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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 183, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2018/183\hfil
Elliptic sectors and Euler discretization]
{Elliptic sectors and Euler discretization}
\author[N. Mohdeb, A. Fruchard, N. Mehidi \hfil EJDE-2018/183\hfilneg]
{Nadia Mohdeb, Augustin Fruchard, Noureddine Mehidi}
\address{Nadia Mohdeb \newline
Laboratoire de Math\'ematiques Appliqu\'ees,
Universit\'e A. Mira, Bejaia, Alg\'erie}
\email{n\_mohdeb@hotmail.com}
\address{Augustin Fruchard \newline
Laboratoire de Math\'ematiques, Informatique et Applications,
Universit\'e de Haute Alsace, Mulhouse, France}
\email{augustin.fruchard@uha.fr}
\address{Noureddine Mehidi \newline
Laboratoire de Math\'ematiques Appliqu\'ees\\
Universit\'e A. Mira, Bejaia, Alg\'erie}
\email{manogha@yahoo.fr}
\dedicatory{Communicated by Adrian Constantin}
\thanks{Submitted September 1, 2018. Published November 14, 2018.}
\subjclass[2010]{34C25, 34C37, 34A34, 39A05}
\keywords{Elliptic sector; nonhyperbolic equilibrium point;
homoclinic orbit;
\hfill\break\indent S-invertible; Euler discretization}
\begin{abstract}
In this work we are interested in the elliptic sector of
autonomous differential systems with a degenerate equilibrium point
at the origin, and in their Euler discretization.
When the linear part of the vector field at the origin has two zero
eigenvalues, then the differential system has an elliptic sector,
under some conditions. We describe this elliptic sector and we show
that the associated Euler discretized system has an elliptic sector
converging to that of the continuous system when the step size of the
discretization tends to zero.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks
\section{Introduction}
In this work we consider the planar differential system
\begin{equation}
\begin{gathered}
\dot{x}= P(x,y) \\
\dot{y}= Q(x,y)
\end{gathered} \label{1}
\end{equation}
where $P$ and $Q$ are analytic functions from $\mathbb{R}^2$ to
$\mathbb{R}$. Also we consider their Euler discretization
\begin{equation}
\begin{gathered}
x_{n+1}=x_n+hP(x_n,y_n) \\
y_{n+1}=y_n+hQ(x_n,y_n)
\end{gathered} \label{discretise}
\end{equation}
where $h>0$ is the step size of the discretization.
We assume that the origin is an isolated equilibrium point of \eqref{1}
and that system \eqref{1} has an elliptic sector $S_0$.
The main aim of this work is to explore to what extent the discrete
system \eqref{discretise} presents also an elliptic sector $S_{h}$
and whether $S_{h}$ tends to $S_0$ in the sense of the Hausdorff
distance, when $h$ tends to zero. \newline
In order to define an elliptic sector of \eqref{1}, we consider a
circle $C$ with center $(0,0)$ and radius $r$, containing no other
equilibria than the origin. We will assume that there exist
two solutions $\gamma _1$
and $\gamma _2$ of system \eqref{1} tending to the origin;
we assume for example that $\gamma _1$ tends to the origin when $t$
tends to $+\infty $ and $\gamma _2$ tends to the same point when $t$
tends to $-\infty $.
We denote by $\gamma _1^{\ast }$ and $\gamma _2^{\ast }$ the
respective corresponding orbits to the solutions
$\gamma _1^{\ast }$ and $\gamma _2^{\ast }$.
Let $M_1$ and $M_2$ be the respective intersection points of
$\gamma _1^{\ast }$ and $\gamma _2^{\ast }$ with the circle
$C$ such that, taking into account the direction of the orbits,
$M_1$ is the last intersection point of $\gamma _1^{\ast }$
with the circle $C$ and $M_2$ is the first intersection point of
$\gamma_2^{\ast }$ with $C$. Let $\sigma $ be the closed curve made
up of the two segments $OM_1$ and $OM_2$ parts of the two orbits
$\gamma_1^{\ast }$ and $\gamma _2^{\ast }$, the origin and the
oriented arc of $C$ joining $M_1$ to $M_2$ in the forward direction.
