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planar vector fields,uniqueness of limit cycles
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%&LaTeX

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% ENVIRONMENTS ET AL. %
\theoremstyle{plain} %% This is the default
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
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\begin{document}

\title{Uniqueness of limit cycles for a 
class of planar vector fields.}

\author{Timoteo Carletti}

\date{\today}

\address[Timoteo Carletti]{Scuola Normale Superiore, piazza dei Cavalieri 7,
 56126  Pisa, Italy}

\email[Timoteo Carletti]{t.carletti@sns.it}

\keywords{planar vector fields,uniqueness of limit cycles}

\begin{abstract}
In this paper we give sufficient conditions to ensure uniqueness of limit cycles
for a class of planar vector fields. We also exhibit a class of examples with exactly one
limit cycle.
\end{abstract}

\maketitle

\section{Introduction}
\label{sec:intro}

In this paper we consider the problem of determine the number of limit cycles,
i.e. isolated closed trajectories, for planar vector fields. This is a classical
problem, included as part of the XVI Hilbert's problem. The literature is huge
and still growing, for a review see~\cite{Sansone} and the more recent~\cite{Zhang,Ye}.

An important subproblem is to study systems with a unique limit cycle, in fact
in this case the dynamics of the cycle can ''dominate'' the global dynamics of the
whole system.

Let us consider planar vector fields of the form:
%
\begin{equation}
\label{eq:vf}
\begin{cases}
\dot x &= \beta(x)\left[ \phi(y)-F(x,y)\right] \\
\dot y &= -\alpha(y)g(x) \, ,
\end{cases}
\end{equation}
%
under the regularity assumptions (to ensure the existence and
uniqueness of 
the Cauchy initial problem) there exists: $-\infty \leq a<0
<b \leq +\infty$, such that:
\begin{enumerate}
\item[A1)] $\beta\in \Lip{a,b}$ and $\alpha \in \Lip{\R}$;
\item[A2)] $\phi \in \Lip{\R}$, $g\in \Lip{a,b}$ and 
$F\in \Lip{(a,b)\times \R}$.
\end{enumerate}
%

Without loss of generality we can assume $\alpha$ and $\beta$ to be positive in
their respective domains of definition, in fact the existence of $x_0$ such that
$\beta(x_0)=0$ (or $y_0$ s.t. $\alpha(y_0)=0$), gives rise to 
{\em invariant lines}, which cannot intersect a limit cycle. Hence we can 
reparametrize time, by dividing the vector field by: $\alpha(y)\beta(x)$. 
The transformed system is:
%
\begin{equation}
\label{eq:vecfiel}
\begin{cases}
\dot x &= \tilde{\phi}(y)-\tilde{F}(x,y) \\
\dot y &= -\tilde{g}(x) \, ,
\end{cases}
\end{equation}
%
where $\tilde{\phi}(y)=\phi(y)/\alpha(y)$, $\tilde{F}(x,y)=F(x,y)/\alpha(y)$ and
$\tilde{g}(x)=g(x)/\beta(x)$. In the following we will drop out the 
$\tilde \,$--mark and consider the general system of previous type.

These systems can be though as ''non--Hamiltonian perturbations'' of Hamiltonian
 ones, with Hamilton function: $H(x,y)=\Phi(y)+G(x)$, where $\Phi(y)=
\int_0^y \phi(s)\, ds$ and $G(x)=\int_0^xg(s)\, ds$, being $F(x,y)$ the
''perturbation''. 

One can also consider~\eqref{eq:vecfiel} as ''generalized''
Li\'enard equations:
\begin{equation*}
\ddot x+f(x)\dot x+g(x)=0 \, ,
\end{equation*}
 which in the Li\'enard plane can be rewritten as:
%
\begin{equation}
\label{eq:lien}
\begin{cases}
\dot x &= y-F(x) \\
\dot y &= -g(x) \, ,
\end{cases}
\end{equation}
%
where $F^{\prime}(x)=f(x)$, hence our systems
 generalize~\eqref{eq:lien} by allowing a dependence of $F$ also on $y$.


Let $\lambda >0$ and let us consider the energy level $\mathcal{H}_{\lambda}=
\{ (x,y)\in \R^2: \Phi(y)+G(x)~=~\lambda \}$, the knowledge 
of the flow through $\mathcal{H}_{\lambda}$ can give informations about the
 existence of limit cycles. Because
\begin{equation*}
<\nabla \mathcal{H}_{\lambda},
X(x,y)>\Big \rvert_{\mathcal{H}_{\lambda}}=-F(x,y)g(x) \, ,
\end{equation*}
 where $X(x,y)= (\phi(y)-F(x,y),-g(x))$, no limit cycles can 
be completely contained in a 
region where $gF$ doesn't change sign. We will see in a while that the set
of zeros of $F$ will play a fundamental role in our construction.

For Li\'enard systems the set of zeros of $F$ is given by vertical lines
$x=x_k$ s.t. $F(x_k)=0$. In a recent 
paper~\cite{SabatiniVillari} authors, using ideas taken from Li\'enard 
systems~\cite{CarlettiVillari}, proved a uniqueness result for 
systems~\eqref{eq:vecfiel} assuming that $F(x,y)$ vanishes only at three vertical
lines $x=x_-<0$, $x=0$ and $x=x_+>0$. We generalize this condition by assuming that
zeros of $F(x,y)$ lie on (quite) general curves. More precisely let us assume 
there exist 
 $\psi_j:\R \rightarrow \R$, $j\in \{1,2\}$, such 
that~\footnote{We remark that our main result
still holds, even if one assume there exist 
$\alpha_j<0<\beta_j$, $j\in \{1,2\}$,
 and the functions $\psi_j$ to be defined in $[\alpha_j,\beta_j]$ and verify
hypothesis B) on their new domain of definition.}:
%% $-\infty \leq \alpha_j<0<\beta_j \leq +\infty$ and functions
%%  $\psi_j:[\alpha,\beta]\rightarrow \R$, $j\in \{1,2\}$, such that:
\begin{enumerate}
\item[B0)] $F(0,y)=0$ for all real $y$;
\item[B1)] $y\mapsto\psi_1(y)$, is {\em positive} for all $y\in \R $,
increasing for negative $y$, decreasing for positive $y$, $\psi_1(0) < b$;
\item[B2)] $y\mapsto\psi_2(y)$, is {\em negative} for all $y\in \R$,
decreasing for negative $y$, increasing for positive $y$, $\psi_2(0) > a$;
\item[B3)] for all $y\in \R$, $j\in \{ 1,2 \}$, we have:
%% \item[B1)] $y\mapsto\psi_1(y)$, is {\em positive} for all $y\in [\alpha_1,\beta_1]$,
%% increasing for negative $y$, decreasing for positive $y$, $\psi_1(0) < b$;
%% \item[B2)] $y\mapsto\psi_2(y)$, is {\em negative} for all $y\in [\alpha_2,\beta_2]$,
%% decreasing for negative $y$, increasing for positive $y$, $\psi_2(0) > a$;
%% \item[B3)] for all $y\in [\alpha_j,\beta_j]$, $j\in \{ 1,2 \}$, we have:
%
\begin{equation*}
F(\psi_j(y),y)\equiv 0 \, ,
\end{equation*}
%
these curves will be called ''non--trivial zeros'' of $F(x,y)$ (in opposition with
the trivial zeros given by $x=0$).
\end{enumerate}

Let us divide the strip $(a,b)\times \R $
 into four distinct domains:
%% Let us divide the (possible unbounded) rectangles $(a,b)\times [\alpha_j,\beta_j]$
%% into four distinct domains:
\begin{itemize}
\item $D_1^{>}:=\{ (x,y) \in (a,b)\times \R : x > \psi_1(y) \}$;
\item $D_1^{<}:=\{ (x,y) \in (a,b)\times \R : 0<x < \psi_1(y) \}$;
\item $D_2^{>}:=\{ (x,y) \in (a,b)\times \R : \psi_2(y) < x < 0 \}$;
\item $D_2^{<}:=\{ (x,y) \in (a,b)\times \R : x < \psi_2(y) \}$.
%% \item $D_1^{>}:=\{ (x,y) \in (a,b)\times [\alpha_1,\beta_1]: x > \psi_1(y) \}$;
%% \item $D_1^{<}:=\{ (x,y) \in (a,b)\times [\alpha_1,\beta_1]: 0<x < \psi_1(y) \}$;
%% \item $D_2^{>}:=\{ (x,y) \in (a,b)\times [\alpha_2,\beta_2]: \psi_2(y) < x < 0 \}$;
%% \item $D_2^{<}:=\{ (x,y) \in (a,b)\times [\alpha_2,\beta_2]: x < \psi_2(y) \}$.
\end{itemize}

The following assumptions generalize ''standard sign ones'':
\begin{enumerate}
\item[C1)] $y\phi(y)>0$ for all $y\neq 0$ and $xg(x) >0$ for all 
$x\in (a,b)\setminus \{ 0 \}$; 
\item[C2)] $g(x)F(x,y)<0$ for all $(x,y)\in D_1^< \cup D_2^>$.
\end{enumerate}

We remark that hypothesis C2) can be weakened into:
\begin{enumerate}
\item[C2')] $g(x)F(x,y)\leq 0$ for all $(x,y)\in D_1^< \cup D_2^>$
 except at some $(x_0,y_0)$ where strictly inequality holds.
\end{enumerate}

With these hypotheses we ensures that $(0,0)$ is the only singular
point of system~\eqref{eq:vecfiel} and trajectories wind clockwise around it.

We are now able to state our main result
%
\begin{theorem}
\label{thm:main}
Let us consider system~\eqref{eq:vecfiel} and let us assume Hypotheses A), B) 
and C) to hold. Then there is at most one limit cycles which intersect both curves
$x=\psi_1(y)$ and $x=\psi_2(y)$ contained in $(a,b)\times \R$, provided:
\begin{enumerate}
\item[D1)] the function $y\mapsto F(x,y)/\phi(y)$ is strictly increasing
for $(x,y) \in D_1^{<}$ and $y\neq 0$;
\item[D2)] the function $y\mapsto F(x,y)/\phi(y)$ is strictly decreasing
for $(x,y) \in D_2^{>}$ and $y\neq 0$;
\item[E)] the function $x\mapsto F(x,y)$ is positive in $D_1^{>}$, negative
in $D_2^{<}$ and increasing in $D_1^{>} \cup D_2^{<}$;
\item[F)] let $A_j(y)=\left[\phi(y)\partial_x F(x,y)-g(x)\partial_y 
F(x,y)\right]\Big \rvert_{x=\psi_j(y)}$, then $A_j(y)y>0$ for $y\neq 0$, $j\in \{1,2\}$.
\end{enumerate}

\end{theorem}
%
Hypotheses D) and E) naturally generalize hypotheses used in the Li\'enard 
case~\cite{CarlettiVillari} or in the more general situation studied 
in~\cite{SabatiniVillari}. Also hypothesis F) is very natural: each closed 
trajectory intersects the non--trivial zeros of $F(x,y)$ at most once in any
 quadrant. We remark that this condition is trivially verified if the functions
$\psi_j$ are indentically constant, namely in the case considered 
in~\cite{SabatiniVillari}.

