Bifurcation Analysis In Five Variables Calcium Oscillation Model

In this paper, we theoretically analyze the bifurcations in a five variables intracellular calcium oscillation model by applying the center manifold theorem and the bifurcation theory. We theoretically analyze the conditions of the occurrence of Hopf bifurcation of the equilibrium point of the system and its types. In addition, we also present numerical simulations and show that there are 3 different types of calcium oscillation in this model---periodic oscillation, quasi periodic oscillation and chaos.The thesis consists of four chapters as the following:The first chapter is an introduction:we described the background of calcium oscillations mainly from nonlinear dynamics and calcium oscillations. And a brief review of the center manifold theorem and Hopf bifurcation theorem are presented, also on the chaotic motion of a brief description.The second chapter, we study the nonlinear dynamics of a five variables calcium oscillation model which taking k3 as the bifurcation parameter, including the type and the stability of the equilibrium point of the system. The theoretical analysis shows that the equilibrium point will produce subcritical Hopf bifurcations when the bifurcation parameter k3 change. And it is the first time we got the system orbits bifurcation diagram. Numerical simulations verify the theoretical analysis and also find that the system of periodic oscillations disappears due to the occurrence of a torus bifurcation and also can produce quasi-periodic oscillation. It also finds that there are oscillations of the system itself.The third chapter, we still research the five variables calcium oscillation model. We modified some parameter values and choose k3 as the bifurcation parameter and study the classification and stability of the equilibrium point. The theoretical analysis shows that the equilibrium point will produce twice supercritical Hopf bifurcations, which can lead to periodic oscillation phenomenon. And we got the system orbits bifurcation diagram at the same time. Numerical simulations verify the theoretical analysis and also find that the system of periodic oscillations disappears due to the occurrence of a torus bifurcation and also can produce quasi-periodic oscillation. And find that the existence of chaotic phenomena in the system.The fourth chapter is a general summary of the whole thesis, and we also determine the focus of the future research at the same time.