Special Cubic Perturbations of the Duffing Oscillator $$x'=x-x^3$$ Near the Eight-Loop

We find an upper bound for the number of limit cycles, bifurcating from the 8-loop of the Duffing oscillator $x"= x-x^{3}$ under the special cubic perturbation $$ x"= x-x^{3}+\lambda_{1}y+\lambda_{2}x^{2}+\lambda_{3}xy+\lambda_{4}x^{2}y . $$
Comment: 4 figures, minor corrections, references added. arXiv admin note: text overlap with arXiv:1306.2340