International Journal of Mathematical AnalysisVol. 8, 2014, no. 53, 2611 - 2617HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijma.2014.49280

Nonlinear Control Method of Chaos Synchronization

for Arbitrary 2D Quadratic Dynamical Systems

in Discrete-Time

Adel Ouannas

Department of Mathematics and Computer ScienceUniversity of Tebessa, Algeria

Copyright c© 2014 Adel Ouannas. 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.

Abstract

In this paper, an arbitrary full state hybrid projective synchroniza-tion (AFSHPS) for general discrete quadratic chaotic systems is in-vestigated. Based on Lyapunov stability theory and nonlinear controlmethod, a new controller is designed for achieving the AFSHPS for 2Dquadratic chaotic systems in discrete-time. Numerical simulation vali-date the main result of this paper.

Keywords: Arbitrary full state hybrid projective synchronization, discrete-time, quadratic systems, chaos, nonlinear control

1 Introduction

Chaos synchronization in discrete-time has been extensively studied, due toits potential applications for secure communication [1, 2, 3]. Different typesand various powerful methods and techniques of chaos synchronization havebeen reported to investigate chaos synchronization in discrete dynamical sys-tems [4, 5, 6, 7, 8, 9].

Recently, a novel type of discrete chaos synchronization, known as arbitraryfull state hybrid projective synchronization, has been introduced and appliedto chaotic systems [10], which includes projective synchronization, hybrid pro-jective synchronization. In AFSHPS, each master system state synchronizes

2612 Adel Ouannas

with a linear combination of slave system states and the scaling constantsare chosen arbitrary. In this paper, an AFSHPS scheme is proposed, for 2Dquadratic chaotic systems in discrete-time, using new synchronization method.To verify the effectiveness of the proposed approach, the new scheme is appliedbetween the drive Lorenz discrete-time system [9] and the controlled responsesystem Henon map.[11].

This paper is organized as follows. In section 2, definition of AFSHPSis introduced. In section 3, description of drive-response chaotic systems ad-dressed in this paper are provided. In section 4, a new synchronization methodis proposed. In section 5, numerical example is used to verify the effectivenessof the proposed scheme. Finally, conclusion is given in section 6.

2 Definition of AFSHPS

We consider the chaotic systems in the following forms

X(k + 1) = F (X(k)), (1)

Y (k + 1) = G(Y (k)) + U, (2)

where X(k) = (x1(k), x2(k), ..., xn(k))T , Y (k) = (y1(k), y2(k), ..., yn(k))T ∈ Rn

are the state vectors, F : Rn −→ Rn, G : Rn −→ Rn and U ∈ Rn is thecontroller to be determined. Those two systems are in arbitrary full-statehybrid projective synchronization (AFSHPS) when, for an initial condition,there is a matrix Λ = (Λij) ∈ Rn×n such that

limk→∞‖e (k)‖ = lim

k→∞‖Y (k)− ΛX (k)‖ = 0, (3)

where e (k) is the erros dynamical systems and ‖.‖ is the euclidean norm.

3 Drive-response systems description

We consider the drive system described by the following quadratic chaoticsystem {

x1 (k + 1) =∑2

j=1 a1jx1 (k) +XT (k)C1X (k) + α1,

x2 (k + 1) =∑2

j=1 a2jx1 (k) +XT (k)C2X (k) + α2,(4)

where X (k) = (x1 (k) , x2 (k))T ∈ R2, is the state vector of drive system,(aij) ∈ R2×2, Ci =

(ciij)∈ R2×2 (i = 1, 2) and (α1, α2, ) ∈ R2 are real numbers.

As the response system, we consider the following quadratic chaotic systemdescribed by

Nonlinear control method of chaos synchronization 2613

{y1 (k + 1) =

∑2j=1 b1jyj (k) + Y T (k)D1Y (k) + β1 + u1,

y2 (k + 1) =∑2

j=1 b2jyj (k) + Y T (k)D2Y (k) + β2 + u2,(5)

where Y (k) = (y1 (k) , y2 (k))T ∈ R2 is the state vector of response system,(bij) ∈ R2×2, Di =

(diij)∈ R2×2 (i = 1, 2) , (β1, β2) ∈ R2 are real numbers and

(u1, u2)T ∈ R2 is the vector controller to be determined.

