\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 198, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/198\hfil Stability and bifurcation analysis]
{Stability and bifurcation analysis for a discrete-time bidirectional
ring neural network model with delay}
\author[Y.-K. Du, R. Xu, Q.-M. Liu \hfil EJDE-2013/198\hfilneg]
{Yan-Ke Du, Rui Xu, Qi-Ming Liu}
\address{Yan-Ke Du \newline
Institute of Applied Mathematics,
Shijiazhuang Mechanical Engineering College,
Shijiazhuang. 050003, China}
\email{yankedu2011@163.com}
\address{Rui Xu \newline
Institute of Applied Mathematics,
Shijiazhuang Mechanical Engineering College,
Shijiazhuang. 050003, China}
\email{rxu88@163.com}
\address{Qi-Ming Liu \newline
Institute of Applied Mathematics,
Shijiazhuang Mechanical Engineering College,
Shijiazhuang. 050003, China}
\email{lqmmath@163.com}
\thanks{Submitted January 3, 2012. Published September 5, 2013.}
\subjclass[2000]{92B20, 34K18, 34K20, 37G05}
\keywords{Neural network; time delay; stability; bifurcation}
\begin{abstract}
We study a class of discrete-time bidirectional ring neural
network model with delay. We discuss the asymptotic stability
of the origin and the existence of Neimark-Sacker bifurcations,
by analyzing the corresponding characteristic equation.
Employing M-matrix theory and the Lyapunov functional method,
global asymptotic stability of the origin is derived.
Applying the normal form theory and the center manifold theorem,
the direction of the Neimark-Sacker bifurcation and the stability
of bifurcating periodic solutions are obtained. Numerical simulations
are given to illustrate the main results.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
Since Hopfiled's pioneering work
\cite{s3,s4}, the dynamic behavior (including stability,
periodic oscillatory and chaos) of continuous-time Hopfield
neural networks has received much attention
due to their applications in optimization, signal processing, image
processing, solving nonlinear algebraic equation, pattern
recognition, associative memories and so on (see, \cite{s9,s10,s13,s6} and
references therein).
It is well known that time delays in the information
processing of neurons exist. The delayed axonal signal transmissions in
the neural networks make the dynamic behaviors more
complicated, and may destabilize stable equilibria and give rise
to periodic oscillation, bifurcation and chaos (see
\cite{s11,s12,s13,s14}). Therefore, the delay is inevitable and cannot
be neglected.
For computer simulations, experimental or computational purposes, it
is common to discretize the continuous-time neural networks. In some
sense, the discrete-time model inherits the dynamical
characteristics of the continuous-time networks. We refer the reader to
\cite{s15,s8,s16,s7} for related discussions on the need and
importance of discrete-time analogues to reflect the dynamics of
their continuous-time counterparts.
In the field of neural networks, rings are studied to gain insight
into the mechanisms underlying the behavior of recurrent networks.
In \cite{s5}, Wang and Han investigated the following continuous-time
bidirectional ring network model with delay
\begin{equation}\label{a1}
\begin{gathered}
\dot{x}=-x+\alpha f(y(t-\tau))+\beta f(z(t-\tau)),\\
\dot{y}=-y+\alpha f(z(t-\tau))+\beta f(x(t-\tau)),\\
\dot{z}=-z+\alpha f(x(t-\tau))+\beta f(y(t-\tau)),
\end{gathered}
\end{equation}
where $\tau$ denotes the synaptic transmission delay, $\alpha$ and
$\beta$ are connection strengths, $f:$ $R\to R$ is the
activation function. In \cite{s5}, some conditions on the linear
stability of the trivial solution of system \eqref{a1} were given
and Hopf bifurcation, including its direction and stability, were
investigated.
Motivated by the work of Wang and Han \cite{s5} and the discussions
above, in the present paper, for simplicity, assuming the neurons in
the network be identical (see \cite{s17}), we are concerned
with the stability and bifurcation analysis of the following discrete-time
bidirectional ring neural network model with delay
\begin{equation}\label{a2}
\begin{gathered}
x(n+1)=ax(n)+\beta f(y(n-k))+\beta f(z(n-k)),\\
y(n+1)=ay(n)+\beta f(z(n-k))+\beta f(x(n-k)),\\
z(n+1)=az(n)+\beta f(x(n-k))+\beta f(y(n-k)),\\
\end{gathered}
\end{equation}
where $a\in(0,1)$ is the internal decay of the neurons, $\beta$
is the connection strength, $k\in N$ is the time delay.
