Perturbed Taylor expansion for bifurcation of solution of singularly parameterized perturbed ordinary differentia equations and differential algebraic equations

Abstract

In This paper deals with the study of singularity perturbed ordinary differential equation, and is considered the basis for obtaining the system of differential algebraic equations. In this study the we use implicit function theorem to solve for fast variable y to get a reduced model in terms of slow dynamics locally around x. It is well known that solving nonlinear algebraic equations analytical is quite difficult and numerical solution methods also face many uncertainties since nonlinear algebraic equations may have many solutions, especially around bifurcation points. We have used singularly perturbed ODE to study the bifurcation problem in Differential algebraic system. So for the first step we need to investigate the bifurcation problem in our original system when , for this purpose the known kinds bifurcations such as saddle node, transtritical and pitch fork has been studied by using Taylor expansion for one dimensional system. And for higher dimension we apply Sotomayor Theorem. The second step is going to study bifurcation problem in DAE: Where is bifurcation parameter. by converting such system to singularly perturbed ODE to make use the study in the first step: The method we used to convert DAEs to singular perturbed ODEs is PTE method. The bifurcation in index one DAEs is investigated by reduced the system to system with lower dimension by implicit function theorem. And for higher dimension index two DAEs we used Sotomayor Theorem. Also the singularity induced bifurcation for which this kind of bifurcation occurred in DAEs is studied by PTE method.