Bifurcation Of Numerical Discretization In Several Types Of Delay Differential Equation

Delay (partial) differential equations (DDE,DPDE) have been widely applied in many fields of science. In these equations, only a few special types can be solved ex-plicitly, therefore it is necessary to construct appropriate numerical methods. However, the proper numerical method that can maintain with the nature of the original system is the most practical valuable. Therefore, in this paper different numerical schemes are applied to study bifurcation in several types of delay differential equation.In this thesis, we investigate bifurcation of numerical discretization in several types of delay (partial) differential equation through finite difference and nonstandard fininte difference methods. The main results are as follows:Firstly, by a nonstandard finite-difference (NSFD) scheme we study the dynamics of the delay differential equation with unimodal feedback. Under three cases local stability of the equilibria is discussed according to Schur polynomial and Neimark-Sacker bifur-cation theory. Through the normal form method and center manifold theorem, we obtain the explicit algorithms for judging the direction of the Neimark-S acker bifurcation and the stability of the bifurcating periodic solutions. It demonstrates significant superiority of nonstandard finite-difference scheme over Euler method under the means of describing approximately the dynamics of the original system.Secondly, we consider a complex autonomously driven single limit cycle oscillator with delayed feedback. The original model is translated to a two-dimensional system by separating real and imaginary parts. Through a nonstandard finite-difference (NSFD) scheme we study the bifurcation of this resulting system. The stability of the equilib-rium of the model is investigated by analyzing the characteristic equation. In the two-dimensional discrete model, stability switches on the time delay are obtained.Thirdly, we study the dynamics of a diffusive Nicholson’s blowflies equation with a finite delay and Dirichlet boundary condition by the forward and backward Euler dif-ference schemes and Crank-Nicolson scheme. The local stability of the equilibria and the existence of Neimark-S acker bifurcation are obtained. Through analyzing the distri-bution of the eigenvalues we find that the unstable equilibrium state without dispersion may become stable with dispersion under certain conditions. Our results show that the forward Euler difference scheme requires a stronger condition than the backward Euler and Crank-Nicolson difference schemes on time and spatial stepsizes.Finally, by the backward Euler difference scheme, nonstandard finite difference scheme and Crank-Nicolson scheme, we study the dynamics of a diffusive food-limited model with a finite delay and Dirichlet boundary condition. We obtain the local stability of the equilibria and the existence of Neimark-Sacker bifurcation. Our results show that nonstandard finite difference scheme is superior to the backward Euler difference scheme under the means of describing approximately the dynamics of the original system. At the same time, Crank-Nicolson scheme is superior to nonstandard finite difference scheme. And the phenomenon that the unstable equilibrium state without dispersion may become stable with dispersion under certain conditions is found through analyzing the distribution of the eigenvalues.