Uniform global asymptotic stability for oscillators with nonlinear damping and nonlinear restoring terms

Abstract

Second-order nonlinear differential equations with a time-varying coefficient describe various damped oscillators to explain the actual phenomena observed in numerous research areas. In actual phenomena, it is often important to immediately suppress the vibration of an object. For this purpose, the equilibrium of the oscillator needs to be uniformly globally asymptotically stable. This study examines uniform global asymptotic stability for a damped nonlinear oscillator. A first-order nonlinear differential equation, referred to as the characteristic equation, related to the second-order differential equation plays an important role in uniform global asymptotic stability for the oscillator. To be more precise, the property of particular solutions of this characteristic equation becomes a criterion for determining whether the equilibrium of the nonlinear oscillator is uniformly globally asymptotically stable. Careful mathematical analysis is used for the proof. In addition, a few examples are presented to show that the property of the particular solutions can be examined by drawing solution curves via a simple numerical simulation. Furthermore, detailed explanations are provided for the assumptions used to obtain the results. Our results clarify the appropriate damping coefficient required for the equilibrium to become uniformly globally asymptotically stable and have the advantage of being able to predict the speed at which a vibrating object returns to its original equilibrium state.