A new qualitative method is presented for studying the bifurcation problems of nonlinear systems with time-dependent parameters. The concept of stability on nite 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 dierent types of delayed bifurcation transitions and jump phenomena of time-dependent Dungvan der Polse quation 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.

A period adding bursting sequence without bursting-chaos in the Chay neuron model is studied by bifurcation analysis. The gent, sis of each periodic bursting is separately evoked by thc corrcsponding periodic spiking patterns through two period doubling bifurcations, except for the period-1 bursting occurring via Hopf bifurcation. Hence, it is concluded that this period adding bursting bifhrcation without chaos has a compound bifurcation structure closely related to pcriod-doubling bifurcations of periodic spiking in essence.

Complete synchronization of coupled Hindmarsh Rope (HR) neurons with a ring structure is studied. The criterion for complete synchronization of coupled neurons with the ring structure is obtained through the stability analysis of the iincarizcd synchronization error system, Numerical simulation is given to test the critcrion for a system with three coupled HR neurons.

The firing activities of Hindmarsh-Rose (HR) neurons are studied by means of numerical simulation and bifurcation analysis. A single HR neuron exhibits various firing patterns, such as quiescent state, periodic spiking, periodic bursting and chaos, when the external current input is changed. The fast/slow dynamical analysis is applied to explore the dynamical behaviour of the HR model. The complete synchronization of two coupled identical HR neurons with electrical coupling mimicking gap junctions can be realized in certain ranges of the coupling strength, whenever each individual neuron shows quiescency, periodic firing and chaos. The criteria for complete synchronization are analysed theoretically, and the corresponding numerical simulation is presented as well. The persistence of the interspike intervals bifurcation structure of the coupled HR neuronal system under electrical coupling is also discussed.