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 x.=-y+εP(x,y,z)+h1(t), y.=x+εQ(x,y,z)+h2(t), and z.=az+εR(x,y,z)+h3(t), where P, Q, and R are polynomials in the variables x, y, and z of degree n, hi(t)=hi(t+2π) with i=1,2,3 being periodic functions, a is a real number, and ε is a small parameter.