The synchronization of chaotic systems with different dimensions by a robust generalized active control

Accepted Manuscript Title: The Synchronization of Chaotic Systems with Different Dimensions by a Robust Generalized Active Control Author: Israr Ahmad Azizan Bin Saaban Adyda Binti Ibrahim Mohammad Shahzad Nawazish Naveed PII: DOI: Reference:

S0030-4026(15)02052-5 http://dx.doi.org/doi:10.1016/j.ijleo.2015.12.134 IJLEO 57072

To appear in: Received date: Accepted date:

3-10-2015 29-12-2015

Please cite this article as: I. Ahmad, A.B. Saaban, A.B. Ibrahim, M. Shahzad, N. Naveed, The Synchronization of Chaotic Systems with Different Dimensions by a Robust Generalized Active Control, Optik - International Journal for Light and Electron Optics (2016), http://dx.doi.org/10.1016/j.ijleo.2015.12.134 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

The Synchronization of Chaotic Systems with Different Dimensions by a Robust Generalized Active Control Israr Ahmad 1*, Azizan Bin Saaban 2, Adyda Binti Ibrahim 3, Mohammad Shahzad 4, Nawazish Naveed 5 1, 2, 3

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School of Quantitative Sciences, College of Arts & Sciences, UUM, Malaysia. College of Applied Sciences Nizwa, Ministry of Higher Education, Sultanate of Oman. 5 College of Applied Sciences Ibri, Ministry of Higher Education, Sultanate of Oman.

1, 4

[email protected], [email protected], [email protected], [email protected],

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[email protected]

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Abstract. Active control strategy is a powerful control technique in synchronizing chaotic/hyperchaotic systems. Until now, active control techniques have been employed to synchronize chaotic systems with the same orders. The present study overcomes the limitations of synchronization of chaotic systems of similar dimensions using active control. In this article, the authors investigate the synchronization problem for a drive-response chaotic system with different orders under the effect of both unknown model uncertainties and external disturbance. Based on the Lyapunov stability theory and Routh-Hurwitz criterion, a robust generalized active control approach is proposed and sufficient algebraic conditions are derived to compute a suitable linear controller gain matrix that guarantees the globally exponentially stable synchronization. Two examples are presented to illustrate the main results, namely reduced-order synchronization between the hyperchaotic Lu and the unified chaotic systems and the increased-order synchronization between the unified chaotic and the hyperchaotic Lu systems. There are three main contributions of the present study: (a) generalization of the active control for synchronization of a class of chaotic systems with different orders; (b) a recursive approach is proposed to compute a suitable linear controller gain matrix and (c) reduced (increased) order synchronization under the effect of both unknown model uncertainties and external disturbances. A comparative study has been done with our results of the previously published work in terms of synchronization speed and quality. Finally, numerical simulations are given to verify the effectiveness of the proposed reduced (increased) order active synchronization approach. Future applications of the proposed reduced (increased) order synchronization approach is discussed.

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Keywords. Chaos synchronization, Lyapunov stability Theory, Active control, Lu hyperchaotic system Unified chaotic system _____________________________________________________________________________________________

1. Introduction

Chaos synchronization has remained an area of active research for the successful applications in different scientific fields [1-4]. In this line, various kinds of synchronization control techniques and methods have been developed such as sliding mode control [5], adaptive control [6], backstepping design [7], phase synchronization [8], projective synchronization [9], active control [10] and the nonlinear active control techniques [11] (are worth citing here among others). Synchronization of chaotic systems can be classified into two basic categories, namely; synchronization between chaotic systems with the same orders and the synchronization between chaotic systems with different orders. In this direction, two types of chaos synchronization between chaotic systems with different orders have been addressed in the literature, namely; reduced-order synchronization and increased-order synchronization. In reduced/increased order synchronization, the dimension of the drive system is greater (smaller) than that of the response system. Synchronization of chaotic systems with different orders can be found in many natural systems; thalamic neurons; in the human brain; synchronization between the heart and lungs; chaotic laser communications, synchronization in the cells of paddlefish [12-14] and elsewhere. In this line, there are a few interesting

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results in the literature dealing with synchronization of chaotic systems with different orders. Femat et al [12] achieved reduced order synchronization between a Chua oscillator and a Duffing system based on the time derivative of system output along the master-slave system vector field. Using the linear errors of corresponding variables and parameters, Ho et al [13] investigated reduced-order synchronization between the generalized uncertain Lorenz hyperchaotic and the uncertain Lu chaotic systems. Alsawalha and Noorani [14] proposed adaptive control to achieve reduced-order anti-synchronization between the uncertain hyperchaotic Lorenz and the uncertain chaotic Lorenz systems and anti-synchronization between the uncertain hyperchaotic Lu and the uncertain chaotic Chen systems. In 2008, based on the Lyapunov stability theory [15] and using the adaptive control technique, Ge and Yang [16] presented reduced-order synchronization between the Quantum-CNN with the Lorenz chaotic and regular time function synchronization between the Quantum-CNN and the Chen systems. Recently, the authors [17] investigated increased-order synchronization and anti-synchronization between the hyperchaotic Lu and the chaotic Lu systems with identification of uncertain parameters. The controller functions and the parameter update laws are derived via an adaptive control method for the uncertain parameters.

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Nevertheless, the synchronization methods and control techniques [12-14, 16-17] only focused on the chaotic systems in simple or ideal condition. In real engineering applications, chaotic systems are perturbed by unknown model uncertainties and external disturbance unavoidably, and practically, this makes the control problem more complicated. These results [12-14, 16-17] would have been more interesting if it had included the reduced/increased order synchronization behavior under the effect of both unknown model uncertainties and external disturbances.

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The active control technique for chaos synchronization was first proposed by Bai et al [18] based on the linear control theory and further studied by Agiza and Yaseen [19], Ho and Hung [20], Chen [21], Vincent [22], Njah [23] and Ahmad et al [10] (to name but a few). The active control techniques have received considerable attention during the last two decades due to the potential applications to various chaotic systems. These include, nonlinear gyros [20]; a permanent magnet reluctance machine [22]; an extended Bonhoffer-van der Pol oscillator [23]; the electronic circuits which model a third-order ‘‘Jerk’’ equation [24]; the RCL-shunted Josephson junction [25]; and most recently the spin orbit problem of Enceladus [26] (to name but a few). On theoretical bases, some of the advantages of active control techniques include, the applicability of the technique to certain chaotic systems (either identical or nonidentical) whether the systems contain external excitation or not [27].

