Measure of stability 
Is there a measure of stability of different systems? What I mean by stability is the ability of a system to return to its original position, even when inputted with high amounts of energy. The more energy a system can handle, the more 'stable' it is. For example, in case (a), just the tiniest amount of energy would be needed to send the ball flying. Whereas for a pendulum, even large amounts of energy will still see the system return to its original state. 
I was thinking about using a measure of stability to classify different types of orbits into stable or non-stable orbits, but I couldn't find a measure of stability online. Perhaps it isn't called stability? 
 A: In control systems engineering we often speak in terms of stability margins for linear systems or otherwise nonlinear systems approximated as linear about some range of operation. A linear system can be expressed in terms of a rational polynomial function in complex frequency called a transfer function. The transfer function relates output response to input excitation. 
Instability for the linear system occurs when the denominator polynomial of the transfer function, called the characteristic function, to become singular, and since the function is expressed in terms of complex frequency, singularity can happen when one or more parameters cause the phase or magnitude of the characteristic polynomial, to go to zero. For closed loop systems, the parameter is often taken as the open loop gain or delay. But any physical parameter can cause this to happen if the structure permits it.
So for the linear system, or at least the linear model that is used to describe it, the amount of energy is not a factor in determining stability but rather the structure of the dynamics. But energy can play a role if the system is nonlinear or begins to behave in a nonlinear way when an energy level is met. Saturation is one type of nonlinearity. In nonlinear systems, stability margins are a bit more difficult to determine. Nyquist plots are one way, Lyapunov analysis another. And there are different types of stability such as asymptotic stability where the system reaches a point of equilibrium, and bounded input, bounded output stability where for an input of a bounded range, the system is output is expected to be also bounded within some range.
But if you search outside of control systems engineering you'll find that other application areas define their own version of stability. For example the stability of surface ships. For a vessel, stability is reduced to a measure called metacentric height. The metacentric height gives the naval engineer an idea of how stable the ship is in its roll axis. The ship stability is a nonlinear system, and it can be reduced to the same type of analysis as described above.
