\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
Tenth MSU Conference on Differential Equations and Computational
Simulations. \newline
\emph{Electronic Journal of Differential Equations},
Conference 23 (2016),  pp. 175--187.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document} \setcounter{page}{175}
\title[\hfilneg EJDE-2016/Conf/23 \hfil Bogdanov-Takens singularity]
{Bogdanov-Takens singularity of a neural network model with delay}

\author[X. P. Wu \hfil EJDE-2016/conf/23 \hfilneg]
{Xiaoqin P. Wu}

\address{Xiaoqin P. Wu \newline
Department of Mathematics, Computer \& Information Sciences\\
Mississippi Valley State University\\
Itta Bena, MS 38941, USA}
\email{xpwu@mvsu.edu}


\thanks{Published March 21, 2016}
\subjclass[2010]{92B20, 34F10}
\keywords{Tree-neuron model with time delay; BT singularity;  Hopf bifurcation;
\hfill\break\indent double limit cycle bifurcation; triple limit cycle bifurcation}

\begin{abstract}
 In this article, we study Bogdanov-Takens (BT) singularity of a tree-neuron
 model with time delay. By using the frameworks of Campbell-Yuan \cite{C_Y}
 and Faria-Magalh\~aes \cite{F_M:1,F_M:2}, the normal form on the center manifold
 is derived for this singularity and hence the corresponding bifurcation
 diagrams such as Hopf, double limit cycle, and triple limit cycle bifurcations
 are obtained. Examples are given to verify some theoretical results.
\end{abstract}

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


\section{Introduction}

The objective of this manuscript is to study codimension-2 (Bogdanov-Takens (BT))
 bifurcation of the  tree-neuron model with delay
\begin{equation}\label{neural0}
\begin{gathered}
\frac{dv_1(t)}{dt}=-v_1(t)+f_1(v_3(t)-bv_3(t-\tau)),\\
\frac{dv_2(t)}{dt}=-v_2(t)+f_2(v_1(t)-bv_1(t-\tau)),\\
\frac{dv_3(t)}{dt}=-v_3(t)+f_3(v_2(t)-bv_2(t-\tau)).
\end{gathered}
\end{equation}
Here $f_i$ is a $C^4$ functions with $f_i(0)=0$ ($i=1,2,3$), $a_i=f_i'(0)>0$
corresponds to the range of the continuous variable $v_i$, $b>0$ is
the measure of the inhibitory influence of the past history,
and $\tau>0$ is the time delay due to the time for other neurons to respond.
This model is a little bit different from the ones studied in
\cite{B_W, D_J_Y, G_L, H, L_G_L} in which our functions $f_i(x)$ ($i=1,2,3$)
can be different.

Neural networks or neural nets have been studied by many researchers since Hopfield
\cite{H} constructed a simplified neural network model of a linear circuit
consisting of a resistor and a capacitor connected to other neurons via
nonlinear sigmoidal activation functions and have been applied to artificial
neural network and artificial brain and other fields.
In this article, we focus on System \eqref{neural0}.

Let $u_i(t)=v_i(t)-bv_i(t-\tau)$ ($i=1,2,3$). Then \eqref{neural0} can
be written as
\begin{equation} \label{neural1}
\begin{gathered}
\frac{du_1(t)}{dt}=-u_1(t)+f_1(u_3(t))-bf_1(u_3(t-\tau)),\\
\frac{du_2(t)}{dt}=-u_2(t)+f_2(u_1(t))-bf_2(u_1(t-\tau)),\\
\frac{du_3(t)}{dt}=-u_3(t)+f_3(u_2(t))-bf_3(u_2(t-\tau)).
\end{gathered}
\end{equation}
Clearly $(0,0,0)$ is an equilibrium point of \eqref{neural1} and hence the
linearized system at $(0,0,0)$ is
\begin{gather*}
\frac{du_1(t)}{dt}=-u_1(t)+a_1u_3(t)-ba_1u_3(t-\tau),\\
\frac{du_2(t)}{dt}=-u_2(t)+a_2u_1(t)-ba_2u_1(t-\tau),\\
\frac{du_3(t)}{dt}=-u_3(t)+a_3u_2(t)-ba_3u_2(t-\tau),
\end{gather*}
whose corresponding characteristic equation is
\begin{equation} \label{cheq}
\Delta(\lambda)=(\lambda+1)^3-a^3(1-be^{-\lambda\tau})^3=0
\end{equation}
where $a^3=a_1a_2a_3$ and $a_i=f_i'(0)$ ($i=1,2,3$).

The dynamical behavior and bifurcation of \eqref{neural1} have been studied
 extensively \cite{B_W, D_J_Y, G_L, L_G_L}. In \cite{B_W, G_L, L_G_L},
 Hopf singularity was studied for $f_i(x)=\tanh(x)$ by using $\tau$ as
bifurcation parameter. In \cite{D_J_Y}, the authors found critical values
of $b$ and $\tau$ such that a zero-Hopf singularity occurs.

