Hopf Bifurcation And Numerical Analysis Of Delay Differential Systems

This paper deals with the stability of the equilibrium point, Hopf bifurcation andT-B singularity of the delay differential system, and numerical discrete system producedby numerical method applying to the original system.Although the delay differential equations(DDEs) have been widely used in scienceand engineering fields, most DDEs arising in practice are nonlinear and can not be solvedexplicitly. Numerical calculation therefore become one of important methods to studythe properties of dynamics. In the numerical calculation, it is interesting to investigatewhether the corresponding numerical discrete systems preserve the dynamical behaviorof the original systems.A class of d dimensional delay differential system with parameter is considered. Ap-plying the strictly zero-stable linear multistep methods to the system, the correspondingnumerical discrete system is obtained. It is proved that if the original system undergoes aHopf bifurcation, then the corresponding numerical discrete system undergoes a numeri-cal Hopf bifurcation (Neimark-Sacker bifurcation). It is shown that the direction and thestability of the invariant curve of the numerical discrete system are the same as that of theoriginal system. Some numerical tests are given.A predator-prey model with two delays is considered. The numerical discrete systemproduced by the Runge-Kutta methods is obtained. The stability of the positive equilib-rium point and the properties of the numerical Hopf bifurcation are investigated. Thedirection of the bifurcation and the stability of the invariant curve are studied. Someexperiments are given to illustrate the analytical results found.A two-neuron network system with two delays is considered. The stability of itsequilibrium point and existence of the Hopf bifurcation is discussed. Some sufficientconditions for stability and instability are obtained, and a sufficient condition for the exis-tence of the Hopf bifurcation is given. Some numerical examples illustrate the correctnessof the conclusions.A second-order differential equations with multi-delays is considered. The stabilityof its equilibrium point is investigated. Some sufficient conditions for stability, instability and the existence of Hopf bifurcation are given. At last, the T-B singularity of a specificequation is discussed. Some properties of the numerical discrete system produced by theexplicit Euler method of the original system are also studied.