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?

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    $\begingroup$ Lots of nonlinear problems don't have exact solutions.. $\endgroup$ – mattos Jan 21 '15 at 12:45
  • $\begingroup$ non linear equations are just...*too difficult!* $\endgroup$ – marco trevi Jan 21 '15 at 13:53
  • $\begingroup$ I guess Poincaré would be the best person to answer. I think he was the person who started to not care about explicit solution, but to look at the behaviour. Indeed, the main problem is that nonlinear systems are rarely solvable. Still, we do not quit in trying to understand them. So the next best thing to do is to study a simplified and local version of the nonlinear problem. For this, we may use the flow-box theorem away from equilibria, and so, it remains to study what happens near equilibria. $\endgroup$ – PepeToro Feb 4 '15 at 6:34

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