\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 195, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/195\hfil Discontinuous Lienard polynomial equations]
{Limit cycles for discontinuous generalized Lienard
polynomial differential equations}

\author[J. Llibre, A. C. Mereu \hfil EJDE-2013/195\hfilneg]
{Jaume Llibre, Ana Cristina Mereu}  % in alphabetical order

\address{Jaume Llibre \newline
Departament de Matem\`atiques,
Universitat Aut\`onoma de Barcelona, 08193 Bellaterra, Barcelona,
Catalonia, Spain}
\email{jllibre@mat.uab.cat}

\address{Ana Cristina Mereu \newline
Department of Physics, Chemistry and Mathematics,
UFSCar, 18052-780, Sorocaba, SP, Brazil}
\email{anamereu@ufscar.br}

\thanks{Submitted  May 7, 2013. Published September 3, 2013.}
\subjclass[2000]{34C29, 34C25, 47H11}
\keywords{Limit cycles; non-smooth Li\'enard systems; averaging theory}

\begin{abstract}
 We divide $\mathbb{R}^2$ into sectors $S_1,\dots ,S_l$, with $l>1$ even,
 and define a discontinuous differential system such that in each sector,
 we have a smooth generalized Lienard polynomial differential equation
 $\ddot{x}+f_i(x)\dot{x} +g_i(x)=0$, $i=1, 2$ alternatively, where
 $f_i$ and $g_i$ are polynomials of degree $n-1$ and $m$ respectively.
 Then we apply the averaging theory for first-order discontinuous differential
 systems to show that for any $n$ and $m$ there are non-smooth Lienard polynomial
 equations having at least $\max\{n,m\}$ limit cycles.
 Note that this number is independent of the number of sectors.

 Roughly speaking this result shows that the non-smooth classical
 $(m=1)$ Lienard polynomial  differential systems can have at least
 the double number of limit cycles than the smooth ones, and that the
 non-smooth generalized Lienard polynomial  differential systems can
 have at least one more limit cycle than the smooth ones. 
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

A large number of problems from mechanics and electrical
engineering, theory of automatic control, economy, impact systems
among others cannot be described with smooth dynamical systems. This
fact has motivated many researchers to the study  of qualitative
aspects of the phase space of non-smooth dynamical systems.

One of the main problems in the qualitative theory of real planar
continuous and discontinuous differential systems is the
determination of their limit cycles. The non-existence, existence,
uniqueness and other properties of limit cycles have been studied
extensively by mathematicians and physicists, and more recently also
by chemists, biologists, economists, etc (see for instance the books
\cite{Bo, CLW, Ye}). This problem restricted to continuous planar
polynomial differential equations is the well known 16th Hilbert's
problem \cite{Hi}.  Since this Hilbert's problem turned out a
strongly difficult one Smale \cite{S}  particularized it to Lienard
polynomial differential equations in his list of problems for the
present century.

The classical Lienard polynomial  differential equations
\begin{equation}\label{Lienard1}
\ddot{x}+f(x) \dot{x}+g(x)=0,
\end{equation}
where $f(x)$ and $g(x)=x$  goes back to \cite{Li}. The dot denotes
differentiation with respect to the time $t$. This second-order
differential equation \eqref{Lienard1}  can be written as the
following first-order differential system in $\mathbb{R}^2$
\begin{equation}\label{Lienard2}
\begin{gathered}
\dot{x}= y-F(x),\\
\dot{y}= -g(x),
\end{gathered}
\end{equation}
where $F(x)=\int_0^x f(s)ds$.

