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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 275, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/275\hfil Bifurcation for non linear ODEs]
{Bifurcation for non linear ordinary differential equations with singular
perturbation}

\author[S. A. Spitalier, R. Bebbouchi \hfil EJDE-2016/275\hfilneg]
{Safia Acher Spitalier, Rachid Bebbouchi}

\address{Safia Acher Spitalier \newline
Universit\'e des Sciences et de la Technologie Houari Boumediene, \newline
 Facult\'e des Math\'ematiques,
BP 32 EL ALIA 16111 Bab Ezzouar Alger, Alg\'erie}
\email{safia.acher.spitalier@gmail.com}

\address{Rachid Bebbouchi \newline
Universit\'e des Sciences et de la Technologie Houari Boumediene, \newline
Facult\'e des Math\'ematiques,
 BP 32 EL ALIA 16111 Bab Ezzouar Alger, Alg\'erie}
\email{rbebbouchi@hotmail.com}


\thanks{Submitted July 10, 2016. Published October 17, 2016.}
\subjclass[2010]{34A26, 34C05, 34D15, 34F10}
\keywords{Singular perturbation; reduced equation with non-uniqueness;
\hfill\break\indent bifurcation; non-standard analysis}

\begin{abstract}
 We study a family of singularly perturbed ODEs with one parameter
 and compare their solutions to the ones of the corresponding reduced
 equations.  The interesting characteristic here is that the reduced
 equations have more  than one solution for a given set of initial conditions.
 Then we consider how those solutions are organized for different values of the
 parameter. The bifurcation associated to this  situation is studied
 using a minimal set of tools from non standard analysis
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

We study the singularly perturbed equation
\begin{equation}
\varepsilon \ddot{y}(t) +\dot{y}^3(t) - y^2(t) + a = 0, \quad t \in \mathbb{R},
\label{ePea}
\end{equation}
with the control parameter $a \ge 0$ finite (see \cite{Rob}
for the terminology in non standard analysis)
and $\varepsilon>0$ infinitely small (i.e. $\forall x\in \mathbb{R}_+^*,
|\varepsilon| < x$, and we note $\varepsilon \simeq 0$).
For a given $a$, we compare its solutions to the ones of the reduced equation
\begin{equation}
\dot{y}^3 - y^2 + {}^{\circ}a = 0 \quad \label{eP0a}
\end{equation}
(note that for any limited hyperreal $x$ there is a unique standard real
 noted ${}^{\circ}x$ infinitely close to $x$, i.e. ${}^{\circ}x - x \simeq 0$).

As this equation has different properties for $a=0$ and $a>0$, we
study the two cases separately then try to understand what happens when
$a \to 0$.


\section{Properties of \eqref{ePea}}


Equation \eqref{ePea} is equivalent to the differential system
\begin{equation} \label{eSea}
\begin{gathered}
\dot{y} = u,\\
\varepsilon \dot{u} = y^2 - u^3 -a.
\end{gathered}
\end{equation}
This is a slow-fast system and its slow manifold has the equation
\begin{equation}
y^2 - u^3 - a = 0 \quad \label{eCa}
\end{equation}
In the phase plane, the field is infinitely large outside of \eqref{eCa},
as $1/\varepsilon$ is infinitely large
(i.e. its inverse is infinitely small)
and inward. On \eqref{eCa} the field is transverse and has the
same sign as $u$. The slow manifold is attractive everywhere.

 \subsection{Case $a=0$}
 This case has been studied in \cite{Ach}. Here, the system has one
saddle-node equilibrium point in $(0,0)$.

 In this case, the reduced equation $ \dot{y}^3 -y^2 = 0 $ has an
infinity of solutions, for an initial condition $y(0)\le0$, such that
\[
y(t) =  \begin{cases}
  \big( \frac{t}{3} + \sqrt[3]{y(0)} \big)^3 , & 0 \le t \le -3\sqrt[3]{y(0)}, \\[4pt]
  0 , & -3\sqrt[3]{y(0)} \le t \le -3\sqrt[3]{y(0)} + \delta, \\[4pt]
  \big( \frac{t - \delta}{3} + \sqrt[3]{y(0)} \big)^3 ,
& -3\sqrt[3]{y(0)} + \delta \le t.
 \end{cases}
\]
where $\delta \ge 0$ is the time spent on the $t$-axis by the solution.