The region $R_{\sigma }$ delimited by $\sigma $ is said to be a
\textit{sector}. An \textit{elliptic sector} is a sector containing only
nested homoclinic or parts of nested homoclinic orbits
(Figure \ref{sectellip}) \cite{Andronov}.
We recall that a solution is said to be homoclinic if it is defined on
$\mathbb{R}$ entirely and tends to the origin when $t$
tends to $+\infty $ as well as when $t$ tends to $-\infty $.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm, height=3.2cm]{fig1a} % secteur.png
\quad
\includegraphics[width=4.cm, height=3.3cm]{fig1b} % elliptique.png
\end{center}
\caption{A sector and an elliptic sector}
\label{sectellip}
\end{figure}
To pose the problem raised above correctly, we must define the notion
of homoclinic orbits of system \eqref{discretise}, even when the map
$$
F_{h}{:}( x,y) \mapsto (x+hP(x,y),y+hQ(x,y))
$$
is not invertible. This leads us to define the S-invertibility notion.
The map $F_{h}$ is said to be S-invertible if, for any $m_0$ in
$\mathbb{R}^2$, there exist two reals $R>0$ and $h_0>0$ such that
for any $h$ in $]0,h_0]$, there exists a unique $m_{-1,h}$ in $\mathbb{R}^2$
for which $\| m_{-1,h}\| \leq R$ and $F_{h}(m_{-1,h})=m_0$.
In this case, we have necessarily
\[
\underset{h\to 0}{\lim }~m_{-1,h}=m_0.
\]
In other words, $F_{h}$ is S-invertible if any point of the plane has a unique
``good predecessor'' by the application $F_{h}$. The notation S comes from
nonstandard analysis (NSA): in the context of NSA, a planar map is
said to be S-invertible if any limited point of the plane has a
unique limited predecessor; for the standard functions, this notion
coincides with the usual invertibility.
Let $U$ be an open set of the disc $D(0; r_0)$ and simply connected.
We say that $U$ is an elliptic sector of the discrete system
\eqref{discretise} if any solution emanating from $U$ is homoclinic.
The works of Beyn \cite{bey87 le bon}. Fiedler and Scheurle
\cite{Fied96} and that of Zou \cite{Zou2003} deal with the problem of
the persistence of the non-degenerate homoclinic orbits of an autonomous
system after the discretization of the latter. They give an error estimate
of order $O(h^{d})$, for the difference between the homoclinic solution
of the differential equation and that of the associated discrete equation,
where $h$ is the step size of the method of discretization and $d$ its order.
They also give the length $l(h)$ of the parameter interval over which the
homoclinic orbit persists. On the other hand, given an autonomous
differential system which has an equilibrium point at $(0,0)$ not
necessarily hyperbolic and an unstable center manifold $W_c^{u}$,
it is shown in \cite{beyn87ab} that, in a small neighborhood of the origin,
the discrete system associated by a one step method has, under some
conditions, an invariant manifold close to $W_c^{u}$.
In the hyperbolic case, it is shown in \cite{beyn87} that, under some
conditions, the phase portrait of the differential system is correctly
reproduced in the associated discretization by a one step method,
on an arbitrary time interval. In \cite{Eirola3}, the author
examines the conditions which make the solution of the differential system
close to that of the associated discretized one on an infinite time interval.
A similar study has been made in \cite{Garay1} for the structurally stable
systems without periodic solutions. On the other hand, it has been
shown in \cite{Feckan2} that in the hyperbolic case, the local stable
manifold of the discrete system tends to that of the corresponding
differential system when the step size tends to zero
(see also \cite{Feckan4,Garay5}). An extension of this result to nonautonomous
differential systems is given in \cite{Garay6}. A Taylor expansion
approximation of this manifold is given in \cite{Eirola1}.
In \cite{Feckan3} it is shown that, in the hyperbolic case,
the maps defining the vector fields and the associated discretized
system by the one step method are uniformly topologically equivalent.
Other results about numerical calculus of homoclinic, heteroclinic
and periodic orbits are established in
\cite{beyn87b}-\cite{beyn2002}, \cite{Doed89}-\cite{Fried93},
\cite{Zou97} and \cite{Zou2003}.
