\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 228, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/228\hfil Arbitrary number of limit cycles]
{Arbitrary number of limit cycles for  planar discontinuous piecewise 
 linear differential systems with two zones}

\author[D. d. C. Braga, L. F. Mello \hfil EJDE-2015/228\hfilneg]
{Denis de Carvalho Braga, Luis Fernando Mello}

\address{Denis de Carvalho Braga \newline
Instituto de Matem\'atica e Computa\c{c}\~ao,
Universidade Federal de Itajub\'a,
Avenida BPS 1303, Pinheirinho, CEP 37.500--903,
Itajub\'a, MG, Brazil}
\email{braga@unifei.edu.br}

\address{Luis Fernando Mello \newline
Instituto de Matem\'atica e Computa\c{c}\~ao,
Universidade Federal de Itajub\'a,
Avenida BPS 1303, Pinheirinho, CEP 37.500--903,
Itajub\'a, MG, Brazil.\newline
Tel: 00-55-35-36291217, Fax: 00-55-35-36291140}
\email{lfmelo@unifei.edu.br}

\thanks{Submitted April 28, 2015. Published September 2, 2015.}
\subjclass[2010]{34C05, 34C25, 37G10}
\keywords{Piecewise linear differential system; limit cycle;
\hfill\break\indent non-smooth differential system}

\begin{abstract}
 For any given positive integer $n$ we show the existence of a class of
 discontinuous piecewise linear differential systems with two zones in the
 plane having exactly $n$ hyperbolic limit cycles. Moreover, all the points
 on the separation boundary between the two zones are of sewing type,
 except the origin which is the only equilibrium point.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction and statement of main results}\label{S:1}

One of the most challenging problems in the qualitative theory of planar 
ordinary differential equations is the second part of the classical 
16th Hilbert problem: the determination of an upper bound for the number 
of limit cycles (and their relative positions) for the class of polynomial 
vector fields of degree $n$. This problem remains unsolved if $n \geq 2$.
 The case $n = 1$, that is for the class of planar linear vector fields the 
problem has a trivial answer. However, this problem presents a surprising 
richness when adapted to the class of the planar piecewise linear systems.

Planar piecewise linear differential systems are widely studied nowadays because of 
their applicability in several branches of science. 
A landmark of such study was the work of Andronov et al. \cite{A}. 
There is an expectation that these systems can present all the dynamical 
behaviors of the classical nonlinear differential systems.

In this article, we study the existence, number, stability and distribution 
of limit cycles for a class of piecewise linear differential systems in the plane. 
These issues must be studied taking into account the following aspects: 
the number and stability of equilibrium points as well as their locations 
with respect to the separation boundary $\mathcal{L}$ (which defines the number
of zones) and the behavior of the linear vector fields on $\mathcal{L}$.
Usually, the points of discontinuity on the separation boundary $\mathcal{L}$
are classified as sewing, sliding or tangency points.
Here the term sliding is used in a broad sense meaning sliding or escaping points. 
See \cite{LTT} and the references therein for more details.

Without being exhaustive we present below some
results relative to the study of limit cycles in piecewise linear differential 
systems in the plane. For the simplest case, that is for planar piecewise 
linear differential systems with two zones separated by a straight line 
$\mathcal{L}$ and with only one equilibrium point $p \in \mathcal{L}$,
we have the following results:
\begin{itemize}
\item[(i)] With the additional hypothesis of the continuity on 
$\mathcal{L} \backslash \{ p \}$, there is at most one limit cycle \cite{FPRT};

\item[(ii)] With the additional hypothesis that the points on 
$\mathcal{L} \backslash \{ p \}$ are of sewing type, there is at most one 
limit cycle \cite{MT};

\item[(iii)] With the additional hypothesis of the existence of a sliding 
segment on $\mathcal{L} \backslash \{ p \}$, there are examples with two 
limit cycles surrounding the sliding segment \cite{P}.
\end{itemize}

The answer to the existence and the number of limit cycles in piecewise 
linear systems with more than two zones in the plane was given in \cite{LP0}. 
In that article the authors provided an example of a planar piecewise linear
Li\'enard system with an arbitrary number of limit cycles all of them
hyperbolic. More precisely, given a positive integer $n$ the authors
showed, using the averaging theory, that the system under consideration 
can have at least $n$ limit cycles in the strip $|x| \leq 2n+2$ for a 
parameter $\epsilon \in \mathbb R$ sufficiently small. 
Llibre, Ponce and Zhang \cite{LPX} proved a conjecture of \cite{LP0} in which 
the upper bound for the number of limit cycles can be achieved for a fixed $n$.