This paper contributes to understanding of neural networks as follows:
(1) There is a large body of work discussing the stability and bifurcation
of neural networks with delays, but most of them deal only with
continuous-time neural network models,
or discrete-time neural network models of two neurons with or without time
delays (\cite{s8,s7}).
Here we discuss the dynamic behavior of a tri-neuron discrete-time
bidirectional ring neural network with delay.
The characteristic equation of the neural network is a polynomial equation
with high order terms. Using a new approach, sufficient and necessary
conditions are derived to ensure that all the roots of the characteristic
equation stay inside or on the unit circle.
(2) We remove some restrictions on the conditions required by \cite{s5},
and in a sense, our results on the asymptotic stability of the origin
are less restrictive than those for the corresponding continuous system in
\cite{s5}.
(3) Employing M-matrix theory and the Lyapunov functional
method, global asymptotic stability of
the origin is derived, which was not taken into account in \cite{s5}.
The stability criterion is simple and can be easily checked.
The rest of this paper is organized as follows. In Section 2, we
analyze the location of roots of a class of polynomial equation. In
Section 3, the local stability of the origin and the existence of
Neimark-Sacker bifurcations are discussed by analyzing the
corresponding characteristic equation, and global asymptotic
stability is derived using the method of M-matrix and Lyapunov
function. In Section 4, we discuss the stability and direction of
the Neimark-Sacker bifurcation by employing the normal form method and
the center manifold theorem. Some numerical simulations are carried
out in Section 5 to illustrate the main results. In Section 6, a
brief discussion is given to conclude the work of this paper.
\section{analysis of polynomial equations}
In this section, we analyze the location of the roots
of the polynomial equation
\begin{equation}\label{c1}
\lambda^{k+1}-a\lambda^k-b=0,~~a\in (0,1),~b\in R,
\end{equation}
which will be used to determine the asymptotic stability of
system \eqref{a2}.
Suppose that $\lambda=\mathrm{e}^{i\theta}$ is a root
of \eqref{c1}. Substituting it into \eqref{c1} and separating the
real and imaginary parts, we have
\begin{equation}\label{c2}
\begin{gathered}
\cos ((k+1)\theta)-a\cos (k\theta)=b,\\
\sin ((k+1)\theta)-a\sin (k\theta)=0.
\end{gathered}
\end{equation}
Using the identities
$\cos ((k+1)\theta)=\cos (k\theta)\cos \theta
-\sin (k\theta)\sin \theta$ and
$ \sin ((k+1)\theta)=\sin (k\theta)\cos \theta
+\cos (k\theta)\sin \theta$,
we rewrite \eqref{c2} as
\begin{gather*}
\sqrt{a^2+1-2a\cos \theta}
\big[
\frac{\cos \theta-a}{\sqrt{a^2+1-2a\cos \theta}}\cos (k\theta)
-\frac{\sin \theta}{\sqrt{a^2+1-2a\cos \theta}}\sin (k\theta)
\big]=b,\\
\sqrt{a^2+1-2a\cos \theta}\big[
\frac{\cos \theta-a}{\sqrt{a^2+1-2a\cos \theta}}\sin (k\theta)
+\frac{\sin \theta}{\sqrt{a^2+1-2a\cos \theta}}\cos (k\theta)
\big]=0.
\end{gather*}
It is easy to see that
if $\theta\in (0,\pi)$, \eqref{c2} is equivalent to the
equations
\begin{equation}\label{c3}
\begin{gathered}
\sqrt{a^2+1-2a\cos \theta}\cdot\cos (h(\theta))=b,\\
\sin (h(\theta))=0,
\end{gathered}
\end{equation}
where
$$
h(\theta)=\operatorname{arccot}\frac{\cos \theta-a}{\sin \theta}+k\theta.
$$
Since
$$
h'(\theta)=\frac{1}{1+\big(\frac{\cos \theta-a}{\sin \theta}\big)^2}
\cdot \frac{1-a\cos \theta}{\sin ^2\theta}+k>0
$$
for $\theta\in(0,\pi)$ and
$$
\lim_{\theta\to 0^+}h(\theta)=0,\quad
\lim_{\theta\to \pi^-}h(\theta)=(k+1)\pi,
$$
we derive that $h(\theta):(0,\pi)\to(0,(k+1)\pi)$ is an
increasing bijective function.
