Cubic systems of nonlinear oscillations with seven limit cycles

Abstract

Consider the system of differential equations ˙x = y, ˙y = −x + λy + Ax2 + 3Bxy + Cy2 + Kx3 + 3Lx2y + M xy2 + N y3, (1) where λ, A, B, C, K, L, M , and N are real constants and |λ| < 2. The center-focus problem for system (1) was considered for the first time in [1]. Some center cases for system (1) with N 6 = 0 were missing in [1]; this was shown in [2], where necessary and sufficient center conditions of algebraic character were also presented for this system. Similar necessary and sufficient conditions were considered in [3]. The center-focus problem was solved in [4, 5] for N = 0; moreover, it was shown in [5] that, for N = 0, there exist cubic systems with five limit cycles. The center-focus problem for system (1) with B = 0 was solved in [6–8] on the basis of the analysis of focus quantities. It was shown in [7, 8] that if B = 0, then there exist cubic systems with six limit cycles. A detailed study of necessary center conditions for system (1) on the basis of the analysis of focus quantities was performed in [9], where all center cases of system (1) were obtained; however, the necessity of these conditions was not completely justified there. The center-focus problem for system (1) was solved in [12] on the basis of the Cherkas method [10; 11, p. 70]. In the present paper, on the basis of the analysis of focus quantities, we solve the center-focus problem for system (1) with BN 6 = 0 and prove the existence of cubic systems of nonlinear oscillations of the form (1) with seven limit cycles.