Adaptive Modified Function Projective Synchronization between Two Different Hyperchaotic Dynamical Systems

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Volume 2012, Article ID 810626,16pages doi:10.1155/2012/810626

Research Article

Adaptive Modified Function Projective Synchronization between Two Different Hyperchaotic Dynamical Systems

M. M. El-Dessoky,

^{1, 2}

M. T. Yassen,

²

and E. Saleh

²

1Department of Mathematics, Faculty of Science, King AbdulAziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Correspondence should be addressed to M. T. Yassen,[email protected] Received 11 November 2011; Accepted 17 December 2011

Academic Editor: Jun-Juh Yan

Copyrightq2012 M. M. El-Dessoky et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This work investigates modified function projective synchronization between two different hyperchaotic dynamical systems, namely, hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between two diffierent hyperchaotic dynamical systems. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.

1. Introduction

During the last three decades, synchronization of chaotic systems has attracted increasing attention from scientists and engineers and has been explored intensively both theoretically and experimentally. Since Pecora and Carrol 1 introduced a method to synchronize two identical systems with diﬀerent initial conditions, many approaches have been proposed for the synchronization of chaotic or hyperchaotic systems such as complete synchronization 1, phase synchronization 2, generalized synchronization 3, lag synchronization 4, intermittent lag synchronization5, time-scale synchronization6, intermittent generalized synchronization 7, projective synchronization 8, modified projective synchronization 9,10, and function projective synchronization11,12. Most of them are based on exactly knowing the system structure and parameters, but in practice, some or all of the system’s parameters are unknown. Moreover, these parameters change from time to time. A lot of works have been done to solve this problem using adaptive synchronization13–16. Most

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of the methods mentioned above synchronize two identical chaotic systems. Hyperchaotic system is usually classified as a chaotic system with more than one positive Lyapunov exponent, indicating that the chaotic dynamics of the system are expanded in more than one direction giving rise to a more complex attractor. The method of the synchronization of two diﬀerent hyperchaotic systems is far from being straightforward. There is little work about this challenging problem because it consists of diﬀerent structures and parameter mismatch of the two hyperchaotic systems. Complete synchronization is characterized by the equality of state variables while evolving in time. Antisynchronization is characterized by the vanishing of the sum of relevant variables. Projective synchronization occurs when the drive and response system could be synchronized up to a scaling factor. Function projective synchronization is the most general definition of projective synchronization. It means that the derive and response systems could be synchronized up to a scaling function.

A focused problem in the study of chaos synchronization is how to design a physically available and simple controller to guarantee the realization of high-quality synchronization in coupled chaotic systems. Linear feedback is of course a practical technique, but the shortcoming is that it needs to find the suitable feedback constant. Recently, Huang proposed a simple adaptive feedback control method, which neednot to estimate or find feedback constant, to eﬀectively synchronize two almost arbitrary identical chaotic systems in his series paper17–19.

In this work, we investigate modified function projective synchronization MFPS between hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. This work is organized as follows. InSection 2the modified function projective synchronization MFPS scheme is presented. Section 3 briefly describes hyperchaotic Lorenz system and hyperchaotic Chen system. Section 4 proposes adaptive control laws and parameter update rules for the modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Chen system. InSection 5, numerical examples are given to demonstrate the eﬀectiveness of the proposed method. Finally, the conclusions are given inSection 6.

2. Adaptive Modified Function Projective Synchronization (MFPS) Scheme

Consider the following master and slave system:

˙

xfx, t, 2.1

˙ yg

y, t u

x, y, t

, 2.2

wherex, y∈Rⁿare the state vector of the system2.1and2.2, respectively;f, g:Rⁿ → Rⁿ are two continuous nonlinear vector functions,ux, y, tis the vector controller. We define the error dynamical system as

et y−M htx, 2.3

whereMis a constant diagonal matrixMdiag{m1, m2, . . . , mn} ∈ R^n×nandhta continuous diﬀerentiable function withht/0 for allt.