Note that all the results mentioned above depend on the distribution of roots
of the characteristic equation \eqref{cheq}. If \eqref{cheq} has a pair of
 purely imaginary roots, a Hopf singularity occurs and hence a limit cycle
may bifurcate from the equilibrium point. If \eqref{cheq} has a simple zero
root and a pair of purely imaginary roots, a zero-Hopf singularity occurs.
However, under certain conditions, the characteristic equation may have
double zero root and this has not been studied in the literature. For a double
zero eigenvalue, the corresponding Jordan matrix is either
$\left(\begin{smallmatrix}
0&0\\
0&0
\end{smallmatrix}\right)$ or
$\left(\begin{smallmatrix}
0&1\\
0&0
\end{smallmatrix}\right)$.
 Our study shows that only the latter case occurs for \eqref{neural1}.
More specifically, we use ($b, \tau)$ as bifurcation parameter to obtain
the critical value $(b^*, \tau^*)$ such that the characteristic equation has a
double zero and then investigate its corresponding dynamical behaviors.
Note that we can find the conditions such that the equilibrium point is
asymptotically stable. But this is not practical since cyclic behaviors
are very common in real world. This leads to study Hopf singularity and
 in many cases the condition for Hopf singularity is not always satisfied.
We show that, for double zero singularity, we still can obtain limit cycles
under small perturbations of $(b^*,\tau^*)$ and under certain conditions
despite of the fact that the condition for Hopf singularity is violated.
It turns out that double zero singularity has rich dynamical behaviors.
We use the frameworks of Campbell-Yuan \cite{C_Y} and
Faria-Magalh\~aes \cite{F_M:1,F_M:2} to conduct the center manifold
reduction to obtain the normal form for this singularity and hence the
corresponding bifurcation diagrams such as Hopf, double limit cycle, and
triple limit cycle bifurcations.

The rest of this manuscript is organized as follows.
In Section \ref{sect2}, the detailed conditions are given for the linear part
of \eqref{neural1} at an equilibrium point in the $(b,\tau)$-parameter space
to have a triple zero eigenvalue and other eigenvalues with negative real parts.
In Section \ref{sect3}, the normal form of double zero singularity for
\eqref{neural1} is obtained on the center manifold by using the frameworks
from \cite{C_Y} and \cite{F_M:1, F_M:2}. In Section \ref{sect4}, the normal
form in Section \ref{sect3} is used to obtain bifurcation diagrams of the original
\eqref{neural1} such as Hopf and homoclinic bifurcations, and two examples
are presented to confirm some theoretical results.

\section{Distribution of eigenvalues}\label{sect2}

In the rest of this manuscript, we assume $b=b^*\equiv\frac{a-1}{a}$ and $a>1$.
 Clearly if $\tau=\tau^*\equiv\frac{1}{a-1}$, we have
$\Delta(0)=\Delta'(0)=0$ and $\Delta''(0)=\frac{3}{a-1}\neq0$.
Namely $\Delta(\lambda)=0$ has a zero root of with multiplicity 2 if
$\tau=\tau^*$. Clearly \eqref{cheq} is equivalent to the  equations
\[
\lambda-(a-(a-1)e^{-\lambda\tau})e^{2k\pi i/3}=0, \quad k=0,1,2.
\]
For $\omega>0$, letting $\Delta(i\omega)=0$, we have
\begin{equation} \label{cube1}
1+i\omega-(a-(a-1)e^{-i\omega\tau})=0,
\end{equation}
or
\begin{equation} \label{cube2}
1+i\omega-(a-(a-1)e^{-i\omega\tau})e^{\frac{2\pi}{3}i}=0,
\end{equation}
or
\begin{equation} \label{cube3}
1+i\omega-(a-(a-1)e^{-i\omega\tau})e^{\frac{4\pi}{3}i}=0.
\end{equation}
From\eqref{cube1}, after separating the real part from imaginary part, we have
\[
\cos(\omega\tau)=1,\quad \sin(\omega\tau)=\frac{\omega}{a-1}
\]
which give $\omega=0$. Similarly, from \eqref{cube2}, we have
\[
\cos(\omega\tau)=\frac{1+2a-\sqrt{3}\omega}{2(a-1)},\quad
\sin(\omega\tau)=-\frac{\sqrt{3}+\omega}{2(a-1)}.
\]
Using $\cos^2(\omega\tau)+\sin^2(\omega\tau)=1$, we obtain
\[
4\omega^2-a\sqrt{3}\omega+3a=0.
\]
Clearly if $a<4$, this equation does not have positive roots.
If $a>4$, it has two different positive roots
$$
\omega_{\pm}=\frac{\sqrt{3}}{2}(a\pm\sqrt{a(a-4)}).
$$
In this case, define
\[
\tau_j^{\pm}(a)=\frac 1{\omega_{\pm}}[2(j+1)\pi
-\arccos \frac{1+2a-\sqrt{3}\omega_\pm}{2(a-1)}], \quad j=0,1,2,\dots.
\]
If $a=4$ it has a positive root $\omega=\omega^*\equiv2\sqrt3$ with
multiplicity 2. In this case, define
$$
\tau_j=\frac 1{2\sqrt3}[2(j+1)\pi-\frac{\pi}{3}], \quad j=0,1,2,\dots.
$$
Note that if $\omega i$ is a root of \eqref{cube2}, then $-\omega i$ is a
 root of \eqref{cube3}. Now define
\begin{gather*}
\gamma=\{(a,\tau): \tau=\frac{1}{a-1},a>1\}, \quad
l=\{(a,\tau): a=4, \tau>0\}, \\
\Gamma_j^\pm=\{(a,\tau): \tau=\tau_j^\pm(a),a>4\},
\end{gather*}
and $P_j=(4,\tau_j), j=0,1,2,\dots$. Thus we obtain the following result
(see Figure \ref{fig0} for $j=0$).

