\documentclass[12pt,twoside]{article}
\usepackage{graphicx}
\renewcommand{\baselinestretch}{1}

\setcounter{page}{1}

\setlength{\textheight}{21.6cm}

\setlength{\textwidth}{14cm}

\setlength{\oddsidemargin}{1cm}

\setlength{\evensidemargin}{1cm}

\pagestyle{myheadings}

\thispagestyle{empty}

\markboth{\small{Mohammad Ali Khan}}{\small{Adaptive synchronization with uncertain parameter}}

\date{}

\begin{document}

{\scriptsize
\emph{\textbf{International Journal of Basic and Applied Sciences},  xx (xx) (20xx) xxx-xxx}

\emph{\copyright Science Publishing Corporation}

\emph{www.sciencepubco.com/index.php/IJBAS}

\emph{}}

\centerline{}
\centerline{}
\centerline{}

\centerline {\Large{\bf Adaptic  Synchronization of two coupled Newton-Leipnik   }}



\centerline{\Large{\bf System  with Uncertain Parameter} }

%% My definition
\newcommand{\mvec}[1]{\mbox{\bfseries\itshape #1}}

\centerline{\bf {Mohammad Ali Khan}}

\centerline{}

\centerline{Department of Mathematics, Garhbeta Ramsundar Vidyabhaban }
\centerline{Paschim Medinipur, West Bengal, India.}
\centerline{Email:mdmaths@gmail.com}



\centerline{}

\newtheorem{Theorem}{\quad Theorem}[section]

\newtheorem{Definition}[Theorem]{\quad Definition}

\newtheorem{Corollary}[Theorem]{\quad Corollary}

\newtheorem{Lemma}[Theorem]{\quad Lemma}

\newtheorem{Example}[Theorem]{\quad Example}

\begin{abstract}
In this letter, an adaptive synchronization and parameters identification scheme is proposed for two coupled chaotic Newton-Leipnik systems. Based on Lyapunov stability theory an adaptive controller is designed to make the states of two identical chaotic Newton-Leipnik system  with unknown system parameters asymptotically synchronized. Especially when some unknown parameters are positive, we can make the controller more simple, beside the controller is independent of those positive uncertain parameters. Numerical simulation results are presented to visualize the effectiveness and feasibility of the developed approaches.
\end{abstract}

{\bf Keywords:}  \emph{Chaotic systems, Chaos control, Chaos synchronization, Uncertain parameters, Newton-Leipnik system. }

%=============================
\section{Introduction}
%=============================
$~~~~$Since the pioneer works by Ott, Grebogi and Yorke$(1990)$ [1] and Pecorra and Carroll $(1990)$ [2], chaos control and synchronization  has received increasing 
attention due to its theoretical challenges and its  potential application to various disciplines. Synchronization in biological systems is one of the fascinating area that has attracted a lots of renewed attention. Various modern synchronization methods, such as adaptive control [3,4], backstepping design [5], active control [6], nonlinear feedback control [7] , impulsive control [8] has been successfully applied to obtain chaos synchronization in recent years. Basically, chaos synchronization problem can be formulated as follows:
given a chaotic system, which is considered as the master system, and we have to formulate another system which is considered as the slave system which synchronizes
to the master system. Tarai et.al. studied synchronization of bidirectionally coupled chaotic Chen's system with delay in 2007 [9] and they also observed synchronization of generalised linearly bidirectionally coupled unified chaotic system in 2009 [10]. Synchronization via adaptive control has been successfully 
tested in variety of non-linear dynamical systems including Unified chaotic system, Lu system and Chen system [11,12,13].
In 1994 J.K.John and R.E.Amritkar have studied 
synchronization of unstable orbits using adaptive control [14]. Adaptive synchronization of Lu system with uncertain parameters was studied by Ellabasy et.al. [15]. Dai et.al. have investigated chaos synchronization 
by using intermittent parametric adaptive control method in 2001 [16]. Chen
et.al. have studied parameters identifications and synchronizations of chaotic systems based upon adaptive control [17]. Adaptive control for the synchronization of Chen system via a single variable was studied by Wang
et.al [18]. In 2004, adaptive control of uncertain Lu system was studied by Wu et.al. [19]. 
Aim of this work is to study chaos synchronization of two coupled
Newton-Leipnik system with former case where few parameters are unknown and the latter case where  all parameters
are unknown. In this paper, an adaptive controller is derived based on Lyapunov stability theory where both master and slave Newton-Leipnik system  have unknown 
parameters. We introduce the parameters update law into the design of the 
adaptive synchronization controller based on Lyapunov stability theorem.
Especially, when some unknown uncertain parameters are positive, we can make 
the controller more simple, besides the controller is independent of those positive uncertain parameters.