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From the literature survey, it has been observed that the active control techniques [10, 19-27] have been utilized for the synchronization of chaotic systems with the same orders. The stability of the closed-loop system has been established by assigning all the eigenvalues of the coefficient matrix of the error system to the left half of the complex plane. The convergence time can be reduced by choosing large controller gains. In real applications, this may lead to automatic signal saturations. There is no recursive approach [10, 19-27] to compute a suitable linear controller gain matrix. In fact, the above notable results [10, 1927] affect each other mutually and need a systematic approach to compute a suitable linear controller gain matrix. In spite of the several reported active control strategies in the concerned literature, a considerable attention is still paid to addressing the above issues for a better synchronization behavior. Thus, it is a theoretically as well as practically significant to propose such an active feedback control strategy that synchronizes chaotic systems with different orders under the effect of both unknown model uncertainties and external disturbances with the computation of a suitable linear controller gain matrix. Motivated by the above discussions, the authors will study, the reduced (increased) order synchronization phenomena under the effect of both unknown model uncertainties and external disturbances. Using the drive-response system synchronization scheme, a generalized active control

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approach will be proposed and sufficient algebraic conditions will be derived to compute a suitable linear controller gain matrix that would guarantee the globally exponential stable reduced (increased) order synchronization. Two illustrative examples will be given to verify the robustness and performance of the proposed approach; reduced order synchronization between the hyperchaotic Lu [28] and the unified chaotic [29] systems and increased order synchronization between the unified chaotic and the hyperchaotic Lu systems. To the best of the authors’ knowledge, the proposed generalized active control technique and the reduced/increased order synchronization problem under the effect of both unknown model uncertainties and external disturbances have not been addressed in the literature and this has remained an open problem. There are three main contributions of the present study; (a) generalization of the active control for a class of chaotic systems with different orders (b) a recursive approach is proposed to compute the linear controller gain matrix and (c) the proposed generalized active control approach is robust against the effect of both unknown model uncertainties and external disturbances that increases security of the synchronized systems and improved the reduced/increased order synchronization performance in comparison with notable results. In this paper, we also give some future potential applications of the proposed active reduced/increased order synchronization for the first time and aim that the corresponding research will own more practical value. The rest of the paper is organized as follows: Section 2 presents the problem statement and a theory for the proposed generalized active control for reduced order and increased order synchronization schemes are given. In section 3, descriptions of the unified chaotic and the hyperchaotic Lu systems are given and solved the problem of reduced-order synchronization between the unified chaotic and the hyperchaotic Lu systems. Section 4 is devoted to solve the increased order synchronization problem between the hyperchaotic Lu and the unified chaotic systems with some few potential applications of the proposed synchronization approach. At the end, this paper is concluded in section 5. Theory for the proposed generalized active control

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In this section of the paper, the authors briefly describe a theory for the proposed generalized active control to establish the reduced order and increased order synchronization schemes under the effect of both unknown model uncertainties and external disturbances.

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Definition 1. A real constant matrix P is said to be positive definite matrix (PDM) if: i). All the ordered principal minor determinants of Error! Bookmark not defined. being positive ii). P is symmetric and X T  t  PX  t   0 for all X  t   0 . 2.1 Reduced-order synchronization scheme 2.1.1

Problem statement

Most of the synchronization problem belongs to the drive-response system arrangement, in which the response system is forced to track the properties of the drive system under some coupling force. Let us consider a drive hyperchaotic system that is defined as follows:

x  t   F1  x  t   A  f1  x  t    h  x  t      t  ,

(1)

where x  t   R n is the state vector, F1  R n , A  R nn is the vector of the system parameters, f1  R n1 is the nonlinear continuous vector function without the parameter vector and h  x  t   and   t 

respectively, represent the model uncertainties and external disturbances acting on the drive system (1).

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Remark 1. Many chaotic/hyperchaotic systems can be transferred in the form of system (1). For instance, one may see the two chaotic systems (15) and (16). Likewise, the response chaotic system is defined as follows:

y  t   F2  y  t   B  f 2  y  t    g  y  t   
 t   u  t  ,

(2)

f 2  R m1

ip t

where y  t   R m is the state vector  n  m  , F2  R m , B  R mm is the vector of the system parameters, is the nonlinear continuous vector function without the parameter vector, and

g  y  t   and
 t  respectively, represent the model uncertainty and external disturbance present in the

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response system (2), where u  t   R m1 is the control input which is yet to be determined. Actually, the reduced-order synchronization is the problem of synchronizing the response system to be the projection part of the drive system [13]. Therefore, divide the drive system into two parts as follows: The projection part is given as follows: x p  t   F1 p  x p  t   Ap  f1 p  x p  t    h  x p  t     p  t  ,

(3)

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where x p  t   R m , F1 p  R m , f1 p  R m1 and Ap  R mm . And the rest of the system is given as follows: xr  t   F1r  xr  t   Ar  f1r  xr  t    h  xr  t     r  t  ,

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where xr  t   R r , F1r  R r , Ar  R r r , f1r  R r 1 and the orders m and r satisfy, m  r  n . Thus, the reduced-order synchronization scheme between the projection part of the drive system (1) and response system (2) that is described as follows:

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Drive system: x p  t   F1 p  x p  t   Ap  f1 p  x p  t    h  x p  t     p  t  Response system : y  t   F2  y  t   B  f 2  y  t    g  y  t   
 t   u  t 

(4)

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The error dynamics for the reduced-order synchronization scheme (4) can be defined as follows:

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e t   y t   xp t  , e t   Rm .

