A geometric approach to global-stability problems. (English) Zbl 0873.34041

Li, Michael Y.; Muldowney, James S.

SIAM J. Math. Anal. 27, No. 4, 1070-1083 (1996).

An equilibrium of an autonomous differential system is said to be globally stable with respect to an open set \(D\) if it is asymptotically stable and its basin of attraction contains \(D\). The global stability problem asks to find additional conditions under which the only equilibrium in a simply connected domain is globally stable provided that it is locally stable. The paper gives such conditions using higher dimensional generalizations of the criteria of Bendixson and Dulac of planar systems and a local version of the \(C^1\) closing lemma of Pugh. An epidemiological model is studied as an application.

Reviewer: L.Hatvani (Szeged)

MSC:

34D20	Stability of solutions to ordinary differential equations
37-XX	Dynamical systems and ergodic theory
34D45	Attractors of solutions to ordinary differential equations

Keywords:

attractor; equilibrium of an autonomous differential system; globally stable; asymptotically stable; criteria of Bendixson and Dulac of planar systems; lemma of Pugh