Abstract
We study in this paperC 1 two-dimensional dynamical systems of the formx=P(x,y), y=Q(x,y). We analyse the properties of the vanishing set of inverse integrating factors V, which are defined asC 1 solutions of the equation\(P\frac{{\partial V}}{{\partial x}} + Q\frac{{\partial V}}{{\partial y}} = Vdiv(P,Q)\). 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.
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Berrone, L.R., Giacomini, H.J. On the vanishing set of inverse integrating factors. Qual. Th. Dyn. Syst 1, 211–230 (2000). https://doi.org/10.1007/BF02969478
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DOI: https://doi.org/10.1007/BF02969478