Resumen de On the Vanishing Set of Inverse Integrating Factors

Hector J. Giacomini , Lucio R. Berrone

• We study in this paper $\mathcal{C}^{1}$ two-dimensional dynamical systems of the form $\; \stackrel{.}{x} = P(x,y),\; \;\stackrel{.}{y} = Q(x,y).$ We analyse the properties of the vanishing set of inverse integrating factors V, which are defined as $\mathcal{C}^{1}$ solutions of the equation \[ P\frac{\partial V}{\partial x}+Q\frac{\partial V}{\partial y}=V\limfunc{div}(P,Q) \nonumber. \] Isolated zeros of V are studied and their relationships with critical points of the system is evidenced. We show how the knowledge of an inverse integrating factor in a neighborhood of a critical point provides useful information on the local dynamics of the system. A general result is proved on vanishing of V on the separatrix curves of a saddle-point. Finally, the problem of vanishing on graphs of inverse integrating factors is discussed. It is shown that a bounded graph is contained in the vanishing set of an inverse integrating factor when the critical points of the graph are non-degenerate.