 In such a configuration, the main question that arises is which
of these solutions of \eqref{ePea} with $\varepsilon=0$ and $a=0$,
 starting from $y(0)$ will be the closest to the unique solution
of \eqref{ePea} with $a=0$, starting from a given point infinitely
close to that $y(0)$. The following theorem gives the answer to that
question for the solutions of \eqref{ePea} with $a=0$  starting
from $y(0)<0$ not infinitely small (that we call slow paths).

 \begin{theorem} \label{thm1.1}
 The standard part of a slow path for \eqref{ePea} with $a=0$ and
$\varepsilon \simeq 0$ is the solution of \eqref{ePea} with
$\varepsilon=0$ and $a=0$  that starts from ${}^{\circ}y(0)$
and does not spend time on the $t$-axis.
 \end{theorem}

For a proof of the above theorem see \cite{Ach}.
 This phenomenon is shown in Figure \ref{fig1}.

 \begin{figure}[htb]
  \begin{center}
  \includegraphics[width=0.8\textwidth]{fig1}
%ReducedSol_y0=-50_and_sol_y0=-50_a=0_eps=cv.png
\end{center}
\caption{Convergence of the solution of \eqref{ePea} with $a=0$
towards the ``fastest'' solution of \eqref{ePea} with $\varepsilon=0$ and $a=0$ 
 (in red) for $y(0)=-50$,
shown here (in green) for $\varepsilon \in \{1, 0.1, 0.01 \}$.}
\label{fig1}
 \end{figure}


 \subsection{Case $a>0$}
  The field is shown in Figure \ref{fig2}.

 \begin{figure}[htb]
  \begin{center}
 \includegraphics[width=0.8\textwidth]{fig2}
%slowman_field_a=1.png
\end{center}
\caption{Slow manifold $(\mathcal{C}_1)$ such that: $y^2 - u^3 - 1 = 0$
and vector field, for $\varepsilon=1$.}
\label{fig2}
 \end{figure}

 Let us now discuss  the nature of the equilibria that appear here.

 \begin{theorem} \label{thm1.2}
 The system has two equilibrium points:
 \begin{itemize}
  \item $(\sqrt{a},0)$ which is a saddle point;
  \item $(-\sqrt{a},0)$ which is a stable sink.
 \end{itemize}
 \end{theorem}

\begin{proof} The equilibria are:
$(y, u) = (\pm \sqrt{a}, 0)$. The Jacobian matrix of \eqref{eSea} at these
points is
 \[
J_{\varepsilon,a} =  \begin{pmatrix}
 0 & 1 \\
2y/\varepsilon & 0\\
 \end{pmatrix}.
 \]

\subsection*{Nature of the equilibrium $(\sqrt{a},0)$}
The eigenvalues of $J_{\varepsilon,a}$ are
 \[
\lambda = \pm \sqrt{2\sqrt{a}/\varepsilon}.
\]
 This equilibrium is a saddle point (attractive-repulsive) which is
structurally stable;  this means that this point will have the same
dynamic in the initial non-linearised system \eqref{eSea}.

 \begin{figure}[htb]
   \begin{center}
 \includegraphics[width=0.34\textwidth]{fig3a} \quad
% pt-selle-lin_a=1_eps=1.png}
 \includegraphics[width=0.56\textwidth]{fig3b} % pt-selle_a=1_eps=1.png
 \end{center}
\caption{Equilibrium $(\sqrt{a},0)$ still acts as saddle point on the paths of
\eqref{ePea}.}
\label{fig3}
 \end{figure}