Recent studies suggest that three is the maximum number of limit cycles 
for planar discontinuous piecewise linear differential systems with two 
zones separated by a straight line $\mathcal{L}$ and only one equilibrium 
point $p \notin \mathcal{L}$. Numerical examples which support this statement 
can be found in \cite{DL} and \cite{LP}.

The separation boundary $\mathcal{L}$ between the two zones plays an important 
role in planar discontinuous piecewise linear differential systems with only 
one equilibrium point $p \notin \mathcal{L}$. The article \cite{DL1} 
exhibits an example of such a system with seven limit cycles having 
$\mathcal{L}$ as a polygonal curve.

Here we show that a planar discontinuous piecewise linear
differential system with two zones can have an arbitrary number of
limit cycles, that is, the main idea in \cite{LPX} still remains for
only two zones. So, we study the following class of discontinuous
piecewise linear differential with two zones in the plane
\begin{equation}\label{eq:SLPP}
X'=\begin{cases}
G^{-}X,& \mathcal{H}(X)<0,  \\
G^{+}X,& \mathcal{H}(X) \geq 0,
\end{cases}
\end{equation}
where the prime denotes derivative with respect to the independent
variable $t$, called here the time, $X = (x,y) \in \mathbb{R}^2$ and
\begin{equation}\label{eq:A}
G^{\pm}=\begin{pmatrix}
g_{11}^{\pm} & g_{12}^{\pm}  \\
g_{21}^{\pm} & g_{22}^{\pm}
\end{pmatrix},
\end{equation}
are matrices with real entries satisfying the following assumptions:
\begin{itemize}
\item[(A1)] $g_{12}^{\pm} <0$;

\item[(A2)] $G^{-}$ has complex eigenvalues with negative real
parts, $\lambda_{1,2}^{-}=\gamma^{-}\pm i\omega^{-}$,
while $G^{+}$ has complex eigenvalues with positive real parts,
$\lambda_{1,2}^{+}=\gamma^{+}\pm i\omega^{+}$, where
$\gamma^{\pm},\omega^{\pm} \in \mathbb{R}$ and $\omega^{\pm}>0$.
\end{itemize}
In addition,
\begin{itemize}
\item[(A3)] the function $\mathcal{H}$ is at least continuous and the set
$\mathcal H^{-1}(0)$ divides the plane in two unbounded components,
that is the function $\mathcal H$ implicitly defines a simple planar
curve homeomorphic to the real line and whose trace is unbounded.
\end{itemize}

A member of the class \eqref{eq:SLPP} will be denoted by
$(G^{-},G^{+},\mathcal{H})$. Note that the hypothesis (A3)
ensures the existence of only two zones whose separation (boundary) set
is defined by $\mathcal{L}_{\mathcal{H}}=\{X \in \mathbb{R}^2:\mathcal{H}(X)=0\}$.

Our goal is to build a suitable function $\Psi$ satisfying the
assumptions on $\mathcal{H}$ and to choose matrices $G^{-}$ and
$G^{+}$ in the Jordan normal forms $J^{\pm}$ such that, for any
positive integer $n$, the system $(J^{-},J^{+},\Psi)$ has exactly
$n$ hyperbolic limit cycles. Furthermore, $0 = (0,0) \in \mathcal{H}^{-1}(0)$
and $\mathcal{L}_{\mathcal{H}}\backslash \{0\}$ is a sewing set. As far
as we know, it is the first example of such systems. We prove the
following main theorem.

\begin{theorem}\label{thm:main}
Given any positive integer $n$ there is a planar piecewise linear
differential system with two zones $(J^{-}, J^{+}, \Psi)$ having $n$
hyperbolic limit cycles.
\end{theorem}

Novaes and Ponce \cite{DN} obtained examples of planar
piecewise linear differential systems with two zones having $n$ limit
cycles, for every positive integer $n$. However, the limit cycles
obtained can be non-hyperbolic.

This article is structured as follows. 
In Section \ref{S:2} we present the proof of Theorem \ref{thm:main} 
and we give an example of such a system with ten limit cycles. 
In the last section we make some concluding remarks.