The proof of this result will be given in the next section. In the last
section (\S~\ref{sec:example}) we will provide a family of systems with
 exactly one limit cycle.


\section{Proof of Theorem~\ref{thm:main}}
\label{sec:proof}

The aim of this section is to prove our main result Theorem~\ref{thm:main}. 
The proof is based on the following remark,
%
\begin{remark}
\label{rem:integral}
Along any closed curve $\gamma:[0,T]\rightarrow (a,b)\times \R$ one has:
%
\begin{equation*}
\int_0^T \frac{d}{dt}H\Big \rvert_{\text{flow}}\circ \gamma(s) \, ds = H\circ \gamma (T)- H\circ \gamma (0)
=0 \, ,
\end{equation*}
%
moreover if $\gamma$ is an integral curve of system~\eqref{eq:vecfiel} we can 
evaluate the integrand function to obtain:
%
\begin{equation}
\label{eq:intzero}
I_{\gamma}:=\int_0^T g(x_{\gamma}(s)) F(x_{\gamma}(s),y_{\gamma}(s)) \, ds = 0 \, ,
\end{equation}
%
where $\gamma(s)=(x_{\gamma}(s),y_{\gamma}(s))$.
\end{remark}
%
The uniqueness result will be proved by showing that the existence of two
limit cycles, $\gamma_1$ contained~\footnote{By this we mean $\gamma_1$
is properly contained in the compact set whose boundary is $\gamma_2$.}
 in $\gamma_2$, both intersecting
$x=\psi_1(y)$ and $x=\psi_2(y)$, will imply: 
$I_{\gamma_1}< I_{\gamma_2}$, which contradicts~\eqref{eq:intzero}.

From now on we will assume the existence of two limit cycles,
 $\gamma_1$ contained
in $\gamma_2$, which intersect both non--trivial zeros of $F$.

 
Let us now consider the set of zeros of the equation: $\phi(y)-F(x,y)=0$ inside
$D_1^< \cup D_2^>$; we claim that this set is the graph of some function $x\mapsto
\zeta(x)$. Moreover this function vanishes for $x\in \{\psi_2(0),0,\psi_1(0) \}$,
it is positive for $x\in (\psi_2(0),0)$ and negative for $x\in (0,\psi_1(0))$.

The first statement follows from hypotheses D). In fact, let assume the claim to be 
false, then there exist $(\bar x,y_k)\in D_2^>$, such that:
%
\begin{equation*}
y_1 < y_2 \quad \text{and} \quad \phi(y_k)-F(\bar x,y_k)=0 \quad k\in\{1,2\} \, .
\end{equation*}
%
Then using the decreasing property (hypothesis D)) we obtain a contradiction:
%
\begin{equation*}
0=\left[F(\bar x,y_1)-\phi(y_1)\right]/\phi(y_1)>
\left[F(\bar x,y_2)-\phi(y_2)\right]/\phi(y_2)=0\, .
\end{equation*}
%
We can do the same construction for $D_1^<$, and the statement is proved.
The sign properties of $\zeta$ can be proved as follows. Let
$\psi_2(0)<x<0$, 
then by C2) $0<F(x,\zeta(x))=\phi(\zeta(x))$,
 using now C1) we conclude that $\zeta(x)>0$. The case $0<x<\psi_1(0)$
 can be handle similarly and we omit. By continuity we get the result
 about the zeros of $\zeta(x)$.

Hypothesis F) guarantees that a closed trajectory can intersect the non--trivial
zeros of $F(x,y)$ only once in each quadrant, in fact $A_j(y)$ gives a measure of 
the angle between the vector field and the normal to 
$\mathcal{F}_0=\{ (x,y): x=\psi_1(y) \} \cup \{ (x,y): x=\psi_2(y) \}$
 at $(\psi_j(y),y)$:
%
\begin{equation*}
<\nabla \mathcal{F}_{0},
X(x,y)\Big \rvert_{\mathcal{F}_{0}}>=\left[\phi(y)\partial_x F(x,y)
-g(x)\partial_y F(x,y)\right]\Big\rvert_{\mathcal{F}_0} =A_j(y)\quad j\in
\{1,2 \} \, .
\end{equation*}
%
For instance, because the angle between the vector field and 
$\{ (x,y): x=\psi_1(y) \}\cap \{ y >0 \}$ is in absolute value smaller than $\pi/2$,
 a trajectory starting at $(0,\bar{y})$, for some $\bar{y}>0$, which will
intersect $\{ (x,y): x=\psi_1(y) \}\cap \{ y >0 \}$, could not meet anew 
$\{ (x,y): x=\psi_1(y) \}\cap \{ y >0 \}$.
 
From hypotheses D) and the sign of $F$ on $D_2^< \cup D_1^>$, it follows easily
 that for all $(x,y)\in D_2^< \cup D_1^>$ one has: 
$\left(y-\zeta(x)\right)\left(\phi(y)-F(x,y)\right)>0$. Hence a cycle intersects
$\left(D_2^< \cup D_1^>\right)\cap \{ y>0 \}$ in a region where $\phi(y)-F(x,y)>0$,
whereas the intersection with $\left(D_2^< \cup D_1^>\right)\cap \{ y<0 \}$ holds
where $\phi(y)-F(x,y)<0$. This remark allows us to divide the path of integration
needed to evaluate $I_{\gamma_j}$, $j\in \{1,2\}$, in two parts: an ''horizontal''
one where $\dot x >0$ and a vertical one,
 where $\dot x$ vanishes.


Let us define (see Figure~\ref{fig:figura1}), for $j\in \{1,2\}$, $A_j$ 
(respectively $B_j$) the intersection 
point of $\gamma_j$ with $x=\psi_1(y)$ for $y>0$ (respectively $y<0$), and 
$C_j$ (respectively $D_j$) the intersection 
point of $\gamma_j$ with $x=\psi_2(y)$ for $y<0$ (respectively $y>0$).
Let also introduce, $A_*$ being the intersection point of $\gamma_1$ and the
line $x=x_{A_2}$ contained in the first quadrant, and $A_{**}$ being the
 intersection point of $\gamma_2$ and the
line $y=y_{A_1}$ contained in the first quadrant. Similarly we introduce points:
$B_*$, $B_{**}$, $C_*$, $C_{**}$  and $D_*$, $D_{**}$ 
(see Figure~\ref{fig:figura1}).

\begin{center}
  \begin{figure}[ht]
   \makebox{
   \epsfig{file=figura1.eps,height=12cm,width=12cm,angle=-90}}
  \caption{The non--trivial zeros of $F$ (thick), the limit cycles $\gamma_1$
and $\gamma_2$ (thin) intersecting both non--trivial zeros of $F$ 
and their subdivision into arcs.}
  \label{fig:figura1}
  \end{figure}
\end{center}

According to this subdivision of the arcs of limit cycles, we evaluate 
$I_{\gamma_j}$ as follows:
%
\begin{eqnarray}
\label{eq:sudiv}
I_{\gamma_1}=\int_{ {D_*A_*}}+\int_{{A_*A_1}}+
\int_{ {A_1B_1}}+
\int_{ {B_1B_*}}+
\int_{ {A_*C_*}}+
\int_{ {C_*C_1}}+
\int_{ {C_1D_1}}+
\int_{ {D_1D_*}}  \\
I_{\gamma_2}=\int_{ {D_2A_2}}+\int_{ {A_2A_{**}}}
+\int_{ {A_{**}B_{**}}}+\int_{ {B_{**}B_2}}
+\int_{ {B_2C_2}}+\int_{ {C_{2}C_{**}}}
+\int_{ {C_{**}D_{**}}}+\int_{ {D_{**}D_2}} \notag\, .
\end{eqnarray}
%

Let now show that $I_{\gamma_1}<I_{\gamma_2}$, which prove the contradiction
and conclude the proof.

\subsection{Integration along ''horizontal arcs''.}
\label{ssec:intoriz}

Because along horizontal arcs we have $\dot x \neq 0$, we can change integration
variable from $t$ to $x$, hence for example:
%
\begin{equation*}
\int_{ {D_jA_j}}g(x)F(x,y)\, dt= \int_{x_{D_j}}^{x_{A_j}} \frac{g(x)
F(x,y_j(x))}{\phi(y_j(x))-F(x,y_j(x))} \, dx \, ,
\end{equation*}
%
where $y_j(x)$, $j\in \{1,2 \}$, is the parametrization of $\gamma_j$ as 
graph over $x$ for $x\in(x_{D_j},x_{A_j})$.

Because $y_2(x)>y_1(x)$ for all $x\in (x_{D_*},x_{A_*})$, using hypotheses D) 
and the sign assumptions C) we get: 
%
\begin{equation*}
\frac{g(x)F(x,y_1(x))}{\phi(y_1(x))-F(x,y_1(x))} <
\frac{g(x)F(x,y_2(x))}{\phi(y_2(x))-F(x,y_2(x))} \, ,
\end{equation*}
%
hence:
%
\begin{eqnarray}
\label{eq:d1a1}
\int_{x_{D_{1}}}^{x_{A_{1}}} \frac{g(x)
F(x,y_1(x))}{\phi(y_1(x))-F(x,y_1(x))} \, dx & < 
\int_{x_{D_{1}}}^{x_{D_{*}}} \frac{g(x)
F(x,y_1(x))}{\phi(y_1(x))-F(x,y_1(x))} \, dx+\int_{x_{A_*}}^{x_{A_1}} \frac{g(x)
F(x,y_1(x))}{\phi(y_1(x))-F(x,y_1(x))} \, dx \notag \\ 
&+\int_{x_{D_2}}^{x_{A_2}} \frac{g(x) F(x,y_2(x))}{\phi(y_2(x))-F(x,y_2(x))} \, dx 
\leq \int_{x_{D_2}}^{x_{A_2}} \frac{g(x)
F(x,y_2(x))}{\phi(y_2(x))-F(x,y_2(x))} \, dx \, ,
\end{eqnarray}
%
the last step follows because the integrand function is negative by hypothesis C2)
and from the previous discussion on the sign of $\phi(y)-F(x,y)$.


