Periodic Solutions of Some Polynomial Differential Systems in Dimension 3 via Averaging Theory

Abstract

We provide sufficient conditions for the existence of periodic solutions of the polynomial third order differential system $\stackrel{.}{x}=-y+\epsilon P(x,y,z)+h_{\mathrm{1}}(t)$, $\stackrel{.}{y}=x+\epsilon Q(x,y,z)+h_{\mathrm{2}}(t)$, and $\stackrel{.}{z}=az+\epsilon R(x,y,z)+h_{\mathrm{3}}(t)$, where $P$, $Q$, and $R$ are polynomials in the variables $x$, $y$, and $z$ of degree $n,$$h_i(t)=h_i(t+\mathrm{2}\pi )$ with $i=\mathrm{1,2},3$ being periodic functions, $a$ is a real number, and $\epsilon$ is a small parameter.