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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 136, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/136\hfil Centers on center manifolds]
{Centers on center manifolds in a quadratic system obtained from
a scalar third-order differential equation}

\author[W. F. da Cunha, F. S. Dias, L. F. Mello
 \hfil EJDE-2011/136\hfilneg]
{Warley Ferreira da Cunha, Fabio Scalco Dias, Luis Fernando Mello}
% in alphabetical order

\address{Warley Ferreira da Cunha \newline
Instituto de Ci\^encias Exatas, 
Universidade Federal de Itajub\'a\\ 
Avenida BPS 1303, Pinheirinho, CEP 37.500-903,
Itajub\'a, MG, Brazil}
\email{warleycunha@unifei.edu.br}

\address{Fabio Scalco Dias \newline
Instituto de Ci\^encias Exatas, 
Universidade Federal de Itajub\'a\\ 
Avenida BPS 1303, Pinheirinho, CEP 37.500-903,
Itajub\'a, MG, Brazil}
\email{scalco@unifei.edu.br}

\address{Luis Fernando Mello \newline
Instituto de Ci\^encias Exatas,
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 September 29, 2011. Published October 19, 2011.}
\subjclass[2000]{34C40, 34C15, 34C60, 34C25}
\keywords{Center; center manifold; invariant algebraic surface;
quadratic system}

\begin{abstract}
 We give affirmative answers to two questions
 concerning the existence of centers on
 local center manifolds at equilibria of a
 quadratic system in the three dimensional space.
 These questions were posed by Dias and Mello \cite{DM}
 when studying a scalar third-order differential equation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{question}[theorem]{Question}

\section{Introduction}\label{S:1}

Dias and Mello \cite{DM} studied the stability and
bifurcations in the dynamics of the  third-order
differential equation
\begin{equation}\label{eq:01}
x''' + f(x) \: x'' + g(x) x' + h(x) = 0,
\end{equation}
where $f, g, h : \mathbb{R} \to \mathbb{R}$ are
\begin{equation}\label{eq:02}
f(x) = a_1 x + a_0, \quad g(x) = b_1 x + b_0, \quad
h(x) = c_2 x^2 + c_1 x + c_0,
\end{equation}
with $a_1, a_0, b_1, b_0, c_2, c_1, c_0 \in \mathbb{R}$,
$c_2 \neq 0$. From the natural definition of the variables
$y = x'$ and $z = x''$, differential equation \eqref{eq:01} can
be written as the system of nonlinear differential equations
\begin{equation}\label{eq:03}
\begin{gathered}
x' = P(x,y,z) = y,\\
y' = Q(x,y,z) = z,\\
z' = R(x,y,z) = - \big( (a_1 x + a_0) z + (b_1 x + b_0) y + c_2 x^2
+ c_1 x + c_0 \big),
\end{gathered}
\end{equation}
where $(x,y,z) \in \mathbb{R}^3$ are the state variables and
$(a_0, a_1, b_0, b_1, c_0, c_1, c_2) \in \mathbb{R}^7$, $c_2 \neq 0$,
are real parameters. The choice of real affine functions $f$ and $g$
and a quadratic function $h$ implies that the vector field that
defines \eqref{eq:03},
\begin{equation}\label{eq:04}
\mathcal{X} (x,y,z) = \left( P(x,y,z), Q(x,y,z), R(x,y,z) \right),
\end{equation}
is a quadratic vector field. So, system \eqref{eq:03} is a quadratic
system of differential equations in $\mathbb{R}^3$.

Despite its simplicity,  \eqref{eq:03} has a rich local
dynamical behavior presenting several degenerate bifurcations. See
\cite{DM} for more details. Define the following two curves in the
space of parameters of system \eqref{eq:03} (see
\cite[figures 1  and 2]{DM})
\begin{gather*}
\mathcal{L}_2 = \{ a_0 = 1/b_0, a_1 = 0, b_0 > 0, b_1 = 2 b_0, c_0
=0, c_1 = c_2 =1 \}, \\
\mathcal{L}_3 = \{ a_0 = 0, a_1 > 0, b_0 = 1/a_1, b_1 = 0, c_0 =0,
c_1 = c_2 =1 \}.
\end{gather*}

