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Phase portraits of a new class of integrable quadratic vector fields. (English) Zbl 0980.34022

Let \(X\) be a planar polynomial differential equation of degree \(m.\) In the papers [Proc. R. Soc. Edinb., Sect. A 124, No. 6, 1209-1229 (1994; Zbl 0821.34023) and Stud. Math. 114, No. 2, 117-126 (1995; Zbl 0826.34003)], C. J. Christopher and H. Zoladek, respectively, have proved that if \(X\) has \(q\) real or complex algebraic invariant curves, \(f_1,f_2,\ldots,f_q\) such that \(\sum_{i=i}^q\deg(f_i)=m+1,\) the curves have no critical points and they also satisfy some genericity conditions in the projective plane (see the above references for more details) then \(X\) writes as \[ X=\sum_{i=1}^q\alpha_i\left(\prod_{j=1,j\neq i}^qf_j\right)\left(-{{\partial f_i}\over {\partial y}}{{\partial }\over {\partial x}}+{{\partial f_i}\over{\partial x}}{{\partial }\over {\partial y}}\right), \] and therefore it has the Darboux first integral \(|f_1|^{\alpha_1}\ldots|f_q|^{\alpha_q}.\)
Here, the authors study the phase portraits of the above family of integrable vector fields when \(m=2\) and prove the following result: If \(X\) is a quadratic integrable polynomial vector field of the above form, then its phase portrait in the Poincaré sphere is topologically equivalent to one of a list of 46 phase portraits. These phase portraits are given in the paper. The proof is obtained from a case by case study. They split the vector fields into: vector fields having an irreducible algebraic solution of degree three, vector fields having two irreducible algebraic solutions (of degrees one and two) and vector fields having three algebraic solutions of degree one. Afterwards, the authors obtain in each case some reduced expressions of \(X\) with few parameters and finally they get their phase portraits.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37C10 Dynamics induced by flows and semiflows
34A26 Geometric methods in ordinary differential equations
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