Jf. Jiang, CONVERGENCE TO TRAP ALMOST EVERYWHERE FOR FLOWS GENERATED BY COOPERATIVE AND IRREDUCIBLE VECTOR-FIELDS, Chinese annals of mathematics. Ser. B, 14(2), 1993, pp. 165-174
This paper is concerned with the asymptotic behavior of cooperative sy
stems in W subset-of R(n). For a C2 cooperative system whose Jacobian
matrices are irreducible, it is proved that the forward orbit converge
s to an equilibrium for almost every point having compact forward orbi
t closure and the set of all points which have compact forward orbit c
losure's and do not converge to a semi-asymptotically stable equilibri
um is meager in W if the equilibrium set cannot contain a simply order
ed curve. The invariant function and the geometry of the stable manifo
ld of an unstable equilibrium are considered.