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10 0 20 10 0

20 10 20 30 40 50

−10 x(t)

−20−30

−10 −20

y(t)

z(t)

Figure 1: The attractor of hyperchaotic Lorenz dynamical system atα10, β28, γ8/3, andr0.1 in x,y,zsubspace.

The system2.1and2.2i said to be in modified function projective synchronization if there exists a constant diagonal matrixMand functionht, such that Limt→ ∞et0.

It is easy to see that the definition of modified function projective synchronization encompasses function projective synchronization when the scaling matrixMequalsI.

3. System Description

The hyperchaotic Lorenz system is described as follows20–22:

˙ xα

y−x ,

˙

yβx y−xz−w,

˙

zxy−bz,

˙ wryz,

3.1

wherex, y, z, andw are state variables andα,β,γ, andr are real constant parameters. In 21,22, it has been shown that the system3.1has two positive Lyapunov exponents when α 10, β 28, γ 8/3, andr 0.1, the system3.1exhibits hyperchaotic behavior, see Figure 1.

Hyperchaotic Chen system is described as23,24

˙ xa

y−x w,

˙

ydx cy−xz,

˙

zxy−bz,

˙

wlw yz,

3.2

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5 0

15 10 0

20 10 10

15 20 25 30 35

x(t)

−10−20 y(t)

z(t)

−5 −10 −15

Figure 2: The attractor of hyperchaotic Chen dynamical system ata35,b3,c12,d7, andl0.5 in x,y,zsubspace.

wherex,y,zandware state variables anda,b,c,dandhare real constant parameters. When a35, b 3, c12, d7, 0.798≤l≤0.9, system3.2is periodic, whena35, b 3, c 12, d7, 0≤l≤0.085, system3.2is chaotic; whena35, b3, c12, d7, 0.085≤l≤ 0.798, system3.2exhibits hyperchaotic behavior seeFigure 2.

4. Adaptive MFPS between Hyperchaotic Lorenz System and Chen System

In order to achieve the synchronization behavior between hyperchaotic Lorenz system and hyperchaotic Chen system, we assume that hyperchaotic Lorenz system is the drive system whose four variables are denoted by subscript 1 and hyperchaotic Chen system is the response system whose variables are denoted by subscript 2. The drive and response systems are described by the following equations, respectively,

˙ x1α

y1−x1

,

˙

y₁βx₁ y₁−x₁z₁−w₁,

˙

z1x1y1−γz1,

˙

w₁ry₁z₁,

4.1

˙ x2a

y2−x2

w2 u1,

˙

y₂dx₂ cy₂−x₂z₂ u₂,

˙

z2x2y2−bz2 u3,

˙

w2lw2 y2z2 u4,

4.2

whereU u1, u₂, u₃, u₄^T is the nonlinear controller functions which are to be determined later. The term “synchronization,” in general, means that the signal eﬀect upon the synchronized system is very small in comparison with the amplitude of proper oscillations of the

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system but, nevertheless, it is enough to change the system behavior and to “impose” the rhythm of external influence to it. The two hyperchaotic dynamical systems can be synchronized in the sense that

t→ ∞lim|x2−m1htx1|0,

t→ ∞limy₂−m₂hty10,

t→ ∞lim|z2−m3htz1|0,

tlim→ ∞|w2−m₄htw1|0,

4.3

wheremi,i1,2,3,4is the scaling factor andhtthe scaling function.

We have the following error dynamical system:

˙

e_xx˙₂−m₁htx˙₁−m₁htx˙ 1,

˙

eyy˙2−m2hty˙1−m2 hty˙ 1,

˙

e_zz˙₂−m₃htz˙₁−m₃htz˙ 1,

˙

e_ww˙₂−m₄htw˙₁−m₄htw˙ 1,

4.4

wheree_xx₂−m₁htx1,e_yy₂−m₁hty1,e_zz₂−m₁htz1, ande_ww₂−m₁htw1. Substitution of4.1and4.2in4.4yields following error dynamical system

˙ exa

y2−x2

w2 u1−m1htα y1−x1

−m1htx˙ 1,

˙

e_ydx₂ cy₂−x₂z₂ u₂−m₂ht

βx₁ y₁−x₁z₁−w₁

−m₂hty˙ 1,

˙

ezx2y2−bz2 u3−m3ht

x1y1−γz1

−m3htz˙ 1,

˙

e_wlw₂ y₂z₂ u₄−m₄htry1z₁−m₄htw˙ 1.

