How do we choose epsilon in the definition of stability?

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Q How do we choose epsilon in the definition of stability (slide 6)?

A The answer is really easy: if we can choose epsilon arbitrarily small, then stability of the equilibrium point is verified. It means that once I pick such epsilon, I can always find an initial condition that will guarantee that the trajectory x(t) (solution of the ODE describing my dynamical system of interest) will stay close to the equilibrium x_e, in a volume somehow delimited by the epsilon I chose (i.e. arbitrarily close to the equilibrium).

To verify if your equilibrium point is asymptotically stable, then according to the slide you are actually looking for an epsilon that now delimits the distance between the initial condition and your equilibrium point. If as time increases the solution tends to the equilibrium, then lim_{t->inf}|| x-x_e|| =0 and the equilibrium is asymptotically stable.

Another issue is how big epsilon can be. This depends on the specific system: if epsilon can also be arbitrarily large, one talks about global stability (asymptotic stability) of the equilibrium. If there are bounds on how big the epsilon can be, then stability (asymptotic stability) is a local property.