In control theory, we talk about the direct and indirect Lyapunov methods, applied to the stability analysis of nonlinear systems.
There are systems that are locally unstable for specific operating points, but that are globally stable.
What i would like to know is if there is any system which is locally stable, for any linearized point, but is globally unstable.
In other words, I would like to know if a system that is guaranteed to be locally stable, thorough linearization, for any point, can be said to be globally stable as well.
In the case it is positive, I would like to know if uniform stability and exponential stability holds as well.