Stability And Bifurcation Analysis Of Several Nonlinear Delayed Differential Equations

Delayed diferential equations are widely used in many fields since it can describethe natural phenomena much better than ordinary diferential equations. Meanwhile, bi-furcation means that for a system with unstable structure, when a certain parameter varies,the topological structure of the solutions changes, which is very common in our dailylives. So here we study the bifurcation of delayed diferential equations.Fixed point bifurcation means the number or the stability of the equilibria change asthe parameter passes through critical value. Hopf bifurcation means that a small ampli-tude periodic solution comes out with the changes of stability for equilibrium point as acertain parameter varies and passes through a critical value. Since the appearance of theHopf bifurcation always accompanies with the change of the stability of the equilibria,so it is inevitable to discuss the stability of the equilibria of the equation when we studybifurcations firstly, and then we study the properties of the bifurcations including the pat-terns of the fixed point bifurcation, the direction of Hopf bifurcation and the stability ofthe bifurcating periodic solutions.We mainly study four practical problems, including a unidirectionally coupled sys-tem with a nonlinear function of general form, delayed Rosenweig-MacArthur modelwith constant rate prey immigration, a delayed-feedback optical loop system with Mach-Zehnder modulator and coupled Lang-Kobayashi rate equation. By analysing the dis-tribution of the characteristic roots of the linearized equation and using the method ofasymptotic semi-flow of limit equation, we study the local stability of the equilibria andget the conditions under which fixed point bifurcation and Hopf bifurcation occurs. Then,using Lyapunov function and Lassel invariant set principle we investigate the global sta-bility of the equilibria. Basing on the Hopf bifurcation theorem and the coincide degreecontinuation theorem, we prove the existence of the periodic solutions. Furthermore, wecalculate the normal form of the fixed point bifurcation and the Hopf bifurcation on thecenter manifold by the method from Faria and Hassard. Finally, we verify the globalexistence of the periodic solutions by global Hopf bifurcation theorem of Wu.