Bifurcation Analysis Of Several Differential Systems With Delay

The research on the bifurcation of delay differential equations is still a hot topic.DDEs exist widely in many fields such as natural science,engineering technology and social science.They develop the scientific study in various subjects.Bifurcation problems are an important part of nonlinear differential equations and dynamic systems.Structurally unstable system is its main investigate.The exploration of various bifurcation phenomena not only improves the relevant theories and methods in mathematics,but also plays an significant role in promoting the research in biomathematics and life sciences.In order to analyze the dynamic behaviors of several types of realistic biological models and obtain the orbital topological structure near equilibria of these systems,the bifurcation problems of three kinds of differential systems with time delay are addressed by using the central manifold theorem,the normal form theory,the Lyapunov-Schmidt reduction method,the stability theory and the bifurcation theory.Firstly,the dynamic behavior of a class of phase structure model of spruce aphid population with time delay is concerned with.In the course of the study,the existence and uniqueness of non-trivial equilibrium point are discussed,and the local stability of equilibria is decided.Moreover,some sufficient conditions for Hopf bifurcation are established.Furthermore,by virtue of the central manifold theorem and the normal form theory,the stability and direction of bifurcation periodic solution are determined.The validity of the conclusions is verified by numerical simulation.Secondly,by means of Lyapunov-Schmidt reduction method and singularity theory,the dynamic behaviors of a neutral neural network model are explored.It is proved that Hopf bifurcation occurs at the equilibrium point,and the approximate analytical expression of the periodic solution of the bifurcation is obtained,and the error analysis is carried out.Finally,a class of gene regulatory network model with two delays is considered.The existence of interior equilibrium point is dealt,and some conditions for B-T bifurcation appearing at the positive equilibrium point are given.By using the relevant theories such as universal unfolding,normal form and central manifold,the dynamic behaviors near the positive equilibrium point are transformed into the dynamic characteristics of the normal form confined to the central manifold.These results are numerically simulated and the bifurcation curves near the B-T bifurcation point are describled,and the corresponding bifurcation diagrams are displayed.The outcomes drawn in this thesis are vital for maintaining ecological balance,curbing the outbreak caused by the unrestricted growth of aphid population and providing effective control measures.They improve the dynamic properties of gene regulatory network system,deepen the understanding of gene function,and have potential significance for engineering applications,as well as offer an effective solution for some complex nonlinear systems with high or infinite dimensions when it is difficult to solve the exact solution.