Complex Dynamics Analysis In Houart-Dupont Calcium Oscillations Models

In this thesis, we study the nonlinear dynamics of the Houart-Dupont calcium oscillation model by applying theories in dynamical systems and feedback dynamical systems theory, including the existence, type and stability, bifurcation of the equilibrium points. We prove that the system owns Hopf bifurcations by using center manifold theory, the time domain and frequency domain bifurcation theory, which cause phenomena of the calcium oscillation. We also find one way to control the amplitude of limit cycle generated by the Hopf bifurcation. In additional, we present the numerical simulation results to verity our theoretical analysis and to display new phenomena.The thesis consists of five chapters as the following.In chapterⅠ, we introduce the nonlinear dynamical system and calcium oscillations reaction system.In chapterⅡ, a brief review of dynamical systems theory are presented, including the center manifold theorem, the local bifurcation theory, Hopf bifurcation theory in frequency domain and stability criterion of limit cycle.In chapterⅡ, we study the nonlinear dynamics of the Houart-Dupont calcium oscillation model, including the stability, classification and bifurcation of equilibrium points. The results for the model show that supercritical Hopf bifurcations play very important roles in the calcium oscillation. Numerical simulations confirm the theoretical analysis results. By combining the existing numerical results with the theoretical analysis results in this paper, a complete description of the dynamics of the Houart-Dupont calcium oscillation model is obtained.In chapterⅣ, we still study the Houart-Dupont calcium oscillation model, change the parameter value and analysis of the type of equilibrium point and stability. We state that the system exist two Hopf points by using Hopf bifurcation theory in frequency domain. We obtained the approximate analytical expression of limit cycle generated by the Hopf bifurcation, estimated the value of the frequency and amplitude scale of the limit cycle by using second-order harmonic balance approximation method, and determined the stability of the limit cycle. Based on the above, we controlled the limit cycles' amplitude.In chapterⅤ, we summarize the whole thesis, and made some prospects for the future researches.