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}  % bt-neural-bif.eps
\end{center}
 \caption{Bifurcation diagrams in $(a,\tau)$-plane for $b=b^*$}
 \label{fig0}
\end{figure}


\begin{theorem}\label{thm21}
Let $b=\frac{a-1}{a}$ and $a>1$. In the $(a,\tau)$-plane, we have
\begin{itemize}
\item[(i)] if $(a,\tau)$ is on the curve $\gamma$, the characteristic
equation \eqref{cheq} has a double zero root and hence a BT (double zero)
singularity occurs;

\item[(ii)] if $(a,\tau)$ is on one of the curves
 $\Gamma_j^+=\{(a,\tau): \tau=\tau_j^+(a)\}$
(or $\Gamma_j^-=\{(a,\tau): \tau=\tau_j^-(a)\}$) $(j=0,1,2,\dots)$,
the characteristic equation \eqref{cheq} has a simple zero root and a
 pair of purely imaginary roots $\pm\omega_{+}i$ ($\pm\omega_{-}i$) and
hence a zero-Hopf singularity occurs;

\item[(iii)] if $(a,\tau)$ is one of the points $(4,\tau_j)$ ($j=0,1,2,\dots$),
the characteristic equation \eqref{cheq} has a simple zero root and a pair of
 purely imaginary roots $2\sqrt3$ with multiplicity 2 and hence zero-Hopf 1:1
singularity occurs.

\item[(iv)] if $(a,\tau)$ is not one of the above, then the characteristic
equation \eqref{cheq} has a simple zero root and hence a zero
(or fold) singularity occurs.
\end{itemize}
\end{theorem}

For the distribution of the rest of eigenvalues for $\tau>0$, we use the following
lemma.

\begin{lemma}[Ruan and Wei \cite{R_W}] \label{lemma1}
Consider the transcendental polynomial
$$
P(\lambda, e^{-\lambda\tau_1}, e^{-\lambda\tau_2})
=p(\lambda)+q_1(\lambda)e^{-\lambda\tau_1}+q_2(\lambda) e^{-\lambda\tau_2},
$$
where $p,q_1,q_2$ are real polynomials such that $\max\{\deg q_1, \deg q_2\}<\deg(p)$
and $\tau_1,\tau_2\ge 0.$ Then as $(\tau_1,\tau_2)$ varies, the sum of the orders
of the zeros of $P$ in the open right half plane can change only if a zero appears
on or crosses the imaginary axis.
\end{lemma}

Note that
\[
\Delta(\lambda)_{b=b^*,\tau=0}=\lambda(\lambda^2+3\lambda+3)
\]
whose non-zero roots have negative real parts. By using Lemma \ref{lemma1},
 we obtain the following result regarding the rest of eigenvalues.

\begin{lemma} \label{lem2}
If $b=b^*$, all roots of the characteristic equation \eqref{cheq} have negative
real parts except zero-root and purely imaginary roots.
\end{lemma}

We remark that in this manuscript, we only study BT singularity.

\section{Computation of the normal form of double zero singularity}
\label{sect3}

