Polynomial inverse integrating factors for quadratic differential systems
The authors consider the real planar quadratic polynomial system \[ \dot x = P(x,y), \quad \dot y = Q(x,y), \tag{1} \] where the dot denotes the derivative with respect to the time variable, and \(P,Q\) are quadratic polynomials. A function \(R(x,y)\) is called integrating factor of system (1), if \(R\) is a solution of the equation \(\text{div}(RP,RQ) = 0\); and a function \(V(x,y)\) is called the inverse integrating factor of (1) if \(V^{-1}\) is an integrating factor of (1), i.e., \(V\) is a solution of the equation \(PV_x + QV_y = V (P_x+Q_y)\). One of the main problems in the qualitative theory of planar polynomial vector fields is to characterize the integrable ones. In this paper, the authors characterize all the quadratic polynomial systems (1) having a polynomial inverse integrating factor of degree \(k>0\) and provide explicit normal forms for such systems and for their associated first integrals. Also they prove that these families of quadratic systems have no limit cycles.
scientific article; zbMATH DE number 5736416
scientific article