Many results on the number of limit cycles has been obtained for the
continuous generalized polynomial differential equations
\eqref{Lienard1} being $f(x)$ and $g(x)$ polynomials in the variable
$x$ of degrees $n-1$ and $m$ respectively. The continuous classical
Lienard polynomial  differential equations \eqref{Lienard1}  were
studied in 1977 by Lins, de Melo and Pugh \cite{LMP} who stated the
conjecture:
\begin{quote}
If $f(x)$ has degree $n - 1 >0$ and
$g(x)=x$, then \eqref{Lienard1} has at most $[(n-1)/2]$ limit
cycles.
\end{quote}
Here $[z]$ denotes the integer part function of
$z\in \mathbb{R}$. They also proved the conjecture for $n=2, 3$. 
For $n=4$ this conjecture has been proved  in 2012 (see \cite{LiLli}). 
For $n \geq 7$ Dumortier, Panazzolo and Roussarie proved that this
conjecture is not true  in \cite{DPR}, they show that these
differential equations can have $[(n-1)/2]+1$ limit cycles. Recently
De Maesschalck and Dumortier proved in \cite{MD} that the classical
Lienard equation of degree $n \geq 6$ can have $[(n-1)/2]+2$ limit
cycles.  The conjecture for $n=5$ is still open.

Results on the number of limit cycles for continuous generalized
Lienard polynomial  differential equations can be found in
\cite{LMT} where the authors  show that  there are differential
equations \eqref{Lienard1} having at least $[(n+m-2)/2]$ limit
cycles. See also \cite{BM, BC, Cl, GT, L3, L1, L2, LC, Y}.

The objective of this work is to star the study of the number of
limit cycles for a kind of discontinuous generalized Lienard
polynomial differential systems. Here we shall play with many
straight lines of discontinuities through the origin of coordinates
and with two different continuous generalized Lienard polynomial
differential systems located alternatively in the sectors defined by
these straight lines.

A similar work but with only one classical Lienard polynomial
differential system and only one straight line of discontinuity was
studied in  \cite{LT} obtaining  $[n/2]$ limit cycles, instead of
the $[(n-1)/2]$ of the continuous classical Lienard polynomial
differential equation obtained in \cite{LMP}.

Now we shall define the discontinuous generalized Lienard polynomial
differential system that we will study. We consider the function
$h:\mathbb{R}^2\to \mathbb{R}$ defined by
$$
h(x,y)=\prod_{k=0}^{\frac{l}{2}-1}
\Big(y-\tan\big(\alpha+\frac{2k\pi}{l}\big)x\Big),
$$
where $l>1$ even. The set
$$
h^{-1}(0)= \cup_{k=0}^{\frac{l}{2}-1} \{(x,y):y
= \tan \big(\alpha +\frac{2k\pi}{l}\big)x\},
$$
divides $\mathbb{R}^2$ into $l$ sectors, $S_1, S_2, \dots , S_l$, i.e.
$h^{-1}(0)$ is the product of $l/ 2$ straight lines passing through
the origin of coordinates dividing the plane in sectors of angle
$2\pi / l$.

In this work  we study the maximum number of limit cycles given by
the averaging theory of first order, which can bifurcate from the
periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$,
perturbed inside the following class of discontinuous Lienard
polynomial differential systems
\begin{equation}\label{lep}
\dot X = Z(x,y) = \begin{cases}
Y_1(x,y) &  \text{if }  h(x,y) > 0,\\
Y_2(x,y) &  \text{if }  h(x,y) < 0,
\end{cases}
\end{equation}
where
\begin{equation}\label{X}
Y_1(x,y)=\begin{pmatrix}
 y-\varepsilon F_1(x) \\
-x-\varepsilon g_1(x)
\end{pmatrix}, \quad
Y_2(x,y)=\begin{pmatrix}
 y-\varepsilon F_2(x) \\
-x-\varepsilon g_2(x)
\end{pmatrix},
\end{equation}
where $\varepsilon$ is a small parameter, and $F_i(x)$ and $g_i(x)$, for $i=1, 2$
are polynomials in the variable $x$, and degrees $n$ and $m$
respectively. System \eqref{lep} can be written using the sign
function as
\begin{equation}\label{sgn}
\dot{X} = Z(x,y) = G_1(x,y) + \operatorname{sgn}(h(x,y))G_2(x, y),
\end{equation}
where $G_1(x,y) =\frac{1}{2} (Y_1 (x, y) + Y_2 (x, y))$ and 
$G_2(x,y) = \frac{1}{2}(Y_1(x, y) - Y_2 (x, y))$.