In this section, we use the theory of center manifold reduction for general
 delay differential equations (DDEs) (see the detail in \cite{F_M:1, F_M:2})
to compute the normal form of BT singularity. In the rest of this manuscript,
we always assume that the assumption (H1) holds. Now we treat $(b,\tau)$ as a
bifurcation parameter near $(b^*,\tau^*)$.
By scaling $t\to t/\tau$, \eqref{neural1} can be written as
\begin{gather*}
\frac{du_1(t)}{dt}=\tau(-u_1(t)+f_1(u_3(t))-bf_1(u_3(t-1))),\\
\frac{du_2(t)}{dt}=\tau(-u_2(t)+f_2(u_1(t))-bf_2(u_1(t-1))),\\
\frac{du_3(t)}{dt}=\tau(-u_3(t)+f_3(u_2(t))-bf_3(u_2(t-1))).
\end{gather*}
Let
\[
f_i(x)=a_ix+\frac{1}{2}f''_i(0)x^2+\frac{1}{3!}f'''(0)x^3+O(x^4).
\]
Define $C:=C([-1,0],\mathbb{R}^3)$, $C^*:=C([0,1],\mathbb{R}^{3*})$ and $C^1=C^1([-1,0],\mathbb{R}^3)$.
Let $\mu_1=b-b^*$, $\mu_2=\tau-\tau^*$.
Then on $C$ we have
\begin{equation} \label{neural2}
\begin{gathered}
\begin{aligned}
\frac{du_1(t)}{dt}
&= (\tau^*+\mu_2)\Big[-u_1(0)+a_1u_3(0)-a_1(b^*+\mu_1)u_3(-1)
 +\frac{1}{2}f_1''(0)u_3^2(0)\\
&\quad -\frac{1}{2}(b^*+\mu_1)f_1''(0)u_3^2(-1)+\frac{1}{6}f_1'''(0)u_3^3(0)\\
&\quad -\frac{1}{6}(b^*+\mu_1)f_1'''(0)u_3^3(-1)\Big]
+O(\|\mu\|^2+\|\mu\|\|y\|^3) ,
\end{aligned}\\
\begin{aligned}
\frac{du_2(t)}{dt}
&= (\tau^*+\mu_2)\Big[-u_2(0)+a_2u_1(0)-a_2(b^*+\mu_1)u_1(-1)
 +\frac{1}{2}f_2''(0)u_1^2(0)\\
&\quad -\frac{1}{2}(b^*+\mu_1)f_2''(0)u_1^2(-1)
 +\frac{1}{6}f_2'''(0)u_1^3(0)\\
&\quad -\frac{1}{6}(b^*+\mu_1)f_2'''(0)u_1^3(-1)\Big]
 +O(\|\mu\|^2+\|\mu\|\|y\|^3) ,
\end{aligned}\\
\begin{aligned}
\frac{du_3(t)}{dt}
&= (\tau^*+\mu_2)\Big[-u_3(0)+a_3u_2(0)-a_3(b^*+\mu_1)u_2(-1)
 +\frac{1}{2}f_3''(0)u_2^2(0)\\
&\quad -\frac{1}{2}(b^*+\mu_1)f_3''(0)u_2^2(-1)
 +\frac{1}{6}f_3'''(0)u_2^3(0)\\
 &\quad -\frac{1}{6}(b^*+\mu_1)f_3'''(0)u_2^3(-1)]
  +O(\|\mu\|^2+\|\mu\|\|y\|^3).
\end{aligned}
\end{gathered}
\end{equation}
Let
\[
\mathbb{A}=\begin{pmatrix}
-\frac{1}{a-1}&0&\frac{a_1}{a-1}\\
\frac{a_2}{a-1}&-\frac{1}{a-1}&0\\
0&\frac{a_3}{a-1}&-\frac{1}{a-1}
\end{pmatrix}, \quad
\mathbb{B}=\begin{pmatrix}
0&0&-\frac{a_1}{a}\\
-\frac{a_2}{a}&0&0\\
0&-\frac{a_3}{a}&0
\end{pmatrix}.
\]
Define
\[
\Delta(\lambda)=\lambda I-(\mathbb{A}+\mathbb{B}e^{-\lambda}),
 \]
and the linear operator
\[
 \mathcal{L}X_t= \mathbb{A}X(t)+\mathbb{B}X(t-1), \quad \text{for }X\in C.
\]
From Section \ref{sect2}, we see that $\mathcal{L}$ has a double zero eigenvalue
and all the other eigenvalues have negative real parts. It is easy to see that
\[
\Delta(0)=-(\mathbb{A}+\mathbb{B}), \quad \Delta'(0)=I+\mathbb{B}.
\]
Let $u=(u_1,u_2,u_3)^T\in C$, $\mu=(\mu_1,\mu_2)^T$, and
\[
F(u_t,\mu)=(F^1(u_t,\mu),F^2(u_t,\mu),F^3(u_t,\mu))^T,
\] 
where
\begin{gather*}
\begin{aligned}
F^1(u_t,\mu)
&= -\mu_1a_1\tau^*u_3(-1)+\mu_2[-u_1(0)-a_1b^*u_3(-1)]+\frac{1}{2}f_1''(0)u_3^2(0)\\
&\quad -\frac{1}{2}b^*f_1''(0)u_3^2(-1)+\frac{1}{6}f_1'''(0)u_3^3(0)
 -\frac{1}{6}b^*f_1'''(0)u_3^3(-1)\\
&\quad +O(\|\mu\|^2+\|\mu\|\|y\|^3),
\end{aligned}\\
\begin{aligned}
F^2(u_t,\mu)
&= -\mu_1a_2\tau^*u_1(-1)+\mu_2[-u_2(0)-a_2b^*u_1(-1)]+\frac{1}{2}f_2''(0)u_1^2(0)\\
&\quad -\frac{1}{2}(b^*+\mu_1)f_2''(0)u_1^2(-1)
 +\frac{1}{6}f_2'''(0)u_1^3(0)-\frac{1}{6}b^*f_2'''(0)u_1^3(-1)\\
&\quad +O(\|\mu\|^2+\|\mu\|\|y\|^3),
\end{aligned}\\
\begin{aligned}
F^3(u_t,\mu)
&= -\mu_1a_3\tau^*u_2(-1)+\mu_2[-u_3(0)-a_3b^*u_2(-1)]+\frac{1}{2}f_3''(0)u_2^2(0)\\
&\quad -\frac{1}{2}b^*f_3''(0)u_2^2(-1)+\frac{1}{6}f_3'''(0)u_2^3(0)
 -\frac{1}{6}b^*f_3'''(0)u_2^3(-1)\\
&\quad +O(\|\mu\|^2+\|\mu\|\|y\|^3).
\end{aligned}
\end{gather*}
Then \eqref{neural2} can be written as
\begin{equation} \label{absTriple}
\dot{u}(t)=\mathcal{L}u_t+F(u_t,\mu)
\end{equation}
whose corresponding linear part at $0$ is
\begin{equation}\label{absDDEL}
\dot{u}(t)=\mathcal{L}u_t.
\end{equation}
From \cite{C_Y}, the bilinear form between $C$ and $C^*$ can be expressed as
\begin{equation}\label{bilinear}
\left(\psi,\varphi\right)=\psi(0)\cdot\varphi(0)
+\int_{-1}^0\psi(\xi+1)\mathbb{B}\varphi(\xi)d\xi.
\end{equation}
Then $\mathcal{L}$ has a generalized eigenspace $P$ which is invariant under
the flow \eqref{absDDEL}. Let $P^*$ be the space adjoint with $P$ in $C^*$.
Then $C$ can be decomposed as
$C=P\oplus Q$  where
$Q=\{\varphi\in C: \langle \psi,\varphi \rangle =0, \forall \psi\in P^*\}$.
 Furthermore, we can choose the bases $\Phi$ and $\Psi$ for $P$ and $P^*$,
respectively, such that
\[
(\Psi,\Phi)=I, \quad  \dot{\Phi}=\Phi J, \quad \dot{\Psi}=-J \Psi,
\]
where $I$ is the identity matrix and $J=\left(\begin{smallmatrix}
0&1\\
0&0\end{smallmatrix}\right)$ the Jordan matrix associated with
 the double zero eigenvalue with geometric multiplicity $1$.

