On Some Topological Properties of the Structural Graph of Dynamical Systems

Within the scope of the qualitative study of dynamical systems via graph-based methods, we extend the notion of structural graph to dynamical systems generated by actions of monoids, as abstract algebraic structures. The setting considers the action of an ordered monoid (not necessarily \(\mathbb{N},\ \mathbb{Z}\) or \(\mathbb{R}\)) on a metric space in a general sense, and therefore includes a wider class of dynamical systems than those described solely by ordinary differential or difference equations. We then investigate several fundamental properties of the structural graph, such as connectedness and degeneracy. In particular, it is shown that in the finite case, the structural graph associated to a \(C^{1}\)-vector field on a closed differential manifold is necessarily connected. As illustrative examples, we consider the classical van der Pol oscillator on the Poincaré sphere, the mathematical pendulum on the phase plane and the nonanalytic case of the center-focus. And as a perspective, we suggest work about what could be considered as a new form of linearization of dynamical systems via their structural graph, and address the question about a possible use for investigating global qualitative properties such as structural stability.