Thus, the time varying error dynamics for the reduced-order synchronization scheme (4) can be described as follows: e  t  





 





F2  y  t   B  F1 p x p  t  Ap  f 2  y  t    f1 p x p  t    g  y  t   
 t   h x p  t     t   u  t  



e  t   Ae  t   G  x p  t  , y  t    g  y  t    h  x p  t   
 t     t   u  t  ,

(5)

where G  x p  t  , y  t  , e  t     B  B  F2  y  t     Ap  Ap  F1 p  x p  t    f 2  y  t    f1 p  x p  t   is the function of residual terms and B  Ap  A is the

 m  m

coefficient matrix of the error system (5),

alternatively [30]. In these situations, the main goal is to design such a feedback control law that lead the chaotic attractor of the response system to trace the geometrical properties of projection of the drive

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system and the error vector e  t   e1  t  , e2  t  , . . ., em  t   tends to zero globally exponentially as t  .

lim e  t   lim y  t   x p  t   0 ,

i.e.,

t 

t 

ip t

where . represents the Euclidean norm in R m . Remark 2. Equation (5) is reasonable. For instance, one may see equations (18) and (24).

Assumption 1. The chaotic/hyperchaotic trajectories are always bounded [6]. Therefore, the model





hi  x p  t    i , g i  y  t    i ,

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gi  y  t    hi  x p  t     i ,

i  1, 2,  , m ,

(6) (7)

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where  i , i and  i are unknown positive constants

i  1, 2,  , m

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uncertainties h  x  t   & g  y  t   are unknown function of time and are always bounded.

Assumption 2. It is assumed that the unknown external disturbances    t  &
 t   are norm-bounded

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in C1 [6].

i  t   d i and


i.e,

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i

 t   Di ,

 t   i  t   i ,

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i

i  1, 2,  , m

(8)

i  1, 2,  , m ,

(9)

where di , Di and  i are unknown positive constants. Using Eqs. (7) and (9), yields:

2.1.2

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where i   i   i .

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gi  y  t    hi  x p  t   


i

 t    i  t   i ,

i  1, 2,  , m ,

(10)

Controller design

Lemma 1. For the arbitrary initial conditions

 x 0  R ipd

n

 yir  0   R m  of the drive and response

chaotic systems (4) respectively, the two coupled chaotic systems (4) are globally exponential synchronized by the following active feedback control law: u  t   u1  t  , u2  t  , . . ., um  t  

(11)

Proof of lemma 1. In view of the active control strategy, the control function (11) will be constructed in two parts. The first part will eliminate the residual terms and the terms which are not expressed in the form of e  t  to obtain a linear system and the second part will regulate the strength of the feedback controller to achieve reduced-order synchronization globally exponentially. Let us construct the following control function as: u  t   G x p  t  , y  t  , e  t   i  v  t  (12)





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Definition 2. The matrix v  t   vi  t   is defined as follows: T

 vi  t    W  ei  t   , i  1, 2, . . ., m and W is a  m  m  matrix. are the linear controller gains which are to be computed. Using systems of Eqs. (5) and (12), T

e  t   Ae  t   We  t  e  t    W  A  e  t   0 ,

 where W  A  is a  m  m  matrix.

(13)

ip t

Where wi ' s yields:

T

At this stage, the problem is reduced to show that if the linear controller gains wi ' s are properly selected

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such that the real part of all eigenvalues of the coefficient matrix W  A  becomes positive, then, by the Routh-Hurwitz criterion [31] and Lyapunov stability theory [15], the closed-loop system (13) is globally exponentially stable and hence, the reduced-order synchronization scheme (4) is globally exponentially establish.

2.2 2.2.1

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Remark 3. Various choices are available for the linear controller gain matrix W  R mm such that the feed-back system of elements of the coefficient matrix W  A  must have all of its eigenvalues with positive real parts which obey the Routh-Hurwitz criterion [18-29]. But with this assumption, the message signal can be easily extracted from the communications channel during the transmission and the same signal could be re-generated with any possible choice of a positive definite matrix. This may lead to security issue. Therefore, it is necessary and typically significant from a theoretical as well practical point of view to compute a suitable linear controller gain matrix that guarantee globally exponential stability of the closed-loop system in the presence of both model uncertainties and external disturbances which will be presented in section 3. Increased order synchronization scheme Problem statement

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Let us consider the following drive-response system increased order synchronization scheme that is described as: Drive system: x  t   F1  x  t   A  f1  x  t    h  x  t      t  (14) Response system : y  t   F2  y  t   B  f 2  y  t    g  y  t   
 t   u  t  , where x  t   R n and y  t   R m are the state vectors with n  m , F1  R n and F2  R m , A  R nn and B  R mm are the vectors of the drive and response systems parameters alternatively, and f1  R n1 and f 2  R m1 are the nonlinear continuous vector functions without the parameter vectors of the

drive and response systems respectively. h  x  t   and   t  represent the unknown model uncertainty and external disturbance acting on the drive system and g  y  t   and
 t  represent the unknown model uncertainty and external disturbances acting on the response system alternatively, u  t   R m1 is the control input which is yet to determined.

where

Remark 4. The rest of the procedure for increased order synchronization is similar to that which is given in sub-section 2.1.1 but the details are omitted here.

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3. 3.1

The reduced-order synchronization scheme Description of the hyperchaotic Lu system

The vector form of the hyperchaotic Lu [28] system is described as follows: 0   y1  t    0      1   y2  t     y1  t  y3  t    , 0 b 0   y3  t    y1  t  y2  t      0 1 1  y  t    0   4 

a c

0 0

(15)

ip t

 y1  t    a     y 2  t    0    y3  t    0  y  t    0  4 

cr

4 where  y1  t  , y2  t  , y3  t  , y4  t    R are the state variables and a, b and c are the corresponding T

Fig. 1 (b). Time series of the state variables

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Fig. 1 (a). 3D phase portrait of the hyperchaotic Lu attractor

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positive parameters of the system (15). The hyperchaotic Lu system (15) exhibits a chaotic attractor with the parameter values: a  15, b  5 and c  10 as shown in Fig. 1.

Time units

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3.2 Description of the unified chaotic system The vector form of the 3-D unified chaotic system [29] is described as below:    x1  t      25  10   25  10   0   x1  t    0         0   x2  t      x1  t  x3  t   ,  x2  t      28  35   29  1  x3  t      8   x3  t    x1  t  x2  t       0 0  3  

(16)

3 where  x1  t  , x2  t  , x3  t    R are the state variables and    0, 1 is the corresponding parameter T

of the system (11). If   0 , then, the chaotic system (16) is called the generalized Lorenz chaotic system. If   1 , then, the chaotic system (16) is reduced to the generalized Chen chaotic system and if   0.8 , then, the chaotic system (16) becomes the generalized Lu chaotic system. The unified chaotic system (16) exhibits a chaotic attractor with the parameter values:   0, 0.8 and 1 as shown in Fig. 2.

