In studying nonlinear systems of differential equations, unlike linear systems, it turns out that we are more interested in equilibrium points rather than general solutions themselves.
I mean, look at all these techniques, linearization, Liapunov functions, Lassalle's invariance principle, etc. All of this, only to study the behavior of the system at the equilibria, but why? What makes them so special? Is it the applications? Or is it merely our frustration in explicitly stating the general solutions of such systems that makes us tinkle our fancy with only the equilibria?