A sufficient condition for the super-linearization of polynomial systems

We provide in this paper a sufficient condition for a polynomial dynamical system ẋ(t)=f(x(t)) to be super-linearizable, i.e., to be such that all its trajectories are linear projections of the trajectories of a linear dynamical system. The condition is expressed in terms of the hereby introduced weighted dependency graph G, whose nodes v_i correspond to variables x_i and edges v_iv_j have weights [Formula presented]. We show that if the product of the edge weights along any cycle in G is a constant, then the system is super-linearizable. The proof is constructive, and we provide an algorithm to obtain super-linearizations and illustrate it on an example. Our result also provides a partial answer to an open question about polyflows.