The Applications Of Bifurcation Theories For Some Models

In this thesis, we study the behaviors of some continuous systems and discretesystems. Using local bifurcation theorem, we proof that there exists a sequence ofHopf bifurcations for Physiological Control systems. And from Center Manifold The-ory and theory of normal form we get the explicit expression directions of bifurcation.Meanwhile, we obtain the stability of bifurcation periodic solutions. We proof thatthe local periodic solution implies that the global Hopf bifurcation solutions. As fornonlinear integro-differential systems with two delays, we deduce the sufficiency ofconditions of existing of bifurcation.In the following, our work are major in studying discrete system of Nicholson'sblowflies. At first, we discrete the system with forward Euler methods, and thenknow that there exists a sequence of Hopf bifurcations with the theories of discretelocal Hopf bifurcation. Last, we give the explicit expression of these and stability ofbifurcation periodic solutions.