Next, we obtain the explicit expressions of $\Phi$ and $\Psi$.
 According to Campbell and Yuan \cite{C_Y}, the basis $\Phi$ for $P$ can be
chosen as
\[
\Phi=[\varphi_1,\varphi_2]=[v_1,v_2+\theta v_1]
\]
and the basis $\Psi$ for $P^*$ as
\[
\Psi=\begin{pmatrix}\psi_1\\
\psi_2
\end{pmatrix}=\begin{pmatrix}
-w_1s+w_2\\
w_1
\end{pmatrix}
\]
where $v_1,v_2\in \mathbb{R}^3$ and $w_1,w_2\in \mathbb{R}^{3*}$ satisfy
\begin{gather}
\label{v123} \Delta(0)v_1=0,\Delta'(0)v_1+\Delta(0)v_2=0, \\
\label{w123} w_1\Delta(0)=0,w_1\Delta'(0)+w_2\Delta(0)=0.
\end{gather}
Note that \eqref{v123} is  equivalent to
\[
(\mathbb{A}+\mathbb{B})v_1=0, (\mathbb{A}+\mathbb{B})v_2=(I+\mathbb{B})v_1,
\]
from which we obtain
$$
v_1=\begin{pmatrix}1\\ a_2/a \\ a/a_1\end{pmatrix},\quad
v_2=\begin{pmatrix}1\\ a_2/a \\ a/a_1 \end{pmatrix}
$$
Similarly, \eqref{w123} is  equivalent to, respectively,
\[
w_1(\mathbb{A}+\mathbb{B})=0, w_2(\mathbb{A}+\mathbb{B})=w_1(I+\mathbb{B}).
\]
In fact, we have
$ w_1=k_1(1,a/a_2,a_1/a)$,
$w_2=k_2(1,a/a_2,a_1/a)$.
We can choose $k_1=2/3$, $k_2=-4/9$ such that
\[
(\Phi,\Psi)=I.
\]
Thus we obtain the bases $\Phi$ and $\Psi$ of $P$ and $P^*$ such that
$\dot{\Phi}=\Phi J$ and $\dot{\Psi}=-J\Psi$.

Next we compute the corresponding normal form. Let $u=\Phi x+y$
(here $x=(x_1,x_2)^T\in\mathbb{R}^2$ and $y=(y_1,y_2,y_3)^T\in C$); namely
\begin{gather*}
u_1(\theta)= x_1+\theta x_2+y_1(\theta), \\
u_2(\theta)= \frac{a_2}{a}x_1+\frac{a_2(1+\theta)}{a}x_2+y_2(\theta),\\
u_3(\theta)= \frac{a}{a_1}x_1+\frac{a(1+\theta)}{a_1}x_2+y_3(\theta).
\end{gather*}
Then, on the center manifold $y=g(x(t),\theta)$, \eqref{absTriple} becomes
\begin{equation} \label{ode}
\begin{aligned}
\dot{x}
&= Jx+\Psi(0)F(\Phi x+g(x,\theta),\mu)  \\
&= Jx+\frac{1}{2}f_2^1(x,0,\mu)+\frac{1}{3!}f_3^1(x,0,\mu)
 +O(|\mu||x|^2+|\mu|^2|x|+|x|^4)
\end{aligned}
\end{equation}
where
\begin{gather*}
\begin{aligned}
\frac{1}{2}f_2^1(x,0,\mu)
&= \Big(\frac{4a}{3(a-1)}\mu_1x_1-\frac{4}{3}(a-1)\mu_2x_2,
 -\frac{2a}{a-1}\mu_1x_1+2(a-1)\mu_2x_2\Big)\\
&\quad +\frac{1}{9 a_1^2 a_2 (a-1)a^4}(a_2 a^5f_1''(0)
 +a_1^2a^4f_2''(0)\\
&\quad +a_1^3a_2^3f_3''(0))(x_1^2+2 a x_1 x_2+ax_2^2)\binom{-2}{3},
\end{aligned}\\
\begin{aligned}
&\frac{1}{3!}f_3^1(x,0,0)\\
&= \frac{2(a^7a_2f_1'''(0)+a^5a_1^3f_2'''(0)
 +a_1^4a_2^4f_3'''(0))}{27(a-1)a^5a_1^3}(x_1^3
 +3ax_1^2x_2+3ax_1x_2^2+x_2^3)\binom{-2}{3}.
\end{aligned}
\end{gather*}
Using the result in \cite{C_Y} to project on the center manifold up to
the second order and after long calculation (we omit the detail),
then \eqref{ode} can be transformed as the following normal form,
\begin{eqnarray*}
\dot{x}=Jx+\frac{1}{2!}g_2^1(x,0,\mu)+O(|\mu|^2|x|+|x|^3),
\end{eqnarray*}
or
\begin{equation} \label{zero-normal_2nd}
\begin{gathered}
\dot{x}_1=x_2,\\
\dot{x}_2=\chi_1x_1+\chi_2x_2+A_{20}x_1^2+A_{11}x_1x_2+O(|\mu|^2|x|+|x|^3),
\end{gathered}
\end{equation}
in which $\chi_j$ and $A_{jk}$ are given by
\begin{gather*}
\chi_1= -\frac{2a}{a-1}\mu_1,\\
\chi_2= \frac{4a}{3(a-1)}\mu_1+2(a-1)\mu_2,\\
A_{20}= \frac{a^5a_2f_1''(0)+a^4a_1^2f_2''(0)+a_1^3a_2^3f_3''(0)}{3a^4a_1^2a_2(a-1)},\\
A_{11}= \frac{2(3a-2)(a^5a_2f_1''(0)+a^4a_1^2f_2''(0)
+a_1^3a_2^3f_3''(0))}{9a^4a_1^2a_2(a-1)},
\end{gather*}
if $a^5a_2f_1''(0)+a^4a_1^2f_2''(0)+a_1^3a_2^3f_3''(0)\neq 0$.
Since
\[
\big|\frac{\partial\chi}{\partial \mu}\big|
= \det\begin{pmatrix}
\frac{\partial\chi_1}{\partial \mu_1}&\frac{\partial\chi_1}{\partial \mu_2}\\
\frac{\partial\chi_2}{\partial \mu_1}&\frac{\partial\chi_2}{\partial \mu_2}
\end{pmatrix}=-4a\neq 0,
\]
we have that $(\mu_1,\mu_2)\to(\chi_1,\chi_2)$ is regular and hence the
transversality condition holds.
If $a^5a_2f_1''(0)+a^4a_1^2f_2''(0)+a_1^3a_2^3f_3''(0)=0$,
then \eqref{ode} can be transformed as the following normal form with
the third order,
\[
\dot{x}=Jx+\frac{1}{2!}g_2^1(x,0,\mu)+\frac{1}{3!}g_3^1(x,0,\mu)
+O(|\mu|^2|x|+|x|^4),
\]
or
\begin{equation} \label{zero-normal_3rd}
\begin{gathered}
\dot{x}_1=x_2,\\
\dot{x}_2=\chi_1x_1+\chi_2x_2+A_{30}x_1^3
 +A_{21}x_1^2x_2+O(|\mu|^2|x|+|x|^4),
\end{gathered}
\end{equation}
in which $\chi_j$ and $A_{jkl}$ are given by
\begin{gather*}
A_{30}= \frac{a^7a_2f_1'''(0)+a^5a_1^3f_2'''(0)
 +a_1^4a_2^4f_3'''(0)}{9a^5a_1^3a_2(a-1)},\\
A_{21}= \frac{(3a-2)(a^7a_2f_1'''(0)+a^5a_1^3f_2'''(0)
 +a_1^4a_2^4f_3'''(0))}{9a^5a_1^3a_2(a-1)}.
\end{gather*}

