Uniform global asymptotic stability for nonautonomous nonlinear dynamical systems

Abstract

Research on uniform global asymptotic stability has the advantage of being able to predict the asymptotic velocity to the origin for all solutions of equations describing nonlinear phenomena. This study elucidates the sufficient conditions under which the zero solution of a nonautonomous nonlinear dynamical system of second order is uniformly globally asymptotically stable. From the results obtained, a certain first-order nonlinear differential equation associated with the second-order dynamical system plays a vital role in uniform global asymptotic stability. More precisely, under several reasonable assumptions, if the integral from σ to t+σ of a particular solution of the first-order differential equation diverges uniformly with respect to σ, then the zero solution of the dynamical system is guaranteed to be uniformly globally asymptotically stable. For the proof, the behavior of the solutions of the dynamical system is carefully tracked. An example that also includes a linear case is provided to illustrate the main result. Simulations are also presented to facilitate understanding of the example, and a local theorem corresponding to the main result and its application to an ecosystem model are discussed.