Our main result reads as follows.

\begin{theorem}\label{t1}
Assume that for $i=1, 2$ the polynomials $F_{i}(x)$ and $g_{i}(x)$
have degree $n\geq 1$ and $m\geq 1$ respectively, and that $l>1$ is
even.  Then for $|\varepsilon|$ sufficiently small there are discontinuous
Lienard polynomial  differential systems \eqref{lep} having at least
$\max\{n,m\}$ limit cycles bifurcating from the periodic orbits of
the linear center $\dot{x}=y$, $\dot{y}=-x$.
\end{theorem}

Taking into account Theorem \ref{t1} and  roughly
speaking,  we can say that the non-smooth classical Lienard polynomial
differential systems can have at least $\max\{n,1\}$ limit cycles;
i.e.  the double number of limit cycles than the
smooth ones which at least have $[(n-1)/2]+2$ for $n\geq 6$.
Comparing the mentioned result from \cite{LMT}, that smooth
generalized Lienard polynomial differential systems have at least
$[(n+m-1)/2]$ limit cycles with Theorem \ref{t1}, we can say that
the non-smooth generalized Lienard  polynomial differential systems
can have at least one more limit cycle than the smooth ones. Of
course all these comparisons are done with the present known
results.

\section{Averaging theory for first-order discontinuous
differential systems} \label{s:averaging}

The first-order averaging theory  developed for discontinuous
differential systems in \cite{DLM} is presented in this section. It
is summarized as follows.

\begin{theorem}\label{averaging}
We consider the  discontinuous differential system
\begin{equation}\label{MRs1}
x'(t)=\varepsilon F(t,x)+\varepsilon^2R(t,x,\varepsilon),
\end{equation}
with
\begin{gather*}
F(t,x)=F_1(t,x)+\operatorname{sgn}(h(t,x))F_2(t,x),\\
R(t,x,\varepsilon)=R_1(t,x,\varepsilon)+\operatorname{sgn}(h(t,x))R_2(t,x,\varepsilon),
\end{gather*}
where $F_1,F_2:\mathbb{R}\times D\to\mathbb{R}^n$,
$R_1,R_2:\mathbb{R}\times D\times(-\varepsilon_0,\varepsilon_0)\to\mathbb{R}^n$ and
$h:\mathbb{R}\times D\to \mathbb{R}$
are continuous functions, $T$--periodic in the variable $t$ and $D$
is an open subset of $\mathbb{R}^n$. We also suppose that $h$ is a $C^1$
function having $0$ as a regular value. Denote by $\mathcal{M}= h^{-1}(0)$,
by $\Sigma=\{0\}\times D\nsubseteq \mathcal{M}$, by
$\Sigma_0= \Sigma\backslash\mathcal{M} \neq \varnothing$, and its elements by
$z\equiv(0,z)\notin \mathcal{M}$.

Define the averaged function $f:D\to\mathbb{R}^n$ as
\[\label{MRf1}
f(x)=\int_0^T F(t,x) dt.
\]
We assume the following three conditions.
\begin{itemize}
\item[(i)] $F_1,\,F_2,\,R_1,\,R_2$ and $h$ are locally $L$-Lipschitz
with respect to $x$;

\item[(ii)] for $a\in \Sigma_0$ with $f(a)=0$, there exist a neighborhood
$V$ of $a$ such that $f(z)\neq 0$ for all $z\in\overline{V}
\backslash\{a\}$ and $d_B(f,V,a)\neq 0$, (i.e. the Brouwer degree of
$f$ at $a$ is not zero).

\item[(iii)] If $\partial h/\partial t(t_0,z_0)=0$ for some  $(t_0,z_0)\in \mathcal{M}$,
then
\[
\big(\langle\nabla_x h,F_1\rangle^2-\langle\nabla_x h,F_2\rangle^2\big)(t_0,z_0)> 0.
\]
\end{itemize}
Then, for $|\varepsilon|>0$ sufficiently small, there exists a $T$-periodic
solution $x(\cdot,\varepsilon)$ of system \eqref{MRs1} such that $x(t,\varepsilon)\to
a$ as $\varepsilon\to 0$.
\end{theorem}

\begin{remark} \label{rmk1} \rm
We note that if the function $f(z)$ is $C^1$ and the Jacobian of $f$
at $a$ is not zero, then $d_B(f,V,a)\neq 0$. For more details on the
Brouwer degree see \cite{B} and \cite{Ll}.
\end{remark}