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Fig. 2 (d). Time series of the state variables

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ip t

Fig. 2. 3D phase portrait of the unified chaotic attractor when (a)   0, (b)   1, (c)   0.8

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Time units

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The hyperchaotic Lu and unified chaotic systems have different parameter values and initial conditions. The traces change of x  t  and y  t  with time is different. There is a difference in the chaotic attractors and topological properties of the two systems. Thus, it is invited to synchronize these different order chaotic systems under the effect of both unknown model uncertainties and external disturbances which is a challenging task. In the following sub-sections, we present both theoretical and numerical simulation results for the reduced-order synchronization scheme with our proposed generalized active control approach. 2.3 Problem statement

To achieve reduced order synchronization between the hyperchaotic Lu and unified chaotic systems, it is assumed that the projection part of the hyperchaotic Lu system is considered as the drive system and the unified chaotic system as the response system. Let us consider the following drive-response system reduced order synchronization scheme is described as:

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 Drive system 

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cr

ip t

   y1  t     y1  t    a a 0 0       g1  y1  t    D1  t   0      y2  t         y 2  t     0 c 0 1       y1  t  y3  t     g 2  y2  t    D2  t        0 0 b 0  y3  t    y  t  y  t      y t 2   1   g3  y3  t    D3  t    3       y4  t      Response system         h x t d t         1 1  1  x1  t      25  10   25  10  0   x1  t    0               x t x t x t h x t d t u t         28 35  29  1 0 x t                    3 2 2 2  2   1  2        x3  t      8   x3  t    x1  t  x2  t    h  x  t    d  t       3  0 0   3 3   3  

(17)

Where  y1  t  , y2  t  , y3  t  , y4  t    R 4 and  x1  t  , x2  t  , x3  t    R 3 are the state vectors, a, b, c and  are the positive parameters of the drive and response systems (17) respectively. g j  y j  t    D j  t  , j  1, 2, 3, and hi  xi  t    di  t  , i  1, 2, 3, are the unknown model uncertainties T

an

T

and external disturbances acting on the drive and response systems respectively, and u  t   R is the control input. To solve the reduced order synchronization problem, the error dynamics between the drive and response systems (17) can be defined as ei  t   xi  t   yi  t  , i  1, 2, 3 . Thus, the error dynamics for the

ed

M

3

reduced order synchronization scheme (17) can be described as follows: e1  t    25  10   e2  t   e1  t     25  a  10   y1  t   x1  t    h1  x  t    g1  y  t  

Ac ce

pt

    d1  t   D1  t   u1  t   e2  t    28  35  e1  t   ce2  t    28  35  x1  t    29  1 x2  t   cy2  t   y4  t   y1  t  y3  t     (18)  x1  t  x3  t   h2  x  t    g 2  y  t    d 2  t   D2  t   u2  t     8  e3  t   be3  t    b   x3  t   y1  t  y2  t   x1  t  x2  t   h3  x  t    g3  y  t    d3  t   3      D3  t   u3  t  

Objective 1. lim ei  t   lim xi  t   yi  t   0, i  1, 2, 3 . x 

x 

Objective 2. The closed-loop (21) is globally exponentially stable. Lemma 2. For the arbitrary initial conditions, 4 3  y 1  0  , y 2  0  , y 3  0  , y 4  0    R   x 1  0  , x 2  0  , x 3  0    R ; of the drive and response systems respectively, the globally exponential reduced order synchronization is accomplished with the following proper active control functions: T

T

Page 9 of 18

u1  t    a  25  10   y1  t   x1  t    1  v1  t  u2  t    35  28  x1  t   1  29  x2  t   cy2  t   y4  t   y1  t  y3  t   x1  t  x3  t   2  v2  t 

(19)

 8   b  x3  t   y1  t  y2  t   x1  t  x2  t   3  v3  t  u3  t     3 

Definition 3. According to definition 2, the matrix v  t   vi  t   , i  1, 2, 3, is described as follows: v  t   W ei  t   T with W  dig  wi  , i  1, 2, 3 ,

(20)

cr

where wi ' s are the linear controller gains which will be computed.

ip t

T

Proof of lemma 2. Combining systems of Eqs. (18), (19) and (20), that yields:

 e1  t     0  e2  t    0  b  w3   e3  t  

us

0

(21)

an



 e1  t    25  10  w1    25  10      w2  c   e2  t        35  28     0 0  e3  t  

Thus, the system (21) to be controlled is a linear system with control inputs v  t    v1  t  , v2  t  , v3  t  

as a function of the error system e  t    e1  t  , e2  t  , e3  t   . As long as these feedbacks stabilize the

M

system (21), the error vector e  t    e1  t  , e2  t  , e3  t   converges to zero as time tends to infinity. At

ed

this stage, the problem is reduced to show that the coefficient matrix in (21) is PDM. From definition 1, the matrix in (21) will be PDM, if all ordered principal minor determinants being positive. That is:  25  10   35  28
1 : w1    25  10  ,
2 : w2   c,
3 : w3  b (22) w1   25  10 

pt

Thus, the reduced order synchronization scheme (17) is globally exponentially established. Hence, the two objectives 1 and 2 are accomplished.

Ac ce

3.4 Numerical simulation and discussions

In this sub-section of the paper, numerical experimental results using mathematica 10v are provided to verify the robustness and effectiveness of the proposed generalized active control approach. Parameters of the hyperchaotic Lu (15) and the unified chaotic (16) systems which are set as a  15, b  5, c  10 and   0, 1 and 0.8 alternatively. The initial states of the drive and response systems being taken as  x 1  0  , x 2  0  , x 3  0     2, 1,  1 and  y 1  0  , y 2  0  , y 3  0  , y 4  0    1, 2, 3, 4 respectively. According to the conditions (22), the linear controller gains are selected as w1  25, w2  30 and w3  6 . With this choice of w1 , w2 and w3 , the three eigenvalues in the coefficient T

T

T

T

1 1  matrix (21) are   55  i 29.92  ,  55  i 29.92  , 6  which confirms that the closed-loop system (21) is 2 2  globally exponentially stable. Moreover, from (21), we can see that the computation of the linear gain w2 depends on w1 which may confirm security for the communications scheme. The following model uncertainties and external disturbances in the drive and response systems are considered in the simulation.