4.5

Our aim is to find control lawsui i1,2,3,4for stabilizing the error variables of the system at the origin. For this end, we propose following control law:

u₁m₁htα1

y₁−x₁

m₁htx˙ 1−a₁

x₂−y₂

−w₂−k₁e_x, u2m2ht

y1 β1x1−w1−x1z1

x2z2 m2hty˙ 1−d1x2−c1y2−k2ey, u₃m₃ht

x₁y₁−γ₁z₁

m₃htz˙ 1 b₁z₂−x₂y₂−k₃e_z, u4m4htr1y1z1 m4htw˙ 1−l1w2−y2z2−k4ew,

4.6

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and the update laws for the unknown parametersα1, β1, γ1, r1, a1, b1, c1, d1, andl1are

˙ α1

x1−y1

m1htex−k5α1−α, β˙1−x1m2htey−k6

β1−β ,

˙

γ₁z₁m₃htez−k₇ γ₁−γ

,

˙

r1 −y1z1m4htew−k8r1−r,

˙ a₁

y₂−x₂

e_x−k₉a1−a, b˙₁−z2e_z−k₁₀b1−b,

˙

c1y2ey−k11c1−c, d˙₁x₂e_y−k₁₂d1−d,

l˙₁w₂e_w−k₁₃l1−l,

4.7

whereki>0 i1,2,3, . . . ,13.

Theorem 4.1. For given constant scaling matrixMand scaling functionht, the MFPS between two systems4.1and4.2will occur by the control law4.6and update law4.7, and satisfy

t→ ∞lim|α1−α| lim

t→ ∞β₁−β lim

t→ ∞γ₁−γ lim

t→ ∞|r1−r| lim

t→ ∞|a1−a|

lim

t→ ∞|b1−b| lim

t→ ∞|c1−c| lim

t→ ∞|d1−d| lim

t→ ∞|l1−l|0.

4.8

Proof. Define a Lyapunov function, Ve 1

2

e²_x e²_y e_z² e²_w e²_α e_β² e²_γ e_r² e²_a e_b² e²_c e²_d e²_l

, 4.9

where

e_αα₁−α, e_ββ₁−β, e_γ γ₁−γ, e_rr₁−r, e_aa₁−a,

e_b b₁−b, e_cc₁−c, e_dd₁−d, e_ll₁−l. 4.10 The time derivative of the Lyapunov function along the trajectory of error system4.9 is

dVe

dt exe˙x eye˙y eze˙z ewe˙w eαe˙α eβe˙β eγe˙γ

ere˙r eae˙a ebe˙b ece˙c ede˙d ele˙l

e_xe˙_x e_ye˙_y e_ze˙_z e_we˙_w e_αα˙₁ e_ββ˙₁ e_γγ˙₁ err˙1 eaa˙1 ebb˙1 ecc˙1 edd˙1 ell˙1.

4.11

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Inserting4.6and4.7into4.11yields the following:

dVe

dt −k1e_x²−k₂e²_y−k₃e²_z−k₄e²_w−k₅e²_α−k₆e²_β−k₇e²_γ−k₈e_r²

−k₉e²_a−k₁₀e²_b−k₁₁e²_c−k₁₂e²_d−k₁₃e²_l −Ke²,

4.12

wheree ex, e_y, e_z, e_w, e_α, e_β, e_γ, e_r, e_a, e_b, e_c, e_d, e_l^T andK diagk1, k₂, k₃, k₄, k₅, k₆, k₇, k₈, k₉, k₁₀, k₁₁, k₁₂, k₁₃^T.

SincedVe/dt ≤0, we havee_x, e_y, e_z, e_w, e_α, e_β, e_γ, e_r, e_a, e_b, e_c, e_d, e_l → 0 ast → ∞, lim_{t→ ∞}e0.