In a very similar way we can prove that:
%
\begin{equation}
\label{eq:b1c1}
\int_{x_{B_{1}}}^{x_{C_{1}}} \frac{g(x)
F(x,y_1(x))}{\phi(y_1(x))-F(x,y_1(x))} \, dx <
\int_{x_{B_2}}^{x_{C_2}} \frac{g(x) F(x,y_2(x))}{\phi(y_2(x))-F(x,y_2(x))} \, dx 
\, .
\end{equation}
%

\subsection{Integration along ''vertical arcs''.}
\label{ssec:intvert}

Along vertical arcs $\dot y$ never vanishes, hence we can perform the
integration w.r.t. the $y$ variable and getting for example:
%
\begin{equation*}
\int_{ {A_jB_j}}g(x)F(x,y)\, dt= \int_{y_{B_j}}^{y_{A_j}} F(x_j(y),y)\, dy
 \, ,
\end{equation*}
%
where $x_j(y)$, $j\in \{1,2 \}$, is the parametrization of $\gamma_j$ as 
graph over $y$ for $y\in(y_{B_j},y_{A_j})$.

Because $x_2(y)>x_1(y)$ for all $y\in (y_{A_{**}},y_{B_{**}})$, from hypotheses E) 
and the definition of $A_{**}$ and $B_{**}$, we get:
%
\begin{equation*}
\int_{y_{B_{1}}}^{y_{A_{1}}} F(x_1(y),y)\, dy < 
\int_{y_{B_{**}}}^{y_{A_{**}}} F(x_2(y),y)\, dy  \, .
\end{equation*}
%
Again from the sign assumption on $F$ in $D_1^{>}$, we get:
%
\begin{equation*}
\int_{y_{A_{**}}}^{y_{A_{2}}} F(x_2(y),y)\, dy  >0 \quad\text{and}\quad
\int_{y_{B_{2}}}^{y_{B_{**}}} F(x_2(y),y)\, dy >0 \, ,
\end{equation*}
%
hence we obtain:
%
\begin{equation}
\label{eq:a2b2}
\int_{y_{B_1}}^{y_{A_1}} F(x_1(y),y)\, dy 
<\int_{y_{B_2}}^{y_{A_2}} F(x_2(y),y)\, dy \, .
\end{equation}
%
Analogously we can prove that:
%
\begin{equation}
\label{eq:c2d2}
\int_{y_{C_1}}^{y_{D_1}} F(x_1(y),y)\, dy 
<\int_{y_{C_2}}^{y_{D_2}} F(x_2(y),y)\, dy \, .
\end{equation}
%

\subsection{Conclusion of the proof.}
\label{ssec:concpro}

We are now able to complete our proof. In fact from~\eqref{eq:d1a1} 
and~\eqref{eq:b1c1} of \S~\ref{ssec:intoriz}, from~\eqref{eq:a2b2} 
and~\eqref{eq:c2d2} of \S~\ref{ssec:intvert} and the subdivision~\eqref{eq:sudiv}
we get:
%
\begin{equation*}
I_{\gamma_1}<I_{\gamma_2} \, ,
\end{equation*}
%
which contradicts~\eqref{eq:intzero}, and so the Theorem is proved.

\section{A system with exactly one limit cycle}
\label{sec:example}

In this last part we present an important class of examples exhibiting exactly
one limit cycle. Let us
assume that $F$ has the following ''special form'': 
\begin{equation*}
F(x,y)=x\left[x-\psi_1(y)\right]\left[x-\psi_2(y)\right] \, ,
\end{equation*}
where $\psi_j$ verify 
hypotheses B). Then hypothesis F) is equivalent to:
\begin{enumerate}
\item[F')] the function $y\mapsto \Phi(y)+G(\psi_j(y))$ is strictly increasing
for positive $y$ and strictly decreasing for negative ones, $j \in \{ 1,2 \}$.
\end{enumerate}
In fact we have:
%
\begin{eqnarray*}
A_1(y)&=\psi_1(y)\left[ \psi_1(y) - \psi_2(y) \right] \left[ \phi(y)-g(\psi_1(y)) 
\psi_1^{\prime}(y)\right]\\
&=\psi_1(y)\left[\psi_1(y)-\psi_2(y)\right]\frac{d}{dy}\left[
\Phi(y)-G(\psi_1(y))\right] \, ,
\end{eqnarray*}
%
and the claim follows from the sign properties of $\psi_j$ and the definitions of
$\Phi$ and $G$. Similarly for $A_2$.

A concrete example is given by choosing:
\begin{equation*}
\phi(y)=y \, , \, g(x)=x \, , \, \psi_1(y)=c_1 e^{-d_1 y^2}+e_1 \quad \text{and}
 \quad \psi_2(y)=-c_2 e^{-d_2 y^2}-e_2 \, ,
\end{equation*}
 with $c_j,d_j,$ and $e_j$ positive real
numbers such that:
\begin{enumerate}
\item $c_1+e_1=c_2+e_2$,
\item let $r=c_1+e_1$, then $c_jd_j\max\{ r, r^2\}<1/2$ for $j\in \{ 1,2 \}$.
\end{enumerate}

\begin{center}
  \begin{figure}[ht]
   \makebox{
   \epsfig{file=figura2.eps,height=12cm,width=12cm,angle=-90}}
  \caption{An example with $c_1=d_1=e_1=1/2$, $c_2=1/4$, $d_2=1$, and $e_2=3/4$.
 We numerically compute the attracting limit cycle (thick) and three attracted
 trajectories (thin), we also plot the non--trivial zeros of $F$
 (thick) and the direction of
the vector field (arrows).}
  \label{fig:figura2}
  \end{figure}
\end{center}


The example of Figure~\ref{fig:figura2} verifies all the hypotheses of Theorem~\ref{thm:main}. 
Hypotheses A), B), C) and E) are readily verified, whereas hypotheses 
D) and F') (hence F)) follow~\footnote{Using $c_jd_jr<1/2$ hypothesis
F') holds. Whereas (1) and $c_jd_jr^2<1/2$ ensure that the circle $x^2+y^2=r^2$
is contained inside $D_1^< \cup D_2^>$ and it is tangent to $x=\psi_j(y)$ at
$y=0$, for $j\in \{ 1,2\}$, thus any limit cycle must intersect both
curves $x=\psi_j(y)$, $j\in \{ 1,2\}$.}
 from (1) and (2) plus some easy calculations that
we omit.


\begin{thebibliography}{XXX}
\bibitem[CV2003]{CarlettiVillari} T. Carletti and G. Villari:
{\it A note on existence and uniqueness of limit cycles for Li\'enard systems}, 
preprint, Univ. Firenze (2003), {\bf 16}.

\bibitem[SV2004]{SabatiniVillari} M. Sabatini and G. Villari: {\it About
limit cycle's uniqueness for a class of generalized Li\'enard systems},
preprint, Univ. Trento (2004), {\bf 672}.

\bibitem[CS1964]{Sansone} R. Conti e G. Sansone: {\it Non--linear differential
 equations}, Pergamon Press, (1964).

\bibitem[Ye1986]{Ye} Ye Yanquia et al.: {\it  Theory of limit cycles}, 
 Trans. Math. Monog., Vol. 66, Americ. Math. Soc., Providence, Rhode Island,
 (1986).

\bibitem[Zh1992]{Zhang} Zhang Zhi-fen et al.: {\it Qualitative Theory
of differential equations}, Trans. Math. Monog., Vol. 101,
  Americ. Math. Soc., Providence, Rhode Island, (1992).

