Existence of Periodic Solutions for a Class of the Generalized Liénard Equations

Abstract

We study analytically the existence of periodic solutions of the generalized Liénard differential equations of the form $\ddot{x}+f\left(x,\dot{x}\right)\dot{x}+n^{2}x+g\left(x\right)={\epsilon }^2{p}_1\left(t\right)+{\epsilon }^3{p}_2\left(t\right),$ where n $\in {\mathbb{N}}^{*}$, the functions $f,g$ are of class ${\mathcal{C}}^3,{\mathcal{C}}^4$ in a neighborhood of the origin, respectively, the functions $p_i$ are of class ${\mathcal{C}}^0$, $2\pi -$periodic in the variable t, with $i=$$1,2$, and $\epsilon$ is a small parameter as usual. The mathematical tool that we have used is the averaging theory of dynamical systems of second order.

1. Introduction and Statement of the Main Results

One of the most important second-order differential equation is the so-called Liénard equation [1], which is of the form

$$\ddot{x}+f\left(x\right)\dot{x}+g\left(x\right)=0,$$

where f and g are arbitrary real functions of the variable $x=x\left(t\right)$, and the dot denotes derivative with respect to t. The Liénard equations are of great importance in various fields, such as physics, mechanics [2,3], biology and engineering [4]. There are also many examples in ecology and epidemiology where these differential equations are applied [5].

The Liénard equation as well as its generalization of the form

$$\ddot{x}+f\left(x,\dot{x}\right)\dot{x}+g\left(x\right)=0$$

were established by Levinson-Smith [6].

One of the main problems in the qualitative theory of ordinary differential equations is the study of the periodic orbits. In general, the study of the periodic orbits analytically is a very complicated problem. The averaging theory of dynamical systems is a very powerful method of studying the periodic orbits of an ordinary differential equations when it can be applied and the periodic orbits of the system studied are isolated.

In the present paper, we are going to use the mentioned averaging theory of dynamical systems. This theory has the role of strong symmetry because it provides the periodic solutions which live in the each symmetrical level of the energy when such periodic points are isolated. On the other hand, on many occasions, since this theory needs to evaluate a Jacobian of a matrix, the symmetry of such a matrix simplifies the checking of the hypothesis of this theory.

In the literature, there are many manuscripts dealing with analytical studies of periodic solutions using the averaging theory; see [7,8].

In [9], the authors studied analytically the existence of periodic solutions for the perturbed Liénard equations of the form

$$\ddot{x}+f\left(x\right)\dot{x}+n^{2}x+g\left(x\right)={\epsilon }^2{p}_1\left(t\right)+{\epsilon }^3{p}_2\left(t\right).$$

Our goal in the present work is to study analytically the existence of the periodic solutions for the perturbed generalized Liénard equations of the form

$$\ddot{x}+f\left(x,\dot{x}\right)\dot{x}+n^{2}x+g\left(x\right)={\epsilon }^2{p}_1\left(t\right)+{\epsilon }^3{p}_2\left(t\right),$$

(1)

where $\epsilon$ is a small parameter, n$\in {\mathbb{N}}^{*}$, f is of the class ${\mathcal{C}}^3\left({\mathbb{R}}^2,\mathbb{R}\right)$ function in a neighborhood of the origin, g is of the class ${\mathcal{C}}^4\left(\mathbb{R},\mathbb{R}\right)$ function in a neighborhood of the origin, $p_i$ is of class ${\mathcal{C}}^0\left(\mathbb{R},\mathbb{R}\right)$ functions, $2\pi -$periodic in the variable t, with $i=$$1,2$.

