Analysis of nonlinear systems by their linearizations

We present a result which allows us to completely analyze the behavior of a nonlinear system only based on the properties of its linearizations. More precisely, we show that the exponential stability of all the linearizations of a nonlinear system ensures suitable global properties. We point out that a nonlinear system for which the linearizations are exponentially stable, has the unique steady state property, and has a periodic (constant) motion for any periodic input. Finally through some reachable assumption of the system state space from the initial condition, we deduce that all its unperturbed motions which are associated to bounded inputs are uniformly globally asymptotically stable.