Phase Diagram - growth model
In control systems engineering, there was the "Aizerman Conjecture" (the transliteration of Aizerman varies) that argued that a linear time-varying system was stable if the parameters (state matrices) at all times corresponded to stable systems. However, counter-examples were found to this conjecture.

About the only way to show that a time-varying system is stable (for example) is to find a Lyapunov function that holds at all times. This is generally going to be difficult.

Something like graphical analysis would be technically invalid, as you are violating assumptions that are in place. You are pretty much going to have to derive any results from first principles.