Stabilization in a gradient system with a conservation law

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 \[ {\dot u_i} = F’({u_i}) - \sum \limits _j {{\mu _j}F’({u_j}) = 0} ,\] 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.