It is known that, when system \eqref{1} has an elliptic sector,
then the associated Jacobian matrix $M$ of the function
$(x,y)\mapsto (P(x,y),Q(x,y))$ at point $(0,0)$, has two zero eigenvalues
(cf. \cite[p. 241]{lefschetz}); the origin is a non-hyperbolic equilibrium
point. Two situations are possible. The first one is that where the
matrix $M$ is null. In this case, the behavior of the solutions near
the origin is very complex;
If the smallest degree of the nonlinear terms of \eqref{1} is $m$,
then the neighborhood of the origin will be split into $2(m+1)$
parabolic, hyperbolic or elliptic sectors. The number of elliptic
sectors existing in system \eqref{1} depends on the index
of the equilibrium point (cf. \cite{perko}, p 151): let $C$ be a
Jordan curve containing $(0,0)$ and no other critical point of \eqref{1}
in its interior; then the index of the equilibrium point $(0,0)$ with
respect to \eqref{1} is given by
\[
I_{\eqref{1}}(0,0) = I_{\eqref{1}}(C)
=\frac{1}{2\pi}\oint_C \frac{PdQ-QdP}{P^2+Q^2}.
\]
The second situation is that where the matrix $M$ can be reduced by a
linear transformation to $M'=\begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix}$; it is in this situation that we are interested in this work.
In this case, system \eqref{1} is reduced by a sequence of analytic
changes of variables to the form (see \cite{Andronov})
\begin{equation}
\begin{gathered}
\dot{x}= y \\
\dot{y}= ax^{r}\big(1+k(x)\big)+bx^{p}y\big(1+g( x) \big)
+y^2f( x,y)
\end{gathered} \label{norm}
\end{equation}
where $f$, $g$ and $k$ are analytic functions, such that
$k(0)=g(0)=f( 0,0) =0$, $a$ and $b$ are real parameters,
$r$ and $p$ are integer parameters, satisfying $r=2m+1$,
$m\geq 1$, $a<0$, $b\neq 0$, $p\geq 1$, $p$ odd and either
\begin{itemize}
\item $p=m$ and $\Delta =b^2+4(m+1)a\geq 0$, or
\item $p0$. These assumptions are relatively natural.
Indeed, the assumption $p=m=1$ is generic in some sense.
The assumption $\Delta\geq0$ is necessary to get an elliptic sector,
and the assumption $\Delta\neq0$ is generic.
The affine transformation
$(x,y)\mapsto (x,\sqrt{-a}y)$ together with the time change
$\tau =\sqrt{-a}t$ and the parameter change $b\mapsto b/\sqrt{-a}$,
allow us to assume that $a=-1$. Thus we will be interested in the
elliptic sector of system
\begin{equation}
\begin{gathered}
\dot{x}= y \\
\dot{y}= -x^{3}( 1+k( x) ) +bxy\big( 1+g(x) \big) +y^2f( x,y)
\end{gathered} \label{sytcontmodif}
\end{equation}
and in its persistence in the associated Euler discretized system
\begin{equation}
\begin{gathered}
x_{n+1}=x_n+hy_n \\
y_{n+1}=y_n+h\Big(-x_n^{3}\big(1+k(x_n)\big)+bx_ny_n\big(1+g(x_n)
\big)+y_n^2f( x_n,y_n) \Big)
\end{gathered} \label{sytcont33}
\end{equation}
The main result of this work is the following.
\begin{theorem} \label{theorx3copy1}
There exists $h_0>0$ such that for any $h$ in $]0,h_0]$, there exists
a subregion $S_{h}$ of the elliptic sector $S_0$ of
system \eqref{sytcontmodif} with the following properties:
\begin{itemize}
\item Any solution of system \eqref{sytcont33} starting from
$S_{h}$ is homoclinic.
\item When $h$ tends to zero, $S_{h}$
tends to $S_0$ in the sense of the Hausdorff distance.
\end{itemize}
\end{theorem}
We do not try to obtain a maximal region of homoclinic orbits
of \eqref{sytcont33}, whose structure may be highly complicated.
In particular we do not study the behavior of orbits of \eqref{sytcont33}
starting close to the boundary of $S_0$.
A more precise study of these orbits remains to be done.
The speed of convergence of $S_h$ to $S_0$ as $h$ tends to $0$ is another
interesting problem which is not discussed here.
The article is organized as follows.
In section two, we give a local description of the elliptic sector
of system \eqref{sytcontmodif} whose existence is stated above.
