Is My System of ODEs k-Cooperative?

A linear dynamical system is called positive if its flow maps the non-negative orthant to itself. More precisely, it maps the set of vectors with zero sign variations to itself. A linear dynamical system is called k-positive if its flow maps the set of vectors with up to k -1 sign variations to itself. A nonlinear dynamical system is called k-cooperative if its variational system, which is a time-varying linear dynamical system, is k-positive. These systems have special asymptotic properties. For example, it was recently shown that strong 2-cooperative systems satisfy a strong Poincaré-Bendixson property. Positivity and k-positivity are easy to verify in terms of the sign-pattern of the matrix in the dynamics. However, these sign conditions are not invariant under a coordinate transformation. A natural question is to determine if a given n-dimensional system is k-positive up to a coordinate transformation. We study this problem for two special kinds of transformations: permutations and scaling by a signature matrix. For any n ≥ 4 and k ε {2, ...,n - 21}, we provide a graph-theoretic necessary and sufficient condition for k-positivity up to such coordinate transformations. We describe an application of our results to a specific class of Lotka-Volterra systems.