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Isolated singularities of binary differential equations of degree \(n\). (English) Zbl 1282.37017

Summary: We study isolated singularities of binary differential equations of degree \(n\) which are totally real. This means that at any regular point, the associated algebraic equation of degree \(n\) has exactly \(n\) different real roots (this generalizes the so-called positive quadratic differential forms when \(n = 2\)). We introduce the concept of index for isolated singularities and generalize Poincaré-Hopf theorem and Bendixson formula. Moreover, we give a classification of phase portraits of the \(n\)-web around a generic singular point. We show that there are only three types, which generalize the Darbouxian umbilics \(D_1, D_2\) and \(D_3\).

MSC:

37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A34 Nonlinear ordinary differential equations and systems
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A60 Differential geometry of webs
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