Limit Cycles for the Class of D-Dimensional Polynomial Differential Systems

Abstract

We perturb the differential system ${\dot{x}}_{\mathrm{1}}=-x_{\mathrm{2}}(\mathrm{1}+x_{\mathrm{1}})$, ${\dot{x}}_{\mathrm{2}}=x_{\mathrm{1}}(\mathrm{1}+x_{\mathrm{1}})$, and ${\dot{x}}_k=\mathrm{0}$ for $k=\mathrm{3},\dots ,d$ inside the class of all polynomial differential systems of degree $n$ in ${\mathbb{R}}^d$, and we prove that at most $n^{d-\mathrm{1}}$ limit cycles can be obtained for the perturbed system using the first-order averaging theory.