From the second equation in \eqref{c3}, we know that
$h(\theta)=j\pi$, $j=1,2,\dots, k$. Denote
$\theta_j=h^{-1}(j\pi)$, $j=1,2,\dots, k$. Then $\theta_j$ satisfies
the equation
$$
j\pi=\operatorname{arccot}\frac{\cos \theta-a}{\sin \theta}+k\theta,
$$
which yields $f(\theta)=0$, where
$$
f(\theta)=\sin ((k+1)\theta)-a\sin (k\theta).
$$
Obviously,
$$
f(0)=0,~f'(0^+)=(1-a)k+1>0,\quad
f(\frac{j\pi}{k+1})=\begin{cases}
a\sin \frac{j\pi}{k+1}>0,&\text{if $j$ is even}\\
-a\sin \frac{j\pi}{k+1}<0, &\text{if $j$ is odd}
\end{cases}
$$
$j=1,2,\dots,k$.
Therefore, we can deduce that
$\theta_j\in(\frac{(j-1)\pi}{k+1},\frac{j\pi}{k+1})$,
$j=1,2,\dots,k$.
From the first equation in \eqref{c3}, we get that
\begin{equation}\label{c4}
b=b_j=(-1)^j\sqrt{a^2+1-2a\cos \theta_j},~j=1,2,\dots, k.
\end{equation}
If $\theta=0$, then $b=b_0=1-a>0$; if $\theta=\pi$,
then $b=b_{k+1}=(-1)^{k+1}(1+a)$.
Obviously, if $\theta$ is a root of \eqref{c2}, $-\theta$ is also
a root of \eqref{c2}. Hence, we only need to consider the roots
$\lambda=\mathrm{e}^{i\theta}$ of \eqref{c1} in $[0,\pi]$.
Further, from \eqref{c4}, we
deduce that
\begin{equation}\label{c5}
\dots|b_{k+1}|=1+a$, all roots of \eqref{c1} are
outside the unit circle;
\item[(vi)] $\frac{\mathrm{d}|\lambda|^2}{\mathrm{d}b}\big|_{b=b_j}\neq 0$
for $j=0,1,\dots,k+1$,
where $b_j=(-1)^j\sqrt{a^2+1-2a\cos \theta_j}$, and $\theta_j$ is the
unique solution in
$(\frac{(j-1)\pi}{k+1},\frac{j\pi}{k+1})$ of the equation
$\sin ((k+1)\theta)-a\sin (k\theta)=0$ for
$j=1,2,\dots, k$.
\end{itemize}
\end{theorem}
\section{Stability analysis and existence of bifurcations}
Throughout this paper, we assume that
\begin{itemize}
\item[(H1)] $f(0)=0,~f(\cdot)\in C^3(R)$.
\end{itemize}
Denote $x_0(n)=x(n)$, $x_j(n+1)=x_{j-1}(n)$; $y_0(n)=y(n)$,
$y_j(n+1)=y_{j-1}(n)$; $z_0(n)=z(n)$, $z_j(n+1)=z_{j-1}(n)$, $j=1,2,\dots,k$.
Then we can transform system \eqref{a2} into the
following system of $3k+3$ difference equations without delays
\begin{equation}\label{b1}
\begin{gathered}
x_0(n+1)=ax_0(n)+\beta f(y_k(n))+\beta f(z_k(n)),\\
y_0(n+1)=ay_0(n)+\beta f(z_k(n))+\beta f(x_k(n)),\\
z_0(n+1)=az_0(n)+\beta f(x_k(n))+\beta f(y_k(n)),\\
x_j(n+1)=x_{j-1}(n),\\
y_j(n+1)=y_{j-1}(n),\\
z_j(n+1)=z_{j-1}(n),\quad j=1,2,\dots,k.
\end{gathered}
\end{equation}
For convenience, we denote $c=\beta f'(0)$. The Jacobian matrix of
system \eqref{b1} at the equilibrium $E=(0,\dots,0)$ is as follows
\begin{equation}\label{b2}
A=\begin{bmatrix}
B & 0 & 0 & c & 0 & c\\
I_k & 0 & 0 & 0 & 0 & 0\\
0 & c & B & 0 & 0 & c\\
0 & 0 & I_k & 0 & 0 & 0\\
0 & c & 0 & c & B & 0\\
0 & 0 & 0 & 0 & I_k & 0
\end{bmatrix},
\end{equation}
where $B=(a,0,\dots, 0)_{1\times k}$, $I_k$ is a $k\times k$
identity matrix, $0$ is a zero matrix of appropriate size.
The associated characteristic equation of system \eqref{b1} is
\begin{equation}\label{b3}
(\lambda^{k+1}-a\lambda^k+c)^2(\lambda^{k+1}-a\lambda^k-2c)=0.
\end{equation}
Applying Theorem \ref{thm2.1} to \eqref{b3} and noting that
$d_1\leq -d_0$, we can obtain the following results.
\begin{theorem} \label{thm3.1}
Assume {\rm (H1)} and that $00$ such
that $|f'(\cdot)|\leq L$.
