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We consider a $C^{1,\alpha}$ smooth flow in $\mathbb{R}^n$ which is "stronglymonotone" with respect to a cone $C$ of rank $k$, a closed set that contains alinear 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 thephase space are either pseudo-ordered or convergent to equilibria. This coversthe celebrated Hirsch's Generic Convergence Theorem in the case $k=1$, yields ageneric Poincar\'{e}-Bendixson Theorem for the case $k=2$, and holds true witharbitrary dimension $k$. Our approach involves the ergodic argument using the$k$-exponential separation and the associated $k$-Lyapunov exponent (thatreduces to the first Lyapunov exponent if $k=1$).

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