Systems of ordinary differential equations which generate an order preserving flow. A survey of results.

*(English)*Zbl 0674.34012From the author’s summary: “This article consists of a survey of results concerning the qualitative behaviour of solutions of systems of ordinary differential equations which generate an order preserving flow. We restrict our consideration to partial orderings on \(R^ n\) induced by any one of its orthants; a flow preserves ordering if any two solutions x(t) and y(t) are ordered, x(t)\(\leq y(t)\), for all \(t>0\) whenever x(0)\(\leq y(0)\). Many of the important results for such systems have only recently been obtained, principally by M. W. Hirsch, who pointed out the tendency of their solutions to converge to equilibrium. Less well known are some global geometric constraints on the stable manifold of an equilibrium and the existence of heteroclinic orbits connecting ordered equilibria. A particularly striking result for this class of systems is the easily computable necessary and sufficient condition for stability of an equilibrium.

One of our main goals is to show that by allowing partial orderings on \(R^ n\) generated by orthants other than the positive one, the usual restrictive Kamke (quasimonotone) condition (all “off-diagonal” feedbacks are positive) which results from the standard ordering is modified in such a way as to allow (selectively) some negative feedback. As a consequence, there are many interesting and nontrivial applications of the theory.”

One of our main goals is to show that by allowing partial orderings on \(R^ n\) generated by orthants other than the positive one, the usual restrictive Kamke (quasimonotone) condition (all “off-diagonal” feedbacks are positive) which results from the standard ordering is modified in such a way as to allow (selectively) some negative feedback. As a consequence, there are many interesting and nontrivial applications of the theory.”

Reviewer: J.O.C.Ezeilo