Bifurcation of periodic responses in forced dynamic nonlinear circuits: Computation of bifurcation values of the system parameters

In the computer-aided experimental analysis of dynamic nonlinear circuits, the determination of the bifurcation value of system parameters for various types of periodic response is one of the central problems. A bifurcation diagram composed by the sets of bifurcation values exhibits various nonlinear phenomena, such as the coexistence of several periodic responses which are correlated with the jump and hysteresis behaviors, the frequency entrainment, the appearance of quasi-periodic responses and chaotic states, etc. In engineering application the bifurcation diagram can be used for designing dynamic nonlinear circuit with prescribed characteristics. In this paper, we present some computational algorithms which determine the bifurcation values of periodic responses. Our algorithms are based on the geometric approach of ordinary differential equations. Newton's method is effectively used for finding the bifurcation value. The Jacobian matrix is evaluated by the solutions of variational equations. Numerical examples for the second-order systems are illustrated. Some global properties of bifurcation sets of periodic responses are discussed.