\section{Proof of Theorem \ref{t1}}\label{proof1}

The discontinuous Lienard differential systems \eqref{lep} in polar
coordinates $(r, \theta)$ become
\begin{gather*}
\dot{r} = -\varepsilon (\cos \theta\,F_i (r \cos \theta) + \sin \theta\,g_i(r \cos
\theta)),\\ 
\dot{\theta} = -1+\frac{\varepsilon}{r} (\sin \theta\,F_i (r
\cos \theta) - \cos \theta\,g_i(r \cos \theta)),
\end{gather*}
with $i=1$ if $\operatorname{sgn} (h(r\cos\theta, r\sin \theta))>0$ and $i=2$
if $\operatorname{sgn} \left(h(r\cos\theta, r\sin \theta)\right)<0$. 
Taking the angle $\theta$ as new independent variable  the discontinuous 
differential systems become
\begin{equation}\label{8}
\dot{r} = \varepsilon (\cos \theta\,F_i (r \cos \theta) + \sin \theta\,g_i(r \cos
\theta))+O(\varepsilon^2).
\end{equation}
This discontinuous differential system  is studied under the
assumptions of Theorem \ref{averaging}, taking
$$
t = \theta, \quad T = 2\pi, \quad x = r, \quad
\mathcal{M}= h^{-1}(0) = \cup_{k=0}^{\frac{l}{2}-1}
\{(\theta,r):\theta = \alpha +\frac{2k\pi}{l}, r>0\}.
$$
So according to Theorem \ref{averaging} we must study the zeros 
of the averaged function
\begin{equation}\label{ave}
\begin{aligned}
f(r)&=\sum_{k=1}^{l}\Big[\int_{\alpha+
\frac{2(k-1)\pi}{l}}^{\alpha+\frac{(2k-1)\pi}{l}}(
\cos \theta\,F_1(r\cos \theta)+\sin \theta\,g_1(r\cos \theta))\,d\theta
\\
&\quad+ \int_{\alpha+\frac{(2k-1)\pi}{l}}^{\alpha+\frac{2k\pi}{l}}
(\cos \theta\,F_2(r\cos \theta)+\sin \theta\,g_2(r\cos \theta))\,d\theta
\Big].
\end{aligned}
\end{equation}
Denoting
\[
F_1(x)=\sum_{i=0}^na_i x^i, \quad
F_2(x)=\sum_{i=0}^n b_i x^i, \quad
g_1(x)=\sum_{i=0}^{m}c_i x^i,\quad 
g_2(x)=\sum_{i=0}^{m}d_i x^i
\]
we have
\begin{align*}
f(r)&=\sum_{k=1}^{l}\Big[\int_{\alpha+
\frac{2(k-1)\pi}{l}}^{\alpha+\frac{(2k-1)\pi}{l}}\Big(
\sum_{i=0}^{n }a_i r^i\cos^{i+1} \theta+ \sum_{i=0}^{m}c_i r^i\cos^{i}
\theta \sin \theta \Big)\,d\theta \\
&\quad + \int_{\alpha+\frac{(2k-1)\pi}{l}}^{\alpha+
\frac{2k\pi}{l}}\Big(\sum_{i=0}^nb_i
r^i\cos^{i+1} \theta+ \sum_{i=0}^{m}d_i r^i\cos^{i} \theta \sin \theta
\Big)\,d\theta \Big].
\end{align*}

