Global Stability Of Differential Equations With Piecewise Continuous Arguments And The Convergence Of Exponential Runge-Kutta

This paper deals with the global stability of analytic and numerical solutions of a class of differential equations with piecewise continuous arguments (EPCA), which are widely used to describe population models. The analysis of the global properties is of important theoretical value and practical significance. Some fundamental concepts such as Stability, Oscillation, Global Stability, Periodic solutions and Boundedness are reviewed in the first part.In the second part we investigate the global stability conditions of EPCA with several piecewise constant arguments in both constant coefficients and variable coefficients cases. The global stability conditions are really improved Runge-Kutta methods in the exponential form is applied to solve this class of corresponding ordinary differential equations. The convergence order of the exponential explicit Euler method, the exponential implicit Euler method and the exponential midpoint method is investigated.The last part we apply the exponential Runge-Kutta methods to the class of EPCA models. It is shown that the exponential explicit Euler method, the exponential implicit Euler method and the exponential midpoint method preserve their original convergence order. We also do some experiments to illustrate whether the methods preserve the global stability of the models.