%==============================
\section{Newton-Leipnik system}
%==============================


$~~$The Newton-Leipnik system is charaterized by the following differential equation 
\begin{eqnarray}
\dot{x} & = &- ax+y+10yz\nonumber\\
\dot{y} & = & -x-0.4y+5xz\nonumber\\
\dot{z} & = & bz-5xy
\end{eqnarray}
where x, y and z are state variables and $a$, $b$ are unknown positive constant parameters. The Newton-Leipnik system exhibits a chaotic attractor at the parameter
values $a=0.4$ and $b=0.175$ shown in Fig.1(a)
\begin{figure}
\begin{center}
\includegraphics[width=1.1\textwidth]{nlunc.eps}
\caption{ Fig.1(a). Chaotic attractor of Newton-Leipnik system at $a=0.4$ and $b=0.175$, Fig.1(b). Time evolution of $x_{1}$ and $x_{2}$ states, Fig.1(c). Time evolution of $y_{1}$ and $y_{2}$ states and Fig.1(d). Time evolution of $z_{1}$ and $z_{2}$ states when the control functions are deactivated. }
\end{center}
\end{figure}
%==============================
\section{Formulation of the problem}
%==============================
In order to observe the synchronization behavior in the Newton-Leipnik system, we have two Newton-Leipnik systems where the drive system with three state variables denoted by the subscript 1 and the response system having identical equations denoted by the subscript 2. The initial conditions for the drive system are different from that of the response system. The drive and response systems are defined as follows. 
\begin{eqnarray}
\dot{x_{1}} & = & -ax_{1}+y_{1}+10y_{1}z_{1}\nonumber\\
\dot{y_{1}} & = & -x_{1}-0.4y_{1}+5x_{1}z_{1}\nonumber\\
\dot{z_{1}} & = & bz_{1}-5x_{1}y_{1}
\label{E_01}
\end{eqnarray}
and
\begin{eqnarray}
\dot{x_{2}} & = & -ax_{2}+y_{2}+10y_{2}z_{2}+u_{1}(t)\nonumber\\
\dot{y_{2}} & = & -x_{2}-0.4y_{2}+5x_{2}z_{2}+u_{2}(t)\nonumber\\
\dot{z_{2}} & = & bz_{2}-5x_{2}y_{2}+u_{3}(t)
\label{E_02}
\end{eqnarray}
where $U=[u_{1}(t),u_{2}(t),u_{3}(t)]^{T}$ is the controller function
introduced in the response system. The controller is determined for the purpose of synchronizing of two coupled identical Newton-Leipnik systems with the same  unknown parameters $a$ and $b$ but with  different in initial conditions.\\
Substracting equation (2) from equation (3) yields the error dynamical system as
\begin{eqnarray}
\dot{e_{1}} & = & -ae_{1}+e_{2}+10(e_{2}e_{3}+e_{2}z_{1}+e_{3}y_{1})+u_{1}(t)\nonumber\\
\dot{e_{2}} & = & -e_{1}-0.4e_{2}+5(e_{1}e_{3}+e_{1}z_{1}+e_{3}x_{1})+u_{2}(t)\nonumber\\
\dot{e_{3}} & = & be_{3}-5(e_{1}e_{2}+e_{2}x_{1}+e_{1}y_{1})+u_{3}(t)
\label{E_03}
\end{eqnarray}
where $e_{1}=x_{2}-x_{1}, e_{2}=y_{2}-y_{1}$ and $e_{3}=z_{2}-z_{1}$. Our goal is to find a controller $ U=[u_{1}(t),u_{2}(t),u_{3}(t)]^{T} $ and a 
