Periodic solutions of Lienard differential equations via averaging theory of order two

Abstract

For $\epsilon \ne 0$sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form

$x^{\mathrm{\prime \prime }}+f(x)x^{\prime }+n^{2}x+g(x)={\epsilon }^2{p}_1(t)+{\epsilon }^3{p}_2(t),$

where $n$ is a positive integer,$f:ℝ\to ℝ$is a $C^3$function,$g:ℝ\to ℝ$is a $C^4$function, and$p_i:ℝ\to ℝ$for $i=1,2$are continuous $2\pi$–periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained.

Key words
periodic solution; Lienard differential equation; averaging theory; bifurcation theory