Cross-nonnegativity and monotonicity analysis of nonlinear dynamical systems

Let K be a closed convex cone in Rn and let F:Rn→Rn be a locally Lipschitz map, not necessarily differentiable. We revisit the problem of analyzing whether the flow generated by an autonomous dynamical system φ˙(t)=F(φ(t)) is order-preserving with respect to K. This issue has to do with the notion of cross-nonnegativity of a nonlinear map relative to a cone. We study this property in-depth and derive various characterizations of it. We deviate from the classical literature in at least two ways. First of all, K is allowed to be unpointed and nonsolid. These nontrivial relaxations are needed to cover some interesting examples arising in applications. On the other hand, taking into account the possible lack of differentiability of F, we bring the machinery of Clarke's nonsmooth analysis into the picture.