4 Synchronization method

The AFSHPS errors between the drive system (4) and the response system (5)are defined by{

e1 (k + 1) = y1 (k + 1)− Λ11x1 (k + 1)− Λ12x2 (k + 1) ,e2 (k + 1) = y2 (k + 1)− Λ21x1 (k + 1)− Λ22x2 (k + 1) ,

(6)

where (Λij) ∈ R2×2 are arbitrary scaling constants. The errors dynamics be-tween (4) and (5), can be derived as{

e1 (k + 1) =∑2

j=1 b1jej (k) +R1 + f1 + g1 + u1,

e2 (k + 1) =∑2

j=1 b2jej (k) +R2 + f2 + g1 + u1,(7)

where

{R1 =

∑2j=1 (a1j − Λ11b1j)xj (k) +

∑2j=1 (a1j − Λ11b1j)xj (k) ,

R2 =∑2

j=1 (a2j − Λ21b2j)xj (k) +∑2

j=1 (a2j − Λ22b2j)xj (k) ,(8)

{f1 = Y T (k)D1Y (k)− Λ11X

T (k)C1X (k)− Λ12XT (k)C2X (k) ,

f2 = Y T (k)D2Y (k)− Λ21XT (k)C1X (k)− Λ22X

T (k)C2X (k) ,(9)

and {g1 = −Λ11α1 − Λ12α2 + β1,g2 = −Λ21α1 − Λ22α2 + β2,

(10)

To achieve synchronization between systems (4) and (5), we choose thevector controller as:

{u1 = (b21 − l1) e1 (k)− (b22 + 2b12 − l2) e2 (k)−R1 − f1 − g1,

u2 = (b11 − l1) e1 (k) + (b12 − l2) e2 (k)−R2 − f2 − g1,(11)

where (l1, l2) ∈ R2 are constants to be determined. Substituting Eq. (11) intoEq. (7), one can simply the synchronization errors into:

2614 Adel Ouannas

{e1 (k + 1) = (b11 + b21 − l1) e1 (k)− (b22 + b12 − l2) e2 (k) .e2 (k + 1) = (b11 + b21 − l1) e1 (k) + (b22 + b12 − l2) e2 (k) .

(12)

Theorem 1 If l1 and l2 are chosen such that the following inequalities:

|b2j + b1j − lj| <1√2, (j = 1, 2) , (13)

holds. Then, the two systems (4) and (5), are globally synchronized.

Proof. We take as a candidate Lyapunov function:

V (e (k)) = e21 (k) + e22 (k) , (14)

we get:

∆V (e(k)) = V (e(k + 1))− V (e(k))

=2∑

i=1

e2i (k + 1)−2∑

i=1

e2i (k)

=2∑

j=1

(2 (b2j + b1j − lj)2 − 1

)e2j (k)

+2 [(b11 + b21 − l1) (b22 + b12 − l2)− (b11 + b21 − l1) (b22 + b12 − l2)] e1 (k) e2 (k) .

By using (13), we obtain: 4V (e (k)) < 0. Thus, by Lyapunov stability it isimmediate that limk→∞ ei(k) = 0, (i = 1, 2), and from the fact limk→∞ ‖e(k)‖ =0. We conclude that the systems (4) and (5) are globally synchronized.

5 Numerical example

To demonstrate the use of chaos synchronization criterion proposed herein, anexample of chaotic systems is considered in this section.

The drive system is described by the 2D Lorenz discrete-time system [9].{x1 (k + 1) = (1 + αβ)x1 (k)− βx2 (k)x1 (k) ,

x2 (k + 1) = (1− β)x2 (k) + βx21 (k) ,(15)

where x1(k), x2(k) are the states and a, b are bifurcation parameters of thesystem.The 2D Lorenz discrete-time system is chaotic when α = 1.25 andβ = 0.75. Figure 1 illustrate the chaotic attractor of system

The response system is described by the controlled Henon map [11].{y1 (k + 1) = y2 (k)− ay21 (k) + 1 + u1,

y2 (k + 1) = by1 (k) + u2,(16)

Nonlinear control method of chaos synchronization 2615

Figure 1: Chaotic attractor of Lorenz discrete-time system

where y1(k), y2(k) are the states, a, b are bifurcation parameters of the con-trolled system and u1, u2 are synchronization controllers. The Henon map ischaotic when a = 1.4 and b = 0.3.

Figure 2 illustrate the chaotic attractor of system (16).

Figure 2: Chaotic attractor of Henon map

Corollary 2 For the two coupled Lorenz discrete-time system and Henon map,if (l1, l2) are chosen such that the following inequalities:{

−0.4 < l1 < 1,0.3 < l2 < 1.7,

(17)

holds. Then, they are globally synchronized.

2616 Adel Ouannas

Finally, if we choose (l1, l2) = (−0.06, 0.75), the conditions of Theorem 1are satisfied and by using Matlab we get the numeric result that is showed inFig. 3.

Figure 3: Synchronization errors between Lorenz discrete-time system andHenon map

6 Conclusion

In this paper, to achieve arbitrary full state hybrid synchronization (AFSHPS),a new nonlinear control method was presented for two coupled of arbitrary 2Dquadratic dynamical systems in discrete-time. It was shown that the proposedsynchronization criterion was based on a simple and effective result. Thenumerical example was utilized to illustrate the effectiveness of the proposedmethod.

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Received: September 1, 2014; Published: November 21, 2014