To calculate the exact expression of $f(r)$ we use 
\cite[formulae 2.513 3 and 2.513 4]{GR}:
\begin{gather*}
 \int \cos^{2m} \theta \,d\theta 
 = \frac{1}{2^{2m}}\begin{pmatrix} 2m\\m\end{pmatrix} \theta 
 + \frac{1}{2^{2m-1}}  \sum_{j=0}^{m-1} \begin{pmatrix} 2m\\j\end{pmatrix}
\frac{\sin(2m-2j)\theta}{2m-2j},\\ 
 \int \cos^{2m+1} \theta \,d\theta 
 = \frac{1}{2^{2m}} \sum_{j=0}^{m} \begin{pmatrix}
2m+1\\j\end{pmatrix} \frac{\sin(2m-2j+1)\theta}{2m-2j+1}.
\end{gather*}
Thus we have
\[
f(r)=  f_1(r)+f_2(r)+f_3(r)+f_4(r),
\]
where
\begin{align*}
f_1(r)
&= \sum_{k=1}^{l}\int_{\alpha+
\frac{2(k-1)\pi}{l}}^{\alpha+\frac{(2k-1)\pi}{l}} 
\Big(\sum_{i=0}^n a_i r^i\cos^{i+1} \theta  \Big)\,d\theta
\\
&=\sum_{k=1}^{l}\Big[\sum_{i=1,\,i\text{ odd}}^n a_ir^i
\Big[\frac{1}{2^{i+1}} \begin{pmatrix} i+1 \\ (i+1)/2 \end{pmatrix} \frac{\pi}{l}
+\frac{1}{2^i} \sum_{j=0}^{\frac{i-1}{2}} \begin{pmatrix} i+1\\j\end{pmatrix}
\varphi_{i,j,k}]
\\
&\quad  + \sum_{i=0,\,i\text{ even}} ^n  \frac{a_ir^i}{2^i}
\sum_{j=0}^{\frac{i}{2}}\begin{pmatrix} i+1\\j\end{pmatrix}
\varphi_{i,j,k}\Big]
\\
&= \sum_{k=1}^{l}[\sum_{i=1,\, i\text{ odd}}^n
\frac{a_ir^i}{2^{i+1}} \begin{pmatrix} i+1 \\
(i+1)/2\end{pmatrix} \frac{\pi}{l}+\sum_{i=0}^n
\frac{a_ir^i}{2^i}\sum_{j=0}^{[\frac{i}{2}]}
\begin{pmatrix} i+1\\j\end{pmatrix} \varphi_{i,j,k}],
\end{align*}
with
\begin{gather*}
\varphi_{i,j,k}
=\frac{\sin\left((i-2j+1)\left(\alpha+\frac{(2k-1)\pi}{l}
\right)\right)
-\sin\left((i-2j+1)\left(\alpha+\frac{2(k-1)\pi}{l}\right)
\right)}
{i-2j+1}\neq 0;
\\
 f_2(r)= \sum_{k=1}^{l}\int_{\alpha+\frac{
2(k-1)\pi}{l}}^{\alpha+\frac{(2k-1)\pi}{l}} \Big(\sum_{i=0}^{m}c_i r^i
\cos^{i} \theta \sin \theta \Big)\,d\theta
= -\sum_{k=1}^{l}\sum_{i=0}^{m}
\frac{c_i r^i}{i+1}\phi_{i,k},
\end{gather*}
with
\begin{gather*}
\phi_{i,k}=\cos^{i+1}\Big(\alpha+\frac{(2k-1)\pi}{l}\Big)
-\cos^{i+1}\Big(\alpha+\frac{2(k-1)\pi}{l}\Big)\neq 0;
\\
\begin{aligned}
f_3(r)&= \sum_{k=1}^{l}\int_{\alpha+
\frac{2(k-1)\pi}{l}}^{\alpha+\frac{(2k-1)\pi}{l}} \Big(
\sum_{i=0}^nb_i r^i\cos^{i+1} \theta  \Big)\,d\theta
\\
&= \sum_{k=1}^{l}\Big[\sum_{i=1,\,i \text{ odd}} ^n
\frac{b_ir^i}{2^{i+1}} \begin{pmatrix} i+1 \\ (i+1)/2\end{pmatrix}
\frac{\pi}{l}+\sum_{i=0}^n  \frac{b_ir^i}{2^i}
\sum_{j=0}^{[\frac{i}{2}]} \begin{pmatrix}
i+1\\j\end{pmatrix} \psi_{i,j,k}\Big],
\end{aligned}
\end{gather*}
with
\begin{gather*}
\psi_{i,j,k}=\frac{\sin\left((i-2j+1)\big(\alpha+\frac{2k\pi}{l}\big)\right)-
\sin\left((i-2j+1)\big(\alpha+\frac{(2k-1)\pi}{l}\big)\right)}{i-2j+1}\neq 0;
\\
f_4(r)= \sum_{k=1}^{l}\int_{\alpha+\frac{(2k-1)
\pi}{l}}^{\alpha+\frac{2k\pi}{l}} \Big(\sum_{i=0}^{m}d_i r^i\cos^{i}
\theta \sin \theta \Big) \,d\theta
= -\sum_{k=1}^{l}\sum_{i=0}^{m}\frac{d_i r^i}{i+1}\zeta_{i,k},
\end{gather*}
with
\[
\zeta_{i,k}=\cos^{i+1}\big(\alpha+\frac{2k\pi}{l}\big)-
\cos^{i+1}\big(\alpha+\frac{(2k-1)\pi}{l}\big)\neq 0.
\]
Thus
\begin{align*}
f(r)&= \sum_{k=1}^{l}\Big[\sum_{i=1,\, i \text{ odd}}^n
\frac{r^i}{2^{i+1}} \begin{pmatrix} i+1\\(i+1)/2\end{pmatrix}
\frac{\pi}{l}(a_i+b_i) 
\\
&\quad + \sum_{i=0}^n \frac{r^i}{2^i} \sum_{j=0}^{[\frac{i}{2}]}
\begin{pmatrix} i+1\\j\end{pmatrix} (a_i\varphi_{i,j,k}+b_i\psi_{i,j,k})
-\sum_{i=0}^{m}\frac{r^i}{i+1}(c_i \phi_{i,k} +d_i \zeta_{i,k})\Big].
\end{align*}