It was shown in \cite{DM} that for parameters in $\mathcal{L}_2$ the
Jacobian matrix of $\mathcal{X}$ at the equilibrium point $E_0 =
(0,0,0)$ presents one negative real eigenvalue and a pair of purely
imaginary eigenvalues,
\[
\lambda_1 = - \frac{1}{b_0}, \quad \lambda_{2,3}= \pm i \sqrt{b_0},
\]
and the first four Lyapunov coefficients vanish. Analogously, for
parameters in $\mathcal{L}_3$ the Jacobian matrix of $\mathcal{X}$ at
the equilibrium point $E_1 = (-1, 0, 0)$ presents one positive real
eigenvalue and a pair of purely imaginary eigenvalues,
\[
\theta_1 = a_1, \quad \theta_{2,3}= \pm i /\sqrt{a_1},
\]
and the first four Lyapunov coefficients vanish too.


In the study of local and global bifurcations of
system \eqref{eq:03} in  \cite{DM},  the following two questions
were posed.

\begin{question}\label{q:01} \rm
Consider system \eqref{eq:03} with parameters in $\mathcal{L}_2$. Is
the equilibrium point $E_0$ a center for the flow of system
\eqref{eq:03} restricted to the center manifold?
\end{question}

\begin{question}\label{q:02} \rm
Consider system \eqref{eq:03} with parameters in $\mathcal{L}_3$. Is
the equilibrium point $E_1$ a center for the flow of system
\eqref{eq:03} restricted to the center manifold?
\end{question}

The study of stability of equilibrium points is an interesting
subject of research; for recent developments see \cite{MPS,MC}.
However, the stability of degenerate equilibrium points
is very difficult.
The present article may contribute to the understanding
of degenerate equilibrium points of system \eqref{eq:03},
by giving affirmative answers the two questions above.

\section{Answers to Questions \ref{q:01} and \ref{q:02}}\label{S:2}

For parameters in $\mathcal{L}_2$ ($\mathcal{L}_3$, respectively)
system \eqref{eq:03} has a nonhyperbolic equilibrium point at
$E_0$ ($E_1$, respec.). By the Center Manifold Theorem,
see \cite{kuznet}, there
is a two dimensional invariant manifold $W_0^{c}$
($W_1^{c}$, respec.) in a neighborhood of $E_0$
($E_1$, respec.) that is tangent
to the center eigenspace $E_0^c$ at $E_0$
($E_1^c$ at $E_1$, respec.) and contains all the local recurrent
behavior of the system. The center manifold $W_0^{c}$
($W_1^{c}$, respec.) is attracting (repelling, respec.)
since $\lambda_1 < 0$ ($\theta_1 >0$, respec.).

Our answers to Questions \ref{q:01} and \ref{q:02} are based on the
existence of invariant algebraic surfaces for system \eqref{eq:03}:
a polynomial $F(x,y,z)$ defines an invariant algebraic surface
$\mathcal{A} = F^{-1}(0)$ for system \eqref{eq:03} if and only if
there exists a polynomial $K(x,y,z)$, called the cofactor of $F$,
such that $\mathcal{X} F = K F$. See \cite{llibre} and the references
therein.

\begin{theorem}\label{thm:01}
For parameters in $\mathcal{L}_2$ system \eqref{eq:03} has an invariant
algebraic surface $\mathcal{A}_{b_0} = F_{b_0}^{-1}(0)$, $b_0 > 0$,
where
\begin{equation}\label{eq:05}
F_{b_0} (x,y,z) = b_0 x + z + b_0 x^2.
\end{equation}
Furthermore, $W_0^{c} \subset \mathcal{A}_{b_0}$ and the flow of
system \eqref{eq:03} restrict to $\mathcal{A}_{b_0}$ has a center at
$E_0$.
\end{theorem}