\section{Proof of main results}\label{S:2}

We consider the case in which the matrices $G^{+}$ and $G^{-}$ are
in the Jordan normal forms; that is,
\begin{equation}\label{eq:J}
J^{\pm}=\begin{pmatrix}
\gamma^{\pm} & -\omega^{\pm} \\
\omega^{\pm} & \gamma^{\pm}
\end{pmatrix}\,.
\end{equation}
The solutions of $X' = J^{-} X$ will be denoted by
\[
(t,X_0) \mapsto X^{-}(t,X_0)=(x^{-}(t,X_0),y^{-}(t,X_0)),
\]
while the solutions of $X' = J^{+} X$ will be denoted by
\[
(t,X_0) \mapsto X^{+}(t,X_0)=(x^{+}(t,X_0),y^{+}(t,X_0)),
\]
where
\begin{equation}\label{eq:JF}
\begin{gathered}
x^{\pm}(t,X_0)=e^{\gamma^{\pm} t}\left(\cos(\omega^{\pm} t)x_0
 -\sin(\omega^{\pm} t)y_0\right), \\
y^{\pm}(t,X_0)=e^{\gamma^{\pm} t}\left(\sin(\omega^{\pm} t)x_0
 +\cos(\omega^{\pm} t)y_0\right).
\end{gathered}
\end{equation}

In Lemma \ref{lem:rho} we present a result associated with the
system $(J^{-},J^{+},\Phi)$, where
\[
X \mapsto \Phi(X)=x-\phi(y)
\]
and
\begin{equation}\label{eq:Fk}
y \mapsto \phi(y)=\rho v(y),
\end{equation}
with $\rho>0$, $s \in \mathbb{R} \mapsto v(s)=s \: u(s)$ and
\[
s \in \mathbb{R} \mapsto u(s)= 
\begin{cases}
0,& s< 0, \\
1,& s \geq 0
\end{cases}
\]
is the unit step function.

\begin{figure}
\begin{center}
 \includegraphics[width=0.6\textwidth]{fig1}
\end{center}
\caption{Displacement function $(y_0,\rho) \mapsto \delta_{\phi}(y_0,\rho)$.}
\label{fig:DF}
\end{figure}

\begin{lemma}\label{lem:rho}
Let $\eta^{-}<0$ and $\eta^{+}>0$ be numbers satisfying
$-\eta^{-}<\eta^{+}<-3\eta^{-}$,
where $\eta^{\pm}=\gamma^{\pm}/\omega^{\pm}$, and define
\[
\rho_c=\tan\Big(\pi\frac{\eta^{+}+\eta^{-}}{\eta^{+}-\eta^{-}}\Big).
\]
Then, the origin of the system $(J^{-},J^{+},\Phi)$ is:
\begin{itemize}
\item[(a)] An unstable focus, if $0<\rho<\rho_c$,
\item[(b)] A stable focus, if $\rho>\rho_c$,
\item[(c)] A center, if $\rho=\rho_c$.
\end{itemize}
\end{lemma}

\begin{proof} 
Let $X_0=(x_0,y_0)=(\phi(y_0),y_0) \in \mathcal{L}_{\Phi}$ be any
initial condition with $y_0 > 0$. The displacement function is defined by
\begin{equation}\label{eq:DF}
(y_0,\rho) \mapsto \delta_{\phi}(y_0,\rho)=
y^{+}(-\tau^{+},X_0)-y^{-}(\tau^{-},X_0),
\end{equation}
where $\tau^{-}>0$ is the smallest time such that
$X^{-}(\tau^{-},X_0)\in \mathcal{L}_{\Phi}$,
and $\tau^{+}>0$ is the smallest time such that
$X^{+}(-\tau^{+},X_0) \in \mathcal{L}_{\Phi}$.
See Figure \ref{fig:DF}.