The function $f(r)$ is a polynomial in the variable $r$ of degree
$\max\{n,m\}$ therefore $f(r)$ has at most $\max\{n,m\}$ positive
roots. If $r^*$ is a simple zero of $f(r)$; i.e. $f(r^*)=0$ and
$\frac{df}{dr}\big|_{r=r^*} \neq 0$, then the Brouwer
degree $d_B(f,V,r^*)\neq 0$ being $V$ a convenient open neighborhood
of $r^*$ (see Remark \ref{rmk1}). We can choose the coefficients
$a_{i}$, $b_i$, $c_i$ e $d_i$ in such a way that $f(r)$ has exactly
$\max\{n,m\}$ simple positive roots. Hence  Theorem \ref{t1} is
proved.


\section{Examples}

In this section we illustrate Theorem \ref{t1} by studying the
existence of $2\pi$-periodic solutions for two non-smooth Lienard
polynomial  differential systems.


\begin{example} \label{examp1} \rm 
 We consider $l=2$ and $\alpha=0$. Thus the
function $h:\mathbb{R}^2\to \mathbb{R}$ is defined by $h(x,y)=y$ and
$h^{-1}(0)=\{(x,y)\in \mathbb{R}^2: y=0\}$. System \eqref{lep} becomes
\begin{equation}\label{lep1}
\dot X = Z(x,y) = \begin{cases}
Y_1(x,y) &   \text{if }  y > 0,\\
Y_2(x,y) &   \text{if }  y < 0,
\end{cases}
\end{equation}
where 
\begin{gather*}
F_1=1+x+x^2+\big(\frac{1}{9\pi}-1\big)x^3, \quad
F_2=1+\big(\frac{11}{12\pi}-1\big)x+x^2+x^3, \\
g_1=\frac{7}{8}+x+\frac{5}{8}x^2,\quad
g_2=1+x+x^2.
\end{gather*}
 Thus we have
\begin{gather*}
Y_1(x,y)=\begin{pmatrix}
 y-\varepsilon \big(1+x+x^2+\big(\frac{1}{9\pi}-1\big)x^3 \big)\\
-x-\varepsilon \big(\frac{7}{8}+x+\frac{5}{8}x^2\big)
\end{pmatrix} ,
\\
Y_2(x,y)=\begin{pmatrix}
 y-\varepsilon \big(1+\big(\frac{11}{12\pi}-1\big)x+x^2+x^3\big) \\
-x-\varepsilon (1+x+x^2)
\end{pmatrix}.
\end{gather*}
The averaging function \eqref{ave} is 
% \label{ave2}
\begin{align*}
f(r)
&=\int_{0}^{\pi}\left( \cos \theta \, F_1(r\cos \theta)
 +\sin \theta \, g_1(r\cos \theta) \right)\,d\theta\\
&\quad +\int_{\pi}^{2\pi}\left(  \cos \theta \,
F_2(r\cos \theta)+\sin \theta \, g_2(r\cos \theta) \right)\,d\theta \\
&=-6 + 11 r - 6 r^2 + r^3.
\end{align*}
The zeros of $f(r)$ are $r=1$, $r=2$ and $r=3$, and they are simple.
Hence, by Theorem \ref{t1} it follows that for $\varepsilon\neq 0$
sufficiently small the discontinuous differential system
\eqref{lep1} has three periodic solutions.
\end{example}