Page 10 of 18

 g1  y1  t    D1  t   0.35cos  y1  t    0.01sin  30t  ,     5  y2  t    0.03cos  20t  ,  g 2  y2  t    D2  t   0.2sin   6      g3  y3  t    D3  t   0.02cos  y3  t    0.04cos  50t   2  

us

cr

    h2  x2  t    d 2  t   0.25cos  x2  t    0.02sin  20t  ,    2  h3  x3  t    d3  t   0.2cos  x3  t    0.03cos  30t     3 

ip t

 5  x1  t    0.1cos 10t  , h1  x1  t    d1  t   0.5sin   6 

(23)

,Subsequently, 1  0.85,  2  0.45 and 3  0.22 .

M

an

Time series of the convergence of the synchronized error signals to the zero state are depicted in Figs. 3, 4 and 5, for   0, 1 and 0.8 alternatively. As expectedly, one can observe the smoothness of the synchronized error signals while converging to the zero state with fast converging rates after the controllers are switched on, which demonstrates the robustness and performance of the control action (19) for the reduced order synchronization scheme. Fig. 6, depicts the synchronization analysis between the drive and response systems which is confirmed by the convergence of the synchronization quality defined by the refinement of the error states:

ed

E  t   e12  t   e22  t   e32  t  Fig. 3. Time series of the synchronized error states when   0

1

pt

0 1

3 4 0.0

1.0

1.5

e1 t e2 t

3

e3 t

Time units

0

2

e2 t

0.5

1

1

e1 t

Ac ce

2

Fig. 4. Time series of the synchronized error states when   1

e3 t

4

2.0

0.0

0.5

1.0

1.5

2.0

Time units

Page 11 of 18

Fig. 5. Time series of the synchronized error states when   0.8

Fig. 6. Convergence of errors when   0

5 1

4

0

2

e2 t

3 2

e3 t

3

1

4

0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

4

1.0

1.5

2.0

Time units

us

Time units

ip t

e1 t

cr

1

Increased order synchronization scheme

an

4.1 Problem statement In this sub-section of the paper, we present increased order synchronization scheme for the chaotic systems (15) and (16). The unified chaotic system is considered as the drive system and the hyperchaotic Lu system is considered as the response system. Let us consider the following drive-response system increased order synchronization scheme that is described as:         h x t  d t  x1  t      25  10   25  10    1  1   1     0   x1  t    0        0   x2  t      x1  t  x3  t     h2  x2  t    d 2  t     29  1  x2  t     28  35         8   x3  t    x1  t  x2  t    h  x  t    d  t     x3  t    3  0 0   3 3  3     Response system       g y t D t        y1  t     a a 0 0   y1  t    1 0   1 1         y  t    y t y t   g  y  t    D  t     y t   0 0 1 c  2   2   2    1   3     2 2    u t      0 0 b 0    y t y t       y t  y t         g y t D t     1 2   3 3 3 3 3            y  t    0 0 1 1  y  t    0   g  y  t    D  t     4   4  4 4 4   

(24)

Ac ce

pt

ed

M

Drive system

 x1  t  , x2  t  , x3  t    R3 and  y1  t  , y2  t  , y3  t  , y4  t    R 4 are the state vectors,  and a, b, c are the positive parameters of the drive and response systems, alternatively,

Where

T

hi  xi  t    di  t  , i  1, 2, 3,

and

T

g j  y j  t    D j  t  , j  1, 2, 3, 4,

are the unknown model

uncertainty and external disturbances present in the drive and response systems respectively and

u  t   R 3 is the control input. The error dynamics between the drive and response systems (24) can be

defined as

ei  t   yi  t   xi  t  , i  1, 2, 3, 4 . Thus, the error dynamics for the increased order

synchronization scheme (24) are described as follows:

Page 12 of 18

(25)

lim ei  t   lim yi  t   xi  t   0, i  1, 2, 3, 4 x 

x 

us

Objective 3.

cr

ip t

  e1  t   a  e2  t   e1  t     a  25  10   y1  t   x1  t    g1  y  t    h1  x  t      D1  t   d1  t   u1  t   e2  t    28  35  e1  t   ce2  t    28  35  x2  t    c  29  1 y1  t   x1  t  x3  t     y1  t  y3  t   g 2  y  t    h2  x  t    D2  t   d 2  t   u2  t     8  e3  t   be3  t     b  z1  t   y1  t  y2  t   x1  t  x2  t   g3  y  t    h3  x  t      3    D3  t   d3  t   u3  t    e4  t   y2  t   y4  t   g 4  y  t    D4  t   u4  t  

Lemma

3.

 x  0 , x  0 , x  0 , 1

2

3

For

an

Objective 4. The closed-loop (28) is globally exponentially stable. the

arbitrary

initial

conditions



 R 3  y1  0  , y2  0  , y3  0  , y4  0   R 4 ; of the drive and response systems

M

respectively, the globally exponential increased order synchronization is accomplished with the following proper active control functions:

u1  t    a  25  10   x1  t   y1  t    1  v1  t 

ed

u2  t    35  28  x2  t    29  c  1 y1  t   y1  t  y3  t   x1  t  x3  t    2  v2  t 

(26)

Ac ce

pt

 8  u3  t    b   z1  t   x1  t  x2  t   y1  t  y2  t   3  v3  t  3   u4  t   y4  t   y2  t   4  D4   v4  t 

Definition 4.

The matrix v  t   vi  t   , i  1, 2, 3, 4 , is defined as follows: T

v  t   W ei  t   T with W  dig  wi  , i  1, 2, 3, 4 .

(27)

Proof of lemma 3. Combining systems of Eqs. (25), (26) and (27), that yields:  e1  t     w1  a      e t    2     28  35      0  e3  t     e  t   0   4 

a

0

 w2  c 

0

0

 w3  b 

0

0

0   e1  t     0   e2  t    0 0   e3  t    w4   e  t    4 

(28)

At this stage, the problem is reduced to show that if the linear controller gains wi ' s , i  1, 2, 3, 4 are selected properly which make all the eigenvalues of the coefficient matrix in (28) with positive real parts.