Let us consider

$${\delta }_1={\int }_0^{2\pi }{p}_1\left(t\right)sin\left(nt\right)dt\mathrm{and}{\delta }_2={\int }_0^{2\pi }{p}_1\left(t\right)cos\left(nt\right)dt,$$

and let the functions

$$\begin{array}{ccc}\hfill f_{21}\left(u,v\right)& =& {\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }\left(\left(\frac{2sin\left(nt\right)a_1{\left(cos\left(nt\right)\right)}^{2}u-sin\left(nt\right)u{a}_1}{n}\right.\right.\hfill \\ & & +\frac{2{\left(sin\left(nt\right)\right)}^2{a}_{1}vcos\left(nt\right)+2{\left(sin\left(nt\right)\right)}^2{b}_{1}cos\left(nt\right)u}{n^2}\hfill \\ & & -\frac{2sin\left(nt\right)b_{1}v{\left(cos\left(nt\right)\right)}^2}{n^3}\hfill \\ & & +\left.{\displaystyle \frac{2sin\left(nt\right)b_{1}v}{n^3}}\right){\int }_0^t{p}_1\left(s\right)cos\left(ns\right)ds\hfill \\ & & +\left(\frac{2{\left(sin\left(nt\right)\right)}^2{a}_{1}cos\left(nt\right)u}{n}+\frac{sin\left(nt\right)a_{1}v}{n^2}\right.\hfill \\ & & -\frac{2sin\left(nt\right){\left(cos\left(nt\right)\right)}^2(b_{1}u+a_{1}v)}{n^2}\hfill \\ & & -\left.\left.\frac{2{\left(sin\left(nt\right)\right)}^2{b}_{1}cos\left(nt\right)v}{n^3}\right){\int }_0^t{p}_1\left(s\right)sin\left(ns\right)ds\right)dt\hfill \\ & & -{\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }\frac{p_2\left(t\right)sin\left(nt\right)}{n}dt-\frac{5{b_1}^2{v}^3}{12n^6}\hfill \\ & & -\frac{v^3{a_1}^2+10{b_1}^{2}v{u}^2-3b_1{v}^{2}u{a}_1}{24n^4}\hfill \\ & & +\frac{3u^3{a}_1{b}_1-v{u}^2{a_1}^2}{24n^2},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill f_{22}\left(u,v\right)& =& {\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }\left(\left(-2{\left(cos\left(nt\right)\right)}^2{a}_{1}usin\left(nt\right)+\frac{2{\left(cos\left(nt\right)\right)}^3{b}_{1}u}{n}\right.\right.\hfill \\ & & -\frac{cos\left(nt\right)a_{1}v}{n}+\frac{2{\left(cos\left(nt\right)\right)}^3{a}_{1}vs}{n}\hfill \\ & & \left.+\frac{2{\left(cos\left(nt\right)\right)}^2{b}_{1}sin\left(nt\right)vs}{n^2}\right){\int }_0^t{p}_1\left(s\right)sin\left(ns\right)ds\hfill \\ & & +\left(-\frac{2{\left(cos\left(nt\right)\right)}^2{b}_{1}usin\left(nt\right)}{n}-\frac{2{\left(cos\left(nt\right)\right)}^2{a}_{1}sin\left(nt\right)v}{n}\right.\hfill \\ & & +cos\left(nt\right)a_{1}u-\frac{2cos\left(nt\right)v{b}_1}{n^2}-2{\left(cos\left(nt\right)\right)}^3{a}_{1}u\hfill \\ & & \left.\left.+\frac{2{\left(cos\left(nt\right)\right)}^3{b}_{1}v}{n^2}\right){\int }_0^t{p}_1\left(s\right)cos\left(ns\right)ds\right)dt\hfill \\ & & +{\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }cos\left(nt\right)p_2\left(t\right)dt+{\displaystyle \frac{1}{24}}{a_1}^2{u}^3+\frac{5v^2{b_1}^{2}u}{12n^4}\hfill \\ & & +\frac{a_1{v}^3{b}_1}{8n^4}+\frac{10{b_1}^2{u}^3+3a_{1}v{b}_1{u}^2+{a_1}^2{v}^{2}u}{24n^2}\hfill \\ & & -\frac{a_3{v}^{3}n+3a_5{v}^3{n}^3+3b_2{u}^3{n}^3+n^5{a}_4{u}^3}{8n^3}\hfill \\ & & -\frac{a_3{u}^{2}v{n}^3+3b_2{v}^{2}un+3n^5{a}_5{u}^{2}v+a_4{v}^{2}u{n}^3}{8n^3}.\hfill \end{array}$$

Our main results for the periodic solutions of the differential Equation (1) are given in the next theorem.