parameter update law for equation (4) such that the states of the response system (3) and the states of the drive system (2) are globally 
synchronized asympotically i.e.
\begin{center}
$\lim_{t\rightarrow\infty}\parallel e(t)\parallel=0~~~\mbox{for all} ~~a\in R~~ and~~ b>0$\\
where  $e(t)=[e_{1},e_{2},e_{3}]^{T}$ 
\end{center}
%==============================
\section{ Adaptive Synchronization of Newton-Leipnik system with single parameter estimation}
%==============================
\begin{eqnarray}
V(e,\tilde{a})=\frac{1}{2} e^{T}e+\frac{1}{2}\tilde{a}^{2}
=\frac{1}{2}(e_{1}^{2}+e_{2}^{2}+e_{3}^{2})+\frac{1}{2}\tilde{a}^{2}
\end{eqnarray}
where $\tilde{a}=a-\hat{a}$,~~$\hat{a}$ is an estimated value of the unknown 
parameter $a$. Now 
\begin{eqnarray}
\dot{V}\leq -w(e)
\end{eqnarray}
where $w(e)=e_{1}^{2}+0.4e_{2}^{2}+e_{3}^{2}$\\
We therefore need to find a controller U and a parameter estimation and  update law $\hat{a}$ to guarantee that for all $e\in R^{3}$ the equality (6) holds.
There are many possible choices for the controller U.
We choose 
\begin{eqnarray}
u_{1} & = & (a-1)e_{1}\nonumber\\
u_{2} & = &-(\hat{a}e_{2}+15e_{1}z_{1})\nonumber\\
u_{3} & = & -be_{3}-10e_{1}e_{2}-5e_{1}y_{1}-e_{3}
\end{eqnarray}
and the parameter estimation update law $\dot{\hat{a}}$.\\
\begin{eqnarray}
\dot{\hat{a}}  =  e_{2}^{2}
\end{eqnarray}
By the choice of the controller (7) the error dynamical systems becomes 
\begin{eqnarray}
\dot{e_{1}} & = & -e_{1}+e_{2}+10(e_{2}e_{3}+e_{2}z_{1}+e_{3}y_{1})\nonumber\\
\dot{e_{2}} & = & -e_{1}-0.4e_{2}+5(e_{1}e_{3}-2e_{1}z_{1}+e_{3}x_{1})-\hat{a}e_{2}\nonumber\\
\dot{e_{3}} & = & -e_{3}-5(3e_{1}e_{2}+e_{2}x_{1}+2e_{1}y_{1})
\end{eqnarray}
Under with this choice 
\begin{eqnarray}
\dot{V} & = & e_{1}\dot{e_{1}}+e_{2}\dot{e_{2}}+e_{3}\dot{e_{3}}-(a-\hat{a})\dot{\hat{a}}\nonumber\\
 & = & -e_{1}^{2}-0.4e_{2}^{2}-e_{3}^{2}\nonumber 
\end{eqnarray}
This leads to 
\begin{center}
$\lim_{t\rightarrow\infty}\parallel e(t)\parallel=0~~~\mbox{for all}~~ a\in R~~ and~~ b(>0)\in R $\\
where  $e(t)=[e_{1},e_{2},e_{3}]^{T}$ 
\end{center}
It is clear that (4) is independent of unknown uncertain parameter b, if
$b>0$\\
Therefore the synchronization of two Newton-Leipnik systems are achieved under
the controller (7) and a parameter estimation and update law equation (8).
\begin{figure}
\begin{center}
\includegraphics[width=1.1\textwidth]{nlvari1.eps}
\caption{ Fig.2(a). Time evolution of $x_{1}$ and $x_{2}$ states, Fig.2(b). Time evolution of $y_{1}$ and $y_{2}$ states, Fig.2(c). Time evolution of $z_{1}$ and $z_{2}$ states and Fig.2(d). Display the error systems tends to zero when the control functions are activated with the parameter estimation update law $\dot{\hat{a}}  =  e_{2}^{2}$.}
\end{center}
\end{figure}