\section{Bifurcation diagrams and computer simulation}
\label{sect4}

In this section, we only give the bifurcation diagrams for
 \eqref{zero-normal_3rd} since it has much richer dynamical behaviors
than \eqref{zero-normal_2nd} does. Remember that, in this situation,
 we have $a^5a_2f_1''(0)+a^4a_1^2f_2''(0)+a_1^3a_2^3f_3''(0)=0$.
 Noting that $a>1$ and that, if
 $a^7a_2f_1'''(0)+a^5a_1^3f_2'''(0)+a_1^4a_2^4f_3'''(0)\neq 0$,
then $A_{30}A_{21}>0$.
\smallskip

\noindent\textbf{Case 1:} $A_{30}<0$ and $A_{21}<0$. Then under the substitution
\[
t\to \frac{A_{21}}{A_{30}}t,\quad
x_1\to -\frac{A_{21}}{\sqrt{|A_{30}|}}x_1, \quad
x_2\to\frac{A_{21}^2}{|A_{30}|^{3/2}}x_2,
 \]
System \eqref{zero-normal_3rd} is transformed into
\begin{equation}
\label{TruncatedBTnormal31}
\begin{gathered}
\dot{x}_1=x_2,\\
\dot{x}_2=\varepsilon_1x_1+\varepsilon_2x_2-x_1^3-x_1^2x_2,
\end{gathered}
\end{equation}
where
\begin{gather*}
\varepsilon_1= \left(\frac{A_{21}}{A_{30}}\right)^2\chi_1
 =-\frac{2a(3a-2)^2}{a-1}\mu_1, \\
\varepsilon_2= \frac{A_{21}}{A_{30}}\chi_2
=\frac{2(3a-2)}{3a^2(a-1)}(2a\mu_1+3(a-1)^2\mu_2).
\end{gather*}
The complete bifurcation diagrams of \eqref{TruncatedBTnormal31}
can be found in \cite{Kuznetsov:1}. Here, we just list two results.

\begin{lemma}
Let
\begin{gather*}
F^1_+= \{(\varepsilon_1,\varepsilon_2):\varepsilon_1=0, \ \varepsilon_2>0\},\\
H^1= \{(\varepsilon_1,\varepsilon_2):\varepsilon_2=0, \ \varepsilon_1<0\},\\
H^2= \{(\varepsilon_1,\varepsilon_2):\varepsilon_2=\varepsilon_1, \ \varepsilon_1>0\},\\
P= \{(\varepsilon_1,\varepsilon_2): \varepsilon_2=\frac{4}{5}\varepsilon_1+o(\varepsilon_1), \ \varepsilon_1>0\}.\\
K= \{(\varepsilon_1,\varepsilon_2): \varepsilon_2=\kappa_0\varepsilon_1+o(\varepsilon_1), \ \varepsilon_1>0\}, \kappa_0\approx 0.752.
\end{gather*}
For small $\varepsilon_1,\varepsilon_2$, then

(i) if $(\varepsilon_1,\varepsilon_2)$ is in the region between the
curves $F^1_+$ and $H^1$ or between $F^1_+$ and $H^2$,
\eqref{TruncatedBTnormal31} has a limit cycle;

(ii) if $(\varepsilon_1,\varepsilon_2)$ is in the region between the curves
 $H^2$ and $P$, \eqref{TruncatedBTnormal31} has three limit cycles: a ``big''
one and two ``small'' ones;

