The limit cycles in a generalized Rayleigh-Liénard oscillator

We compute the cyclicity of open period annuli following generalized Rayleigh-Liénard equation \begin{document}$ \ddot{x}+ax+bx^3-(\lambda_1+\lambda_2 x^2+\lambda_3\dot{x}^2+\lambda_4 x^4+\lambda_5\dot{x}^4+\lambda_6 x^6)\dot{x} = 0 $\end{document} and equivalent planar system $ X_\lambda $, where coefficients perturbation \lambda_j are independent small parameters a, b fixed nonzero constants. Our main tool is machinery so called higher-order Poincaré-Pontryagin-Melnikov functions (Melnikov M_n for short), combined with explicit computation center conditions corresponding Bautin ideal.We consider first arbitrary analytic arcs \varepsilon \to \lambda(\varepsilon) explicitly all possible Melnikov related to deformation X_{ \lambda(\varepsilon)} $. At a second step we obtain exact bounds number zeros (complete elliptic integrals depending on parameter) in an appropriate complex domain, using modification Petrov's method.To deal general case six-parameter deformations \lambda ideal. To do this carefully study up order three, then use Nakayama lemma from Algebraic geometry. The principalization ideal (achieved after blow up) reduces finally } one-parameter