Special cubic perturbations of the Duffing oscillator $x\mathrm{"}=x-x^3$ near the eight-loop

We find an upper bound for the number of limit cycles, bifurcating from the 8-loop of the Duffing oscillator $x\mathrm{"}=x-x^3$ under the special cubic perturbation $$x\mathrm{"}=x-x^3+{\lambda }_{1}y+{\lambda }_2{x}^2+{\lambda }_{3}xy+{\lambda }_4{x}^{2}y\mathrm{.}$$