\end{thebibliography}

\end{document}
---------------0409210533372
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Content-Transfer-Encoding: 7bit
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NP 2369 1249 m 2309 1303 l
S
NP 2435 1197 m 2369 1249 l
S
NP 2508 1150 m 2435 1197 l
S
NP 2585 1106 m 2508 1150 l
S
NP 2667 1066 m 2585 1106 l
S
NP 2752 1031 m 2667 1066 l
S
NP 2839 1000 m 2752 1031 l
S
NP 2926 973 m 2839 1000 l
S
NP 3014 951 m 2926 973 l
S
NP 3102 933 m 3014 951 l
S
NP 3188 919 m 3102 933 l
S
NP 3272 910 m 3188 919 l
S
NP 3355 904 m 3272 910 l
S
NP 3435 903 m 3355 904 l
S
NP 3513 906 m 3435 903 l
S
NP 3588 912 m 3513 906 l
S
NP 3661 922 m 3588 912 l
S
NP 3732 935 m 3661 922 l
S
NP 3800 952 m 3732 935 l
S
NP 3866 973 m 3800 952 l
S
NP 3931 996 m 3866 973 l
S
NP 3993 1023 m 3931 996 l
S
NP 4054 1052 m 3993 1023 l
S
NP 4114 1085 m 4054 1052 l
S
NP 4172 1121 m 4114 1085 l
S
NP 4228 1159 m 4172 1121 l
S
NP 4284 1200 m 4228 1159 l
S
NP 4339 1244 m 4284 1200 l
S
NP 4394 1291 m 4339 1244 l
S
NP 4447 1340 m 4394 1291 l
S
NP 4501 1392 m 4447 1340 l
S
NP 4555 1447 m 4501 1392 l
S
NP 4609 1504 m 4555 1447 l
S
NP 4663 1564 m 4609 1504 l
S
NP 4719 1627 m 4663 1564 l
S
NP 4776 1693 m 4719 1627 l
S
NP 4836 1762 m 4776 1693 l
S
NP 4900 1833 m 4836 1762 l
S
NP 4970 1909 m 4900 1833 l
S
NP 5047 1988 m 4970 1909 l
S
NP 5138 2072 m 5047 1988 l
S
NP 5248 2161 m 5138 2072 l
S
NP 5392 2258 m 5248 2161 l
S
NP 5601 2365 m 5392 2258 l
S
NP 5978 2491 m 5601 2365 l
S
NP 4647 1907 m 4619 1848 l
S
thick setlinewidth
%qui1
NP 4674 1968 m 4647 1907 l
S
NP 4699 2030 m 4674 1968 l
S
NP 4722 2093 m 4699 2030 l
S
NP 4744 2157 m 4722 2093 l
S
NP 4763 2222 m 4744 2157 l
S
NP 4780 2288 m 4763 2222 l
S
NP 4796 2355 m 4780 2288 l
S
NP 4810 2422 m 4796 2355 l
S
NP 4821 2490 m 4810 2422 l
S
NP 4830 2558 m 4821 2490 l
S
NP 4837 2627 m 4830 2558 l
S
NP 4842 2696 m 4837 2627 l
S
NP 4844 2765 m 4842 2696 l
S
NP 4844 2835 m 4844 2765 l
S
NP 4841 2904 m 4844 2835 l
S
NP 4834 2972 m 4841 2904 l
S
NP 4825 3040 m 4834 2972 l
S
NP 4812 3108 m 4825 3040 l
S
NP 4795 3175 m 4812 3108 l
S
NP 4773 3240 m 4795 3175 l
S
NP 4747 3305 m 4773 3240 l
S
NP 4715 3367 m 4747 3305 l
S
NP 4678 3428 m 4715 3367 l
S
NP 4636 3487 m 4678 3428 l
S
NP 4587 3543 m 4636 3487 l
S
NP 4533 3597 m 4587 3543 l
S
NP 4473 3647 m 4533 3597 l
S
NP 4408 3695 m 4473 3647 l
S
NP 4339 3739 m 4408 3695 l
S
NP 4265 3779 m 4339 3739 l
S
NP 4188 3815 m 4265 3779 l
S
NP 4109 3848 m 4188 3815 l
S
NP 4028 3876 m 4109 3848 l
S
NP 3946 3900 m 4028 3876 l
S
NP 3864 3920 m 3946 3900 l
S
NP 3783 3936 m 3864 3920 l
S
NP 3704 3949 m 3783 3936 l
S
%qui1
NP 3626 3957 m 3704 3949 l
S
NP 3549 3961 m 3626 3957 l
S
NP 3475 3962 m 3549 3961 l
S
NP 3404 3959 m 3475 3962 l
S
NP 3334 3953 m 3404 3959 l
S
NP 3268 3943 m 3334 3953 l
S
NP 3203 3930 m 3268 3943 l
S
NP 3141 3914 m 3203 3930 l
S
NP 3082 3895 m 3141 3914 l
S
NP 3024 3873 m 3082 3895 l
S
NP 2969 3849 m 3024 3873 l
S
NP 2916 3822 m 2969 3849 l
S
NP 2864 3792 m 2916 3822 l
S
NP 2815 3759 m 2864 3792 l
S
%qui
NP 2767 3725 m 2815 3759 l
S
NP 2721 3688 m 2767 3725 l
S
NP 2676 3648 m 2721 3688 l
S
NP 2633 3607 m 2676 3648 l
S
NP 2592 3564 m 2633 3607 l
S
NP 2551 3518 m 2592 3564 l
S
NP 2513 3471 m 2551 3518 l
S
NP 2475 3421 m 2513 3471 l
S
NP 2439 3370 m 2475 3421 l
S
NP 2405 3318 m 2439 3370 l
S
NP 2371 3263 m 2405 3318 l
S
NP 2340 3207 m 2371 3263 l
S
NP 2310 3149 m 2340 3207 l
S
NP 2281 3090 m 2310 3149 l
S
NP 2254 3030 m 2281 3090 l
S
%qui2
NP 2228 2968 m 2254 3030 l
S
NP 2205 2906 m 2228 2968 l
S
NP 2183 2842 m 2205 2906 l
S
NP 2163 2777 m 2183 2842 l
S
NP 2144 2711 m 2163 2777 l
S
NP 2128 2644 m 2144 2711 l
S
NP 2114 2577 m 2128 2644 l
S
NP 2102 2509 m 2114 2577 l
S
NP 2092 2441 m 2102 2509 l
S
NP 2084 2372 m 2092 2441 l
S
NP 2078 2303 m 2084 2372 l
S
NP 2075 2233 m 2078 2303 l
S
NP 2074 2164 m 2075 2233 l
S
NP 2076 2095 m 2074 2164 l
S
NP 2081 2026 m 2076 2095 l
S
NP 2090 1957 m 2081 2026 l
S
NP 2102 1889 m 2090 1957 l
S
NP 2118 1822 m 2102 1889 l
S
NP 2138 1756 m 2118 1822 l
S
NP 2164 1691 m 2138 1756 l
S
NP 2195 1628 m 2164 1691 l
S
NP 2233 1566 m 2195 1628 l
S
NP 2276 1507 m 2233 1566 l
S
NP 2325 1450 m 2276 1507 l
S
NP 2381 1396 m 2325 1450 l
S
NP 2442 1345 m 2381 1396 l
S
NP 2509 1298 m 2442 1345 l
S
NP 2580 1254 m 2509 1298 l
S
NP 2656 1213 m 2580 1254 l
S
NP 2734 1177 m 2656 1213 l
S
NP 2815 1145 m 2734 1177 l
S
NP 2897 1116 m 2815 1145 l
S
NP 2979 1092 m 2897 1116 l
S
NP 3061 1072 m 2979 1092 l
S
NP 3141 1056 m 3061 1072 l
S
NP 3221 1044 m 3141 1056 l
S
NP 3299 1036 m 3221 1044 l
S
NP 3374 1032 m 3299 1036 l
S
NP 3448 1031 m 3374 1032 l
S
NP 3519 1034 m 3448 1031 l
S
NP 3587 1040 m 3519 1034 l
S
NP 3654 1050 m 3587 1040 l
S
NP 3718 1063 m 3654 1050 l
S
NP 3780 1079 m 3718 1063 l
S
NP 3839 1098 m 3780 1079 l
S
%qui3
NP 3897 1120 m 3839 1098 l
S
NP 3952 1144 m 3897 1120 l
S
NP 4006 1171 m 3952 1144 l
S
NP 4058 1201 m 4006 1171 l
S
NP 4108 1234 m 4058 1201 l
S
NP 4157 1268 m 4108 1234 l
S
NP 4204 1306 m 4157 1268 l
S
NP 4249 1345 m 4204 1306 l
S
NP 4293 1386 m 4249 1345 l
S
NP 4335 1430 m 4293 1386 l
S
NP 4376 1476 m 4335 1430 l
S
NP 4415 1524 m 4376 1476 l
S
NP 4453 1573 m 4415 1524 l
S
NP 4490 1625 m 4453 1573 l
S
NP 4525 1678 m 4490 1625 l
S
NP 4558 1733 m 4525 1678 l
S
NP 4590 1789 m 4558 1733 l
S
NP 4620 1847 m 4590 1789 l
S
NP 4649 1907 m 4620 1847 l
S
NP 4676 1967 m 4649 1907 l
S
NP 4701 2029 m 4676 1967 l
S
NP 4724 2092 m 4701 2029 l
S
NP 4746 2157 m 4724 2092 l
S
NP 4766 2222 m 4746 2157 l
S
NP 4784 2288 m 4766 2222 l
S
NP 4800 2355 m 4784 2288 l
S
NP 4814 2423 m 4800 2355 l
S
NP 4825 2491 m 4814 2423 l
S
0.368627 G
medium setlinewidth
%qui6 inizio frecce
NP 5862 4056 m 5928 4055 l
S
%qui5
NP 6095 4136 m 5862 4056 l
S
%qui4
NP 5608 4045 m 5674 4050 l
S
NP 5819 4147 m 5608 4045 l
S
NP 5367 4030 m 5428 4046 l
S
NP 5530 4162 m 5367 4030 l
S
NP 5146 4016 m 5194 4044 l
S
NP 5221 4177 m 5146 4016 l
S
NP 4939 4013 m 4963 4052 l
S
NP 4898 4179 m 4939 4013 l
S
NP 4722 4024 m 4721 4066 l
S
NP 4585 4168 m 4722 4024 l
S
NP 4486 4039 m 4466 4079 l
S
NP 4291 4153 m 4486 4039 l
S
NP 4238 4053 m 4205 4090 l
S
NP 4009 4139 m 4238 4053 l
S
NP 3984 4068 m 3940 4100 l
S
NP 3733 4124 m 3984 4068 l
S
NP 3725 4086 m 3672 4111 l
S
NP 3462 4106 m 3725 4086 l
S
NP 3460 4107 m 3398 4123 l
S
NP 3197 4085 m 3460 4107 l
S
NP 3185 4130 m 3119 4135 l
