On the Singularities of the Planar Cubic Polynomial Differential Systems and the Euler Jacobi Formula
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On the Singularities of the Planar Cubic Polynomial Differential Systems and the Euler Jacobi Formula Jaume Llibre1
· Claudia Valls2
Received: 4 September 2020 / Accepted: 20 October 2020 © Springer Nature Switzerland AG 2020
Abstract Using the Euler–Jacobi formula we obtain an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar cubic polynomial differential systems when these systems have nine finite singular points. Keywords Euler–Jacobi formula · Singular points · Topological index · Cubic polynomial differential systems Mathematics Subject Classification Primary 34A05; Secondary 34C05 · 37C10
1 Introduction and statement of the main results Consider in R2 the polynomial differential system x˙ = P(x, y),
y˙ = Q(x, y),
(1)
where P(x, y) and Q(x, y) are real polynomials of degree 3, called a planar cubic polynomial differential system, or simply cubic system. The motivation of our paper comes from the fact that for the planar quadratic polynomial differential systems the characterization of all configurations of the indices
B
Jaume Llibre [email protected] Claudia Valls [email protected]
1
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-00, Lisbon, Portugal 0123456789().: V,-vol
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J. Llibre, C. Valls
of the singular points of all quadratic differential systems that have four singular points is the well-known Berlinskii’s Theorem proved in [2,6], and reproved in [4] using the Euler–Jacobi formula. We say that a quadrilateral is convex if any vertex of it is contained in the convex hull of the other three vertices, otherwise the quadrilateral is called concave. Then the Berlinskii’s Theorem can be stated as follows. Assume that a real quadratic differential system has exactly four real singular points. In this case if the quadrilateral formed by these points is convex, then two opposite singular points are anti-saddles (i.e. nodes, foci or centers) and the other two are saddles. If this quadrilateral is concave, then either the three exterior vertices are saddles and the interior vertex is an anti-saddle, or the exterior vertices are anti-saddles and the interior vertex is a saddle. We want to extend the Berlinskii’s Theorem to the case of cubic systems and obtain all configurations of the singular points together with their topological indices when the cubic systems have the maximum number of finite singular points, i.e. nine singular points. By Bézout’s Theorem (see [7] for a proof of this theorem) the maximum number of singular points of a cubic polynomial differential system is nine. To these cubic polynomial differential systems (1) having nine singular points we can apply the Euler– Jacobi formula (see [1] for a proof of such formul
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