\begin{example} \label{examp2} \rm
We consider $l=4$ and $\alpha=\pi/4$. Thus the
function $h:\mathbb{R}^2\to \mathbb{R}$ is defined by $h(x,y)=(y-x)(y+x)$
and $h^{-1}(0)= \left\{(x,y):y = x\right\}\cup \left\{(x,y):y =
-x\right\}$. System \eqref{lep} becomes
\begin{equation}\label{lep2}
\dot X = Z(x,y) = \begin{cases}
Y_1(x,y) &   \text{if }  (y-x)(y+x) > 0,\\
Y_2(x,y) &   \text{if }  (y-x)(y+x) < 0,
\end{cases}
\end{equation}
where $F_1=x^2$, $F_2=12\sqrt{2}\pi + \frac{72\sqrt{2}}{5}x^2$,
$g_1=1+x+x^2+x^3$ and $g_2=-88\pi x -\frac{32\pi}{3} x^3$. Thus we
have
\[
Y_1(x,y)=\begin{pmatrix}
 y-\varepsilon  x^2 \\
-x-\varepsilon ( 1+x+x^2+x^3)
\end{pmatrix},
\quad
Y_2(x,y)=\begin{pmatrix}
 y-\varepsilon \big(12\sqrt{2}\pi + \frac{72\sqrt{2}}{5}x^2\big) \\
-x-\varepsilon \big(-88\pi x -\frac{32\pi}{3}x^3 \big)
\end{pmatrix}.
\]
The averaging function \eqref{ave} is 
%\label{ave2}
\begin{align*}
f(r)&=\int_{\pi/4}^{3\pi/4} \left(
 \cos \theta \, F_1(r\cos \theta)+\sin \theta \, g_1(r\cos \theta) \right)\,d\theta \\
&\quad +\int_{3\pi/4}^{5\pi/4}\left(
  \cos \theta \, F_2(r\cos \theta)+\sin \theta \, g_2(r\cos \theta) \right)\,d\theta \\
&\quad + \int_{5\pi/4}^{7\pi/4}\left(
 \cos \theta \, F_1(r\cos \theta)+\sin \theta \, g_1(r\cos \theta) \right)\,d\theta \\
&\quad + \int_{7\pi/4}^{9\pi/4}\left(
  \cos \theta \, F_2(r\cos \theta)+\sin \theta \, g_2(r\cos \theta) \right)\,d\theta \\
&=-6 + 11 r - 6 r^2 + r^3.
\end{align*}
The zeros of $f(r)$ are $r=1$, $r=2$ and $r=3$, and they are simple.
Hence, by Theorem \ref{t1} it follows that for $\varepsilon\neq 0$
sufficiently small the discontinuous differential system
\eqref{lep2} has three periodic solutions.
\end{example}

\subsection*{Acknowledgments}
The first author is partially supported by a MICINN/FEDER grant MTM
2008--03437, an AGAUR grant number 2009SGR-0410, an ICREA Academia,
and FP7-PEOPLE-2012-IRSES-316338 and 318999. The second author is
partially supported by a FAPESP-BRAZIL grant 2012/20884-8. Both
authors are also supported by the joint project CAPES--MECD grant
PHB-2009-0025-PC.

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\end{document}