 \subsection*{Nature of the equilibrium $(-\sqrt{a},0)$}
The eigenvalues of $J_{\varepsilon,a}$ are
 \[
\lambda = \pm i \sqrt{2\sqrt{a}/\varepsilon}.
\]
 This equilibrium is a center which is structurally unstable;
this means that it will not necessarily have the same dynamic in the
initial system \eqref{eSea}. We thus need a further analysis to get the
nature of this equilibrium.

 \begin{figure}[htb]
   \begin{center}
   \includegraphics[width=0.34\textwidth]{fig4a} % centre-lin_a=1_eps=1.png}
\quad
   \includegraphics[width=0.56\textwidth]{fig4b} % centre_a=1_eps=1.png}
 \end{center}
\caption{Equilibrium $(-\sqrt{a},0)$ no longer acts as a center on the paths of
$\eqref{ePea}$.}
\label{fig4}
 \end{figure}

 The equilibrium point $(-\sqrt{a},0)$ can be of one of the following types:
center, center-sink, or sink. Both a center and a center-sink have at
least one periodic solution. Let us prove that this is excluded.

 \begin{lemma} \label{lem1.3}
  The equilibrium point $(-\sqrt{a}, 0)$ is asymptotically stable for
$a$ postive and finite.
 \end{lemma}

 \paragraph{Proof}
Let $\gamma$ be a path that starts with initial conditions in the basin
of attraction of $(-\sqrt{a},0)$. We put $y = \bar{y} - \sqrt{a}$
and use the following change of variables, with $\beta > 0$:
\[
\bar{y} = \beta^3 Y, \quad
  u = \beta U,\quad
  t = \beta^2 T,
\]
 to write \eqref{eSea} as
\begin{gather*}
  Y' = U, \\
  U' = \frac{\beta^4}{\varepsilon} \big( - 2\sqrt{a}Y + \beta^3 Y^2- U^3 \big).
 \end{gather*}
For $\beta = \varepsilon^{1/4}$, $\gamma$'s standard part
(i.e. $t \mapsto {}^{\circ} \gamma(t)$.) is solution of the standard system
 \begin{gather*}
  Y' = U, \\
  U' = - 2\sqrt{{}^{\circ}a}Y - U^3,
 \end{gather*}
 as $\beta^3 = \varepsilon^{\frac{3}{4}} \ll 1$.
Note that ${}^{\circ}a$ can be $0$ if $a\simeq0$. Multiplying the second
equation by $U=Y'$ leads to
 \[
\frac{\mathrm{d}}{\mathrm{d}T} [ K(Y,U) ] = UU' + 2 \sqrt{{}^{\circ}a} YY'
= - U^4 < 0
\]
 for $U \neq 0$, with $K(Y,U) = \frac{U^2}{2} + \sqrt{{}^{\circ}a} Y^2 > 0$.
Hence $K$ is a Lyapunov function for the system so $\gamma$
converges towards $(0,0)$.


 The equilibrium point $(-\sqrt{a},0)$ is thus a stable sink.
Every path in its basin of attraction converges towards $-\sqrt{a}$ as
$t \to +\infty$.
\end{proof}

 \begin{definition} \label{def1.1} \rm
 We call slow paths the ones who enter in the slow manifold $\mathcal{C}_a$
neighborhood with an abscissa lower than and non infinitely close to
 $-\sqrt{{}^{\circ}a}$.
 \end{definition}

 In the phase plane, the attractive separatrix of $(\sqrt{a},0)$ goes,
for $a$ appreciable (i.e. a bounded non infinitely small.),
from this point and is almost vertical according to the vector field
description made earlier. Therefore, every slow path is in the basin
of attraction of $(-\sqrt{a},0)$ and thus is firstly increasing
than oscillating around $y=-\sqrt{a}$ to finally converge towards it.
 Their standard part is the only solution of the reduced equation
$\dot{y}^3 - y^2 + {}^{\circ}a = 0 $ starting from ${}^{\circ}y(0)$.