Theorem 1.

Suppose that the functions of the generalized Liénard differential Equation (1) satisfy the following statements.

(a) 

The functions $f,g$ are of class ${\mathcal{C}}^3,{\mathcal{C}}^4$ in a neighborhood of the origin, respectively;

(b) 

$f(0,0)={\partial }_{2}f\left(0,0\right)=g\left(0\right)=$$g^{\prime }\left(0\right)=0$;

(c) 

${\delta }_1={\delta }_2=0.$

Then, for every simple zero $\left(u^{*},v^{*}\right)$ of the system

$$\begin{array}{c}{f}_{21}\left(u,v\right)=0,\hfill \\ f_{22}\left(u,v\right)=0,\hfill \end{array}$$

and for an $\epsilon \ne 0$ which is sufficiently small, the differential Equation (1) has a periodic solution $x(t,\epsilon )$ such that $x(0,\epsilon )\approx \epsilon u^{*}+O\left({\epsilon }^2\right)$ and $x^{\prime }(0,\epsilon )\approx \epsilon v^{*}+O\left({\epsilon }^2\right)$.

Theorem 1 is proved in Section 3.

2. The Averaging Theory of the First and Second Orders for Ordinary Differential System

The averaging theory of the first and second orders for studying periodic orbits can be found from the article [10,11] at any order. For an introduction to the averaging theory, see the book [12], for instance.

Theorem 2.

Let us consider the first order ordinary differential system

$$\dot{x}\left(t\right)=\epsilon {\mathcal{F}}_1\left(t,x\right)+{\epsilon }^2{\mathcal{F}}_2\left(t,x\right)+{\epsilon }^3\mathcal{R}\left(t,x,\epsilon \right),$$

(2)

where ${\mathcal{F}}_1,{\mathcal{F}}_2$$:\mathbb{R}\times D\to$${\mathbb{R}}^n,\mathcal{R}:\mathbb{R}\times D\times (-{\epsilon }_f,{\epsilon }_f)\to {\mathbb{R}}^n$ are of class ${\mathcal{C}}^0$ functions and $T-$periodic in the first variable t, where $D\subseteq$${\mathbb{R}}^n$ is an open subset and ε is a small parameter as usual.

Assume that assumptions $\left(i\right)$–$\left(ii\right)$ hold.

$\left(i\right)$${\mathcal{F}}_1\left(t,.\right)\in {\mathcal{C}}^1\left(D\right)$ for all $t\in \mathbb{R}$, ${\mathcal{F}}_1,{\mathcal{F}}_2,\mathcal{R},D_x{\mathcal{F}}_1$ are locally Lipschitz with respect to x, and $\mathcal{R}$ is differentiable with respect to ε. We define the two functions $f_1,f_2:D\to {\mathbb{R}}^n$ as

$$\begin{array}{c}\hfill f_1\left(z\right)=\frac{1}{T}\underset{0}{\overset{T}{\int }}{\mathcal{F}}_1\left(t,z\right)dt,\end{array}$$

and

$$\begin{array}{c}\hfill f_2\left(z\right)=\frac{1}{T}\underset{0}{\overset{T}{\int }}\left[D_z{\mathcal{F}}_1\left(t,z\right)·y_1(t,z)+{\mathcal{F}}_2\left(t,z\right)\right]dt,\end{array}$$

where

$$y_1(t,z)=\underset{0}{\overset{t}{\int }}{\mathcal{F}}_1\left(s,z\right)ds.$$

$\left(ii\right)$ For $V\subset D$, an open and bounded set, and for each $\epsilon \in (-{\epsilon }_f,{\epsilon }_f)\backslash \left\{0\right\}$, there exist $\mu \in V$ such that $f_1\left(\mu \right)+\epsilon f_2\left(\mu \right)=0$ and

$$d_B(f_1+\epsilon f_2,V,\mu )\ne 0;$$

Then, for an $\epsilon \ne 0$ which is sufficiently small, the differential system (2) admits a T-periodic solution $\psi \left(t,\epsilon \right)$ such that $\psi \left(0,\epsilon \right)\to \mu$ as $\epsilon \to 0$.

