalues associated with the generalized Jacobian matrix appear, the Hopf bifurcation may take place, leading the system to change from periodic motion into the quasi-periodic oscillation, while when zero eigenvalue occurs, it may lead the system to oscillate between different equilibrium points. Combined with the numerical simulations, two typical oscillation behaviors of the system verify the theoretical results.