 \begin{figure}[htb]
  \begin{center}
 \includegraphics[width=0.8\textwidth]{fig5}
% ReducedSol_y0=-50_and_sol_y0=-50_a=1_eps=cv.png}
\end{center}
\caption{Convergence of the solution of \eqref{ePea} with $a=1$ towards
the ``slow'' solution of \eqref{ePea} with $\varepsilon=0$ (in red) for $y(0)=-50$,
shown here (in green) for $\varepsilon \in \{1, 0.1, 0.01 \}$.}
\label{fig5}
 \end{figure}


\section{Case $a$ approaches $0$}

\subsection*{Observations on the behaviors of the solutions}

 When $a \to 0$, the two equilibria collide into one to create a saddle-node
singularity. This saddle-node bifurcation is a classical one and refers
to the topological change that occurs between $a=0$ and any $a>0$.

 \begin{figure}[htb]
  \begin{center}
  \includegraphics[width=0.47\textwidth]{fig6a} % phase_thin_eps=1_a=1.png
\quad
  \includegraphics[width=0.47\textwidth]{fig6b} % phase_thin_eps=1_a=0.png
\end{center}
\caption{Saddle-node bifurcation for $a=1$ and $a=0$, and $\varepsilon = 1$.}
\label{fig6}
 \end{figure}

 Now let us look at what is happening with slow paths.
We saw that as $\varepsilon \simeq 0$, their standard parts are the solutions
of the reduced equation starting from ${}^{\circ}y(0)$ that do not stay
on the time axis for $a=0$ (cf. Theorem \ref{thm1.1}) and the ones that stay
indefinitely on $y=-\sqrt{a}$ for $a>0$ (cf. Theorem \ref{thm1.2}). Let us look
at those solutions for different values of $a$.

 \begin{figure}[htb]
  \begin{center}
  \includegraphics[width=0.47\textwidth]{fig7a} % SlowPathRed_eps=1_a=1.png
\quad
  \includegraphics[width=0.47\textwidth]{fig7b} % SlowPathRed_eps=1_a=0.png
\end{center}
\caption{Slow paths converge towards these solutions as $\varepsilon \to 0$,
shown here for $a=1$ and $a=0$.}
\label{fig7}
 \end{figure}

 The graphs above show that as $a$ goes to $0$, the shift between the ``chosen''
solutions of the reduced equations suddenly jumps at $a=0$ from the ones
that stay indefinitely near the time axis (as $y=-\sqrt{a}$ is getting
closer to $t$-axis here) to the ones that do not spend time on this axis at all.
 From this perspective, a discontinuity appears.


 \subsection*{Exhibiting the bifurcation}

 We  now analyze the natural 1-dimensional foliation of $\mathbb{R}^2$
defined by the vector field of \eqref{ePea}, its leaves being the integral
curves. This foliation evolving continuously with the parameter $a \ge 0$,
the solutions of \eqref{ePea} that converge towards the solutions of the
reduced equation that do not spend time near $t$-axis should leave some
traces around $a=0$. Let $a>0$ be infinitely small along with $\varepsilon$.
For different infinitely small values of $a$, the global aspect of the
foliation is independent on the parameter, both look identical:

 \begin{figure}[htb]
  \begin{center}
  \includegraphics[width=0.47\textwidth]{fig8a} % phase_thin_all_eps=1_a=0,26.png
\quad
  \includegraphics[width=0.47\textwidth]{fig8b} %
% phase_thin_all_eps=1_a=0,2601.png}
\end{center}
\caption{Global aspect for $a = 0.26$ (left), and $a=0.2601$ (right),
for $\varepsilon=1$.}
\label{fig8}
 \end{figure}


 But if we separate the leaves based on whether they are in the basin of
attractions of $(\sqrt{a},0)$ or of $(-\sqrt{a},0)$, we realise that the
foliation composition is actually very dependant on the value of $a$.

 \begin{figure}[htb]
  \begin{center}
  \includegraphics[width=0.47\textwidth]{fig9a}
%phase_thin_upper_eps=1_a=0,26.png
\quad
  \includegraphics[width=0.47\textwidth]{fig9b} \\
% phase_thin_upper_eps=1_a=0,2601.png
  \includegraphics[width=0.47\textwidth]{fig9c} 
% phase_thin_lower_eps=1_a=0,26.png
\quad
  \includegraphics[width=0.47\textwidth]{fig9d}
% phase_thin_lower_eps=1_a=0,2601.png}
\end{center}
\caption{Phase plane foliation composition for those same values.}
\label{fig9}
 \end{figure}

