Abstract
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.
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