Solution of the center-focus problem for a nine-parameter cubic system

Nondegenerate centers for Abel polynomial differential equations of second kind

“…The first goal of this paper is to study system (7) with f 2 = 0 and arbitrary values of c 0 and d 0 , see Theorem 1 below. In fact the particular case f 2 = d 1 = 0 becomes a subfamily of the system studied in [27]. However for completeness present here the discussion for each case found.…”

“…Note that system (27) has the invariant curves f 1 = 1 − 2f 2 y/3 and the inverse integrating factor V = f 3 1 . So, system (27) has a center at the origin.…”

Center conditions for a polynomial differential system

“…We use the Cherkas method and the Lüroth theorem [1; 9; 10, p. 259] to study the center-focus problem for system (1) IV. Q(x)R(x) = 0, R 3 (x) Q 5 (x) = const , P 0 (x)S(x)/R 2 (x) ≡ const .…”

“…where A, B, C, D, H, K, L, M, N, P, Q ∈ C.For H = Q = 0, the solution of the center-focus problem can be found in[1].Letω 1 (x) = Nx 5 (H + Qx) 3 − x 3 (C + Mx)(H + Qx) 2 (1 + Dx + P x 2 ) + 3x 2 (B + Lx)(H + Qx)(1 + Dx + P x 2 ) 2 + (1 − Ax − Kx 2 )(1 + Dx + P x 2 ) 3 , ω 2 (x) = Nx 3 (H + Qx) 2 − x(2C + D + 2(M + P )x)(H + Qx)(1 + Dx + P x 2 )/3 + ((3B + 2H)/3 + (L + Q)x)(1 + Dx + P x 2 ) 2 .…”

Center conditions for generalized polynomial Kukles systems

“…In [24] a straightforward generalization of the Kukles systems is studied and it is given necessary and sufficient conditions to have a center at the origin for the system of the forṁ…”