On stability of integral manifolds of dynamic systems (Q1342561)

Lyapunov's direct method for local asymptotic stability of a non-compact integral manifold \(M\) of the system: \(\dot x= f(x)\) with \(x\in {\mathfrak R}^ n\), is investigated. If locally, there exists a \(C^ 1\) function \(x\mapsto V(x)\) which vanishes on \(M\), which is bounded below and above by functions monotonically increasing from zero with the distance from \(x\) to \(M\) and which satisfies: \[ \dot V< -\alpha(d(x, M), | x|)| f(x)|, \] where \(\alpha\), \(C^ 0\) and positive, satisfies (in order to handle the non-compacity of \(M\)), \(\int^ \infty_ 0 \inf_{u\geq w}(\alpha(u, v))dv= \infty\) for any small positive \(w\), then the distance from a solution to \(M\) tends to zero for large times and appropriate initial conditions. The result is adapted and applied to a model of a consumer market which assumes that individual productivity is a function of the value of money only. The manifold of equilibria relating wages and price index is asymptotically stable.