From \eqref{eq:JF} and \eqref{eq:Fk}, the time $\tau^{-}$
is the solution of the equation $x^{-}(\tau^{-},X_0)=0$ and is given
by
\begin{equation} \label{eq:taum}
\tau^{-}=\tau^{-}(y_0,\rho)=\frac{1}{\omega^{-}}
\Big(\arctan\big(\frac{x_0}{y_0}\big)+\pi\Big),
\end{equation}
and $\tau^{+}$ is the solution of $x^{+}(-\tau^{+},X_0)=0$, and is given by
\begin{equation}\label{eq:tauM}
\tau^{+}=\tau^{+}(y_0,\rho)=-\frac{1}{\omega^{+}}
\Big(\arctan\big(\frac{x_0}{y_0}\big)-\pi\Big)\,.
\end{equation}
Thus,
\begin{equation}\label{eq:ymM}
\begin{gathered}
y^{-}(\tau^{-},X_0)=-e^{\gamma^{-}\tau^{-}}\sqrt{x_0^2+y_0^2}, \\
y^{+}(-\tau^{+},X_0)=-e^{-\gamma^{+}\tau^{+}}\sqrt{x_0^2+y_0^2},
\end{gathered}
\end{equation}
and, therefore,
\[
(y_0,\rho) \mapsto \delta_{\phi}(y_0,\rho)
=\Delta_{\phi}(y_0,\rho)\sqrt{\phi(y_0)^2+y_0^2},
\]
where
\begin{equation}\label{eq:DFC}
(y_0,\rho) \mapsto \Delta_{\phi}(y_0,\rho)
 =  \Big(e^{\gamma^{-}\tau^{-}}-e^{-\gamma^{+}\tau^{+}}\Big)
 =  e^{-\gamma^{+}\tau^{+}}\Big(e^{\gamma^{-}\tau^{-}+\gamma^{+}\tau^{+}}-1\Big).
\end{equation}
From \eqref{eq:taum} and \eqref{eq:tauM}, we have
\begin{equation}\label{eq:AR}
\rho \mapsto a(\rho)
= \gamma^{-}\tau^{-}+\gamma^{+} \tau^{+}
= \pi (\eta^{+}+\eta^{-})-(\eta^{+}-\eta^{-})\arctan(\rho)
\end{equation}
and $a(\rho)=0$ implies
\[
\rho=\rho_c=\tan\Big(\pi \frac{\eta^{+}+\eta^{-}}{\eta^{+}-\eta^{-}}\Big).
\]
If $-\eta^{-}<\eta^{+}<-3\eta^{-}$, then $\rho_c>0$. So, for any
$y_0>0$ such that
\[
x_0=\phi(y_0)=\rho_cy_0u(y_0)=\rho_cy_0,
\]
$\delta_{\phi}(y_0,\rho_c)\equiv 0$.
This proves item (c). Items (a) and (b) follow from
$x_0=\rho v(y_0)$ and \eqref{eq:AR} since $s \mapsto \arctan(s)$ is
a monotonically increasing function.
\end{proof}

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig2}
\end{center}
\caption{The graphs of the functions $y \mapsto \phi(y)$ and 
$y \mapsto \psi(y)$ are illustrated by continuous black and orange lines, 
respectively.}
\label{fig:PP}
\end{figure}

Given an integer number $n \geq 1$, consider the  finite
sequences $\{u_l\}_{l \in N}$ and $\{v_l\}_{l \in N}$, where
\begin{equation}\label{eq:uv}
\begin{gathered}
u_l  =  (2 l-3)r u(l-2), \\
v_l  =  \frac{1}{\rho_c}\left(u_l+(-1)^l \varepsilon  u(l-2)\right),
\end{gathered}
\end{equation}
for $l \in N=\{1,2,\ldots,n+2\}$, $r,\varepsilon \in \mathbb{R}$ 
satisfying $0<\varepsilon<r$. So $v_1<v_2<\cdots<v_{n+2}$.
Consider
\[
X \mapsto \Psi(X)=x-\psi(y),
\]
where
\begin{equation}\label{eq:h}
y \mapsto \psi(y)
=u_1+\sum_{k=1}^{n+1} \alpha_k\left(v(y-v_{k})-\beta_kv(y-v_{k+1})\right),
\end{equation}
and the real numbers $\alpha_k$ and $\beta_k$ are given by
\begin{gather*}
\alpha_k  =  \frac{u_{k+1}-u_k}{v_{k+1}-v_{k}},\quad k=1,2,\ldots,n+1,  \\
\beta_k  =  \begin{cases}
1,& k=1,\ldots,n, \\
0,& k = n+1.
\end{cases}
\end{gather*}
The sets $\mathcal{L}_{\Phi}$ and $\mathcal{L}_{\Psi}$ have
intersections at the points $p_i=(x_i,y_i) \in \mathbb{R}^2$, with
\begin{equation}\label{eq:xy}
\begin{gathered}
x_i  =  2r  i, \\
y_i  =  \frac{1}{\rho_c}x_i=\frac{2r}{\rho_c}  i,
\end{gathered}
\end{equation}
for $i=1,\ldots,n$. In fact,
\begin{align*}
y \mapsto g(y) 
& =  \psi(y)-\phi(y)  \\
& =  u_1+ \sum_{k=1}^{n+1}
\alpha_k\left(v(y-v_{k})-\beta_kv(y-v_{k+1})\right)-\rho_c v(y)
\end{align*}
is a continuous function on the closed interval $[v_j,v_{j+1}]$
satisfying $g(v_1)=0$ and
\[
g(v_j)  =   \sum_{k=1}^{n+1}F(j,k) 
 =   \sum_{k=1}^{j-1}F(j,k)+F(j,j)+\sum_{k=j+1}^{n+1}F(j,k)-\rho_cv(v_j),
\]
for $j=2,\ldots,n+1$, where
\[
F(j,k)=\alpha_k(v(v_j-v_k)-\beta_kv(v_j-v_{k+1})).
\]
Since $\{v_l\}_{l \in N}$ is a monotone increasing finite sequence,
it follows that
\begin{align*}
g(v_j) 
&= \sum_{k=1}^{j-1}F(j,k)-\rho v(v_j)\\
&= \sum_{k=1}^{j-1} \alpha_k (v_{k+1}-v_k)-\rho_c v_j  \\
&= \sum_{k=1}^{j-1} (u_{k+1}-u_k)-\rho_c v_j=(1+2(j-2))r-\rho_c v_j \\
&=  (2j-3)r-\left((2j-3)r+(-1)^j\varepsilon \right)=(-1)^{j+1}\varepsilon,
\end{align*}
for $j=2,\ldots,n+1$.