S
NP 2942 4062 m 3185 4130 l
S
NP 2904 4147 m 2838 4142 l
S
NP 2693 4045 m 2904 4147 l
S
NP 2629 4154 m 2565 4144 l
S
NP 2438 4038 m 2629 4154 l
S
NP 2363 4155 m 2299 4145 l
S
NP 2174 4037 m 2363 4155 l
S
NP 2103 4152 m 2038 4143 l
S
NP 1904 4040 m 2103 4152 l
S
NP 1845 4146 m 1779 4141 l
S
NP 1632 4046 m 1845 4146 l
S
NP 1587 4140 m 1521 4139 l
S
NP 1360 4053 m 1587 4140 l
S
NP 1327 4133 m 1261 4136 l
S
NP 1090 4059 m 1327 4133 l
S
NP 1066 4128 m 1001 4133 l
S
NP 821 4065 m 1066 4128 l
S
NP 5862 3888 m 5928 3887 l
S
NP 6095 3968 m 5862 3888 l
S
NP 5609 3877 m 5674 3882 l
S
NP 5818 3979 m 5609 3877 l
S
NP 5367 3862 m 5428 3878 l
S
NP 5530 3994 m 5367 3862 l
S
NP 5144 3848 m 5193 3876 l
S
NP 5223 4008 m 5144 3848 l
S
NP 4935 3845 m 4961 3883 l
S
NP 4902 4011 m 4935 3845 l
S
NP 4718 3855 m 4719 3897 l
S
NP 4589 4001 m 4718 3855 l
S
NP 4482 3869 m 4465 3909 l
S
NP 4295 3987 m 4482 3869 l
S
NP 4235 3883 m 4205 3920 l
S
NP 4012 3973 m 4235 3883 l
S
NP 3982 3898 m 3940 3931 l
S
NP 3735 3958 m 3982 3898 l
S
NP 3725 3917 m 3672 3942 l
S
NP 3462 3939 m 3725 3917 l
S
NP 3459 3941 m 3398 3956 l
S
NP 3198 3915 m 3459 3941 l
S
NP 3181 3966 m 3115 3968 l
S
NP 2946 3890 m 3181 3966 l
S
NP 2898 3983 m 2833 3975 l
S
NP 2699 3873 m 2898 3983 l
S
NP 2622 3990 m 2559 3977 l
S
NP 2445 3866 m 2622 3990 l
S
NP 2357 3990 m 2294 3977 l
S
NP 2180 3866 m 2357 3990 l
S
NP 2099 3986 m 2035 3976 l
S
NP 1908 3870 m 2099 3986 l
S
NP 1843 3980 m 1777 3974 l
S
NP 1634 3876 m 1843 3980 l
S
NP 1586 3973 m 1519 3971 l
S
NP 1361 3883 m 1586 3973 l
S
NP 1327 3966 m 1261 3968 l
S
NP 1090 3890 m 1327 3966 l
S
NP 1066 3960 m 1000 3966 l
S
NP 821 3896 m 1066 3960 l
S
NP 5862 3720 m 5928 3719 l
S
NP 6095 3800 m 5862 3720 l
S
NP 5609 3709 m 5674 3714 l
S
NP 5818 3811 m 5609 3709 l
S
NP 5366 3694 m 5428 3710 l
S
NP 5531 3826 m 5366 3694 l
S
NP 5142 3680 m 5192 3708 l
S
NP 5225 3840 m 5142 3680 l
S
NP 4932 3676 m 4959 3715 l
S
NP 4905 3844 m 4932 3676 l
S
NP 4714 3685 m 4717 3727 l
S
NP 4593 3835 m 4714 3685 l
S
NP 4479 3698 m 4464 3739 l
S
NP 4298 3821 m 4479 3698 l
S
NP 4233 3712 m 4204 3750 l
S
NP 4014 3808 m 4233 3712 l
S
NP 3981 3727 m 3941 3761 l
S
NP 3736 3792 m 3981 3727 l
S
NP 3724 3747 m 3673 3773 l
S
NP 3463 3773 m 3724 3747 l
S
NP 3459 3775 m 3397 3789 l
S
NP 3198 3745 m 3459 3775 l
S
NP 3177 3803 m 3111 3803 l
S
NP 2950 3716 m 3177 3803 l
S
NP 2890 3821 m 2826 3809 l
S
NP 2707 3699 m 2890 3821 l
S
NP 2614 3827 m 2553 3811 l
S
NP 2453 3693 m 2614 3827 l
S
NP 2351 3826 m 2289 3810 l
S
NP 2186 3694 m 2351 3826 l
S
NP 2095 3821 m 2031 3809 l
S
NP 1912 3699 m 2095 3821 l
S
NP 1840 3814 m 1775 3807 l
S
NP 1637 3706 m 1840 3814 l
S
NP 1584 3806 m 1518 3804 l
S
NP 1363 3714 m 1584 3806 l
S
NP 1326 3799 m 1260 3801 l
S
NP 1091 3721 m 1326 3799 l
S
NP 1066 3793 m 1000 3798 l
S
NP 821 3727 m 1066 3793 l
S
NP 5862 3552 m 5928 3551 l
S
NP 6095 3632 m 5862 3552 l
S
NP 5609 3540 m 5674 3546 l
S
NP 5818 3643 m 5609 3540 l
S
NP 5366 3526 m 5428 3542 l
S
NP 5531 3658 m 5366 3526 l
S
NP 5141 3512 m 5190 3540 l
S
NP 5226 3671 m 5141 3512 l
S
NP 4928 3508 m 4957 3546 l
S
NP 4909 3676 m 4928 3508 l
S
NP 4709 3516 m 4715 3557 l
S
NP 4598 3668 m 4709 3516 l
S
NP 4475 3528 m 4463 3569 l
S
NP 4302 3656 m 4475 3528 l
S
NP 4229 3541 m 4203 3580 l
S
NP 4018 3642 m 4229 3541 l
S
NP 3979 3556 m 3941 3591 l
S
NP 3738 3627 m 3979 3556 l
S
NP 3724 3577 m 3673 3604 l
S
NP 3463 3606 m 3724 3577 l
S
NP 3458 3609 m 3395 3622 l
S
NP 3199 3575 m 3458 3609 l
S
NP 3170 3642 m 3104 3637 l
S
NP 2957 3542 m 3170 3642 l
S
NP 2879 3659 m 2818 3643 l
S
NP 2718 3525 m 2879 3659 l
S
NP 2604 3663 m 2545 3643 l
S
NP 2463 3521 m 2604 3663 l
S
NP 2343 3661 m 2283 3643 l
S
NP 2194 3522 m 2343 3661 l
S
NP 2089 3656 m 2027 3642 l
S
NP 1918 3528 m 2089 3656 l
S
NP 1837 3648 m 1772 3639 l
S
NP 1640 3536 m 1837 3648 l
S
NP 1582 3640 m 1516 3636 l
S
NP 1365 3544 m 1582 3640 l
S
NP 1325 3632 m 1259 3633 l
S
NP 1092 3552 m 1325 3632 l
S
NP 1065 3625 m 999 3630 l
S
NP 822 3559 m 1065 3625 l
S
NP 5862 3383 m 5929 3382 l
S
NP 6095 3464 m 5862 3383 l
S
NP 5609 3372 m 5675 3378 l
S
NP 5818 3475 m 5609 3372 l
S
NP 5366 3358 m 5427 3373 l
S
NP 5531 3489 m 5366 3358 l
S
NP 5139 3345 m 5189 3372 l
S
NP 5228 3503 m 5139 3345 l
S
NP 4924 3340 m 4955 3377 l
S
NP 4913 3508 m 4924 3340 l
S
NP 4705 3346 m 4713 3388 l
S
NP 4602 3501 m 4705 3346 l
S
NP 4470 3358 m 4461 3399 l
S
NP 4307 3490 m 4470 3358 l
S
NP 4226 3370 m 4203 3410 l
S
NP 4021 3477 m 4226 3370 l
S
NP 3976 3385 m 3940 3420 l
S
NP 3741 3463 m 3976 3385 l
S
NP 3723 3407 m 3674 3435 l
S
NP 3464 3440 m 3723 3407 l
S
NP 3457 3444 m 3393 3456 l
S
NP 3200 3403 m 3457 3444 l
S
NP 3159 3482 m 3095 3472 l
S
NP 2968 3366 m 3159 3482 l
S
NP 2864 3497 m 2807 3476 l
S
NP 2733 3351 m 2864 3497 l
S
NP 2591 3500 m 2536 3476 l
S
NP 2477 3348 m 2591 3500 l
S
NP 2333 3497 m 2276 3476 l
S
NP 2204 3350 m 2333 3497 l
S
NP 2083 3491 m 2022 3475 l
S
NP 1924 3356 m 2083 3491 l
S
NP 1833 3483 m 1769 3472 l
S
NP 1644 3365 m 1833 3483 l
S
NP 1580 3473 m 1515 3469 l
S
NP 1367 3374 m 1580 3473 l
S
NP 1324 3465 m 1258 3465 l
S
NP 1093 3383 m 1324 3465 l
S
NP 1065 3458 m 999 3462 l
S
NP 822 3390 m 1065 3458 l
S
NP 5862 3215 m 5929 3214 l
S
NP 6095 3296 m 5862 3215 l
S
NP 5609 3204 m 5674 3210 l
S
NP 5818 3307 m 5609 3204 l
S
NP 5365 3190 m 5427 3206 l
S
NP 5532 3321 m 5365 3190 l
S
NP 5137 3177 m 5188 3204 l
S
NP 5230 3334 m 5137 3177 l
S
NP 4920 3172 m 4953 3208 l
S
NP 4917 3340 m 4920 3172 l
S
NP 4699 3177 m 4711 3218 l
S
NP 4608 3335 m 4699 3177 l
S
NP 4465 3187 m 4459 3229 l
S
NP 4312 3324 m 4465 3187 l
S
NP 4221 3199 m 4201 3239 l
S
NP 4026 3313 m 4221 3199 l
S
NP 3973 3213 m 3940 3250 l
S
NP 3744 3299 m 3973 3213 l
S
NP 3722 3236 m 3674 3265 l
S
NP 3465 3275 m 3722 3236 l
S
NP 3455 3281 m 3390 3290 l
S
NP 3202 3230 m 3455 3281 l
S
NP 3143 3323 m 3082 3307 l
S
NP 2984 3188 m 3143 3323 l
S
NP 2844 3335 m 2793 3308 l
S
NP 2753 3177 m 2844 3335 l
S
NP 2575 3336 m 2525 3308 l
S
NP 2492 3176 m 2575 3336 l
S
NP 2321 3333 m 2268 3308 l
S
NP 2216 3179 m 2321 3333 l
S
NP 2075 3326 m 2016 3307 l
S
NP 1932 3185 m 2075 3326 l
S
NP 1829 3317 m 1765 3305 l
S
NP 1648 3194 m 1829 3317 l
S
NP 1578 3307 m 1512 3302 l
S
NP 1369 3204 m 1578 3307 l
S
NP 1323 3298 m 1257 3298 l
S
NP 1094 3213 m 1323 3298 l
S
NP 1064 3291 m 998 3295 l
S
NP 823 3221 m 1064 3291 l
S