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From definition 1, the coefficient matrix in (28) will be PDM, if all ordered principal minor determinants being positive. That is: a  28  35 

 c,
3 : w3  b,
4 : w4  0 w1  a Thus, the closed-loop system (28) is globally exponentially stable. This completes the proof.

4.2

(29)

ip t


1 : w1   a
2 : w2 

Numerical simulation and discussions

In the drive unified chaotic system (16), the parameters are taken as:   0, 0.8 and 1, and the initial states being set as  x1  0  , x2  0  , x3  0     2, 1, -1 . In the response system (15), the parameters are a  15, b  5, c  10, and the initial states being set as taken as T

cr

T

 y 1  0  , y 2  0  , y 3  0  , y 4  0    1, 2, 3, 4, . According to conditions (29), the linear controller gains are selected as w1  w2  30 and w3  w4  10 . The following model uncertainties and external disturbances are considered in the simulation. T

an

us

T

    f 2  x2  t    d 2  t   0.25cos  x2  t    0.01sin  20t  ,    2  f 3  x3  t    d3  t   0.35cos  x3  t    0.03cos  30t     3 

M

f1  x1  t    d1  t   0.3sin  x1  t    0.02cos 10t  ,

g1  y1  t    D1  t   0.4cos  y1  t    0.01sin  30t  ,

Ac ce

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ed

 (30)    5  y2  t    0.03cos  20t  ,  g 2  y2  t    D2  t   0.25sin   6      g3  y3  t    D3  t   0.25cos  y3  t    0.04cos  50t  ,  2     5  y4  t    0.01sin  30t   g 4  y4  t    D4  t   0.15sin   6   Accordingly, 1  0.7,  2  0.5,  3  0.6 and 4  0.15 . Time series of the convergence of the synchronized error signals to the zero state is depicted in Figs. 7, 8 and 9, for   0, 1 and 0.8 alternatively. As expectedly, one can notice the smoothness of the synchronized error signals while converging to the zero state with fast converging rates, which demonstrates the robustness and performance of the control action (26) for the increased order synchronization scheme. Fig. 10, depicts the synchronization analysis between the drive and response systems which is confirmed by the convergence of the synchronization quality defined by the refinement of the error states: E  t   e12  t   e22  t   e32  t   e42  t 

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Fig. 7. Time series of the synchronized error states when   0

4

Fig. 8. Time series of synchronized error states when   1

e1 t

3

e2 t

2

e3 t e4 t

ip t

1 0

0.0

0.5

1.0

1.5

cr

1 2.0

Time units

Fig. 10. Time series of convergence of errors when   0

4.3

Time units

pt

Time units

ed

M

an

Fig. 9. Time series of the synchronized error states when   0.8

us

Time units

Applications of the proposed synchronization approach

Ac ce

Synchronization between chaotic systems with different orders have potential applications in information processing, in physical, social and biological systems due to the complex behavior in their chaotic attractors. The traces changes in their respective trajectories with time and topological properties of the two chaotic systems are different. These properties increase security in the communications channel. The proposed synchronization approach can be investigated for compression purpose where it is desirable to reduce the dimensions of a system so that it becomes suitable to transfer the data from the transmitter to the receiver. At the receiver end, the data can be decoded again to get the exact size with required dimension. Similar applications can also be seen in very popular Principle Component Analysis (PCA) algorithm [32], where dimension reduction target is achieved by taking only those eigenvectors which have higher values. The second very vital application of the proposed synchronization approach is that it can be used to enhance the security of the data during transmission [33]. Before transmitting the data, the dimensions could be increased or decreased and at the receiver end, the required dimension could be achieved according to the requirements. During transmission, data has the capability to synchronize with different dimension signals which enhance security of the transmitting data.

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5. Conclusions

Ac ce

References

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an

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ip t

Based on the Routh-Hurwitz criterion and Lyapunov stability theory, a generalized robust active control approach is proposed and some easily sufficient conditions are derived to compute a suitable linear controller gain matrix that assured the globally exponentially stable reduced-order and increased order synchronizations for a general class of chaotic systems under the effect of both unknown model uncertainties and external disturbances. Security of the communications channel and future applications of the proposed approach is discussed. In comparison with other conventional synchronization approaches and methods for reduced/increased order synchronization, the main advantages of the proposed approach which are summarized as follows: i. The presented generalized active control approach is simple in design and improves the synchronization quality for reduced/increased order synchronization under the effect of both unknown model uncertainty and external disturbances. ii. The synchronization speed is fast as well as the amplitude of the oscillations is smaller. The amplitude problem is noteworthy in control system theory, because it may specify the amount of energy required to synchronize chaotic systems. iii. The eigenvalues of the coefficient matrix of the closed-loop system plays a vital role in the synchronization stability. In the proposed approach, the eigenvalues of the linear part can be adjusted to have a desirable synchronization time. iv. Most of the chaotic systems in real practical applications have different structures, thus, we believe that the proposed generalized active control approach will be a helpful tool for synchronization of chaotic systems with different dimensions. v. The proposed approach applied successfully to achieve reduced (increased) order synchronization, such as the chaotic Lu and hyperchaotic Chen systems, the Hyperchaotic Lorenz and chaotic Chen system, the hyperchaotic Li and chaotic Lu systems. It has been observed that the error states are synchronized in a time less than 1 second which shows the robustness and effectiveness of the proposed robust generalized active control strategy. vi. Numerical experimental results further validated the robustness and effectiveness of the current study and show that the error signals converged to the zero state rapidly with good synchronization rates and quality.

