What is the proper definition for the Asymptotic stability of a scalar autonomous equation. For eg. $x'=f(x),$ where $x \in \mathbb{R},$ (say) the equilibrium is at a point $x_0.$ So $f(x_0)=0.$ And how do we go and determine whether it is asymptotically stable or not ?
If it's a system you linearize the system, evaluate the Jacobian at the critical points and find the eigen values of the matrix. All eigen values having a negative real part is the criterion for asymptotic stability. Do we have to come up with a proper Lyapunov function to achieve this ? Any help in understanding this is much appreciated.