Time-Dependent Bifurcation: A New Method and Applications

A new qualitative method is presented for studying the bifurcation problems of nonlinear systems with time-dependent parameters. The concept of stability on finite time intervals is introduced and some related stability theorems are established in order to analyze time-dependent bifurcation problems. As some applications of the new method, three different types of delayed bifurcation transitions and jump phenomena of time-dependent Duffing–van der Pol's equation are investigated qualitatively. The bifurcation transition values are predicted by constructing the V-functions or by the linear stability. The sensitivity of motion to the initial values and parameters is also discussed.