NP 5862 3047 m 5929 3046 l
S
NP 6095 3128 m 5862 3047 l
S
NP 5609 3036 m 5674 3042 l
S
NP 5818 3139 m 5609 3036 l
S
NP 5364 3023 m 5426 3038 l
S
NP 5533 3153 m 5364 3023 l
S
NP 5134 3010 m 5186 3036 l
S
NP 5233 3166 m 5134 3010 l
S
NP 4916 3004 m 4950 3040 l
S
NP 4921 3172 m 4916 3004 l
S
NP 4693 3008 m 4708 3049 l
S
NP 4613 3168 m 4693 3008 l
S
NP 4459 3017 m 4457 3059 l
S
NP 4318 3159 m 4459 3017 l
S
NP 4215 3027 m 4199 3068 l
S
NP 4031 3148 m 4215 3027 l
S
NP 3968 3040 m 3939 3078 l
S
NP 3749 3135 m 3968 3040 l
S
NP 3721 3064 m 3675 3095 l
S
NP 3466 3111 m 3721 3064 l
S
NP 3450 3121 m 3384 3126 l
S
NP 3207 3054 m 3450 3121 l
S
NP 3117 3164 m 3064 3140 l
S
NP 3010 3011 m 3117 3164 l
S
NP 2821 3170 m 2779 3138 l
S
NP 2776 3005 m 2821 3170 l
S
NP 2557 3170 m 2514 3138 l
S
NP 2510 3005 m 2557 3170 l
S
NP 2308 3168 m 2259 3139 l
S
NP 2229 3008 m 2308 3168 l
S
NP 2066 3162 m 2010 3140 l
S
NP 1941 3014 m 2066 3162 l
S
NP 1823 3152 m 1761 3138 l
S
NP 1654 3023 m 1823 3152 l
S
NP 1575 3142 m 1510 3134 l
S
NP 1372 3034 m 1575 3142 l
S
NP 1321 3132 m 1255 3131 l
S
NP 1096 3044 m 1321 3132 l
S
NP 1063 3124 m 997 3127 l
S
NP 824 3052 m 1063 3124 l
S
NP 5862 2879 m 5928 2878 l
S
NP 6095 2960 m 5862 2879 l
S
NP 5608 2869 m 5674 2874 l
S
NP 5819 2971 m 5608 2869 l
S
NP 5363 2855 m 5425 2870 l
S
NP 5534 2984 m 5363 2855 l
S
NP 5131 2842 m 5184 2868 l
S
NP 5236 2997 m 5131 2842 l
S
NP 4910 2836 m 4947 2871 l
S
NP 4927 3004 m 4910 2836 l
S
NP 4687 2838 m 4704 2879 l
S
NP 4620 3001 m 4687 2838 l
S
NP 4452 2846 m 4454 2888 l
S
NP 4325 2993 m 4452 2846 l
S
NP 4208 2855 m 4197 2897 l
S
NP 4039 2984 m 4208 2855 l
S
NP 3961 2866 m 3938 2906 l
S
NP 3756 2973 m 3961 2866 l
S
NP 3717 2890 m 3675 2922 l
S
NP 3470 2949 m 3717 2890 l
S
NP 3438 2967 m 3372 2964 l
S
NP 3219 2872 m 3438 2967 l
S
NP 3083 3003 m 3042 2970 l
S
NP 3044 2837 m 3083 3003 l
S
NP 2797 3004 m 2764 2967 l
S
NP 2800 2836 m 2797 3004 l
S
NP 2539 3004 m 2503 2968 l
S
NP 2529 2836 m 2539 3004 l
S
NP 2294 3002 m 2251 2970 l
S
NP 2243 2837 m 2294 3002 l
S
NP 2057 2997 m 2003 2972 l
S
NP 1950 2843 m 2057 2997 l
S
NP 1818 2987 m 1757 2970 l
S
NP 1659 2852 m 1818 2987 l
S
NP 1572 2976 m 1507 2967 l
S
NP 1375 2864 m 1572 2976 l
S
NP 1320 2965 m 1254 2963 l
S
NP 1097 2874 m 1320 2965 l
S
NP 1063 2957 m 996 2959 l
S
NP 825 2883 m 1063 2957 l
S
NP 5862 2712 m 5928 2710 l
S
NP 6095 2792 m 5862 2712 l
S
NP 5607 2701 m 5673 2706 l
S
NP 5820 2802 m 5607 2701 l
S
NP 5361 2688 m 5424 2702 l
S
NP 5536 2815 m 5361 2688 l
S
NP 5127 2676 m 5181 2700 l
S
NP 5240 2828 m 5127 2676 l
S
NP 4903 2668 m 4943 2702 l
S
NP 4934 2835 m 4903 2668 l
S
NP 4678 2669 m 4700 2709 l
S
NP 4629 2834 m 4678 2669 l
S
NP 4443 2675 m 4450 2717 l
S
NP 4334 2828 m 4443 2675 l
S
NP 4199 2683 m 4194 2724 l
S
NP 4048 2821 m 4199 2683 l
S
NP 3951 2691 m 3935 2732 l
S
NP 3766 2812 m 3951 2691 l
S
NP 3710 2712 m 3675 2748 l
S
NP 3477 2791 m 3710 2712 l
S
NP 3399 2823 m 3341 2803 l
S
NP 3258 2681 m 3399 2823 l
S
NP 3045 2835 m 3020 2796 l
S
NP 3082 2668 m 3045 2835 l
S
NP 2773 2834 m 2752 2794 l
S
NP 2824 2669 m 2773 2834 l
S
NP 2521 2835 m 2493 2797 l
S
NP 2546 2668 m 2521 2835 l
S
NP 2280 2835 m 2242 2801 l
S
NP 2257 2668 m 2280 2835 l
S
NP 2046 2831 m 1997 2803 l
S
NP 1961 2672 m 2046 2831 l
S
NP 1812 2822 m 1752 2803 l
S
NP 1665 2682 m 1812 2822 l
S
NP 1569 2810 m 1505 2800 l
S
NP 1378 2693 m 1569 2810 l
S
NP 1318 2799 m 1252 2796 l
S
NP 1099 2705 m 1318 2799 l
S
NP 1062 2789 m 996 2792 l
S
NP 825 2714 m 1062 2789 l
S
NP 5861 2544 m 5928 2543 l
S
NP 6095 2623 m 5861 2544 l
S
NP 5606 2534 m 5672 2538 l
S
NP 5821 2633 m 5606 2534 l
S
NP 5359 2522 m 5422 2534 l
S
NP 5538 2645 m 5359 2522 l
S
NP 5122 2509 m 5178 2532 l
S
NP 5245 2658 m 5122 2509 l
S
NP 4895 2501 m 4938 2533 l
S
NP 4942 2666 m 4895 2501 l
S
NP 4668 2500 m 4695 2539 l
S
NP 4639 2667 m 4668 2500 l
S
NP 4432 2504 m 4445 2546 l
S
NP 4345 2663 m 4432 2504 l
S
NP 4187 2510 m 4189 2552 l
S
NP 4060 2657 m 4187 2510 l
S
NP 3936 2515 m 3929 2557 l
S
NP 3781 2652 m 3936 2515 l
S
NP 3690 2526 m 3671 2566 l
S
NP 3497 2641 m 3690 2526 l
S
NP 3305 2666 m 3282 2627 l
S
NP 3352 2501 m 3305 2666 l
S
NP 3012 2661 m 3004 2619 l
S
NP 3115 2506 m 3012 2661 l
S
NP 2753 2662 m 2741 2621 l
S
NP 2844 2505 m 2753 2662 l
S
NP 2504 2666 m 2485 2625 l
S
NP 2563 2502 m 2504 2666 l
S
NP 2267 2668 m 2234 2631 l
S
NP 2270 2500 m 2267 2668 l
S
NP 2037 2665 m 1990 2635 l
S
NP 1970 2502 m 2037 2665 l
S
NP 1806 2656 m 1748 2635 l
S
NP 1671 2511 m 1806 2656 l
S
NP 1566 2644 m 1502 2632 l
S
NP 1381 2523 m 1566 2644 l
S
NP 1317 2632 m 1251 2628 l
S
NP 1100 2535 m 1317 2632 l
S
NP 1061 2622 m 995 2624 l
S
NP 826 2545 m 1061 2622 l
S
NP 5861 2377 m 5927 2375 l
S
NP 6096 2454 m 5861 2377 l
S
NP 5605 2367 m 5671 2371 l
S
NP 5822 2464 m 5605 2367 l
S
NP 5356 2355 m 5420 2367 l
S
NP 5541 2476 m 5356 2355 l
S
NP 5117 2343 m 5174 2364 l
S
NP 5250 2488 m 5117 2343 l
S
NP 4886 2334 m 4932 2364 l
S
NP 4951 2497 m 4886 2334 l
S
NP 4657 2331 m 4688 2368 l
S
NP 4650 2499 m 4657 2331 l
S
NP 4419 2334 m 4438 2374 l
S
NP 4358 2497 m 4419 2334 l
S
NP 4171 2337 m 4181 2379 l
S
NP 4076 2494 m 4171 2337 l
S
NP 3913 2339 m 3920 2381 l
S
NP 3804 2492 m 3913 2339 l
S
NP 3627 2334 m 3645 2375 l
S
NP 3560 2497 m 3627 2334 l
S
NP 3235 2475 m 3252 2435 l
S
NP 3422 2356 m 3235 2475 l
S
NP 2988 2485 m 2994 2443 l
S
NP 3139 2346 m 2988 2485 l
S
NP 2737 2490 m 2734 2448 l
S
NP 2861 2341 m 2737 2490 l
S
NP 2491 2495 m 2478 2454 l
S
NP 2576 2336 m 2491 2495 l
S
NP 2255 2499 m 2228 2461 l
S
NP 2282 2332 m 2255 2499 l
S
NP 2028 2498 m 1985 2466 l
S
NP 1979 2333 m 2028 2498 l
S
NP 1800 2490 m 1744 2467 l
S
NP 1677 2341 m 1800 2490 l
S
NP 1563 2477 m 1500 2465 l
S
NP 1384 2354 m 1563 2477 l
S
NP 1316 2465 m 1250 2460 l
S
NP 1101 2366 m 1316 2465 l
S
NP 1061 2455 m 994 2456 l
S
NP 827 2376 m 1061 2455 l
S
NP 5860 2209 m 5926 2207 l
S
NP 6097 2285 m 5860 2209 l
S
NP 5604 2200 m 5670 2203 l
S
NP 5823 2295 m 5604 2200 l
S
NP 5353 2189 m 5417 2199 l
S
NP 5544 2306 m 5353 2189 l
S
NP 5111 2177 m 5170 2196 l
S
NP 5256 2318 m 5111 2177 l
S
NP 4877 2168 m 4926 2196 l
S
NP 4960 2327 m 4877 2168 l
S
NP 4643 2164 m 4681 2198 l
S
NP 4664 2331 m 4643 2164 l
S
NP 4403 2164 m 4430 2202 l
S
NP 4374 2331 m 4403 2164 l
S
NP 4151 2165 m 4172 2205 l
S
NP 4096 2330 m 4151 2165 l
S
NP 3881 2165 m 3904 2204 l
S
NP 3836 2330 m 3881 2165 l
S