As is well known, a sufficient condition for the Brouwer degree of a function at a fixed-point $\mu$ being non-zero is that the Jacobian of the function at $\mu$ is non-zero. In this paper, for the function $f=f_1+ϵf_2$, we are going to denote by $d_B(f_1+ϵf_2,V,\mu )$ its Brouwer degree at $\mu$. At this point, we have two possibilities: whether $f_1$ is zero or not. These two options offer two interpretations of Theorem 2. In the first case, if $ϵ$ is small enough, the zeros of f are mainly the zeros of $f_1$; therefore, Theorem 2 provides the averaging theory of the first order approach. In the second case, as $f=ϵf_2$, Theorem 2 provides the averaging theory of the second order.

3. Proof of the Main Results

In this section, we present a proof the main results.

Proof of Theorem 1. 

First, we perform some procedures to write the generalized Liénard Equation (1) in the normal form (2) for applying the averaging theory; see Theorem 2.

By the linear scaling of variable $x=\epsilon z$, the Equation (1) becomes

$$z^{″}+f\left(\epsilon z,\epsilon z^{\prime }\right)z^{\prime }+n^{2}z+\frac{g\left(\epsilon z\right)}{\epsilon }=\epsilon p_1\left(t\right)+{\epsilon }^2{p}_2\left(t\right).$$

(3)

Since the functions $f,g$ are of class ${\mathcal{C}}^3,{\mathcal{C}}^4$ in a neighborhood of the origin, respectively, and $f(0,0)={\partial }_{2}f\left(0,0\right)=g\left(0\right)=$$g^{\prime }\left(0\right)=0$, we can write

$$\begin{array}{ccc}\hfill f\left(\epsilon z,\epsilon w\right)& =& \epsilon \left(a_{1}z\right)+{\epsilon }^2\left(a_3{z}^2+a_{4}zw+a_5{w}^2\right)+O\left({\epsilon }^3\right),\hfill \\ \hfill g\left(\epsilon z\right)& =& {\epsilon }^2{b}_1{z}^2+{\epsilon }^3{b}_2{z}^3+O\left({\epsilon }^4\right).\hfill \end{array}$$

From it, we find that

$$\begin{array}{ccc}\hfill f\left(\epsilon z,\epsilon w\right)w+\frac{g\left(\epsilon z\right)}{\epsilon }& =& \epsilon \left(a_{1}zw+b_1{z}^2\right)+{\epsilon }^2(a_3{z}^{2}w+a_{4}z{w}^2\hfill \\ & & +a_5{w}^3+b_2{z}^3)+O\left({\epsilon }^3\right),\hfill \end{array}$$

with

$$\begin{array}{cc}& a_1={\partial }_{1}f\left(0,0\right),a_2={\partial }_{2}f\left(0,0\right)=0,a_3={\displaystyle \frac{1}{2}}{\partial }_{11}f\left(0,0\right),\\ & a_4={\partial }_{12}f\left(0,0\right),a_5={\displaystyle \frac{1}{2}}{\partial }_{22}f\left(0,0\right),b_1={\displaystyle \frac{1}{2}}{g}^{″}\left(0\right),b_2={\displaystyle \frac{1}{6}}{g}^{‴}\left(0\right).\end{array}$$