\begin{proof}
For parameters in $\mathcal{L}_2$ we have
\begin{equation}\label{eq:051}
\mathcal{X}_{b_0} = \Big( y, z, - \big( x + b_0 y + \frac{1}{b_0} z
+ x^2 + 2 b_0 xy \big) \Big).
\end{equation}
It is simple to see that $\mathcal{X}_{b_0} F_{b_0} = K F_{b_0}$ for
$F_{b_0}$ in \eqref{eq:05} and the cofactor $K(x,y,z) = - 1/b_0$.
Therefore, $\mathcal{A}_{b_0} = F_{b_0}^{-1}(0)$ is an invariant
algebraic surface of the system defined by \eqref{eq:051} for each
$b_0> 0$. It is immediate that $E_0 \in \mathcal{A}_{b_0}$.
The center eigenspace $E_0^c$ at $E_0$ is spanned by the vectors
\[
V_{b_0}^{1} = \big( - 1/b_0, 0, 1 \big), \quad
V_{b_0}^{2} = \big( 0, - 1/\sqrt{b_0}, 0 \big).
\]
The gradient of $F_{b_0}$ at $E_0$ is given by $\nabla F_{b_0}(E_0)
= (b_0, 0, 1)$. Hence $\nabla F_{b_0}(E_0)$ is orthogonal to
$V_{b_0}^{1}$ and $V_{b_0}^{2}$. This implies that
$W_0^{c} \subset \mathcal{A}_{b_0}$.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
\caption{Phase portrait of
system \eqref{eq:06}. The equilibrium $E_0$ is a center while the
equilibrium $E_1$ is a saddle. Note a homoclinic loop at $E_1$
bounding the center region} \label{fig1}
\end{figure}


 Solving $F_{b_0} = 0$ for the variable $z$ in terms of $x$
and substituting into the first and second equations of the system
defined by \eqref{eq:051} we have the  differential
equations
\begin{equation}\label{eq:06}
x' = y, \quad y' = -b_0 x - b_0 x^2,
\end{equation}
which is a Hamiltonian system with Hamiltonian function
\[
H(x, y) = \frac{b_0}{2} x^2 + \frac{1}{2} y^2 + \frac{b_0}{3} x^3.
\]
The phase portrait of this system is illustrated in Figure
\ref{fig1} which can be viewed as the projection in the plane $xy$
of the phase portrait of the system defined by \eqref{eq:051} on the
invariant algebraic surface $\mathcal{A}_{b_0}$ for each $b_0 > 0$.
The phase portrait of the system defined by \eqref{eq:051} on
$\mathcal{A}_{b_0}$ is depicted in Figure \ref{fig2}. The proof is
complete.
\end{proof}

The affirmative answer to Question \ref{q:01} follows from Theorem
\ref{thm:01}.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}
\end{center}
\caption{Phase portrait of the system defined by
\eqref{eq:051} on $\mathcal{A}_{b_0}$ in a neighborhood of the
equilibrium $E_0$} \label{fig2}
\end{figure}

To give an affirmative answer to Question \ref{q:02} we make the
 change of variables
$(\bar{x}, \bar{y}, \bar{z}) = (x,y,z) - (-1, 0, 0)$;
that is, we translate the equilibrium
$E_1 = (-1, 0,0)$ to $\bar{E}_1 = (0,0,0)$.

\begin{theorem}\label{thm:02}
For parameters in $\mathcal{L}_3$ system \eqref{eq:03} with the above
change of variables has an invariant algebraic surface
$\mathcal{A}_{a_1} = F_{a_1}^{-1}(0)$, $a_1 > 0$, where
\begin{equation}\label{eq:07}
F_{a_1} (x,y,z) = x + a_1 z.
\end{equation}
Furthermore, $W_1^{c} \subset \mathcal{A}_{a_1}$ and the flow of
system \eqref{eq:03}, with the above change of variables, restrict
to $\mathcal{A}_{a_1}$ has a center at $\bar{E}_1$.
\end{theorem}