NP 3530 2174 m 3586 2196 l
S
NP 3657 2321 m 3530 2174 l
S
NP 3213 2289 m 3247 2252 l
S
NP 3444 2206 m 3213 2289 l
S
NP 2972 2308 m 2988 2268 l
S
NP 3155 2186 m 2972 2308 l
S
NP 2724 2317 m 2729 2275 l
S
NP 2873 2178 m 2724 2317 l
S
NP 2480 2324 m 2473 2282 l
S
NP 2587 2171 m 2480 2324 l
S
NP 2245 2330 m 2222 2290 l
S
NP 2292 2165 m 2245 2330 l
S
NP 2020 2331 m 1980 2297 l
S
NP 1988 2164 m 2020 2331 l
S
NP 1795 2323 m 1741 2299 l
S
NP 1682 2172 m 1795 2323 l
S
NP 1561 2310 m 1498 2297 l
S
NP 1386 2184 m 1561 2310 l
S
NP 1315 2298 m 1249 2293 l
S
NP 1102 2197 m 1315 2298 l
S
NP 1060 2287 m 994 2289 l
S
NP 827 2208 m 1060 2287 l
S
NP 5859 2042 m 5926 2040 l
S
NP 6097 2116 m 5859 2042 l
S
NP 5602 2034 m 5668 2036 l
S
NP 5825 2125 m 5602 2034 l
S
NP 5350 2023 m 5415 2032 l
S
NP 5547 2136 m 5350 2023 l
S
NP 5105 2012 m 5166 2029 l
S
NP 5262 2147 m 5105 2012 l
S
NP 4866 2002 m 4919 2027 l
S
NP 4971 2157 m 4866 2002 l
S
NP 4629 1997 m 4672 2029 l
S
NP 4678 2162 m 4629 1997 l
S
NP 4385 1995 m 4420 2031 l
S
NP 4392 2163 m 4385 1995 l
S
NP 4128 1995 m 4159 2032 l
S
NP 4119 2163 m 4128 1995 l
S
NP 3843 1996 m 3883 2030 l
S
NP 3874 2163 m 3843 1996 l
S
NP 3486 2030 m 3552 2034 l
S
NP 3701 2129 m 3486 2030 l
S
NP 3205 2110 m 3246 2077 l
S
NP 3452 2049 m 3205 2110 l
S
NP 2962 2133 m 2985 2094 l
S
NP 3165 2025 m 2962 2133 l
S
NP 2714 2144 m 2725 2103 l
S
NP 2883 2014 m 2714 2144 l
S
NP 2471 2153 m 2469 2111 l
S
NP 2596 2005 m 2471 2153 l
S
NP 2236 2161 m 2218 2120 l
S
NP 2301 1998 m 2236 2161 l
S
NP 2013 2163 m 1976 2128 l
S
NP 1994 1996 m 2013 2163 l
S
NP 1791 2156 m 1738 2131 l
S
NP 1686 2002 m 1791 2156 l
S
NP 1559 2143 m 1497 2129 l
S
NP 1388 2015 m 1559 2143 l
S
NP 1314 2130 m 1248 2125 l
S
NP 1103 2028 m 1314 2130 l
S
NP 1060 2120 m 994 2121 l
S
NP 827 2039 m 1060 2120 l
S
NP 5859 1875 m 5925 1872 l
S
NP 6098 1947 m 5859 1875 l
S
NP 5601 1867 m 5667 1868 l
S
NP 5826 1955 m 5601 1867 l
S
NP 5347 1857 m 5412 1865 l
S
NP 5550 1965 m 5347 1857 l
S
NP 5099 1847 m 5161 1861 l
S
NP 5268 1976 m 5099 1847 l
S
NP 4856 1837 m 4913 1859 l
S
NP 4981 1986 m 4856 1837 l
S
NP 4615 1831 m 4663 1860 l
S
NP 4692 1992 m 4615 1831 l
S
NP 4367 1828 m 4409 1861 l
S
NP 4410 1994 m 4367 1828 l
S
NP 4103 1828 m 4145 1861 l
S
NP 4144 1994 m 4103 1828 l
S
NP 3808 1834 m 3860 1859 l
S
NP 3909 1989 m 3808 1834 l
S
NP 3473 1877 m 3539 1872 l
S
NP 3714 1946 m 3473 1877 l
S
NP 3202 1935 m 3247 1905 l
S
NP 3455 1887 m 3202 1935 l
S
NP 2955 1959 m 2983 1921 l
S
NP 3172 1863 m 2955 1959 l
S
NP 2707 1972 m 2723 1931 l
S
NP 2890 1850 m 2707 1972 l
S
NP 2463 1982 m 2466 1941 l
S
NP 2604 1840 m 2463 1982 l
S
NP 2229 1991 m 2214 1950 l
S
NP 2308 1831 m 2229 1991 l
S
NP 2007 1995 m 1972 1959 l
S
NP 2001 1827 m 2007 1995 l
S
NP 1788 1989 m 1736 1963 l
S
NP 1689 1833 m 1788 1989 l
S
NP 1558 1976 m 1496 1961 l
S
NP 1390 1846 m 1558 1976 l
S
NP 1313 1963 m 1248 1957 l
S
NP 1104 1860 m 1313 1963 l
S
NP 1060 1952 m 993 1953 l
S
NP 828 1871 m 1060 1952 l
S
NP 5858 1708 m 5924 1704 l
S
NP 6099 1778 m 5858 1708 l
S
NP 5599 1701 m 5665 1701 l
S
NP 5828 1786 m 5599 1701 l
S
NP 5344 1692 m 5409 1697 l
S
NP 5553 1795 m 5344 1692 l
S
NP 5093 1682 m 5157 1694 l
S
NP 5274 1805 m 5093 1682 l
S
NP 4847 1672 m 4906 1692 l
S
NP 4990 1814 m 4847 1672 l
S
NP 4601 1666 m 4654 1691 l
S
NP 4706 1820 m 4601 1666 l
S
NP 4348 1663 m 4397 1691 l
S
NP 4429 1823 m 4348 1663 l
S
NP 4079 1664 m 4130 1691 l
S
NP 4168 1822 m 4079 1664 l
S
NP 3781 1675 m 3842 1692 l
S
NP 3936 1811 m 3781 1675 l
S
NP 3468 1717 m 3532 1708 l
S
NP 3719 1769 m 3468 1717 l
S
NP 3200 1763 m 3248 1734 l
S
NP 3457 1723 m 3200 1763 l
S
NP 2950 1786 m 2982 1750 l
S
NP 3177 1700 m 2950 1786 l
S
NP 2701 1800 m 2721 1760 l
S
NP 2896 1686 m 2701 1800 l
S
NP 2457 1812 m 2463 1770 l
S
NP 2610 1675 m 2457 1812 l
S
NP 2222 1822 m 2211 1781 l
S
NP 2315 1664 m 2222 1822 l
S
NP 2001 1827 m 1969 1790 l
S
NP 2006 1659 m 2001 1827 l
S
NP 1784 1822 m 1733 1795 l
S
NP 1693 1664 m 1784 1822 l
S
NP 1556 1809 m 1494 1794 l
S
NP 1391 1678 m 1556 1809 l
S
NP 1313 1795 m 1247 1789 l
S
NP 1104 1691 m 1313 1795 l
S
NP 1059 1784 m 993 1785 l
S
NP 828 1702 m 1059 1784 l
S
NP 5857 1541 m 5923 1537 l
S
NP 6100 1609 m 5857 1541 l
S
NP 5598 1534 m 5664 1534 l
S
NP 5829 1616 m 5598 1534 l
S
NP 5341 1526 m 5407 1530 l
S
NP 5556 1624 m 5341 1526 l
S
NP 5088 1517 m 5153 1527 l
S
NP 5279 1634 m 5088 1517 l
S
NP 4839 1508 m 4900 1524 l
S
NP 4998 1642 m 4839 1508 l
S
NP 4589 1502 m 4646 1523 l
S
NP 4718 1648 m 4589 1502 l
S
NP 4332 1499 m 4386 1523 l
S
NP 4445 1651 m 4332 1499 l
S
NP 4059 1502 m 4116 1523 l
S
NP 4188 1649 m 4059 1502 l
S
NP 3764 1516 m 3828 1527 l
S
NP 3953 1634 m 3764 1516 l
S
NP 3465 1554 m 3529 1543 l
S
NP 3722 1596 m 3465 1554 l
S
NP 3199 1592 m 3248 1564 l
S
NP 3458 1558 m 3199 1592 l
S
NP 2946 1614 m 2982 1579 l
S
NP 3181 1536 m 2946 1614 l
S
NP 2696 1629 m 2720 1589 l
S
NP 2901 1522 m 2696 1629 l
S
NP 2451 1641 m 2461 1600 l
S
NP 2616 1509 m 2451 1641 l
S
NP 2216 1652 m 2208 1611 l
S
NP 2321 1498 m 2216 1652 l
S
NP 1995 1659 m 1966 1621 l
S
NP 2012 1491 m 1995 1659 l
S
NP 1781 1655 m 1731 1627 l
S
NP 1696 1496 m 1781 1655 l
S
NP 1555 1642 m 1493 1626 l
S
NP 1393 1509 m 1555 1642 l
S
NP 1312 1628 m 1247 1621 l
S
NP 1105 1523 m 1312 1628 l
S
NP 1059 1616 m 993 1617 l
S
NP 828 1534 m 1059 1616 l
S
NP 5856 1374 m 5922 1369 l
S
NP 6101 1440 m 5856 1374 l
S
NP 5596 1368 m 5662 1366 l
S
NP 5831 1446 m 5596 1368 l
S
NP 5339 1360 m 5405 1363 l
S
NP 5558 1454 m 5339 1360 l
S
NP 5084 1352 m 5149 1360 l
S
NP 5283 1462 m 5084 1352 l
S
NP 4831 1344 m 4894 1357 l
S
NP 5006 1470 m 4831 1344 l
S
NP 4578 1338 m 4638 1356 l
S
NP 4729 1476 m 4578 1338 l
S
NP 4318 1336 m 4376 1356 l
S
NP 4459 1478 m 4318 1336 l
S
NP 4044 1340 m 4105 1356 l
S
NP 4203 1474 m 4044 1340 l
S
NP 3753 1357 m 3818 1362 l
S
NP 3964 1458 m 3753 1357 l
S
NP 3464 1390 m 3527 1377 l
S
NP 3723 1424 m 3464 1390 l
S
NP 3198 1422 m 3249 1395 l
S
NP 3459 1392 m 3198 1422 l
S
NP 2944 1443 m 2981 1408 l
S
NP 3184 1371 m 2944 1443 l
S
NP 2692 1457 m 2719 1419 l
S
NP 2905 1357 m 2692 1457 l
S
NP 2446 1470 m 2459 1429 l
S
NP 2621 1344 m 2446 1470 l
S
NP 2210 1483 m 2206 1441 l
S
NP 2327 1332 m 2210 1483 l
S
NP 1990 1491 m 1963 1452 l
S
NP 2017 1324 m 1990 1491 l
S
NP 1777 1488 m 1729 1459 l
S
NP 1700 1327 m 1777 1488 l
S
NP 1553 1474 m 1492 1458 l
S
NP 1394 1340 m 1553 1474 l
S