To do so, we will use the results obtained during the study of the
global behavior of the solutions of the model example
\begin{equation}
\begin{gathered}
\dot{x}= y \\
\dot{y}= -x^{3}+bxy
\end{gathered} \label{sytcontx3}
\end{equation}
In section three, we give a complete proof of theorem \ref{theorx3copy1}.
In the last section we deal with the elliptic sector of the family
of differential systems which are diffeomorphic to system
\eqref{sytcontmodif} and with its persistence in the corresponding
discretized systems.
\section{Description of the continuous system}
We are first interested in the global behavior of the solutions of system
\eqref{sytcontx3} (Figure \ref{1Champs}).
The following proposition describes the phase portrait of \eqref{sytcontx3}.
The symmetry with respect to the $y$-axis allows to consider only the
solutions of \eqref{sytcontx3} starting at points with positive abscissas.
Let us denote $\Delta=b^2-8$ and $\alpha _1=( b-\sqrt{\Delta }) /4$
and $\alpha _2=( b+\sqrt{\Delta }) /4$ the solutions of the equation
$-2\alpha ^2+b\alpha -1=0$.
\begin{proposition}\label{propx3}
Let $( x_0,y_0) $ be a point of the plane such that
$x_0>0$ and let $\gamma(t,x_0,y_0) $ be the solution of system
\eqref{sytcontx3} emanating from this point.
\begin{itemize}
\item[(1)] If $y_0<\alpha _1x_0^2$, then the solution
$\gamma(t,x_0,y_0) $ is homoclinic.
\item[(2)] If $\alpha _1x_0^20$ such that for any
$u_0$ in the interval $]-\delta ,\delta ] $,
the solution of \eqref{syteclat6} starting at $(u_0,0)$ tends to
$A_2$ when $\tau$ tends to $-\infty $ and enters in the lower
half-plane after some time $\tau_0 >0$ (depending on $u_0$).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm, height=6cm]{fig4a} % portr.png
\includegraphics[width=6cm, height=6cm]{fig4b} % eclat2p.png
\end{center}
\caption{the phase portraits of system \eqref{syteclat6} near
$A_1$ and $A_2$ and of system \eqref{syteclat2}
near $B_1$, $B_2$, $B_3$ and $B_4$ for $f(x,y)=g(x)=k(x)=x$ and $b=3$.}
\label{portrait1}
\end{figure}
System \eqref{syteclat2} has as equilibria (Figure \ref{portrait1})
two saddles on
$B_1=(-\sqrt{2\alpha _1},0)$ and on
$B_2=(\sqrt{2\alpha _1},0)$ and two nodes, stable on
$B_3=(-\sqrt{2\alpha _2},0)$ and unstable on
$B_4=(\sqrt{2\alpha _2},0)$. The solutions in the neighborhoods of
$B_1$, $B_2$, $B_3$ and
$B_4$ having the tangents $w=\sqrt{2\alpha _1}$, $w=-\sqrt{2\alpha _1}$,
$w=\sqrt{2\alpha _2}$ or $w=-\sqrt{2\alpha _2}$, correspond in the
$( x,y) $ plane to the solutions of system \eqref{sytcontmodif}
whose order two approximations near $(0,0)$ are given by
$y=\alpha_1x^2$ and $y=\alpha _2x^2$. \newline
The $w$-axis is invariant under system \eqref{syteclat2}.
By continuity of the solutions of \eqref{syteclat2} with respect to the
initial conditions, those that lie near $B_1$ and $B_2$
sufficiently close to the lines $w=-\sqrt{2\alpha _2}$\ and
$w=\sqrt{2\alpha _2}$, with the initial conditions $(w_0,z_0)$
such that $-\sqrt{2\alpha _2}0$ is fixed infinitesimal.
By the Transfer Principle of NSA, it is sufficient to prove that,
for all limited $(x_0,y_0)\in S_0$ in the $S$-interior of $S_0$,
the solution $( x_n,y_n)_{n\in \mathbb{Z}}$
of \eqref{sytcont33} starting from $(x_0,y_0)$ is homoclinic.
We recall that $(x_0,y_0)$ is in the $S$-interior of $S_0$ if there exists a
standard $r>0$ such that the disk of center $(x_0,y_0)$ and of radius $r$
is in $S_0$.