%==============================
\section{ Adaptive Synchronization of Newton-Leipnik system with all parameters estimation}
%==============================
$~~~~~~$Let the unknown uncertain parameter $b>0$ is cancelled, we take 
a Lyapunov function $V(e,\tilde{a},\tilde{b})$for equation (4) 
\begin{center}
$V(e,\tilde{a},\tilde{b})=\frac{1}{2}ee^{T}+\frac{1}{2}(\tilde{a}^{2}+\tilde{b}^{2})$\\
\end{center}
where $\tilde{a}=a-\hat{a}$, $\tilde{b}=b-\hat{b}$, $\hat{a}$ and $\hat{b}$
are estimate values of the unknown parameters $a$ and $b$ respectively.\\
We have the time derivative  $V(e,\tilde{a},\tilde{b})$ along the solution 
of the equation (4)
\begin{eqnarray}
\dot{V}=-ae_{1}^{2}-0.4e_{2}^{2}+10e_{1}e_{2}e_{3}+15e_{1}e_{2}z_{1}+5e_{1}e_{3}y_{1}+be_{3}^{2}\\\nonumber
+ e_{1}u_{1}+e_{2}u_{2}+e_{3}u_{3}-a\dot{\hat{a}}+\hat{a}\dot{\hat{a}}-b\dot{\hat{b}}+\hat{b}\dot{\hat{b}}
\end{eqnarray}
There are many possible choices for the controller U  but we choose
as follows
\begin{eqnarray}
u_{1}& = & (\hat{a}-1)e_{1}\nonumber\\
u_{2}& = &-(10e_{3}+15z_{1})e_{1}\nonumber\\
u_{3} & = &-(5e_{1}y_{1}+(\hat{b}+1)e_{3})
\end{eqnarray}
 and the parameter  estimation update laws $\dot{\hat{a}}$ and $\dot{\hat{b}}$.
\begin{eqnarray}
\dot{\hat{a}}  =  e_{1}^{2}\nonumber\\
\dot{\hat{b}}  =  e_{3}^{2}
\end{eqnarray}
By the choice of the controller (11) the error dynamical system becomes 
\begin{eqnarray}
\dot{e_{1}} & = & -e_{1}+(\hat{a}-a)e_{1}+e_{2}+10(e_{2}e_{3}+e_{2}z_{1}+e_{3}y_{1})\nonumber\\
\dot{e_{2}} & = & -e_{1}-0.4e_{2}-5(e_{1}e_{3}+2e_{1}z_{1}-e_{3}x_{1})\nonumber\\
\dot{e_{3}} & = & -e_{3}+(b-\hat{b})e_{3}-5(e_{1}e_{2}+e_{2}x_{1}+2e_{1}y_{1})
\end{eqnarray}

Under this choice we have
\begin{center}
$\dot{V} =  -e_{1}^{2}-0.4e_{2}^{2}-e_{3}^{2}$
\end{center}
This leads to 
\begin{center}
$\lim_{t\rightarrow\infty}\parallel e(t)\parallel=0~~~\mbox{for all}~~ a\in R~~ and~~ b\in R $\\
where  $e(t)=[e_{1},e_{2},e_{3}]^{T}$ 
\end{center}
Therefore the synchronization of two Newton-Leipnik systems are achieved under
the controller (11) and a parameter estimation and update law equation (12).
\begin{figure}
\begin{center}
\includegraphics[width=1.1\textwidth]{nlvari2.eps}
\caption{ Fig.3(a). Time evolution of $x_{1}$ and $x_{2}$ states, Fig.3(b). Time evolution of $y_{1}$ and $y_{2}$ states, Fig.3(c). Time evolution of $z_{1}$ and $z_{2}$ states and Fig.3(d). Display the error systems tends to zero when the control functions are activated with the parameter estimation update laws $\dot{\hat{a}}  =  -e_{1}^{2}$ and $\dot{\hat{b}}  =  e_{3}^{2}$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1.1\textwidth]{adappara.eps}
\caption{ Estimated value of the unknown parameters {\bf a} and {\bf b}.}
\end{center}
\end{figure}




