Polynomial Vector Fields on Algebraic Surfaces of Revolution
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Results in Mathematics
Polynomial Vector Fields on Algebraic Surfaces of Revolution Fabio Scalco Dias
and Luis Fernando Mello
Abstract. In the first part of the article we study polynomial vector fields of arbitrary degree in R3 having an algebraic surface of revolution invariant by their flows. In the second part, we restrict our attention to an important case where the algebraic surface of revolution is a cubic surface. We characterize all the possible configurations of invariant meridians and parallels that the vector fields can exhibit. Additionally we shall consider when the invariant parallels can be limit cycles. The results obtained in the second part can be adapted to the general surfaces studied in the first part. Mathematics Subject Classification. Primary 34C07, 34C05, 34C40. Keywords. Polynomial vector field, invariant parallel, invariant meridian, limit cycle, algebraic surface of revolution.
1. Introduction and Statement of the Main Results In the Qualitative Theory of Ordinary Differential Equations great interest is given to the study of invariant algebraic curves of polynomial vector fields in the plane since the pioneering work of Darboux [2]. Considerable attention is paid to the case where an invariant algebraic curve is a limit cycle, especially the problem about the maximum number of such curves in terms of the degree of the polynomial vector field. In Differential Geometry the study of curves on surfaces is traditional even in the particular case when the surface is of revolution. For instance, the parallels and meridians of a surface of revolution are lines of curvature and the great circles, in particular the meridians, are geodesics of a sphere [5]. Surfaces 0123456789().: V,-vol
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F. S. Dias and L. F. Mello
Results Math
of revolution are well known in mathematics and are used as models in many applications, such as in geometric modeling, mainly due to their symmetries. In this article we put together these two ingredients by studying polynomial vector fields of arbitrary degree in R3 having an algebraic surface of revolution invariant by their flows. Denote by K[x, y, z] the ring of the polynomials in the variables x, y and z with coefficients in K = R or K = C. By definition a polynomial differential system in R3 is a system of the form dx dy dz = P1 (x, y, z), = P2 (x, y, z), = P3 (x, y, z), (1) dt dt dt where Pi ∈ R[x, y, z] for i = 1, 2, 3 and t is the independent variable. We denote by (2) X (x, y, z) = (P1 (x, y, z), P2 (x, y, z), P3 (x, y, z)) the polynomial vector field associated to system (1). We say that m = max{mi }, where mi is the degree of Pi , i = 1, 2, 3, is the degree of the polynomial differential system (1) or the polynomial vector field (2). An invariant algebraic surface for system (1) or for the vector field (2) is an algebraic surface h−1 (0) with h ∈ R[x, y, z], such that for some polynomial K ∈ R[x, y, z] we have X h = Kh. This implies that if a solution curve of system (1) has a point on the algebraic surface h−1 (0), then the whole solution curve is containe
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