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Assume a dynamical system $\dot{x}=f(x)$, $x \in R^n$ (with $f$ sufficiently smooth -- see below) satisfies the following:

  1. The box $B=[-1,1]^n$ is forward invariant: any trajectory that starts in $B$ stays in $B$;

  2. The system has a finite number of equilibrium points in $B$;

  3. There is a smooth function $V(x) \ge 0$ such that $\dot{V}(x) < 0$ for all points in $B$ except the fixed points. In particular, the only periodic trajectories are the trivial ones corresponding to the fixed points.

I would like to conclude that any trajectory starting in $B$ must approach in the limit one of the fixed points.

Is this actually correct? What hypotheses on $f$ are necessary/sufficient?

This looks like a basic question, so I would be happy with just a reference to a standard text.

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I don't understand what $V(x)$ has to do with the system. – Gerry Myerson Nov 9 '10 at 22:33
@Gerry: $V$ is what's called a Lyapunov function for the system. Using that as a keyword in Google should provide lots of information. – Mariano Suárez-Alvarez Nov 9 '10 at 22:45
up vote 4 down vote accepted

The function $V$ is constant on the set of limit points of any trajectory, and this set is connected and invariant. It contains the complete forward orbit of every point it contains, so your hypothesis implies that each orbit it contains is a fixed point. Connectedness and the fact that orbits trajectires never leave your compact $B$ then let you conclude.

You can find the needed details in Perko's book Differential equations and dynamical systems.

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