Some Bifurcation Problems For Several Systems Of Ordinary Differential Equations

The bifurcation problem of ordinary differential equation is a problem extensively considered.The main aim of this thesis is to state the bifurcation problem of selected systems of ordinary differential equations,including the saddle node bifurcation,transcritical bifurcation,Hopf bifurcation and center of manifold theory.Continuous dynamical systems consisting of differential equations comprise many parameters.A small fluctuation in a parameter may often significantly influence in many ways.This work appears to be mainly concerned with: How to maintain the equilibrium properties of this system and constant parameter orbits of dynamical systems;How to calculate the stable limits and cycles of the spatial parameter;How to imagine qualitative behaviour changes in systems at some equilibrium points.In the thesis,the categorization of bifurcation in an equilibrium or a periodic orbit is substantially covered.When a parameter varies,some properties of the system can shift.An equilibrium could go from being stable to unstable;A limit cycle can raise or,even a new stable equilibrium can also be found to alter the previous equilibrium’s stability.First,we consider a particular system of differential equations and check the stability and Hopf bifurcation theorem through use of Hopf bifurcation theorem.Then we study another system of differential equations to determine its stability,the direction of stability and period of bifurcated periodic solution at the critical value using the centre manifold reduction and normal form method.