The Bifurcational Consistency Of Numerical Methods For Some Kinds Of Delay Differential Equations

In the past several decades, delay differential equations have been widely appliedin many fields of science, such as in modern physics, biology, medicine, economics, de-mography, chemical reaction engineering, the theory of automatic control etc. Since theseequations can only be solved explicitly in some special cases, it is necessary to developsome appropriate numerical methods. However, a numerical method is valuable only ifthe method can re?ect exactly the property of the original system. Hence, both in the the-ory and in the applications, it is great significant to study whether the numerical methodscan preserve the dynamical behavior of the original system.In this dissertation, the bifurcational consistency of some numerical methods forcertain kinds of delay differential equations is studied, that is, whether the numericalmethods could preserve the bifurcation of the original systems is researched.Firstly, for a class of second order delay differential equations with negative feed-back, the dynamical behavior of the numerical discrete system derived by a differencemethod is investigated. The sufficient conditions under which the Neimark-Sacker bi-furcation exists are derived by analyzing the moving of the characteristic roots withthe changing of the delay parameter and using the Neimark-Sacker bifurcation theorem.Meanwhile, the explicit expressions of determining the direction of the bifurcation and thestability of the closed invariant curve are given by using the normal form theory and thecenter manifold theorem. Through comparing the bifurcation of the origin system withthe numerical discrete system, it is showed that the difference method is bifurcationallyconsistent for the second order delay differential equations.Secondly, for a delay differential equation constructed by M.C. Mackey and L.Glass, which characterizes the regulation of the density of mature cells in blood circu-lation, the bifurcational consistency of a non-standard finite difference method is consid-ered. It follows the similar way as in the above problem. The stability of the positivefixed point of the numerical discrete system is analyzed. The conditions which guaranteethe existence of the Neimark-Sacker bifurcation are given. The explicit expressions fordetermining the direction of the bifurcation and the stability of the closed invariant curveare obtained. Thirdly, for an arterial carbon dioxide control system, a delay differential equation,the Midpoint rule is applied to solve the numerical solutions. It is analyzed the stabilityof the positive fixed point of the numerical discrete system. The conditions under whichthe discrete system undergoes a Neimark-Sacker bifurcation are given. The explicit ex-pressions for determining the direction of the bifurcation and the stability of the closedinvariant curve are obtained. It is illustrated that the Midpoint rule for the system is bi-furcationally consistent, by comparing the results with the dynamical behaviors of thecontrol system.At last, a class of Runge-Kutta methods for some general delay differential equationsare studied. It is proved by employing the implicit function theorem that the Runge-Kuttamethods are bifurcationally consistent for the equations which undergo Hopf bifurcation,and that the Neimark-Sacker bifurcation point converges to the Hopf bifurcation pointwith order p if the Runge-Kutta method is of order p. To illustrate the correctness of theresults, the 2-stage Gauss method is used to solve the delay Logistic equation and it isseen that the Neimark-Sacker bifurcation point converges to the Hopf bifurcation pointwith order 4.Moreover, after the theoretical proof in every chapter, some numerical examples aregiven which illustrate the correctness of the theoretical results.