(iii)
 if $(\varepsilon_1,\varepsilon_2)$ is in the region between the curves
$P$ and $K$, \eqref{TruncatedBTnormal31} has two limit cycles:
the outer one is stable while the inner is unstable.
\end{lemma}

Using the expressions of $\varepsilon_1,\varepsilon_2$, we have the following result.

\begin{theorem} \label{thm41}
Suppose that $b=b^*+\mu_1$ and $\tau=\tau^*+\mu_2$. Let
\begin{gather*}
\bar{F}^1_+= \{(\mu_1,\mu_2): \mu_1=0, \ \mu_2>0\},\\
\bar{H}^1= \{(\mu_1,\mu_2): \mu_2=-\frac{2a}{3(a-1)^2}\mu_1, \ \mu_1>0\},\\
\bar{H}^2= \{(\mu_1,\mu_2): \mu_2=-\frac{a(9a-4)}{3(a-1)^2}\mu_1, \ \mu_1<0\},\\
\bar{P}= \{(\mu_1,\mu_2): \mu_2=-\frac{2a(18a-7)}{15(a-1)^2}\mu_1+o(|\mu|), \ \mu_1<0\}.\\
\bar{K}= \{(\mu_1,\mu_2): \mu_2=-\frac{a(\kappa_0(9a-6)+2)}{3(a-1)^2}\mu_1+o(|\mu_1|), \ \mu_1<0\}.
\end{gather*}
For small $\mu_1,\mu_2$, then

(i') if $(\mu_1,\mu_2)$ is in the region between the curves
$\bar{F}^1_+$ and $\bar{H}^1$ or between $\bar{F}^1_+$ and $\bar{H}^2$,
\eqref{neural0} has a stable limit cycle;

(ii') if $(\mu_1,\mu_2)$ is in the region between the curves $H^2$ and
$P$, \eqref{neural0} has three limit cycles: a ``big'' one and two ``small'' ones;

(iii')
 if $(\mu_1,\mu_2)$ is in the region between the curves $P$ and $K$, \eqref{neural0}
has two limit cycles: the outer one is stable while the inner is unstable.
\end{theorem}

\noindent\textbf{Case 2:}
$A_{30}>0$ and $A_{21}>0$. Then under the substitution
\[
t\to \frac{A_{21}}{A_{30}}t,\quad
x_1\to \frac{A_{21}}{\sqrt{A_{30}}}x_1,\quad
x_2\to-\frac{A_{21}^2}{A_{30}^{3/2}}x_2,
\]
System \eqref{zero-normal_3rd} is transformed into
\begin{equation}\label{TruncatedBTnormal32}
\begin{gathered}
\dot{x}_1=x_2,\\
\dot{x}_2=\varepsilon_1x_1+\varepsilon_2x_2+x_1^3-x_1^2x_2,
\end{gathered}
\end{equation}
where
\begin{gather*}
\varepsilon_1= \Big(\frac{A_{21}}{A_{30}}\Big)^2\chi_1
 =-\frac{2a(3a-2)^2}{a-1}\mu_1, \\
\varepsilon_2= -\frac{A_{21}}{A_{30}}\chi_2
 =\frac{2(3a-2)}{3a^2(a-1)}(2a\mu_1+3(a-1)^2\mu_2).
\end{gather*}
The complete bifurcation diagrams of \eqref{TruncatedBTnormal32}
can also be found in \cite{Kuznetsov:1}. Here, we just list two results.

\begin{lemma}
for small $\varepsilon_1,\varepsilon_2$, we have:

(i) System \eqref{TruncatedBTnormal32} undergoes a Hopf bifurcation for the
trivial equilibrium point on the line
\[
H=\{(\varepsilon_1,\varepsilon_2):\varepsilon_2=0, \, \varepsilon_1<0\}.
\]

(ii) On the curve
\[
C=\{(\varepsilon_1,\varepsilon_2): \varepsilon_2
=-\frac{1}{5}\varepsilon_1+o(\varepsilon_1), \, \varepsilon_1<0\},
\]
\eqref{TruncatedBTnormal32} undergoes a heteroclinic bifurcation.
 Moreover, if $(\varepsilon_1,\varepsilon_2)$ is in the region between
the curves $H$ and $C$ \eqref{TruncatedBTnormal32} has a unique stable
periodic orbit.
\end{lemma}

Using the expressions of $\varepsilon_1,\varepsilon_2$, we have the following result.

\begin{theorem} \label{thm42}
Suppose that $b=b^*+\mu_1$ and $\tau=\tau^*+\mu_2$.
 For small $\mu_1,\mu_2$, we have:

(i') System \eqref{neural0} undergoes a Hopf bifurcation for the trivial
equilibrium point on the line
\[
\bar{H}=\{(\mu_1,\mu_2):\mu_2=-\frac{2a}{3(a-1)^2}\mu_1, \, \mu_1>0\}.
\]

