Operator-theoretic characterization of eventually monotone systems

Eventually monotone systems are dynamical systems whose solutions preserve a partial order in the initial condition after some initial transient. While monotone systems have a characterization in terms of their vector fields, eventually monotone systems have not been characterized in such an explicit manner. In order to provide a characterization, we drew inspiration from the results for linear systems, where eventually monotone (positive) systems are studied using the spectral properties of the system. We extend this spectral characterization to nonlinear systems by employing the Koopman operator framework. We also present a method to certify strong eventual monotonicity with respect to an unknown cone, a tool which to our best knowledge does not exist for monotone systems. These results are illustrated on biologically inspired numerical examples, which highlight the potential applicability of eventual monotonicity.