\begin{proof}
For parameters in $\mathcal{L}_3$, with the change of variables
$(\bar{x}, \bar{y}, \bar{z}) = (x,y,z) - (-1, 0, 0)$ and dropping
the bars we have
\begin{equation}\label{eq:08}
\mathcal{X}_{a_1} = \Big( y, z, - \big( -x + \frac{1}{a_1} y - a_1
z + x^2 + a_1 xz \big) \Big).
\end{equation}
It is simple to see that $\mathcal{X}_{a_1} F_{a_1} = K F_{a_1}$ for
$F_{a_1}$ in \eqref{eq:07} and the cofactor
$K(x,y,z) = a_1 - a_1 x$. Therefore,
$\mathcal{A}_{a_1} = F_{a_1}^{-1}(0)$ is an invariant
algebraic surface of the system defined by \eqref{eq:08} for each
$a_1 > 0$. It is immediate that $\bar{E}_1 \in \mathcal{A}_{a_1}$.
The center eigenspace $E_1^c$ at $\bar{E}_1$ is spanned by the
vectors
\[
V_{a_1}^{1} = ( - a_1, 0, 1 ), \quad V_{a_1}^{2}
= ( 0, - \sqrt{a_1}, 0 ).
\]
The gradient of $F_{a_1}$ at $\bar{E}_1$ is given by
$\nabla F_{a_1}(\bar{E}_1) = (1, 0, a_1)$. Hence
$\nabla F_{a_1}(\bar{E}_1)$ is orthogonal to $V_{a_1}^{1}$ and
$V_{a_1}^{2}$. This implies that
$W_1^{c} \subset \mathcal{A}_{a_1}$.


 Solving $F_{a_1} = 0$ for the variable $z$ in terms of $x$
and substituting into the first and second equations of the system
defined by \eqref{eq:08} we have the  differential
equations
\begin{equation}\label{eq:09}
x' = y, \quad y' = -\frac{1}{a_1} x,
\end{equation}
which is a Hamiltonian linear system with Hamiltonian function
\[
H(x, y) = \frac{1}{2 a_1} x^2 + \frac{1}{2} y^2.
\]
The phase portrait of the system defined by \eqref{eq:08} on
$\mathcal{A}_{a_1}$ is depicted in Figure \ref{fig3}. The proof is
complete.
\end{proof}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig3}
\end{center}
\caption{Phase portrait of the system defined by
\eqref{eq:08} on $\mathcal{A}_{a_1}$ in a neighborhood of the
equilibrium $\bar{E}_1$} \label{fig3}
\end{figure}

The affirmative answer to Question \ref{q:02} follows from
Theorem \ref{thm:02}.

\subsection*{Concluding remarks}\label{S:5}
This paper provides a stability analysis that accounts for the
characterization, in the space of parameters, of the structural as
well as Lyapunov stability of the equilibria of system
\eqref{eq:03}. Concerning the vanishing of the Lyapunov
coefficients in a quadratic system two questions about the stability
of the equilibria $E_0$ and $E_1$ are answered. See Questions
\ref{q:01} and \ref{q:02} and Theorems \ref{thm:01} and
\ref{thm:02}.


Our proofs of Theorems \ref{thm:01} and \ref{thm:02} show that the
local center manifolds of equilibria $E_0$ and $E_1$ are algebraic
ruled surfaces. In particular, the local center manifolds of
equilibrium $E_1$ are planes coincident with the center eigenspaces
$E_1^{c}$ for each parameter $a_1 > 0$. These are unexpected
results.

\subsection*{Acknowledgements} W. F. da Cunha is partially
supported by CAPES.
L. F. Mello is partially supported by grants 304926/2009-4
from CNPq, and PPM-00204-11 from FAPEMIG.
F. S. Dias and L. F. Mello are partially supported by project
APQ-01511-09 from FAPEMIG.


\begin{thebibliography}{0}

\bibitem{DM}
{F. S. Dias and L. F. Mello};
\emph{Analysis of a quadratic system obtained from a scalar third
order differential equation},
Electron. J. Differential Equations,  vol. 2010 (2010), No. 161, 1--25.

\bibitem{kuznet} Y. A. Kuznetsov;
\emph{Elements of Applied Bifurcation Theory},
second edition, Springer-Verlag, New York, 1998.

\bibitem{llibre} J. Llibre;
\emph{On the integrability of the differential
systems in dimension two and of the polynomial differential systems
in arbitrary dimension}, Journal of Applied Analysis and
Computation, {\bf 1} (2011), 33--52.

\bibitem{MPS} A. Mahdi, C. Pessoa, D. S. Shafer;
\emph{Centers on center manifolds in the L\"u system},
 Phys. Lett. A, {\bf 375} (2011), 3509--3511.

\bibitem{MC} L. F. Mello, S. F. Coelho;
\emph{Degenerate Hopf bifurcations in the L\"u system},
Phys. Lett. A, {\bf 373} (2009), 1116-1120.

\end{thebibliography}

\end{document}