Thus $g(v_{i+1})g(v_{i+2})=-\varepsilon^2<0$ for $i=1,\ldots,n$ and
by  Bolzano's Theorem the function $y \mapsto g(y)$ has at
least $n$ zeros. Since the function $y \mapsto \psi(y)$ is piecewise
linear and the union of straight lines of the form
\begin{equation}\label{eq:L}
y \in [v_l,v_{l+1}] \mapsto L_l(y)=\alpha_l(y-v_l)+u_l,
\end{equation}
for $l=1,\ldots,n+1$, there exist exactly $n$ zeros, that is there
exists a unique $y_i \in [v_{i+1},v_{i+2}]$ such that $g(y_i)=0$.
Moreover, it is easy to see that
\[
y_i=\frac{v_{i+1}+v_{i+2}}{2}=\frac{1}{\rho_c}\frac{u_{i+1}+u_{i+2}}{2},
\]
is such as in \eqref{eq:xy} for $i=1,\ldots,n$.

The $n$ points $p_i=(x_i,y_i) \in \mathbb{R}^2$ given by
\eqref{eq:xy} are zeros of the displacement function $y_0 \mapsto
\delta_{\psi}(y_0)$ associated with the system $(J^{-},J^{+},\Psi)$.

Here we studied only the case in which $\eta^{-}$ and
$\eta^{+}$ are such as in Lemma \ref{lem:rho} and in addition
$\eta^{+}<-1/\eta^{-}$ although other choices are possible.
The slope of the straight line $y \mapsto L_1(y)=\alpha_1 y$ is
\[
\alpha_1=\frac{1}{\rho_c}\left(1+\frac{\varepsilon}{r}\right).
\]
Thus, we take $\rho_c$ such that
\[
\frac{v_2-v_1}{u_2-u_1}=\frac{1}{\rho_c}\left(1+\frac{\varepsilon}{r}\right)
<\frac{2}{\rho_c}<-\frac{1}{\eta^{-}}
\]
and
\[
\frac{v_3-v_1}{u_3-u_1}=\frac{1}{\rho_c}\left(1-\frac{\varepsilon}{3r}\right)
>\frac{2}{3\rho_c}>\eta^{+}
\]
for $0 <\varepsilon<r$. Therefore,
\begin{equation}
\begin{gathered}
-\frac{1}{3\eta^{+}}  <  \eta^{-}<0,\\
-2\eta^{-}  <  \tan\Big(\pi\frac{\eta^{+}+\eta^{-}}{\eta^{+}-\eta^{-}}\Big)
<\frac{2}{3\eta^{+}}.
\end{gathered}
\end{equation}

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.6\textwidth]{fig3}
\end{center}
\caption{The set $\mathcal{R}$ is illustrated by the blue region. 
The continuous black, red and brown lines represent the graphs of the sets
$\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}_3$,
respectively.}
\label{fig:R}
\end{figure}