By defining the variable $w=z^{\prime }$, we can write the differential Equation (3) as the differential system

$$\begin{array}{c}{z}^{\prime }=w,\hfill \\ w^{\prime }=-n^{2}z+\epsilon \left(p_1\left(t\right)-a_{1}zw-b_1{z}^2\right)+{\epsilon }^2(p_2\left(t\right)-a_3{z}^{2}w\hfill \\ -a_{4}z{w}^2-a_5{w}^3-b_2{z}^3)+O\left({\epsilon }^3\right).\hfill \end{array}$$

(4)

Now, we write the variables $\left(z,w\right)$ in terms of $\left(u,v\right)$ through the matrix

$$\left(\begin{array}{c}z\\ w\end{array}\right)=\left(\begin{array}{cc}cos\left(nt\right)& {\displaystyle \frac{sin\left(nt\right)}{n}}\\ -nsin\left(nt\right)& cos\left(nt\right)\end{array}\right)\left(\begin{array}{c}u\\ v\end{array}\right).$$

(5)

Thus, in the new variables $(u,v)$, the system (4) is written as follows:

$$\begin{array}{c}{u}^{\prime }=-{\displaystyle \frac{1}{n^4}}\epsilon sin\left(nt\right)(p_1\left(t\right)n^3-b_1{\left(cos\left(nt\right)\right)}^2{u}^2{n}^3\hfill \\ -2b_{1}cos\left(nt\right)u{n}^{2}sin\left(nt\right)v-n{b}_1{v}^2+n{b}_1{v}^2{\left(cos\left(nt\right)\right)}^2\hfill \\ -2a_1{n}^3{\left(cos\left(nt\right)\right)}^{2}uv-a_1{n}^{2}sin\left(nt\right)v^{2}cos\left(nt\right)+a_1{n}^{3}vu\hfill \\ +a_1{n}^{4}cos\left(nt\right)u^{2}sin\left(nt\right))+{\displaystyle \frac{1}{n^4}}{\epsilon }^2(sin\left(nt\right)(-p_2\left(t\right)n^3\hfill \\ -2a_4{n}^3{v}^{2}cos\left(nt\right)u+3a_3{n}^3{\left(cos\left(nt\right)\right)}^3{u}^{2}v+a_5{n}^3{\left(cos\left(nt\right)\right)}^3{v}^3\hfill \\ +b_2{\left(cos\left(nt\right)\right)}^3{u}^3{n}^3-a_5{n}^{6}sin\left(nt\right)u^3+a_{3}n{v}^{3}cos\left(nt\right)\hfill \\ -a_{3}n{v}^3{\left(cos\left(nt\right)\right)}^3+a_4{n}^{5}cos\left(nt\right)u^3-a_4{n}^5{\left(cos\left(nt\right)\right)}^3{u}^3\hfill \\ -b_{2}sin\left(nt\right)v^3{\left(cos\left(nt\right)\right)}^2+b_{2}sin\left(nt\right)v^3-a_3{n}^4{\left(cos\left(nt\right)\right)}^2{u}^{3}sin\left(nt\right)\hfill \\ +3a_3{n}^2{\left(cos\left(nt\right)\right)}^{2}usin\left(nt\right)v^2-3a_4{n}^4{\left(cos\left(nt\right)\right)}^2{u}^{2}sin\left(nt\right)v\hfill \\ +3a_4{n}^3{\left(cos\left(nt\right)\right)}^{3}u{v}^2+a_4{n}^{2}sin\left(nt\right)v^3{\left(cos\left(nt\right)\right)}^2\hfill \\ -3a_5{n}^4{\left(cos\left(nt\right)\right)}^{2}usin\left(nt\right)v^2+3b_2{\left(cos\left(nt\right)\right)}^2{u}^2{n}^{2}sin\left(nt\right)v\hfill \\ +a_5{n}^{6}sin\left(nt\right)u^3{\left(cos\left(nt\right)\right)}^2-a_3{n}^{2}sin\left(nt\right)v^{2}u+a_4{n}^{4}sin\left(nt\right)v{u}^2\hfill \\ +3a_5{n}^{5}cos\left(nt\right)u^{2}v-2a_3{n}^{3}cos\left(nt\right)u^{2}v-3a_5{n}^5{\left(cos\left(nt\right)\right)}^3{u}^{2}v\hfill \\ +3b_2{v}^{2}cos\left(nt\right)un-3b_2{v}^2{\left(cos\left(nt\right)\right)}^{3}un)+O\left({\epsilon }^3\right),\hfill \end{array}$$