NP 1311 1460 m 1246 1453 l
S
NP 1106 1354 m 1311 1460 l
S
NP 1059 1448 m 993 1449 l
S
NP 828 1366 m 1059 1448 l
S
NP 5856 1208 m 5921 1202 l
S
NP 6101 1271 m 5856 1208 l
S
NP 5595 1201 m 5661 1199 l
S
NP 5832 1277 m 5595 1201 l
S
NP 5336 1194 m 5403 1196 l
S
NP 5561 1284 m 5336 1194 l
S
NP 5080 1187 m 5145 1193 l
S
NP 5287 1292 m 5080 1187 l
S
NP 4825 1179 m 4889 1190 l
S
NP 5012 1299 m 4825 1179 l
S
NP 4569 1174 m 4631 1189 l
S
NP 4738 1304 m 4569 1174 l
S
NP 4306 1173 m 4368 1189 l
S
NP 4470 1305 m 4306 1173 l
S
NP 4032 1178 m 4096 1190 l
S
NP 4215 1300 m 4032 1178 l
S
NP 3745 1195 m 3812 1196 l
S
NP 3972 1283 m 3745 1195 l
S
NP 3463 1224 m 3525 1210 l
S
NP 3724 1254 m 3463 1224 l
S
NP 3198 1252 m 3249 1226 l
S
NP 3459 1226 m 3198 1252 l
S
NP 2941 1272 m 2981 1238 l
S
NP 3186 1206 m 2941 1272 l
S
NP 2689 1286 m 2718 1249 l
S
NP 2908 1192 m 2689 1286 l
S
NP 2442 1300 m 2458 1259 l
S
NP 2625 1178 m 2442 1300 l
S
NP 2205 1313 m 2203 1271 l
S
NP 2332 1165 m 2205 1313 l
S
NP 1984 1322 m 1959 1283 l
S
NP 2024 1156 m 1984 1322 l
S
NP 1772 1320 m 1726 1290 l
S
NP 1705 1158 m 1772 1320 l
S
NP 1550 1307 m 1490 1290 l
S
NP 1397 1171 m 1550 1307 l
S
NP 1310 1293 m 1245 1286 l
S
NP 1107 1185 m 1310 1293 l
S
NP 1058 1281 m 992 1281 l
S
NP 829 1197 m 1058 1281 l
S
NP 5855 1040 m 5920 1034 l
S
NP 6102 1101 m 5855 1040 l
S
NP 5594 1035 m 5660 1032 l
S
NP 5833 1107 m 5594 1035 l
S
NP 5334 1028 m 5401 1029 l
S
NP 5563 1114 m 5334 1028 l
S
NP 5077 1021 m 5143 1026 l
S
NP 5290 1121 m 5077 1021 l
S
NP 4820 1015 m 4885 1024 l
S
NP 5017 1127 m 4820 1015 l
S
NP 4562 1010 m 4625 1022 l
S
NP 4745 1132 m 4562 1010 l
S
NP 4298 1010 m 4361 1022 l
S
NP 4479 1132 m 4298 1010 l
S
NP 4023 1016 m 4088 1024 l
S
NP 4224 1126 m 4023 1016 l
S
NP 3741 1033 m 3807 1031 l
S
NP 3976 1109 m 3741 1033 l
S
NP 3463 1058 m 3524 1043 l
S
NP 3724 1084 m 3463 1058 l
S
NP 3197 1083 m 3250 1057 l
S
NP 3460 1059 m 3197 1083 l
S
NP 2940 1101 m 2982 1068 l
S
NP 3187 1041 m 2940 1101 l
S
NP 2686 1116 m 2717 1078 l
S
NP 2911 1026 m 2686 1116 l
S
NP 2438 1129 m 2456 1089 l
S
NP 2629 1013 m 2438 1129 l
S
NP 2199 1143 m 2201 1101 l
S
NP 2338 999 m 2199 1143 l
S
NP 1977 1153 m 1956 1113 l
S
NP 2030 989 m 1977 1153 l
S
NP 1767 1153 m 1722 1122 l
S
NP 1710 989 m 1767 1153 l
S
NP 1548 1141 m 1488 1122 l
S
NP 1399 1001 m 1548 1141 l
S
NP 1309 1126 m 1244 1118 l
S
NP 1108 1016 m 1309 1126 l
S
NP 1058 1113 m 992 1113 l
S
NP 829 1029 m 1058 1113 l
S
NP 5854 873 m 5920 867 l
S
NP 6102 932 m 5854 873 l
S
NP 5593 868 m 5659 864 l
S
NP 5834 938 m 5593 868 l
S
NP 5333 862 m 5399 861 l
S
NP 5564 944 m 5333 862 l
S
NP 5074 856 m 5140 859 l
S
NP 5293 950 m 5074 856 l
S
NP 4816 850 m 4881 857 l
S
NP 5021 956 m 4816 850 l
S
NP 4556 846 m 4621 855 l
S
NP 4751 960 m 4556 846 l
S
NP 4291 846 m 4355 855 l
S
NP 4486 960 m 4291 846 l
S
NP 4017 853 m 4083 858 l
S
NP 4230 953 m 4017 853 l
S
NP 3737 869 m 3803 864 l
S
NP 3980 937 m 3737 869 l
S
NP 3462 891 m 3524 876 l
S
NP 3725 914 m 3462 891 l
S
NP 3197 913 m 3250 888 l
S
NP 3460 892 m 3197 913 l
S
NP 2938 931 m 2982 899 l
S
NP 3189 875 m 2938 931 l
S
NP 2684 945 m 2717 908 l
S
NP 2913 861 m 2684 945 l
S
NP 2434 958 m 2455 918 l
S
NP 2633 848 m 2434 958 l
S
NP 2193 972 m 2199 930 l
S
NP 2344 834 m 2193 972 l
S
NP 1970 984 m 1952 944 l
S
NP 2037 822 m 1970 984 l
S
NP 1761 986 m 1718 953 l
S
NP 1716 820 m 1761 986 l
S
NP 1544 974 m 1486 954 l
S
NP 1403 832 m 1544 974 l
S
NP 1308 959 m 1243 950 l
S
NP 1109 847 m 1308 959 l
S
NP 1057 946 m 991 945 l
S
NP 830 860 m 1057 946 l
S
    0 G
medium setlinewidth
NP 3461 743 m 3461 4256 l
S
thin setlinewidth
NP 3439 854 m 3461 854 l
S
NP 3439 980 m 3461 980 l
S
NP 3439 1106 m 3461 1106 l
S
NP 3417 1232 m 3461 1232 l
S
NP 3439 1358 m 3461 1358 l
S
NP 3439 1484 m 3461 1484 l
S
NP 3439 1610 m 3461 1610 l
S
NP 3439 1736 m 3461 1736 l
S
NP 3417 1861 m 3461 1861 l
S
NP 3439 1987 m 3461 1987 l
S
NP 3439 2113 m 3461 2113 l
S
NP 3439 2239 m 3461 2239 l
S
NP 3439 2365 m 3461 2365 l
S
NP 3417 2491 m 3461 2491 l
S
NP 3439 2617 m 3461 2617 l
S
NP 3439 2743 m 3461 2743 l
S
NP 3439 2868 m 3461 2868 l
S
NP 3439 2994 m 3461 2994 l
S
NP 3417 3120 m 3461 3120 l
S
NP 3439 3246 m 3461 3246 l
S
NP 3439 3372 m 3461 3372 l
S
NP 3439 3498 m 3461 3498 l
S
NP 3439 3624 m 3461 3624 l
S
NP 3417 3750 m 3461 3750 l
S
NP 3439 3875 m 3461 3875 l
S
NP 3439 4001 m 3461 4001 l
S
NP 3439 4127 m 3461 4127 l
S
NP 3439 4253 m 3461 4253 l
S
medium setlinewidth
NP 692 2491 m 6230 2491 l
S
thin setlinewidth
NP 692 2474 m 692 2491 l
S
NP 944 2458 m 944 2491 l
S
NP 1195 2474 m 1195 2491 l
S
NP 1447 2474 m 1447 2491 l
S
NP 1699 2474 m 1699 2491 l
S
NP 1951 2474 m 1951 2491 l
S
NP 2202 2458 m 2202 2491 l
S
NP 2454 2474 m 2454 2491 l
S
NP 2706 2474 m 2706 2491 l
S
NP 2958 2474 m 2958 2491 l
S
NP 3209 2474 m 3209 2491 l
S
NP 3713 2474 m 3713 2491 l
S
NP 3964 2474 m 3964 2491 l
S
NP 4216 2474 m 4216 2491 l
S
NP 4468 2474 m 4468 2491 l
S
NP 4720 2458 m 4720 2491 l
S
NP 4971 2474 m 4971 2491 l
S
NP 5223 2474 m 5223 2491 l
S
NP 5475 2474 m 5475 2491 l
S
NP 5727 2474 m 5727 2491 l
S
NP 5978 2458 m 5978 2491 l
S
NP 6230 2474 m 6230 2491 l
S
medium setlinewidth
(\2611) dup 0 stringbbox 2 idiv 1232 exch sub exch 3398 exch sub exch m show
(\2610.5) dup 0 stringbbox 2 idiv 1861 exch sub exch 3398 exch sub exch m show
(0) dup 0 stringbbox 2491 exch sub exch 3398 exch sub exch m show
(0.5) dup 0 stringbbox 2 idiv 3120 exch sub exch 3398 exch sub exch m show
(1) dup 0 stringbbox 2 idiv 3750 exch sub exch 3398 exch sub exch m show
(y) dup 0 stringbbox 2 idiv 4298 exch sub exch 3408 exch sub exch m show
(\2612) dup 0 stringbbox 2444 exch sub exch 2 idiv 944 exch sub exch m show
(\2611) dup 0 stringbbox 2444 exch sub exch 2 idiv 2202 exch sub exch m show
(1) dup 0 stringbbox 2444 exch sub exch 2 idiv 4720 exch sub exch m show
(2) dup 0 stringbbox 2444 exch sub exch 2 idiv 5978 exch sub exch m show
(x) dup 0 stringbbox 2644 exch sub exch 2 idiv 6200 exch sub exch m show
boundarythick setlinewidth
% The following draws a box around the plot,
% if the variable drawborder is true
% drawborder { 
% /bd boundarythick 2 idiv def
% [] 0 setdash 
% NP bd bd m bd 5000 bd sub l 
% 6923 bd sub 5000 bd sub l 
% 6923 bd sub bd l 
% bd bd l S
% } if % end of if to draw the border

showpage
grestore
end
%%EOF

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