Solution of dynamical systems bounded in norm

Question

Is there a name for those non-linear dynamical systems whose solutions are not just bounded in norm but where the norms have a behaviour similar to this one? Indeed, the norms should increase for a finite time and then converge asymptotically to the zero solution. This is a sort of relaxation of asymptotic global stability of the origin, where instead of having an immediate decay in the norm we allow it to increase for a finite time interval.

Answer 1

You have to be more precise here. Lets say you consider the equilibrium point $x^{\star}=0$. Then, if for each neighbourhood $U_{\varepsilon}\ni x^{\star}$ there exists a neighbourhood $U_{\delta}\ni x^{\star}$ such that the solution $x(t)$ remains in $U_{\varepsilon}$ for any initial condition in $U_{\delta}$, then we speak of (Lyapunov) stability. If $\lim_{t\to \infty}\|x(t)\|=x^{\star}$ we generally speak of $x^{\star}$ being attractive, if $x^{\star}$ is both Lyapunov stable and attractive, we usually say that $x^{\star}$ is asymptotically stable.

Your system sketch displays typical second order behaviour and what you are looking for is most likely asymptotic stability, but to show this we need more information.

See the following paper for a very interesting discussion around these stability notions:
http://www-personal.umich.edu/~dsbaero/library/BhatTopObstrSCL2000.pdf