Generic behavior of flows strongly monotone with respect to high-rank cones

Abstract

We consider a C1,α smooth flow in Rd which is “strongly monotone” with respect to a cone C of rank k, a closed set that contains a linear subspace of dimension k and no linear subspaces of higher dimension. We prove that orbits with initial data from an open and dense subset of the phase space are either pseudo-ordered or convergent to equilibria. This covers the celebrated Hirsch's Generic Convergence Theorem in the case k=1, yields a generic Poincaré-Bendixson Theorem for the case k=2, and holds true with arbitrary dimension k. Our approach involves the ergodic argument using the k-exponential separation and the associated k-Lyapunov exponent (that reduces to the first Lyapunov exponent if k=1).