(ii') On the curve
\[
\bar{C}=\{(\mu_1,\mu_2): \mu_2=-\frac{a(9a+4)}{15(a-1)^2}\mu_1+o(\mu_1),
\, \mu_1>0\},
\]
System \eqref{neural0} undergoes a heteroclinic bifurcation. Moreover,
if $(\varepsilon_1,\varepsilon_2)$ is in the region between the curves
$\bar{H}$ and $\bar{C}$, \eqref{neural0} has a unique stable periodic orbit.
\end{theorem}

\begin{example}\rm
This example  verifies the result in Theorem \eqref{thm42}(iii). Let
\[
f_1(x)=2x+x^3, \quad f_2(x)=x-2.01x^3, \quad f_3(x)=x+x^3.
\]
Then we have $a_1=2$,  $a_2=a_3=1$, $a=\sqrt[3]{2}$,and hence
\[
b^*=\frac{\sqrt[3]{2}-1}{\sqrt[3]{2}}, \quad
\tau^*=\frac{1}{\sqrt[3]{2}-1}.
\]
Since $f_1''(0)=f_2''(0)=f_3''(0)=0$, we have $A_{20}=A_{11}=0$ and
\[
 A_{30}=-3.030700653228997,\quad  A_{21}=-5.393929340342073.
 \]
Thus in Theorem \ref{thm41},
\begin{gather*}
\bar{H}^2= \{(\mu_1,\mu_2): \mu_2=-45.623982067834476\mu_1,\mu_1<0\}, \\
\bar{P}= \{(\mu_1,\mu_2): \mu_2=-38.985746915247205\mu_1,\mu_1<0\}, \\
\bar{K}= \{(\mu_1,\mu_2): \mu_2=-37.392570478626254\mu_1,\mu_1<0\}.
\end{gather*}
If we choose $(\mu_1,\mu_2)=(-0.0001,0.0042)$, then it is easy to check that
$(\mu_1,\mu_2)$ is between $\bar{H}^2$ and $\bar{P}$ (Figure \ref{fig2}(a)).
According to Theorem \ref{thm41}, \eqref{neural0} has three limit
cycles (Figure \ref{fig2}(b), \ref{fig2}(c) and \ref{fig2}(d)).

If we choose $(\mu_1,\mu_2)=(-0.0001,0.0038)$, then it is easy to check
that $(\mu_1,\mu_2)$ is between $\bar{P}$ and $\bar{K}$
(Figure \ref{fig3}(a)). According to Theorem \ref{thm41}(iii),
\eqref{neural0} has two ``big'' limit cycles (Figure \ref{fig3}(b)
 and \ref{fig3}(c)).
\end{example}

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.45 \textwidth]{fig2a} % bt-neural10.eps
 \includegraphics[width=0.45 \textwidth]{fig2b}\\  % bt-neural11.eps
(a) \hfil (b)
\\
 \includegraphics[width=0.45 \textwidth]{fig2c} %  bt-neural12.eps
 \includegraphics[width=0.45  \textwidth]{fig2d} \\%  bt-neural13.eps
(c) \hfil (d)\\
\end{center}

\caption{(a): $(\mu_1,\mu_2)$ is between $\bar{H}^2$ and $\bar{P}$;
(b): Initial: $y_1(t)=1,y_2(t)=-0.14,y_3(t)=-0.011$ for $t\le0$;
(c): Initial: $y_1(t)=0.0001,y_2(t)=-0.001,y_3(t)=-0.001$ for $t\le0$;
(d): Initial: $y_1(t)=0.0178,y_2(t)=0.01417,y_3(t)=0.01$ for $t\le0$.}
\label{fig2}
\end{figure}

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.45 \textwidth]{fig3a} %  bt-neural20.eps}
 \includegraphics[width=0.45  \textwidth]{fig3b} \\ % bt-neural21.eps}
(a) \hfil (b) \\
 \includegraphics[width=0.45 \textwidth]{fig3c} \\ % bt-neural22.eps}
(c)
\end{center}
\caption{ $(\mu_1,\mu_2)=(-0.0001,0.0038)$:
 there is one periodic limit cycle for $y_1(t)=0.1,y_2(t)=0,y_3(t)=0$ when
 $t\le 0$.}
\label{fig3}
\end{figure}

\begin{example}\normalfont
This example  verifies the result in Theorem \ref{thm42}(ii). Let
\[
f_1(x)=\tanh(x),\quad  f_2=3\tanh(x), \quad f_3(x)=9\tanh(x).
\]
Then $a_1=1$, $a_2=3$, $a_3=9$ so that $a=3$.
Thus $b^*=\frac{2}{3}$, $\tau^*=\frac{1}{2}$. Then
\begin{gather*}
\bar{H}=\{(\mu_1,\mu_2): \mu_2=-\frac{1}{2}\mu_1,\, \mu_1>0\},\\
\bar{C}=\{(\mu_1,\mu_2): \mu_2=\frac{11}{20}\mu_1+o(\mu_1), \, \mu_1>0\}.
\end{gather*}
Choose $\mu_1=0.0005,\mu_2=0.0000875$ and it is easy to see that
$(0.0005,0.0000875)$ is in the region between the curves $\bar{H}$
and $\bar{C}$. According to Theorem \ref{thm41}(ii), \eqref{neural0}
has a unique stable periodic orbit (see Figure \ref{fig4}).
\end{example}

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.45 \textwidth]{fig4a} %  bt-neural30.eps
 \includegraphics[width=0.45 \textwidth]{fig4b}\\  % bt-neural31.eps
 \includegraphics[width=0.45 \textwidth]{fig4c} % bt-neural32.eps
 \includegraphics[width=0.45 \textwidth]{fig4d}\\  % bt-neural33.eps
\end{center}
\caption{When $(\mu_1,\mu_2)=(0.0005,0.0000875)$ lies between the curves
$\bar{H}$ and $\bar{C}$, a periodic solution is bifurcated from the origin.}
\label{fig4}
\end{figure}

\subsection*{Conclusion}
Neural networks are important both in theory and in application.
In this article, we discussed BT singularity of a neural network model
and obtained its corresponding normal. Using this normal form,
we obtained interesting dynamical behaviors such as Hopf and double
limit cycle bifurcations. Two examples were given to verify our theoretical
results.

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\end{document}