On the Asymptotic Stability of a Class of Perturbed Ordinary Differential Equations with Weak Asymptotic Mean Reversion

In this paper we consider the global and local stability and instability of solutions of a scalar nonlinear differential equation with non–negative solutions. The differential equation is a perturbed version of a globally stable autonomous equation with unique zero equilibrium where the perturbation is additive and independent of the state. It is assumed that the restoring force is asymptotically negligible as the solution becomes large, and that the perturbation tends to zero as time becomes indefinitely large. It is shown that solutions are always locally stable, and that solutions either tend to zero or to infinity as time tends to infinity. In the case when the perturbation is integrable, the zero solution is globally asymptotically stable. If the perturbation is non–integrable, and tends to zero faster than a critical rate which depends on the strength of the restoring force, then solutions are globally stable. However, if the perturbation tends to zero more slowly than this critical rate, and the initial condition is sufficiently large, the solution tends to infinity. Moreover, for every initial condition, there exists a perturbation which tends to zero more slowly than the critical rate, for which the solution once again escapes to infinity. Some extensions to general scalar equations as well as to finite– dimensional systems are also presented, as well as global convergence results using Liapunov techniques.