Dynamical Systems, Vector Fields and Commutators

We consider linear and nonlinear autonomous systems of first order ordinary differential equations and their corresponding vector fields. Taking the commutator of such two vector fields provides a new vector field and thus a new autonomous system of first order differential equations. We discuss how the properties of the input vector fields such as fixed points, periodic orbits, limit cycles, chaotic behaviour, first integrals, gradient systems, Hamilton systems etc reflect in the dynamical system of the vector field given by the commutator of the two input vector fields.