\item[(H3)] $1-a-2L|\beta|>0$.
\end{itemize}
\end{theorem}
\begin{proof}
Since $1-a-2L|\beta|>0$, the matrix
$$
A=\begin{pmatrix}
1-a & -L|\beta| & -L|\beta| \\
-L|\beta| & 1-a & -L|\beta| \\
-L|\beta| & -L|\beta| & 1-a
\end{pmatrix}
$$
is an M-matrix, and there exists a vector $p=(p_i)_{1\times 3}>0$
such that $pA>0$ (\cite{s18}); that is,
$$
p_i(1-a)-(\sum_{j=1}^{3}p_j-p_i)L|\beta|>0,~i=1,2,3.
$$
Hence, we can choose $\lambda>1$ such that
\begin{equation} \label{b4}
p_i(1-a\lambda)-\Big(\sum_{j=1}^{3}p_j-p_i\Big)L|\beta|\lambda^{k+1}>0,
\quad i=1,2,3.
\end{equation}
Let $U_1(n)=\lambda^n|x(n)|$, $U_2(n)=\lambda^n|y(n)|$,
$U_3(n)=\lambda^n|z(n)|$. From \eqref{a2}, we have
\begin{equation}\label{b5}
U_i(n+1)\leq a\lambda
U_i(n)+L|\beta|\lambda^{k+1}\Big[\sum_{j=1}^{3}U_j(n-k)-U_i(n-k)\Big],\quad
i=1,2,3.
\end{equation}
Define a Lyapunov function
$$
V(n)=\sum_{i=1}^{3}p_iU_i(n)+\sum_{l=n-k}^{n-1}\sum_{i=1}^{3}\Big[
\Big(\sum_{j=1}^{3}p_j-p_i\Big)L|\beta|\lambda^{k+1}U_i(l)\Big].
$$
Then from \eqref{b4} and \eqref{b5}, we deduce that
$$
\Delta V(n)=V(n+1)-V(n)
=-\sum_{i=1}^{3}\Big[p_i(1-a\lambda)-(\sum_{j=1}^{3}p_j-p_i)L|\beta|
\lambda^{k+1}\Big]U_i(n)
\leq 0,
$$
which implies that $V(n)\leq V(0)$.
Note that
\begin{gather*}
V(n)\geq m_0\lambda^n(|x(n)|+|y(n)|+|z(n)|),\\
V(0)=\sum_{i=1}^{3}p_iU_i(0)+\sum_{l=-k}^{-1}\sum_{i=1}^{3}
\Big[(\sum_{j=1}^{3}p_j-p_i)L|\beta|\lambda^{k+1}U_i(l)\Big]:=M_0,
\end{gather*}
where $m_0=\min_{i=1,2,3}\{p_i\}$, $M_0$ is a positive
constant.
Thus,
$$
|x(n)|+|y(n)|+|z(n)|\leq \frac{M_0}{m_0}\lambda^{-n}.
$$
Noting that $\lambda>1$, we get $\lim_{n\to +\infty}x(n)=0$,
$\lim_{n\to +\infty}y(n)=0$,
$\lim_{n\to +\infty}z(n)=0$. Then, the origin of system \eqref{a2} is globally
attractive. On the other hand, it is easy to verify that $|c|0$, then the bifurcation is subcritical;
that is, the closed invariant curve bifurcating from the origin is unstable.
\end{theorem}
\section{Numerical simulations}
In this section, we give two examples to illustrate
the results derived in Sections 3 and 4. In system \eqref{a2},
we choose the activation function as the type of inverse tangent
function or hyperbolic tangent function; i.e., $f(v)=\tanh(v)$,
then $f'(0)=1$, $f''(0)=0$, $f'''(0)=-2$, and $|f'(x)|\leq 1$.
In addition, in the following simulations, the numerical results of
$y(n)$ and $z(n)$ are similar to those
of $x(n)$, so they are omitted.
\begin{example} \label{examp1}\rm
For system \eqref{a2}, If $a=0.5$, $k=2$, then
$b_0=0.5$, $b_1\approx -0.7808$. Choose $\beta=-0.38$ and $0.24$,
respectively, we have
$\max \{-b_0,b_1/2\}=-0.3904 1 − a$, the hypotheses of Theorem \ref{thm3.1}
are less restrictive than those for the corresponding continuous system
($\alpha=\beta$) in \cite{s5}.
