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Stabilization in a gradient system with a conservation law

Author: Robert L. Pego
Journal: Proc. Amer. Math. Soc. 114 (1992), 1017-1024
MSC: Primary 34D05; Secondary 34C30, 58F12
MathSciNet review: 1086340
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Abstract: Suppose $ \sum {{\mu _j} = 1} $ and $ F:{\mathbf{R}} \mapsto {\mathbf{R}}$ is $ {C^1}$ with $ F'$ piecewise $ {C^1}$. For the finite system of ordinary differential equations

$\displaystyle {\dot u_i} = F'({u_i}) - \sum\limits_j {{\mu _j}F'({u_j}) = 0} ,$

I prove that every bounded solution stabilizes to some equilibrium as $ t \to \infty $. For this system, $ \sum {{\mu _j}{u_j}} $ is conserved and the quantity $ \sum {{\mu _j}F({u_j})} $ is nonincreasing and serves as a Lyapunov function, but the set of equilibria can be connected and degenerate. Essential use is made of a result related to one of Hale and Massat that an $ \omega $-limit set that lies in a $ {C^1}$ hyperbolic manifold of equilibria must be a singleton.

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Article copyright: © Copyright 1992 American Mathematical Society

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