and

$$\begin{array}{c}{v}^{\prime }={\displaystyle \frac{1}{n^3}}cos\left(nt\right)\epsilon (p_1\left(t\right)n^3-b_1{\left(cos\left(nt\right)\right)}^2{u}^2{n}^3+a_1{n}^{4}cos\left(nt\right)u^{2}sin\left(nt\right)\hfill \\ -2a_1{n}^3{\left(cos\left(nt\right)\right)}^{2}uv-a_1{n}^{2}sin\left(nt\right)v^{2}cos\left(nt\right)+a_1{n}^{3}vu\hfill \\ +n{b}_1{v}^2{\left(cos\left(nt\right)\right)}^2-2b_{1}cos\left(nt\right)u{n}^{2}sin\left(nt\right)v-n{b}_1{v}^2)\hfill \\ +{\displaystyle \frac{1}{n^3}}cos\left(nt\right){\epsilon }^2(p_2\left(t\right)n^3-b_{2}sin\left(nt\right)v^3\hfill \\ +3a_4{n}^4{\left(cos\left(nt\right)\right)}^2{u}^{2}sin\left(nt\right)v-a_4{n}^{4}sin\left(nt\right)v{u}^2\hfill \\ -a_5{n}^{6}sin\left(nt\right)u^3{\left(cos\left(nt\right)\right)}^2-a_5{n}^3{\left(cos\left(nt\right)\right)}^3{v}^3\hfill \\ -b_2{\left(cos\left(nt\right)\right)}^3{u}^3{n}^3+a_5{n}^{6}sin\left(nt\right)u^3-a_{3}n{v}^{3}cos\left(nt\right)\hfill \\ +a_{3}n{v}^3{\left(cos\left(nt\right)\right)}^3+b_{2}sin\left(nt\right)v^3{\left(cos\left(nt\right)\right)}^2-a_4{n}^{5}cos\left(nt\right)u^3\hfill \\ +a_4{n}^5{\left(cos\left(nt\right)\right)}^3{u}^3+a_3{n}^4{\left(cos\left(nt\right)\right)}^2{u}^{3}sin\left(nt\right)\hfill \\ -3a_3{n}^3{\left(cos\left(nt\right)\right)}^3{u}^{2}v-3a_3{n}^2{\left(cos\left(nt\right)\right)}^{2}usin\left(nt\right)v^2\hfill \\ -3a_4{n}^3{\left(cos\left(nt\right)\right)}^{3}u{v}^2-a_4{n}^{2}sin\left(nt\right)v^3{\left(cos\left(nt\right)\right)}^2\hfill \\ +3a_5{n}^4{\left(cos\left(nt\right)\right)}^{2}usin\left(nt\right)v^2-3b_2{\left(cos\left(nt\right)\right)}^2{u}^2{n}^{2}sin\left(nt\right)v\hfill \\ +a_3{n}^{2}sin\left(nt\right)v^{2}u+2a_3{n}^{3}cos\left(nt\right)u^{2}v+2a_4{n}^3{v}^{2}cos\left(nt\right)u\hfill \\ -3b_2{v}^{2}cos\left(nt\right)un+3b_2{v}^2{\left(cos\left(nt\right)\right)}^{3}un\hfill \\ -3a_5{n}^{5}cos\left(nt\right)u^{2}v+3a_5{n}^5{\left(cos\left(nt\right)\right)}^3{u}^{2}v+O\left({\epsilon }^3\right).\hfill \end{array}$$

(6)

We note that the differential system (6) is in the normal form (2). Hence, we can apply the averaging theory.

