Partial integrals and the first focal value in the problem of centre

Center Problem for Cubic Differential Systems With the Line at Infinity and an Affine Real Invariant Straight Line of Total Multiplicity Four

“…The problem of coexistence in cubic systems of distinct invariant straight lines and critical points of center type was studied in [7,8,21,22,23,24,25]. In these works it was proved that if the cubic system (1) has -four non-homogeneous invariant straight lines, i.e.…”

“…In these works it was proved that if the cubic system (1) has -four non-homogeneous invariant straight lines, i.e. the lines of the form 1 + α j x + β j y = 0, j = 1, 2, 3, 4, then N 0 = 1 [7];…”

“…In the expressions for L j we neglect the denominators and non-zero constant factors. The conditions (7) are the same as in Lemma 3, (i).…”

On some open problems in planar differential systems and Hilbert’s 16th problem

“…During these last years, interesting results relating algebraic solutions and Poincaré-Liapunov constants have been published. For instance, Cozma and S ßubȃ showed in [43] that a weak focus of a polynomial system of degree m P 3 with the first Poincaré-Liapunov constant equal to zero and m(m + 1)/2 À 2 algebraic solutions has a Darboux integral or a Darboux integrating factor. Generalizing the previous result, Shubé [111] proved that if v 2k+1 = 0 for k = 1, .…”

Non-algebraic invariant curves for polynomial planar vector fields

“…These last years, several interesting results linking algebraic invariant curves and the center problem have been published. For instance, Cozma and S ¸ubȃ in [7] have proved that a weak focus of a polynomial system (1.1) of degree m ≥ 3 having the first Liapunov constant zero and m(m + 1)/2 − 2 algebraic invariant curves has a Darboux first integral or a Darboux integrating factor. Related with this result, in [3] it is showed that if a polynomial system (1.1) of degree m with an arbitrary linear part has a center and admits m(m + 1)/2 − [(m + 1)/2] algebraic invariant curves, then this system has a Darboux integrating factor.…”