Solution of the center-focus problem for a cubic system reducible to a lienard system

Center Conditions for Polynomial Liénard Systems

“…These type of symmetry is called generalized symmetry and it has been applied to other more general systems, see for instance [4,6] These results permit to check if a given system has a center at the origin. Moreover several systems that can be transformed to Liénard systems via Cherkas transformation have been studied using these results, see for instance [1,3]. However, in practice from the conditions obtained it is not easy to get the explicit form of the families with center of certain family of Liénard systems with several parameters even for systems of small degree.…”

On Some Reversible Cubic Systems

“…(b) Consider now system (9). Applying transformation (19) and equating the obtained system with (6) we can calculate k and then again we have two cases:…”

“…The only family completely investigated is the quadratic one (n = 2) [3][4][5][6][7][8]. Some partial results have been obtained for the cubic family (when in (1) n = 3), see e.g., [9][10][11][12][13][14][15][16] and references given there; however, it appears that the center problem for the cubic system is still far from resolved. Some partial classifications have also been obtained for higher-degree families; in particular, for systems in the form of a linear center perturbed by homogeneous polynomials of degree 4 and 5 [17,18].…”