[1] Hashtarkhani, B., Aghababa, M.P., Khorrowjerdi, M.J. Design of a robust nonlinear controller for a synchronus generator connected to an infinite bus. Complexity, DOI:10.1002/cplx.21648, (2015). [2] Pisarchik, A. N., Arecchi, F. T., Meucci, R, Garbo A. D. Synchronization of Shilnikov Chaos in CO2 Laser with feedback, Laser Phys., 11(11), 1235-1239 (2014). [3] Lu, L., Li, C., Wang, W., Sun, Y., Wang, Y., Sun, A. Study on spatiotemporal chaos synchronization among complex networks with diverse structures, Nonlinear Dyns. 77(1-2), 145-151 (2014). [4] Goldstein, R. E., Polin, M., Tuval, I. Noise and Synchronization in Pairs of Beating Eukaryotic Flagella. Phys. Rev. Lett. 103 (16), 168103, (2009). [5] Liu, S., Chen, L. Outer synchronization of uncertain small-world networks via adaptive sliding mode, App. Mathematics & Comp., 36(3), 319-328 (2015). [6] Aghababa, M. P., Aghababa, H. P. Synchronization of nonlinear chaotic electromechanical gyrostat systems with uncertainties. Nonlinear Dyn., 67, 2689-2701 (2012). [7] Ojo, K. S., Njah, A. N., Olusola, O. I., Omeike, M. O. Generalized reduced-order hybrid combination synchronization of three Josephson junctions via backstepping technique. Nonlinear Dyn., 77, 583–595 (2014). [8] Fell, J., Axmacher, N. The role of phase synchronization in memory processes. Nature Reviews Neuroscience 12, 105118 (2011) [9] Zheng, S. Partial switched modified function projective synchronization of unknown complex nonlinear systems. Int. Journal for light and electron optics, 2015, DOI: 10.1016/j.ijleo.2015.07.075.

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[10] Ahmad, I., Saaban, A., Ibrahim, A., Shahzad, M. Global Chaos Synchronization of New Chaotic System using Linear Active Control. Complexity. 21(1), 379-386 (2014). [11] Ahmad, I., Saaban, A., Ibrahim, A., Shahzad, M. A Research on the Synchronization of Two Novel Chaotic Systems Based on a Nonlinear Active Control Algorithm. Engineering, Technology & Applied Science Research, 5(1) 739-747 (2015). [12] Femat R, Perales GS. Synchronization of chaotic systems with different order. Physical Rev. E., 65, 036226-1-7 (2002). [13] Ho, M. C., Hung, Y. C., Liu, Z. Y., Jiang, I. M. Reduced-ordered synchronization of chaotic systems with parameters. Phys. Lett. A, 248, 251-259 (2006). [14] Alsawalha, M., Noorani, M. Adaptive reduced-order anti-synchronization of chaotic systems with fully uncertain parameters. Comm Nonlinear Sci. Numr. Simulat., 15, 3022-3034 (2010). [15] Strogatz, S. H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Westview Press, 2011. [16] Ge, A. M., Yang, C. H. The generalized synchronization of a Quantum-CNN chaotic oscillators with different order systems. Chaos, Solitons and Fractals, 35, 980-990 (2008). [17] Alsawalha, M., Noorani, M. Adaptive increasing-order synchronization and anti-synchronization of chaotic systems with fully uncertain parameters. Chin. Phys. Lett., 28 (11) 110507-1-3 (2011). [18] Bai, E. W., Lonngren, K. E. Synchronization of two Lorenz system using Active control, Phy. Rev. Lett., 64, 1199-1196 (1997). [19] Agiza, H. N., Yassen, M. T. Synchronization of Rossler and Chen chaotic dynamical systems using active control, Physics Lett. A, 278(4), 191-197 (2001). [20] Ho, M., Hung, Y. Synchronization of two different systems by using generalized active control, Physics Lett. A, 301, 424– 428 (2001). [21] Chen, H. Chaos synchronization of a symmetric gyro with linear-plus-cubic damping, Journal of Sound and Vibration, 255, 719–740 (2002). [22] Vincent, U. E., Ucar, A. Synchronization and anti-synchronization of chaos in permanent magnet reluctance machine, Far East Journal of Dynamical Systems, 9, 211–221 (2007). [23] Njah, A.N., Vincent, U.E. Synchronization and anti-synchronization of chaos in an extended Bonhoffer-van der Pol oscillator using active control. J Sound Vib., 319, 41–49 (2009). [24] Bai, E.W., Lonngren, K.E., Sprott, J. C. On the synchronization of a class of electronic circuits that exhibit chaos, Chaos, Solitons and Fractals, 13, 1515–1521 (2002). [25] Ucar, A., Lonngren, K.E., Bai, E. W. Chaos synchronization in RCL-shunted Josephson junction via active control, Chaos, Solitons and Fractals, 31, 105–111 (2007). [26] Shahzad, M., Ahmad, I. Experimental study of synchronization & Antisynchronization for spin orbit problem of Enceladus, International. Journal of Control Science and Engineering, 3(2), 41-47 (2013). [27] Elabbasy, E. M., Agiza, H. N., El-Dessoky, M. Anti-synchronization of a novel hyperchaotic system with parameter mismatch and external disturbances, Chaos, Solitons and Fractals, 30(5), 1133-1142 (2006) [28] Ling, L. U., Guo, Z. A., Chao, Z. Synchronization between two different chaotic systems with nonlinear feedback control, Chin. Phys., 16(6), 1603-07 (2007). [29] Ucar, A., Lonngren, K. E., Bai, E. W. Synchronization of the unified chaotic systems via active control, Chaos, Solitons and Fractals, 27(5), 1292–1297 (2006). [30] Stork, M. Adaptive synchronization of chaotic systems with time changing parameters, Proceedings of the International Conference on Circuits, Systems, Signals (CSS), IEEEAM International Conferences, Malta, 96- 100 (2010). [31] Séroul, R. Stable Polynomials. Programming for Mathematician. Berlin: Springer-Verlag, pp. 280-286, 2000. [32] Pearson, K. On lines and planes of Closet Fit to systems of Points in space, Philosophical Magazine, 2 (11): 559–572 (1901). [33] Jang, B., Lee, S., Kwon, K. Perceptual encryption with compression for secure vector map data processing, Digital Signal Processing 25, 224-243, 2014.