 We realise here that there is an analytical bifurcation where the foliation
of the part of the phase plane $y<0$ and $y$ not infinitely small evolves
very quickly but continuously from being included in the basin of
attraction of $(-\sqrt{a},0)$ to being included in the basin of attraction
of $(\sqrt{a},0)$ as $a$ decreases. The first case corresponds to when
 the standard part of $\eqref{ePea}$ slow paths are the solutions of the
reduced equation \eqref{ePea} with $\varepsilon=0$ and $a=0$
(here ${}^{\circ}a = 0$) that stay indefinitely on the time axis and
 the second when they are the ones of the reduced equation that do not
spend time on the $t$-axis.

 As this phenomenon takes place, we can see that as $a \to 0$, the upper
attractive separatrix $\Sigma_0$ of $(\sqrt{a},0)$ will have to continuously
go from a quasi-vertical position above the slow manifold $\mathcal{C}_a$
to crossing it just above $(-\sqrt{a},0)$ and going straight down.
A noteworthy value associated to this bifurcation is when this $\Sigma_0$
is asymptotical to the slow manifold without crossing it, let's call it $a_0$.
This value is necessarily infinitely small.

 The following theorem is the main result of our study and gives an 
expression for $a_0$.

 \begin{theorem} \label{thm3.1}
 The characteristic value associated to the bifurcation described above is
 \[
a_0 = s_0 \varepsilon^{6/5}
\]
 for $\varepsilon \simeq 0$ and $s_0\in \mathbb{R}$ standard.
Simulation gives $0.26 < s_0 < 0.2601$.
 \end{theorem}

\begin{proof} Using the change of variables
 \[
 y = a_0^{1/2} Y,  \quad
 u = a_0^{1/3} U, \quad
 t = a_0^{1/6} T,
 \]
System \eqref{eSea} with $a=a_0$ becomes
 \begin{gather*}
 Y' = U, \\
 U' = \frac{a_0^{5/6}}{\varepsilon} [ Y^2 - U^3 - 1].
 \end{gather*}
 Let us discuss the values of $\beta = \frac{a_0^{5/6}}{\varepsilon}$:
\smallskip

\noindent\textbf{$\beta$ is infinitely small}
The standard part of the slow paths of \eqref{ePea} with $\alpha=a_0$
 are solutions of the trivial system
\begin{gather*}
 Y' = U, \\
 U' = 0.
 \end{gather*}
All its paths are horizontal in the phase plane.
This forces ${}^{\circ}\Sigma_0$ to cross the slow manifold
$\mathcal{C}_{0} = {}^{\circ}\mathcal{C}_{a_0}$ at a bounded abscissa.
This is excluded as $\Sigma_0$ does not cross $\mathcal{C}_{a_0}$ such
that $y^2 - u^3 - a_0= 0$, i.e. $Y^2 - U^3 - 1 = 0$ which is not
horizontal in $(Y,U)$.
\smallskip

\noindent\textbf{$\beta$ is infinitely large}
The initial system is equivalent to $(S_{\varepsilon,1})$.
$\Sigma_0$ being asymptotical to $Y^2 - U^3 - 1 = 0$,
i.e. $\mathcal{C}_{1}$ in $(Y,U)$, a path starting on this separatrix will
enter the slow manifold $\mathcal{C}_{1}$ neighborhood and will stay in
it until it spiral-sinks into the equilibrium $(-1,0)$.
This is impossible as such a path is supposed to follow $\Sigma_0$ until
it reaches $(\sqrt{a_0},0) \simeq (0,0)$.

 Therefore the only value possible is $\beta$ appreciable
(i.e. a bounded real number not infinitely small), and
$a_0 = \beta^{6/5} \varepsilon^{6/5}$.
\end{proof}


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\end{document}