For $n<0$, the points $( x_n,y_n)$ are given
recursively by Lemma \ref{lemx3fgh4}.
The proof of Theorem \ref{theorx3copy1} essentially uses the idea that
for $\alpha _2\lnsim a\lnsim \alpha _1$, the region
$\{(x,y)\in \mathbb{R}^2;y0$ and $00
\]
Also, for any limited $y$,
\[
L'( y) =-1-h( -h\frac{\partial l}{\partial x}(
u-hy,v) +\frac{\partial l}{\partial y}( u-hy,v) ) <0
\]
Since $L$ is continuous, the equation $L( x) =0$ has a unique limited
solution $y$ in $] v-1,v+1[ $.
\end{proof}
We fix a limited point $(x_0,y_0)$.
Let $( x_{-n},y_{-n}) _{n\in \mathbb{N}^\ast }$ be the predecessors
sequence of system \eqref{sytcont33} defined by lemma \ref{lemx3fgh4}.
Then, this sequence is uniquely defined, as long as $( x_{-n},y_{-n})$
is limited, by
\begin{gather*}
x_{-n}=x_{-n-1}+hy_{-n-1} \\
y_{-n}=y_{-n-1}+h\ell(x_{-n-1},y_{-n-1})
\end{gather*}
satisfying $( x_{-1},y_{-1}) \simeq ( x_0,y_0) $
and $( x_{-n-1},y_{-n-1}) \simeq ( x_{-n},y_{-n})$.
\begin{lemma} \label{lemx3fgh5}
Let $a\in \mathbb{R}$ such that $\alpha_2\lnsim a\lnsim
\alpha_1$ and $(X_0,Y_0)$ a limited point in the plane such that $
00$. The solution $( X_{-n},Y_{-n})
_{n\in \mathbb{N}^\ast }$ of system \eqref{sysdiscloupe11} starting at $
(X_0,Y_0)$, does not leave the region of the plane
\[
\{(X,Y)\in \mathbb{R}^2: X>0,\; 01$.
We denote $\delta =-2a^2+ba-1+bag(\varepsilon X_{-1})-k(\varepsilon X_{-1})$
and $\beta = \varepsilon f(\varepsilon X_{-1},\varepsilon ^2Y_{-1})-a\bar{h}$.
We have
\begin{eqnarray*} Y_0-aX_0^2
=(Y_{-1}-aX_{-1}^2) \Big( 1+\bar{h}\big( b-2a+bg(\varepsilon X_{-1})\big)
X_{-1}\Big) + \bar{h} \delta X_{-1}^{3}+\bar{h}\beta
\end{eqnarray*}
We will distinguish two cases:
If $\beta\geq 0$,
since $\delta>0$ we have
\[ \label{ineg1}
0\geq Y_0-aX_0^2\geq ( Y_{-1}-aX_{-1}^2)
(1+\bar{h}(b-2a+bg(\varepsilon X_{-1}))X_{-1}).
\]
This means that $Y_{-1}-aX_{-1}^2<0$.
On the other hand,
\[
\delta X_{-1}^{3}+\beta Y_{-1}^2
= \beta Y_{-1}(Y_{-1}-aX_{-1}^2)+ a\beta X_{-1}^2\Big(Y_{-1}
+ \frac{\delta }{a\beta}X_{-1}\Big).
\]
Since $\delta\not\simeq 0$, $\delta /(a\beta)$ is infinitely large,
then if $\beta <0$,
\[
\frac{\delta }{a\beta}X_{-1}+aX_{-1}^2
= \Big(\frac{\delta }{a\beta}+aX_{-1}\Big)X_{-1} <0.
\]
Thus,
\[
\delta X_{-1}^{3}+\beta Y_{-1}^2\geq\beta (Y_{-1}-aX_{-1}^2)(
Y_{-1}+aX_{-1}^2),
\]
hence
\begin{align*}
&Y_0-aX_0^2 \\
&\geq ( Y_{-1}-aX_{-1}^2) \Big( 1+\bar{h}(
b-2a+bg(\varepsilon X_{-1}) ) X_{-1} + \beta(
Y_{-1}+aX_{-1}^2)\Big)
\end{align*} %\label{ineg2}
It results in this case that $Y_{-1}q, x_n<0, \psi (x_n)0$.
The case $\Delta =0$ is treated separately.
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