The structure of the level surfaces of a Lyapunov function

Abstract

Let f be a real-valued Cl function which is defined on Euclidian space R”. We are interested in characterizing the noncritical level surfaces off near its isolated relative maxima and minima. The technique which is used for this investigation is to study the relationship between the trajectories of a differential equation and its Lyapunov function. As an application of interest, we obtain characterizations of the level surfaces of a Lyapunov function and of the domain of asymptotic stability of an asymptotically stable critical point. The domain of asymptotic stability is diffeomorphic to R”, and the level surfaces are manifolds (as smooth as the defining function) which are homotopically equivalent to the (n 1)-sphere 3-l. It follows from the generalized PoincarC conjecture that the level surfaces are spheres if n # 4, 5. When n = 5, the problem of whether or not the level surface is homeomorphic to the sphere is equivalent to the PoincarC conjecture. The paper concludes with a discussion of similar statements for asymptotically stable sets and nonautonomous systems.