Therefore, the drive system4.1synchronizes the response system4.2in the sense of MFPS.

Remark 4.2. Note that complete synchronization and antisynchronization between two dif- ferent hyperchaotic dynamical systems are special cases of MFPS with the scaling function ht 1 and the scaling factorsm_i1 andm_i−1 i1,2,3,4, respectively.

Remark 4.3. Note that function projective synchronizationFPSbetween two diﬀerent hyperchaotic dynamical systems is special case of MFPS with the scaling factorsm_i 1i 1,2, 3,4, and the scaling functionhtis chosen later.

Remark 4.4. Note that generalized projective synchronization GPS and modified generalized projective synchronization MGPS between two diﬀerent hyperchaotic dynamical systems are special cases of MFPS with the scaling functionht 1 and the scaling factors miare equal andmiare not equali1,2,3,4, respectively.

By suitable choosing forht, we can achieve modified function projective synchronization, complete synchronization, antisynchronization, function projective synchronization, generalized projective synchronization, modified generalized projective synchronization, between two diﬀerent hyperchaotic systemssee Examples5.1–5.6.

5. Numerical Results

In this section, numerical examples are used to demonstrate the eﬀectiveness of the proposed method. By using Maple 12 to solve the systems of diﬀerential equations4.1,4.2,4.6, and 4.7, we assume that the initial conditions of the drive system arex10 2,y10 2,z10 3, andw₁0 1, and the initial conditions of the response system arex₂0 6,y₂0 5, z₂0 3, and w₂0 3. The initial conditions of the estimated parameters are chosen as α10 0, β₁0 0,γ10 0,r10 0,a10 0,b10 0,c10 0,d10 0 andl10 0.

Let the scaling function beht sin0.1πtand the scaling factors are chosen asm₁ 2,m₂3,m₃5, andm₄0.5. The simulation of the error dynamical system between hyperchaotic Lorenz system and hyperchaotic Chen system without control functions is shown inFigure 3adisplays thee_x x₂−m₁htx1,Figure 3bdisplays thee_y y₂−m₁hty1, Figure 3cdisplays thee_zz₂−m₁htz1, andFigure 3ddisplays thee_ww₂−m₁htw1. Example 5.1. Let the scaling function beht sin0.1πtand the scaling factors are chosen as m₁2, m₂3, m₃5, andm₄0.5. Furthermore, the control gains are chosen ask₁ k₂ k₃ k₄ 3, k₅ k₆ k₇ k₈ k₉ k₁₀ k₁₁ k₁₂ h₁₃ 2.Figure 4displays the MFPS between systems4.1and4.2.Figure 5show that the estimatesα1t,β1t,γ1t,r1tof the unknown parameters converge toα10, β 28, γ 8/3, andr 0.1 ast → ∞.Figure 6

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−20

−10 0 10 20 30

ex

5 10 15 20 25 30

t a

−40−30

−20−1010203040500 ey

0 10 20 30

t b

−150

−100−50 0 50 100 150

ez

0 10 20 30 40

t c

−100

−50 0 50 100

ew

0 10 20 30

t d

Figure 3: The behavior of the trajectoriesex,ey,ez, andewof the error system between hyperchaotic Lorenz system and hyperchaotic Chen system without control functions.

−2

−1 0 1 2 3 4 5 6

Error

0 2 4 6 8 10

t ex

ey

ez

ew

Figure 4: The behavior of the trajectoriesex,ey,ez, andewof the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for MFPS.

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0 5 10 15 20 25

0 2 4 6 8 10

α1

β1

γ1

r1

t

Figure 5: The estimatesα1t,β1t,γ1t,r1tof the unknown parameters converges toα10, β28, γ 8/3, andr0.1 asttends to 3.

0 10 20 30

0 2 4 6 8 10

a1

b1

c1

d1

l1

t

Figure 6: The estimatesa1t,b1t,c1t,d1t,l1tof the unknown parameters converge toa35, b 3, c12, d7, andl0.5 asttends to 3.

