Maximum Number of Limit Cycles for Generalized Kukles Polynomial Differential Systems

Abstract

We study the maximum number of limit cycles of the polynomial differential systems of the form $$\begin{aligned} \dot{x}=-y+l(x), \,\dot{y}=x-f(x)-g(x)y-h(x)y^{2}-d_{0}y^{3}, \end{aligned}$$ where \(l(x)=\varepsilon l^{1}(x)+\varepsilon ^{2}l^{2}(x),\)\(f(x)=\varepsilon f^{1}(x)+\varepsilon ^{2}f^{2}(x),\)\(g(x)=\varepsilon g^{1}(x)+\varepsilon ^{2}g^{2}(x),\)\(h(x)=\varepsilon h^{1}(x)+\varepsilon ^{2}h^{2}(x)\) and \(d_{0}=\varepsilon d_{0}^{1}+\varepsilon ^{2}d_{0}^{2}\) where \(l^{k}(x),\)\(f^{k}(x),\)\(g^{k}(x)\) and \(h^{k}(x)\) have degree m, \(n_{1},\)\(n_{2}\) and \(n_{3}\) respectively, \(d_{0}^{k}\ne 0\) is a real number for each \(k=1,2,\) and \(\varepsilon \) is a small parameter. We provide an upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centre \(\dot{x}=-y,\, \dot{y}=x\) using the averaging theory of first and second order.