Can a Lyapunov function be used to show/prove a system is unstable at an equilibrium point similar to the way it can be used to show asymp stability?

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Generally no. A Lyapunov function can only be used to show the stability of a fixed point. However there are some cases when the equilibrium point is a "source", i.e. a fixed point that is asymptotically stable if we run the time backwards. In this case, we can use Lyapunov functions since we can modify the system by reversing time and search for a Lyapunov function for the fixed point of the new system. -Buzi 16:30, 6 October 2008 (PDT)