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I've just begun studying nonlinear systems in my spare time. I'm using 'Nonlinear System Theory' by Rugh. My question is if there is a universal way to find the Lyapunov function of an arbitrary system. It doesn't have to be the easiest way, just a universal method that always works.

This is merely a question of curiosity.

Also, if anyone has any suggestion for good self-study books for nonlinear systems, feel free to mention them.

Perhaps I should rephrase: if a Lyapunov function exists, is there a universal way to find it that usually works. Existence is important.

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    $\begingroup$ "Nonlinear Dynamics and Chaos" by Strogatz is a very classic book. It has very intuitive explanations and easy to read. amazon.com/Nonlinear-Dynamics-And-Chaos-Applications/dp/… $\endgroup$ – KittyL Jun 2 '15 at 15:34
  • $\begingroup$ "My question is if there is a universal way to find the Lyapunov function of an arbitrary system." Unfortunately, no. Plenty of examples on math.se, say this one. $\endgroup$ – Did Jun 2 '15 at 16:07
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If you could always find a Lyapunov function you could always decide questions of stability. That's too much to hope for: there are some problems for which stability is very hard to decide.

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  • $\begingroup$ What if the system is known to be stable, or certain regions are known to be stable, etc.? What about then? $\endgroup$ – Desperate Fluffy Jun 2 '15 at 16:24
  • $\begingroup$ In principle I think (under appropriate assumptions) the existence of a Lyapunov function was proved by J.L. Massera. I don't know whether this produces a very explicit Lyapunov function. $\endgroup$ – Robert Israel Jun 2 '15 at 19:09

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