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−6

−4

−2 0 2 4 6

0 1 2 3 4 5 6

Error

t ex

ey

ez

ew

a

0 5 10 15 20 25

1 2 3 4 5 6

α1

β1

γ1

r1

t

b

0 10 20 30

1 2 3 4 5 6

a1

b1

c1

d1

l1

t

c

Figure 7:aThe behavior of the trajectoriesex,ey,ez, andewof the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for complete synchronization.bthe estimatesα1t,β1t, γ1t,r1tof the unknown parameters converge toα10, β28, γ 8/3, andr 0.1 asttends to 3.

cthe estimatesa1t,b1t,c1t,d1t,l1tof the unknown parameters converges toa35, b3, c 12, d7, andl0.5 asttends to 3.

show that the estimatesa₁t,b₁t,c₁t,d₁t,l₁tof the unknown parameters converge to a35,b3,c12,d7, andl0.5 ast → ∞.

Example 5.2. Let the scaling function be ht 1 and the scaling factors are chosen as m₁ m₂ m₃ m₄1. Furthermore, the control gains are chosen ask₁ k₂k₃ k₄ 3, k5 k6 k7 k8 k9 k10 k11 k12 h13 2.Figure 7adisplays the complete synchronization between systems 4.1 and 4.2. Figure 7b show that the estimates α₁t, β₁t, γ₁t, r₁t of the unknown parameters converge to α 10, β 28, γ 8/3, andr 0.1 ast → ∞.Figure 7cshow that the estimatesa1t,b1t,c1t,d1t,l1tof the unknown parameters converge toa35, b3, c12, d7, and l0.5 ast → ∞.

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−5 0 5 10 15

0 1 2 3 4 5 6

Error

t ex

ey

ez

ew

a

0 10 20 30

1 2 3 4 5 6

α1

β1

γ1

r1

t

b

00 10 20 30

1 2 3 4 5 6

a1

b1

c1

d1

l1

t

c

Figure 8:aThe behaviour of the trajectoriesex,ey,ezandewof the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for anti-synchronization.b: Show the estimatesα1t,β1t, γ1t,r1tof the unknown parameters converges toα10, β28, γ 8/3 andr 0.1 asttends to 3.

c: Show the estimatesa1t,b1t,c1t,d1t,l1tof the unknown parameters converges toa35, b 3, c12, d7 andl0.5 asttends to 3.

Example 5.3. Let the scaling function beht 1 and the scaling factors are chosen asm₁ m₂m₃ m₄−1. Furthermore, the control gains are chosen ask₁k₂k₃ k₄3, k₅ k6 k7 k8 k9 k10 k11 k12 h13 2.Figure 8adisplays the anti-synchronization between systems4.1and4.2.Figure 8bshows the estimatesα₁t,β₁t,γ₁t,r₁tof the unknown parameters converge toα10, β28, γ 8/3, andr0.1 ast → ∞.Figure 8c shows the estimates a1t,b1t,c1t, d1t, l1t of the unknown parameters converge to a35, b3, c12, d7, andl0.5 ast → ∞.

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0 2 4 6 8 10

0 1 2 3 4 5 6

Error

t ex

ey

ez

ew

a

0 5 10 15 20 25

1 2 3 4 5 6

α1

β1

γ1

r1

t

b

0 10 20 30

0 1 2 3 4 5 6

a1

b1

c1

d1

l1

t

c

Figure 9:aThe behavior of the trajectoriesex,ey,ez, andewof the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for FPS.b the estimatesα1t, β1t, γ1t, r1tof the unknown parameters converge toα10, β28, γ 8/3, andr 0.1 asttends to 3.cthe estimates a1t, b1t, c1t, d1t, l1t of the unknown parameters converge toa 35, b 3, c 12, d 7, andl0.5 asttends to 3.

Example 5.4. Let the scaling function beht sin0.1πtand the scaling factors are chosen asm₁ m₂ m₃ m₄ 1. Furthermore, the control gains are chosen ask₁ k₂ k₃ k4 3, k5 k6 k7 k8 k9 k10 k11 k12 h13 2.Figure 9adisplays the FPS between systems4.1and4.2.Figure 9bshows the estimatesα₁t,β₁t,γ₁t,r₁tof the unknown parameters converge toα10, β28, γ 8/3, andr0.1 ast → ∞.Figure 9c shows the estimates a1t,b1t,c1t, d1t, l1t of the unknown parameters converge to a35, b3, c12, d7, and l0.5 ast → ∞.

