The stability and bifurcation of a van der Pol-Duffing oscillator with the delay feedback are investigated, in which the strength of feedback control is a nonlinear function of delay. A geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and Hopf bifurcations. The Hopf bifurcation curves and multi-stable regions are obtained as two parameters vary. Some weak resonant and non-resonant double Hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the "death island" regions simultaneously. By applying the center manifold theory, the normal forms near the double Hopf bifurcation points, as well as classifications of local dynamics are analyzed. Furthermore, some quasi-periodic and chaotic motions are verified in both theoretical and numerical ways.

In this paper, we are concerned with the Rose–Hindmarsh model with time delay. By applying the generalized Sturm criterion, a number of imaginary roots of the characteristic equation are classified. The absolutely stable regions for any value of time delay are detected. By the continuous software DDE-Biftool, both the Hopf bifurcation curves and double Hopf bifurcation points are illustrated in parametric spaces. The normal form and universal unfolding at double Hopf bifurcation points are considered by the center manifold method. Some examples also indicate that the corresponding unique attractor near each double Hopf point is asymptotically stable.

In neural processing information, when we consider the finite propagation speed of signals in synapses, a time delay signal self-feedback mechanism is introduced into the well-known Rose–Hindmarsh Model. The Fold–Hopf bifurcation of this model is analyzed near equilibria. To discuss the singularity of this Codim-2 bifurcation, we discuss the normal form near the Fold–Hopf points on the center manifold. Bifurcations are studied and the stability is investigated by classifying neighborhood regimes near the Fold–Hopf points. Finally, as one of the most important electrical behaviors in neurons, the complex bursting-spiking firing modes associated with bifurcations are illustrated by numerical simulations.

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method.

The effects of time delay on two neurons interaction Morris–Lecar model are investigated. It is assumed the two neurons are coupled via gap junction thus time delay arises naturally since information signals take time transmitting from pre-synaptic to post-synaptic of the neurite. It is found that time delay leads the both neurons to quiescent states at the lower reversal potential value of Ca + channel. However, the bifurcation diagram of inter-spike interval (ISI) uncovers the complex firing behavior as time delay is above a certain critical value. Finally, the complete synchronization and lag synchronization of two neurons are reported under an efficient time delay and reversal potential value of Ca + channel.