Attractors in strongly monotone flows

Abstract

where F is a C’ vector field in R” and all the off-diagonal terms of its Jacobian matrix are nonnegative. The well-known Kamke theorem implies that the flow (4 > I Ir0 generated by (1) has the property: it preserves the partial ordering 0. Another system which has this property is the semilinear parabolic equation on a compact n-dimensional submanifold, solutions of its Dirichlet or Neumann problem determine a monotone flow in appropriate function space (see [ 11 or [ 51). In general, a flow which preserves a partial ordering on the state space is called monotone. M. W. Hirsch has studied this flow in a series of papers [14, IO]. He [2, 31 has studied the limiting behavior and convergence of solutions of system (1); he [ 1,4, lo] has investigated this flow in the considerably abstract state space. In Chapter III of Cl], he has used two sections to discuss the properties of attractors for monotone flows and obtained some very important results. He has shown that a monotone flow cannot have an attracting cycle (see [l, Corollary 2.41) and that every attractor must contain an equilibrium (see Cl, Theorem 3.11). In particular, if the flow is strongly monotone, he has shown that every attractor K contains an order-stable equilibrium (see [ 1, Theorem 4.11). Meanwhile, if we suppose z is attracted to K but is not quasiconvergent, then K contains two order-stable equilibria p, q such that p <o(z) 4 q (see Cl, Theorem 4.31). The purpose of this paper is further to discuss the properties of attractors for strongly monotone flows. We shall study when an o-limit set can be an 210 0022-247X/91 $3.00