Moreover, when the connection weights through the neurons
in a bidirectional ring neural network model are different,
numerical simulations show that the corresponding system still
undergoes Neimark-Sacker bifurcations at the origin. We leave
for future work the study of \eqref{a2} with different connection
weights, activation functions and time delays.
\subsection*{Acknowledgements}
This work was supported by the National Natural Science Foundation of
China (No. 11071254), the Scientific
Research Foundation for the Returned Overseas Chinese Scholars,
State Education Ministry and the Natural Science Foundation of
Young Scientist of Hebei Province (No. A2013506012).
\begin{thebibliography}{00}
\bibitem{s9} J. Cao, W. Yu, Y. Qu;
A new complex network model and convergence dynamics for reputation
computation in virtual organizations, {\it Phys. Lett. A}
\textbf{356} (2006) 414-425.
\bibitem{s11} Q. Gan, R. Xu, W. Hu, P. Yang;
Bifurcation analysis for a tri-neuron discrete-time BAM neural
network with delays, {\it Chaos Solitons Fractals} \textbf{42}
(2009) 2502-2511.
\bibitem{s15} S. Guo, Y. Chen;
Stability and bifurcation of a discrete-time
three-neuron system with delays, {\it Int. J. Appl. Math. Eng. Sci.}
\textbf{1} (2007) 103-115.
\bibitem{s12} S. Guo, X. Tang, L. Huang; Bifurcation
analysis in a discrete-time single-directional network with delays,
{\it Neurocomputing} \textbf{71} (2008) 1422-1435.
\bibitem{s3} J. Hopfield;
Neural networks and physical systems with
emergent collective computational abilities, {\it Proc. Natl. Acad.
Sci.} \textbf{79} (1982) 2554-2558.
\bibitem{s8} E. Kaslik, St. Balint;
Bifurcation analysis for a discrete-time Hopfield
neural network of two neurons with two delays and self-connections,
{\it Chaos Solitons Fractals} \textbf{39} (2009) 83-91.
\bibitem{s1} Y. A. Kuznetsov;
Elements of applied bifurcation theory,
vol. 112 of Applied Mathematical Sciences, Springer, New York,
USA, 1998.
\bibitem{s2} Y. A. Kuznetsov, H. G. E. Meijer;
Numerical normal forms for codim 2 bifurcations of fixed points with at most
two critical eigenvalues, {\it SIAM J. Sci. Comput.}
\textbf{26} (2005) 1932-1954.
\bibitem{s10} S. M. Lee, O. M. Kwonb, Ju H. Park;
A novel delay-dependent criterion for delayed neural
networks of neutral type, {\it Phys. Lett. A} \textbf{374} (2010)
1843-1848.
\bibitem{s13} X. Li, J. Cao;
Delay-dependent stability of neural networks of neutral
type with time delay in the leakage term,
{\it Nonlinearity} \textbf{23} (2010) 1709-1726.
\bibitem{s16} A. Stuart, A. Humphries;
Dynamical systems and numerical analysis,
Cambridge University Press, Cambridge, 1996.
\bibitem{s4} D. Tank, J. Hopfield;
Simple neural optimization networks: an A/D converter,
signal decision circuit and a linear programming circuit, {\it IEEE
Trans. Circ. Syst.} \textbf{33} (1986) 533-541.
\bibitem{s18} R. S. Varga;
Matrix iterative analysis, vol.
27 of Springer Series in Computational Mathematics, Springer,
Berlin, Germany, 2000.
\bibitem{s14} B. Wang, J. Jian;
Stability and Hopf bifurcation analysis on a four-neuron BAM neural
network with distributed delays, {\it Commun. Nonlinear Sci. Numer.
Simul.} \textbf{15} (2010) 189-204.
\bibitem{s5} L. Wang, X. Han;
Stability and Hopf bifurcation
analysis in bidirectional ring network model, {\it Commun. Nonlinear
Sci. Numer. Simul.} \textbf{16} (2011) 3684-3695.
\bibitem{s7} H. Zhao, L. Wang, C. Ma;
Hopf bifurcation and stability analysis on discrete-time Hopfield
neural network with delay, {\it Nonlinear Anal. Real World Appl.} \textbf{9} (2008)
103-113.
\bibitem{s6} J. Zhou, S. Li, Z. Yang;
Global exponential stability of Hopfield neural networks
with distributed delays, {\it Appl. Math. Model.} \textbf{33}
(2009) 1513-1520.
\bibitem{s17} H. Zhu, L. Huang;
Stability and bifurcation in a tri-neuron network model
with discrete and distributed delays, {\it Appl. Math. Comput.}
\textbf{188} (2007) 1742-1756.
\end{thebibliography}
\end{document}