Let ${\mathcal{F}}_i(t,u,v)$$={\left({\mathcal{F}}_{i1}\left(t,u,v\right),{\mathcal{F}}_{i2}\left(t,u,v\right)\right)}^T$ for $i=1,2$, and $T=2\pi$. Now, we compute the function $f_1(u,v)$ defined in the Theorem 2 and we get

$$f_1\left(u,v\right)={\displaystyle \frac{1}{2\pi }}\left(\begin{array}{c}\underset{0}{\overset{2\pi }{\int }}{\mathcal{F}}_{11}\left(t,u,v\right)dt\\ \\ \underset{0}{\overset{2\pi }{\int }}{\mathcal{F}}_{12}\left(t,u,v\right)dt\end{array}\right)=\frac{1}{2\pi }\left(\begin{array}{c}-{\delta }_1\\ {\delta }_2\end{array}\right);$$

since $f_1(u,v)$ is zero, we shall apply the averaging theory of second order.

Therefore, we first compute

$$f_2\left(z\right)=\frac{1}{T}\underset{0}{\overset{T}{\int }}\left[D_z{\mathcal{F}}_1\left(t,z\right)·y_1(t,z)+{\mathcal{F}}_2\left(t,z\right)\right]dt,$$

with

$$y_1(t,z)={\int }_0^t{\mathcal{F}}_1(s,z)\mathrm{d}s.$$

Thus, we start by calculating the two components of $y_1$:

$$\begin{array}{ccc}\hfill y_{11}(t,z)& =& {\int }_0^t{\mathcal{F}}_{11}\left(s,u,v\right)ds\hfill \\ & =& -\frac{{\left(sin\left(nt\right)\right)}^3\left(-a_1{v}^2-2b_{1}uv+a_1{u}^2{n}^2\right)}{3n^3}\hfill \\ & & -\frac{-4b_1{v}^2+2a_{1}vu{n}^2-2b_1{u}^2{n}^2-3a_{2}v{n}^2}{6n^4}\hfill \\ & & -\frac{4a_1{\left(cos\left(nt\right)\right)}^{3}uv{n}^2-2b_1{v}^2{\left(cos\left(nt\right)\right)}^3}{6n^4}\hfill \\ & & -\frac{2b_1{\left(cos\left(nt\right)\right)}^3{u}^2{n}^2+3a_2{\left(cos\left(nt\right)\right)}^{2}v{n}^2}{6n^4}\hfill \\ & & -\frac{6b_1{v}^{2}cos\left(nt\right)-6a_{1}vucos\left(nt\right)n^2}{6n^4}\hfill \\ & & -{\int }_0^t\frac{p_1\left(s\right)sin\left(ns\right)}{n}ds,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill y_{12}(t,z)& =& {\int }_0^t{\mathcal{F}}_{12}\left(s,u,v\right)ds\hfill \\ & =& -\frac{\left(-b_1{v}^2+b_1{u}^2{n}^2+2a_1{n}^{2}vu\right)sin\left(nt\right)\left({\left(cos\left(nt\right)\right)}^2+2\right)}{3n^3}\hfill \\ & & +\frac{v\left(-b_{1}v+a_1{n}^{2}u\right)sin\left(nt\right)}{n^3}\hfill \\ & & -\frac{a_1{v}^2+2b_{1}uv-a_1{u}^2{n}^2-a_1{v}^2{\left(cos\left(nt\right)\right)}^3}{3n^2}\hfill \\ & & -\frac{n^2{a}_1{u}^2{\left(cos\left(nt\right)\right)}^3-2b_{1}uv{\left(cos\left(nt\right)\right)}^3}{3n^2}\hfill \\ & & +{\int }_0^t{p}_1\left(s\right)cos\left(ns\right)ds,\hfill \end{array}$$

and the Jacobian matrix

$$D_{\left(u,v\right)}\left({\mathcal{F}}_1\left(t,u,v\right)\right)=\left(\begin{array}{cc}{\displaystyle \frac{\partial {\mathcal{F}}_{11}\left(s,u,v\right)}{\partial u}}& {\displaystyle \frac{\partial {\mathcal{F}}_{11}\left(s,u,v\right)}{\partial v}}\\ & \\ {\displaystyle \frac{\partial {\mathcal{F}}_{12}\left(s,u,v\right)}{\partial u}}& {\displaystyle \frac{\partial {\mathcal{F}}_{12}\left(s,u,v\right)}{\partial v}}\end{array}\right).$$

