Narrow Results

We study the stability and Hopf bifurcation of a neutral functional differential equation (NFDE) which is transformed from an amplitude equation with multiple-delay feedback control. By analyzing the distribution of the eigenvalues, the stability and existence of Hopf bifurcation are obtained. Furthermore, the direction and stability of the Hopf bifurcation are determined by using the center manifold and normal form theories for NFDEs. Finally, we carry out some numerical simulations to illustrate the results.

A predator–prey system with neutral delay is investigated from the viewpoint of bifurcation analysis on neutral delay differential equations. Stability and Hopf bifurcation of the inner equilibrium are given, by which we show how the neutral terms affect the dynamical behavior of the prey and the predator. To give more detailed information on the periodic oscillations, the direction and stability of Hopf bifurcation are studied by using the normal form theory of neutral equation. We find neutral delay makes the predator–prey system more complicated and usually induces stability switches or double Hopf bifurcations. Near the double Hopf bifurcation we give the detailed bifurcation set by calculating the universal unfoldings. It is shown that the population of prey or predator may exhibit transient quasiperiodic oscillations driven by the neutral delay. Finally, we carry out several groups of illustrations.