On R. A. Smith’s autonomous convergence theorem. (English) Zbl 0841.34052

Li, Michael Y.; Muldowney, James S.

Rocky Mt. J. Math. 25, No. 1, 365-379 (1995).

The authors study the autonomous system \(x'= f(x)\) on a domain \(D\) in \(\mathbb{R}^n\). They are concerned with the \(\alpha\)- and \(\omega\)-limit sets and develop sufficient conditions that these consist only of equilibria. In addition, they give sufficient conditions that any invariant set have Hausdorff dimension \(\leq 1\).

Reviewer: A.Hausrath (Boise)

MSC:

34D05	Asymptotic properties of solutions to ordinary differential equations
34D45	Attractors of solutions to ordinary differential equations
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C30	Manifolds of solutions of ODE (MSC2000)

Keywords:

Bendixson’s criterion; Dulac criterion; center manifold; wandering points; autonomous system; \(\alpha\)- and \(\omega\)-limit sets; invariant set; Hausdorff dimension

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