An algorithm for hopf bifurcation analysis of reaction-diffusion systems with single delay and its applications

In this thesis, we analyze the existence and properties of periodic solutions of a class of systems of reaction-diffusion equations with a single delay and Neumann boundary conditions while the delay parameter changes. For this, the delay term is chosen as the bifurcation parameter and the conditions under which the Hopf bifurcation occurs in the mentioned class of reaction-diffusion systems are determined. To specify the properties of the periodic solutions whose existence guaranteed in as a result of the method described the system is reduced to the center manifold. Moreover, in order to prevent the repetition of the involved steps for different problems belonging to the same class, an algorithm is developed. This algorithm consists of the conditions and formulas that will allow the existence analysis of Hopf bifurcation to be completed only by using the coefficients of the characteristic equation of the system, and will allow the direction analysis of Hopf bifurcation to be completed merely by using the coefficients of the second degree Taylor polynomials of functions in the system. Thus, the considered class of reaction-diffusion equation systems containing a single delay is completely analyzed by this study. Hopf bifurcation analysis of four different problems belonging to this class are performed with the obtained algorithm, and thus the feasibility of the algorithm is presented. Moreover, within the scope of these four problems, effects of diffusion on the dynamics of the systems are also discussed.