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−6

−4

−2 0 2 4 6

0 1 2 3 4 5 6

Error

t ex

ey

ez

ew

a

0 5 10 15 20 25

1 2 3 4 5 6

α1

β1

γ1

r1

t

b

0 10 20 30

0 1 2 3 4 5 6

a1

b1

c1

d1

l1

t

c

Figure 10:aThe behavior of the trajectoriesex,ey,ez, andewof the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for GPS.b the estimatesα1t, β1t,γ1t,r1tof the unknown parameters converges toα10, β28, γ8/3, andr 0.1 asttends to 3.cthe estimates a1t,b1t,c1t,d1t,l1tof the unknown parameters converges toa35, b 3, c12, d7, and l0.5 asttends to 3.

Example 5.5. Let the scaling function beht 1 and the scaling factors are chosen asm₁ m₂m₃m₄0.5. Furthermore, the control gains are chosen ask₁k₂k₃ k₄3, k₅ k6 k7k8k9k10 k11 k12 h13 2.Figure 10adisplays the GPS between systems 4.1 and 4.2. Figure 10b shows the estimates α₁t, β₁t, γ₁t, r₁t of the unknown parameters converge to α 10, β 28, γ 8/3, and r 0.1 as t → ∞. Figure 10c shows the estimatesa1t,b1t,c1t,d1t,l1tof the unknown parameters converges to a35, b3, c12, d7, andl0.5 ast → ∞.

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−4

−2 0 2 4 6 8 10 12

0 1 2 3 4 5 6

Error

t ex

ey

ez

ew

a

−5 0

0 5 10 15 20 25

1 2 3 4 5 6

α1

β1

γ1

r1

t

b

0 10 20 30

0 1 2 3 4 5 6

a1

b1

c1

d1

l1

t

c

Figure 11:aThe behaviour of the trajectoriesex,ey,ez, andewof the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for MGPS.bthe estimatesα1t,β1t,γ1t,r1tof the unknown parameters converge toα10, β28, γ 8/3, andr 0.1 asttends to 3.cthe estimates a1t,b1t,c1t,d1t,l1tof the unknown parameters converge toa35, b3, c12, d7, andl 0.5 asttends to 3.

Example 5.6. Let the scaling function beht 1 and the scaling factors are chosen asm₁ 2, m₂ 3, m₃−2, andm₄0.1. Furthermore, the control gains are chosen ask₁ k₂k₃ k4 3, k5 k6 k7 k8 k9 k10 k11 k12 h13 2.Figure 11adisplays the MGPS between systems4.1and4.2.Figure 11bshow that the estimatesα₁t,β₁t,γ₁t,r₁t of the unknown parameters converge toα 10, β 28, γ 8/3, andr 0.1 ast → ∞.

Figure 11cshow that the estimatesa1t,b1t,c1t,d1t,l1tof the unknown parameters converge toa35, b3, c12, d7, andl0.5 ast → ∞.

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6. Conclusions

This work investigated modified function projective synchronization between the hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters.

Based on Lyapunov stability theory, we design adaptive synchronization controllers˜with corresponding parameter update laws to synchronize the two systems. The MFPS includes complete synchronization, antisynchronization, function projective synchronization FPS, generalized projective synchronizationGPS, and modified generalized projective synchro- nizationMGPS. All the theoretical results are verified by numerical simulations to demonstrate the eﬀectiveness of the proposed synchronization schemes. Thus, our synchronization method is successful for some systems with two positive Lyapunov exponents.

Acknowledgment

The authors would like to thank the editor and the anonymous reviewers for their con- structive comments and suggestions to improve the quality of the paper. The first author ac- knowledges with thanks the Deanship of Scientific ResearchDSR, King Abdulaziz Uni- versity, Jeddah, Saudi Arabia for his support this article.

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