From the inequalities
\begin{gather*}
-\frac{1}{3\eta^{+}}  <  \eta^{-}<0,\quad
\eta^{-}   <  \eta^{+},\\
-\eta^{-}  <  \eta^{+}<-3\eta^{-},\quad
\eta^{+}  <  -1/\eta^{-},  \\
-2\eta^{-}  <  \tan\Big(\pi\frac{\eta^{+}+\eta^{-}}{\eta^{+}-\eta^{-}}\Big)
<\frac{2}{3\eta^{+}},
\end{gather*}
we obtain the following main functions
\begin{gather*}
s \mapsto h_1(s) =  -\frac{\pi-\arctan(2s)}{\pi+\arctan(2s)}s,\\
s \mapsto h_2(s) =  -\frac{\pi-\arctan\big(\frac{2}{3s}\big)}
 {\pi+\arctan\big(\frac{2}{3s}\big)}s,\\
s \mapsto h_3(s) =  -\frac{1}{3s}
\end{gather*}
and the set
\[
\mathcal{R}=\big\{(\eta^{-},\eta^{+}) \in \mathbb{R}^2:\eta^{+}
>h_1(\eta^{-}),\;\eta^{-}<h_2(\eta^{+}) \big\}.
\]
The set $\mathcal{R}$ is illustrated in Figure \ref{fig:R}. The
points $q_1$ and $q_2$ (displayed only with six decimals) are
given by
\begin{gather*}
q_1 =  (\eta_1^{-},\eta_1^{+})=(0,0) \in \mathcal{C}_1 \cap \mathcal{C}_2,\\
q_2 =  (\eta_2^{-},\eta_2^{+})=(-0.454479,0.733439) 
\in \mathcal{C}_1 \cap \mathcal{C}_2 \cap \mathcal{C}_3,
\end{gather*}
where
\begin{gather*}
\mathcal{C}_1 =  \{(\eta^{-},\eta^{+}) \in \mathbb{R}^2:\eta^{+}=h_1(\eta^{-})\},\\
\mathcal{C}_2 =  \{(\eta^{-},\eta^{+}) \in \mathbb{R}^2:\eta^{-}=h_2(\eta^{+})\},\\
\mathcal{C}_3 =  \{(\eta^{-},\eta^{+}) \in \mathbb{R}^2:\eta^{+}=h_3(\eta^{-})\}.
\end{gather*}

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig4}
\end{center}
\caption{The dashed red lines are the nullclines associated with $X'=J^{-}X$ and the
dashed blue ones are the nullclines associated with $X'=J^{+}X$. The
orange and black continuous lines are the graph of the sets
$\Psi^{-1}(0)$ and $\Phi^{-1}(0)$, respectively.}
\label{fig:DCV}
\end{figure}

It follows that the vector fields $J^{-}X$ and $J^{+}X$ are both
transversal to the set $\mathcal{L}_{\Psi}$ if $(\eta^{-},\eta^{+})
\in \mathcal{R}$. This means that the set
$\mathcal{L}_{\Psi}\backslash \{0\}$ is a sewing set according to
Figure \ref{fig:DCV}. Therefore if $X_0=(x_0,y_0)=(\psi(y_0),y_0)
\in \mathcal{L}_{\Psi}$ and employing the same previous notation,
there exists the smallest time $\tau^{-}>0$ such that
$X^{-}(\tau^{-},X_0)\in \mathcal{L}_{\Psi}$ or more precisely
$x^{-}(\tau^{-},X_0)=0$. In the same way there exists the smallest
time $\tau^{+}>0$ such that $X^{+}(-\tau^{+},X_0)\in
\mathcal{L}_{\Psi}$ or $x^{+}(-\tau^{+},X_0)=0$. Moreover, the times
$\tau^{-}$ and $\tau^{+}$ are the same as given in \eqref{eq:taum} and
\eqref{eq:tauM}; that is,
\begin{gather}\label{eq:taumpsi}
\tau^{-}=\tau^{-}(y_0)
=\frac{1}{\omega^{-}}\Big(\arctan\big(\frac{x_0}{y_0}\big)+\pi\Big), \\
\label{eq:tauMpsi}
\tau^{+}=\tau^{+}(y_0)
=-\frac{1}{\omega^{+}}\Big(\arctan\big(\frac{x_0}{y_0}\big)-\pi\Big).
\end{gather}
Thus, the following function is well defined
\begin{equation}\label{eq:DFPsi}
y_0 \mapsto \delta_{\psi}(y_0)=y^{+}(-\tau^{+},X_0)-y^{-}(\tau^{-},X_0),
\end{equation}
where $y^{-}(\tau^{-},X_0)$ and $y^{-}(\tau^{-},X_0)$ are such as 
in \eqref{eq:ymM}.

