Analytical Integrability of Perturbations of Quadratic Systems
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Analytical Integrability of Perturbations of Quadratic Systems Antonio Algaba, Crist´obal Garc´ıa and Manuel Reyes Abstract. We consider analytic perturbations of quadratic homogeneous differential systems having an isolated singularity at the origin. We characterize the systems with an analytic first integral at the origin. We apply the results to two families of degenerate vector fields. Mathematics Subject Classification. 34C20, 34C14.
1. Introduction Given the homogeneous polynomial planar differential system (x, ˙ y) ˙ T = Fn (x, y) = (Pn (x, y), Qn (x, y))T , with Pn and Qn homogeneous polynomials of degree n and whose origin is an isolated singular point, it is interesting to know whether x˙ = Fn + h.o.t., analytic perturbations of Fn , has an analytic first integral at the origin. For n = 1, i.e., when the linear part F1 of the vector field is non-zero, if λ1 and λ2 are the eigenvalues of D(F1 )(0)4, it has the following cases (assuming that the origin is an isolated singular point of F1 ): if λ1 λ2 = 0, the origin is either a saddle, or node or a non-degenerate monodromic singular point (with imaginary eigenvalues). The nodes are not analytically integrable. A non-degenerate monodromic point is analytically integrable if, and only if, the system is orbitally equivalent to (−y, x)T , and a resonant saddle has an analytic first integral around the singular point if, and only if, the system is orbitally equivalent to (px, −qy)T with p, q ∈ N [16,20]. The most studied systems whose origin is a resonant saddle are the Lotka–Volterra systems, see [11–13,15,17,18] and references therein. Therefore, the analytical integrability problem for the vector fields whose origin is an isolated singular point with non-zero linear part is solved theoretically in the following sense: 0123456789().: V,-vol
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Theorem 1.1 [16,20]. The analytic vector field F = F1 + h.o.t., with F1 analytic integrable, is analytically integrable if, and only if, F and F1 are orbitally equivalent. For n = 3, Algaba et al. [5] have proved that Theorem 1.1 is true for the perturbations of cubic Kolmogorov systems. However, in general, Theorem 1.1 is not satisfied for vector fields with zero linear part (the origin is a degenerate singular point). For example, the vector field F = F4 + F5 with 4xy 3 − x4 −3x3 y 2 F4 = , F5 = , 4x3 y − y 4 3x2 y 3 it is a Hamiltonian vector field whose Hamiltonian function is a polynomial and, therefore, is analytically integrable and Algaba et al. [5, Theorem 3.20] prove that is non-orbitally equivalent to its leader term F4 . For vector fields whose origin is a degenerate singular point, the problem remains open. We know only some partial results. The analytical integrability problem when the first quasi-homogeneous component is conservative whose Hamiltonian function h has only simple factors is solved in [2]. A particular case with h having multiple factors is studied in [1]. Recently, Algaba et al. [3] have solved the analytic integrability problem around a nilpotent singulari
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