Numerical dynamics of a nonstandard finite difference method for a class of delay differential equations
A nonstandard finite difference (NSFD) method for a class of delay differential equations is proposed. The method has some remarkable properties for any step size \(h=1/m\) such as: preservation of the qualitative behavior that includes the local stability of the equilibrium, the presence of Hopf bifurcations and their directions, and the stability of the closed invariant curves representing periodic solutions generated by bifurcation. The NSFD method applied to the delay differential equation provides a numerical discrete equation. The existence and the local stability of the Neimark-Sacker bifurcation at the fixed point for the numerical discrete equation is analysed. Moreover, an explicit algorithm for determining the direction of the Neimark-Sacker bifurcation is presented. In order to illustrate the effectiveness of the NSFD method, some numerical tests for a model of the survival of red blood cells in an animal are given, including a comparison with the Euler implicit method.
scientific article; zbMATH DE number 6138101