The function $y_0 \mapsto \delta_{\psi}(y_0)$ can be rewritten as
\[
y_0 \mapsto \delta_{\psi}(y_0)=\Delta_{\psi}(y_0)\sqrt{\psi(y_0)^2+y_0^2},
\]
where
\begin{equation}\label{eq:DPsi}
y_0 \mapsto \Delta_{\psi}(y_0)  
=  \Big(e^{\gamma^{-}\tau^{-}}-e^{-\gamma^{+}\tau^{+}}\Big) 
=  e^{-\gamma^{+}\tau^{+}}\Big(e^{\gamma^{-}\tau^{-}+\gamma^{+}\tau^{+}}-1\Big).
\end{equation}

From \eqref{eq:xy}, \eqref{eq:DPsi} and Lemma \ref{lem:rho}, it
follows that the system $(J^{-},J^{+},\Psi)$ has $n$ limit cycles
since $\delta_{\psi}(y_i)=0$, for $i=1,\ldots,n$.

To show that the $n$ limit cycles are all hyperbolic we take an open
interval $I_i \subset [v_{i+1},v_{i+2}]$ such that $y_i \in I_i$ for
$i=1,\ldots,n$, since the function $y \mapsto \psi(y)$ is not
differentiable at all the points in its domain. Thus, for $y_0 \in I_i$
the derivative of $y_0 \mapsto \delta_{\psi}(y_0)$ with respect to
$y_0$ is
\begin{equation}\label{eq:Ddelta}
\frac{d}{d y_0} \delta_{\psi}(y_0) 
= \frac{d}{d y_0}\Delta_{\psi}(y_0)\sqrt{\psi(y_0)^2+y_0^2}
 +\Delta_{\psi}(y_0) \frac{d}{d y_0}\sqrt{\psi(y_0)^2+y_0^2},
\end{equation}
where
\begin{align*}
y_0 \mapsto \frac{d}{d y_0}\Delta_{\psi}(y_0) 
&=  \frac{d}{d y_0}\Big(e^{\gamma^{-}\tau^{-}}-e^{-\gamma^{+}\tau^{+}}\Big)
 \\
&=   \gamma^{-}e^{\gamma^{-}\tau^{-}}\frac{d}{d y_0}\tau^{-}(y_0)
+\gamma^{+}e^{-\gamma^{+}\tau^{+}}\frac{d}{d y_0}\tau^{+}(y_0).
\end{align*}
From \eqref{eq:L} it results that
\[
x_0=L_{i+1}(y_0)=\alpha_{i+1}(y_0-v_{i+1})+u_{i+1}
\]
and the derivatives of \eqref{eq:taumpsi} and \eqref{eq:tauMpsi} with
respect $y_0$ are
\begin{gather*}
\frac{d}{d y_0}\tau^{-}(y_0) 
=  \frac{\alpha_{i+1}v_{i+1}-u_{i+1}}{\omega^{-}(L_{i+1}(y_0)^2+y_0^2)},\\
\frac{d}{d y_0}\tau^{+}(y_0) 
=  -\frac{\alpha_{i+1}v_{i+1}-u_{i+1}}{\omega^{+}(L_{i+1}(y_0)^2+y_0^2)}.
\end{gather*}
Therefore, for $y_0=y_i \in I_i$ and from \eqref{eq:uv} and \eqref{eq:xy},
\begin{gather*}
\frac{d}{d y_0}\tau^{-}(y_i) 
=  -\frac{1}{\omega^{-}}(-1)^i \varepsilon S(\rho_c,r,\varepsilon,i),\\
\frac{d}{d y_0}\tau^{+}(y_i) 
=  \frac{1}{\omega^{+}}(-1)^i \varepsilon S(\rho_c,r,\varepsilon,i),
\end{gather*}
where
\begin{equation}\label{eq:S}
S(\rho_c,r,\varepsilon,i)
=\frac{\rho_c^2}{2 r i (r+(-1)^i \varepsilon)(1+\rho_c^2)}>0
\end{equation}
for $i=1,\ldots,n$ and $0<\varepsilon<r$. So
\begin{align*}
\frac{d}{d y_0}\Delta_{\psi}(y_i) 
&=  \gamma^{-}e^{\gamma^{-}\tau^{-}}\frac{d}{d y_0}\tau^{-}(y_i)+
\gamma^{+}e^{-\gamma^{+}\tau^{+}}\frac{d}{d y_0}\tau^{+}(y_i) \\
&=  (-1)^i \varepsilon \left(\eta^{+}e^{-\gamma^{+}\tau^{+}}
 -\eta^{-}e^{\gamma^{-}\tau^{-}}\right)S(\rho_c,r,\varepsilon,i) 
\end{align*}
and taking into account the result \eqref{eq:S} and that
$\eta^{+}e^{-\gamma^{+}\tau^{+}}-\eta^{-}e^{\gamma^{-}\tau^{-}}>0$,
$0<\varepsilon<r$ and $\Delta_{\psi}(y_i)=0$ it follows from
\eqref{eq:Ddelta} that
\begin{equation}\label{eq:DDelta}
\frac{d}{d y_0} \delta_{\psi}(y_i) 
= \frac{d}{d y_0}\Delta_{\psi}(y_i)\sqrt{x_i^2+y_i^2}
\end{equation}
is different from zero. This implies that the $n$ limit cycles are
all hyperbolic. Furthermore if $i \in \{1,\ldots,n\}$ is odd (or
even) then the associated limit cycle is stable (or unstable). Note
that with the choice \eqref{eq:uv} the first limit cycle is always
an attractor limit cycle. Also note that the period of each limit cycle is
constant and given by $\tau^-+\tau^+$ (see \eqref{eq:taumpsi} and \eqref{eq:tauMpsi})
and can be made equal to $2\pi$ through a different time rescaling 
in each zone which becomes $\omega^-=\omega^+=1$. 
Theorem \ref{thm:main} is proved.