Appendix

i). Mathematica codes for reduced-order synchronization Needs["PlotLegends`"]; α=0;a=15;b=5;c=10;w1=25;w2=30;w3=6;

a1=NDSolve[{ x1'[t]==a (y1[t]-x1[t])-0.35 Cos [180x1[t]] +0.01 Sin [ 30t],y1'[t]==c y1[t]+ w1[t]- x1[t] z1[t]+0.2 Sin [ 150y1[t]]0.03 Cos [ 20t] ,z1'[t]== -b z1[t]+ x1[t]y1[t]+0.02 Cos [ 90z1[t]] +0.04 Cos [ 50t],w1'[t]== y1[t]- w1[t]+0.01 Sin [ 150w1[t]] +0.001 Sin [ 30t] ,x2'[t]==α (y2[t]-x2[t])+(a-α) (y1[t]-x1[t])- k1 (x2[t]-x1[t] )+0.5 Sin [ 150x2[t]] -0.1 Cos [

Page 17 of 18

ip t

10t],y2'[t]==β x2[t]-β x1[t]- y2[t]+(c+1) y2[t]+ w1[t]- x1[t]z1[t]- k2 (y2[t]-y1[t] )-0.25 Cos [ 180y2[t]]-0.02 Sin [ 20t] ,z2'[t]==-b z2[t]+ x1[t]y1[t]- k3 (z2[t]-z1[t] )-0.2 Cos [ 120z2[t]]+0.03 Cos [ 30t], w2'[t]==0,x1[0]==1,y1[0]==2,z1[0]==3,w1[0]==4,x2[0]==2,y2[0]==1,z2[0]== -1, w2[0]==0},{x1[t],x2[t],y1[t],y2[t],z1[t],z2[t],w1[t],w2[t]},{t,0,50},MaxSteps->∞]; a2=Evaluate[x2[t]-x1[t]/.a1]; a3=Evaluate[y2[t]-y1[t]/.a1]; a4=Evaluate[z2[t]-z1[t]/.a1]; Plot[{a2,a3,a4},{t,0,10},PlotStyle->{Red,Blue,Green,Purple},Frame->True,Axes->False,FrameLabel>{{"",""},{"Figure 2. Time series of the synchronized error states ",""}}, LabelStyle->Directive[Medium,Bold],PlotLegends->LineLegend[{"e1[t]","e2[t]","e3[t]"},LegendFunction>"Frame"],PlotRange->{{0,2},{-4.5,1.5}}]

an

us

cr

a5=Evaluate[Sqrt[(Subscript[x, 2][t]-Subscript[x, 1][t])2+(Subscript[y, 2][t]-Subscript[y, 1][t])2+(Subscript[z, 2][t]-Subscript[z, 1][t])2]/.a1]; Plot[{a5},{t,0,10},Axes->False,Frame->True,FrameLabel->{{"",""},{"Fig 6. Convergence of errors",""}},PlotRange->{{0,2},{-4.5,1}}, LabelStyle->Directive[Medium,Bold],PlotLegends->LineLegend[{" "},LegendFunction>"Frame",LegendMargins->6,RoundingRadius->5],PlotStyle->{Blue}] Note: For Figs. 4 and 5, replace α=0 by α=1 and 0.8 respectively. ii). Mathematica codes for increasing order synchronization

Ac ce

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M

α=0;a=15;b=5;c=10;w1=30;w2=30;w3=10;w4=10; a1=NDSolve[{ a1=NDSolve[{ x1'[t]==α(y1[t]- x1[t])+0.3 Sin [ 180x1[t]] -0.01 Cos [ 10t],y1'[t]==β x1[t]- y1[t]- x1[t] z1[t]-0.25 Cos [ 180y1[t]]0.02 Sin [ 20t],z1'[t]==-ω z1[t]+x1[t] y1[t]-0.35 Cos [ 120z1[t]]+0.03 Cos [ 30t],w1'[t]==0, x2'[t]==a(y2[t]-x2[t])+(α-a) (y1[t]-x1[t])-k2 (x2[t]-x1[t] )-0.4 Cos [150x2[t]] +0.01 Sin [ 30t], y2'[t]==c y2[t]- x1[t] z1[t]+ β x2[t]-(c+1) y1[t]-k2 (y2[t]-y1[t] )+0.25 Sin [ 150y2[t]]-0.03 Cos [ 20t] , z2'[t]==-b z2[t]+ x1[t]y1[t]+(b-ω) z1[t]-k3 (z2[t]-z1[t] )+0.25 Cos [ 90z2[t]] +0.04 Cos [ 50t], w2'[t]==-(k4+1) (w2[t]-w1[t] )+0.15 Sin [ 150w2[t]] +0.01 Sin [ 30t] ,x1[0]==2,y1[0]==1,z1[0]==-1, w1[0]==0,x2[0]==1,y2[0]==2,z2[0]==3,w2[0]==4},{x1[t],x2[t],y1[t],y2[t],z1[t],z2[t],w1[t],w2[t]},{t,0,100}, MaxSteps->∞]; a2=Evaluate[x2[t]-x1[t]/.a1]; a3=Evaluate[y2[t]-y1[t]/.a1]; a4=Evaluate[z2[t]-z1[t]/.a1]; a5=Evaluate[w2[t]-w1[t]/.a1]; Plot[{a2,a3,a4,a5},{t,0,50},PlotStyle->{Red,Blue,Green,Purple},Frame->True,Axes->False, FrameLabel>{{"",""}, {"Fig. 7. Time series of the synchronized error states for α=0",""}}, FrameLabel->{{"",""},{"Figure 2. Time series of the synchronized error states ",""}}, LabelStyle->Directive[Medium,Bold],PlotLegends->LineLegend[{"e1[t]","e2[t]","e3[t]","e4[t]"},LegendFunction>"Frame"],PlotRange->{{0,2},{-1.5,4.5}}] a6=Evaluate[Sqrt[(Subscript[x, 2][t]-Subscript[x, 1][t])2+(Subscript[y, 2][t]-Subscript[y, 1][t])2+(Subscript[z, 2][t]Subscript[z, 1][t])2+(Subscript[w, 2][t]-Subscript[w, 1][t])2]/.a1]; Plot[{a6},{t,0,10},Axes->False,Frame->True,FrameLabel->{{"",""},{"Fig 10. Convergence of errors",""}},PlotRange->{{0,6},{-0.5,6}}, LabelStyle->Directive[Medium,Bold],PlotLegends->LineLegend[{" "},LegendFunction->"Frame",LegendMargins>6,RoundingRadius->5],PlotStyle->{Blue}] Note: For Figs. 8 and 9, replace α=0 by α=1 and 0.8 respectively.

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