Now, we compute the function

$$f_2(u,v)=\left(\begin{array}{c}{f}_{21}(u,v)\\ f_{22}(u,v)\end{array}\right);$$

(7)

then, we solve the following system:

$$\begin{array}{c}{f}_{21}\left(u,v\right)=0,\hfill \\ f_{22}\left(u,v\right)=0.\hfill \end{array}$$

If this system accepts a simple zero $(u^{*},v^{*})$ such that

$${\left.det\left(\begin{array}{cc}{\displaystyle \frac{\partial f_{21}\left(s,u,v\right)}{\partial u}}& {\displaystyle \frac{\partial f_{21}\left(s,u,v\right)}{\partial v}}\\ \\ {\displaystyle \frac{\partial f_{22}\left(s,u,v\right)}{\partial u}}& {\displaystyle \frac{\partial f_{22}\left(s,u,v\right)}{\partial v}}\end{array}\right)\right|}_{\left(u,v\right)=(u^{*},v^{*})}\ne 0,$$

then, for a $\epsilon \ne 0$ which is sufficiently small, the differential Equation (6) has a periodic solution $\left(u\left(t,\epsilon \right),v\left(t,\epsilon \right)\right)$ such that

$$\left(u\left(0,\epsilon \right),v\left(0,\epsilon \right)\right)\to \left(u^{*},v^{*}\right)\mathrm{as}\epsilon \to 0;$$

from it we conclude that the differential system (4) has a periodic solution

$$\begin{array}{ccc}\hfill z\left(\epsilon ,t\right)& =& cos\left(nt\right)u(t,\epsilon )+\frac{1}{n}sin\left(nt\right)v\left(t,\epsilon \right),\hfill \\ \hfill w\left(\epsilon ,t\right)& =& -nsin\left(nt\right)u(t,\epsilon )+cos\left(nt\right)v\left(t,\epsilon \right),\hfill \end{array}$$

such that

$$\left(z\left(0,\epsilon \right),w\left(0,\epsilon \right)\right)\to \left(u^{*},v^{*}\right)\mathrm{as}\epsilon \to 0.$$

Finally, since $x=\epsilon z$, we conclude that the generalized Liénard differential Equation (1) has a periodic solution

$$x\left(t,\epsilon \right)=\epsilon \left(cos\left(nt\right)u(t,\epsilon )+\frac{1}{n}sin\left(nt\right)v\left(t,\epsilon \right)\right),$$

such that

$$x\left(0,\epsilon \right)\approx \epsilon u^{*}+O\left({\epsilon }^2\right).$$

Moreover, since $x^{\prime }=\epsilon w$ we conclude that

$$x^{\prime }\left(0,\epsilon \right)\approx \epsilon v^{*}+O\left({\epsilon }^2\right).$$

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4. Discussion and Conclusions

The analytical study of the periodic solutions of non-linear differential equations is a very difficult matter, and in many cases it cannot be successful. The reason is that the main method used, the so-called averaging theory of dynamical systems, needs to perturbate the system with a small parameter and to transform it into a particular normal form. This task is, in general, complicated and involves chains of variables and the use of particular coordinates, (e.g., action angle or Delonau ones) and to reach the theorem’s hypothesis (e.g., the non-annulation of the determinant of the Jacobian matrix of the transformations). The nice part of this approach is that the periodic solutions found for the perturbed system tend to the solutions of the unperturbed systems, which were not possible to identify by other sources. We underline that this approach is analytic and not numerical. In the present paper, we have studied the existence of periodic solutions for a class of generalized Liénard equations using the described averaging theory.