An analytical approach of applying the Lyapunov direct method to polynomial differential systems with discrete time delays

Abstract

The Lyapunov direct method is a common and effective tool for discussing the global stability of dynamical systems. In this paper, we propose an analytical approach to apply this method for the global stability of the positive equilibrium of a polynomial differential system with discrete time delays, which includes how to construct an applicable Lyapunov functional and to verify the negative (semi-)definiteness of its derivative. Here the linear combination of the Volterra-type functions/functionals and their integrals plays an important role. Moreover, when the delayed system also has a boundary equilibrium, we show that the Lyapunov functional for the global stability of the positive equilibrium can be reformulated to establish the global stability of the boundary equilibrium. As an illustration, we apply this approach to a virus infection model with two delays.