\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.6\textwidth]{fig5}
\end{center}
\caption{The set $\mathcal{R}$ is illustrated by the blue region.
The black dot represents the pair 
$(\eta^{-},\eta^{+})=(-0.3,0.5) \in \mathcal{R}$.}
\label{fig:ExemP}
\end{figure}

\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig6}
\end{center}
\caption{Phase portrait of the system $(J^{-},J^{+},\Psi)$
with ten limit cycles for $(\eta^{-},\eta^{+})=(-0.3,0.5) \in
\mathcal{R}$, $r=1$, $\varepsilon=0.5$ and $n=10$. The stable
(unstable) limit cycles are illustrated by blue (red) lines.}
\label{fig:ExemLC}
\end{figure}

\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig7}
\end{center}
\caption{The continuous black line is the graph of the displacement function $y_0
\mapsto \delta_{\psi}(y_0)$ of $(J^{-},J^{+},\Psi)$ for
$(\eta^{-},\eta^{+})=(-0.3,0.5) \in \mathcal{R}$, $r=1$,
$\varepsilon=0.5$ and $n=10$. The stable limit cycles are
illustrated by blue dots and the unstable by red dots.}
\label{fig:ExemDF}
\end{figure}


Now we present an example with ten limit cycles.

\begin{example} \label{examp3} \rm
In this example we consider the case  $\eta^{-}=-0.3$ and
$\eta^{+}=0.5$; that is, $\rho_c=1$ and the pair 
$(\eta^{-},\eta^{+}) \in \mathcal{R}$
(see Figure \ref{fig:ExemP}).


For $r=1$, we choose $\varepsilon=0.5$. According to Theorem \ref{thm:main}, 
with these values and $n=10$, there exists a system $(J^{-},J^{+},\Psi)$ such
that its phase portrait has exactly ten hyperbolic limit cycles.
This result is summarized in Figures \ref{fig:ExemLC} and
\ref{fig:ExemDF}.
\end{example}

\subsection{Concluding remarks} %\label{S:3}

In this article, we study one of the main
problems in the qualitative theory of planar differential
equations: the number and distribution of limit cycles in piecewise linear
differential systems with two zones in the plane.

We give a rigorous proof of the existence of an arbitrary number of limit
cycles in piecewise linear differential systems
with two zones in the plane. See Theorem \ref{thm:main}. Based on
our results we prove the conjecture about the existence of a piecewise
linear differential system with two zones in the plane with exactly
$n$ limit cycles, for any $n \in \mathbb N$. See \cite{DL1}.


\subsection*{Acknowledgements} 
The authors are partially supported by CNPq
grant 472321 /2013-7 and by CAPES CSF--PVE grant 88881.030454/2013-01.
The second author is partially supported by CNPq grant
301758/2012-3 and by FAPEMIG grant PPM-00092-13.

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\end{document}