%==========================
\section{ Results and Discussions}
%==========================
$~~~~~~~$Fourth order Runge-Kutta method is used to solve the systems of differential equations $(2), (3)$ and $(4)$. Time step of size .001 is employed. The initial states of the drive system are taken as $x_{1}(0)=-5$, $y_{1}(0)=8$ and $z_{1}(0)=10$ and the initial states of the response system are  
$x_{2}(0)=40$, $y_{2}(0)=-4$ and $z_{2}(0)=-5$. Hence the error system has the initial values $e_{1}(0)=45$, $e_{2}(0)=-12$ and $e_{3}(0)=-15$. 
The parameters are chosen to be  $a=0.4$ and $b=0.175$ in all simulations so that the Newton-Leipnik system exhibits the chaotic behavior (Fig.1(a)). In this paper
we discuss adaptive synchronization scheme between two coupled chaotic Newton-Leipnik systems. Lyapunov direct method is used to prove the stability of the synchronized state. Secure communication utilizing the synchronization of chaos has been an intersting issue in recent years. Adaptive synchronization of Newton-Leipnik system can also be used for chaotic masking, chaotic modulation and chaotic shift. Numerical results are presented to show the effectiveness of the proposed scheme.\\  


{\bf ACKNOWLEDGEMENTS}

I am grateful to the Editor for careful reading and constructive comments.


\begin{thebibliography}{99}
\bibitem{1} Ott, Grebogi, Yorke, Controlling chaos, Phys.Rev.Lett, 64 (1990)1196-1199.      
\bibitem{2} Pecorra and Carroll, Synchronization in chaotic systems, Phys.Rev.Lett., 64 (1990)821-824.
\bibitem{3} Lu.Chen, Synchronization of an uncertain unified chaotic system
via adaptive control, Chaos, Soliton and Fractals, 14 (2002)643.
\bibitem{4} S.poria, A.Tarai, Adaptive synchronization of two coupled chaotic Neuronal systems.
 Rev.Bull.Cal.Math.Soc., 15 (2007)53-60.
\bibitem{5} Zhang, Yu, Controlling uncertain Lu system using back stepping design, Chaos Solitons and Fractals., 15 (2003)897.
\bibitem{6} Agiza and Yassen, Synchronization of Rossler and Chen chaotical 
dynamical systems using active control, Physics Letters A., 278 (2001)191.
\bibitem{7} Park, Controlling chaotic systems via nonlinear feedback control, Chaos Solitons and Fractals., 23 (2005)1049.
\bibitem{8} Yang,YangYang, Impulsive control of Lorenz system, Physica D., 110 (1997)18.
\bibitem{9} A. Tarai, S.Poria and P. Chatterjee,  Synchronization of bidirectionally coupled chaotic Chen's system with delay, 
Chaos Solitons and Fractals. 2007, doi: 10.1016/j.chaos.2007.11.025.
\bibitem{10} A. Tarai, S.Poria and P. Chatterjee, Synchronization of generalised linearly bidirectionally coupled unified system,
Chaos Solitons and Fractals., 40 (2009)885-892.
\bibitem{11} Lu Chen, Synchronization of an uncertain unified chaotic system via adaptive control, Chaos Solitons and Fractals., 14 (2002)643.
\bibitem{12} E.M.Labbasy, H.N.Agiza and M.M.El-Dessoky, Adaptive synchronization 
of Lu system with uncertain parameters,  Chaos Solitons and Fractals., 21 (2004)657-667.
\bibitem{13} Y.Wang, Zhi-Hong Guan and Hua O. Wang, Feedback and adaptive control for the synchronization of Chen system via a single variable, 
Physics Letters A., 312, (2003)34-40.
\bibitem{14} J.K.John and R.E.Amritkar, Synchronization of unstable orbits using adaptive control, Physical Review E., 49 (1994)4843-4848.
\bibitem{15} E.M.Labbasy, H.N.Agiza and M.M.El-Dessoky, Adaptive synchronization 
of Lu system with uncertain parameters, Chaos Solitons and Fractals., 21 (2004)657-667.
\bibitem{16} Dong DAi, Xi-Kui Ma, Chaos synchronization by using intermittent
parametric adaptive control method, Physics Letters A., 288 (2001)23-28.
\bibitem{17} Shihua Chen, Jinhu Lu, Parameters identifications and synchronization
of chaotic systems based upon adaptive control, Physics Letters A., 299 (2002)353-358.
\bibitem{18} Y.Wang, Zhi-Hong Guan and Hua O.Wang, Feedback and adaptive control for the synchronization of Chen system via a single variable, 
Physics Letters A., 312, (2003)34-40.
\bibitem{19} X.Wu, J.Lu, Adaptive control of uncertain Lu system, Chaos, Solitons
and Fractals., 22,(2004)375-381.

\end{thebibliography}

\end{document}
