A novel sort of adaptive complex synchronizations of two indistinguishable chaotic complex nonlinear models with uncertain parameters and its applications in secure communications

Accepted Manuscript A novel sort of adaptive complex synchronizations of two indistinguishable chaotic complex nonlinear models with uncertain parameters and its applications in secure communications Emad E. Mahmoud, Fatimah S. Abood PII: DOI: Reference:

S2211-3797(17)30987-7 http://dx.doi.org/10.1016/j.rinp.2017.07.050 RINP 823

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Results in Physics

Received Date: Revised Date: Accepted Date:

6 June 2017 20 July 2017 20 July 2017

Please cite this article as: Mahmoud, E.E., Abood, F.S., A novel sort of adaptive complex synchronizations of two indistinguishable chaotic complex nonlinear models with uncertain parameters and its applications in secure communications, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.07.050

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A novel sort of adaptive complex synchronizations of two indistinguishable chaotic complex nonlinear models with uncertain parameters and its applications in secure communications Emad E. Mahmoud(a),(b)1 , Fatimah S. Abood (a) (b) (c)

(c)

Department of Mathematics, College of Science, Sohag University, Sohag 82524, Egypt.

Department of Mathematics, College of Science, Taif University, Kingdom of Saudi Arabia.

Department of Mathematics, College of Science, King Khalid University, Abha, Kingdom of Saudi Arabia. E-mail: emad [email protected] E-mail: [email protected]

Abstract In this paper, we will demonstrate the adaptive complex anti-lag synchronization (CALS) of two indistinguishable complex chaotic nonlinear systems with the parameters which are uncertain. The significance of CALS is not advised well in the literature yet. The CALS contains or consolidate two sorts of synchronizations (anti-lag synchronization ALS and lag synchronization LS). The state variable of the master system synchronizes with an alternate state variable of the slave system. Depending on the function of Lyapunov, a plan is orchestrated to achieve CALS of chaotic attractors of complex systems with unverifiable parameters. CALS of two indistinguishable complexes of L¨ u systems is viewed as, for example, an occasion for affirming the likelihood of the plan exhibited. In physics, we can see complex chaotic systems in numerous different applications, for example, applied sciences or engineering. With a specific end goal to affirm the proposed synchronization plan viability and demonstrate the hypothetical outcomes, we can compute the numerical simulation. The above outcomes will give the hypothetical establishment to the secure communication applications. CALS of complex chaotic systems in which a state variable of the master system synchronizes with an alternate state variable of the slave system is an encouraging sort of synchronization as it contributes excellent security in secure communication. Amid this secure communication, the synchronization between transmitter and collector is shut and message signals are recouped. The 1

Corresponding author. Tel.: +201221323657; fax: +20934601159.

E-mail address: emad [email protected].

1

encryption and restoration of the signals are simulated numerically. Keywords: Complex anti -lag synchronization; Chaotic system; Synchronization; Uncertain; Error function; Lyapunov function; Complex.

1

Introduction

A dynamical system is called chaotic on the off chance that it is deterministic, as a long-time a periodic conduct, and shows sensitive reliance on the initial conditions. In the event that the system has one positive Lyapunov type, then the system is called chaotic [1]. A chaotic system displays a sensitive reliance on the initial conditions implying that two trajectories emerging out of two diverse close-by beginning conditions isolate exponentially throughout the time. Thus, chaotic system characteristically resists synchronization, in light of the fact that even two indistinguishable systems beginning from marginally extraordinary starting conditions would develop in time in an unsynchronized way. This is an important viable issue since exploratory introductory conditions are never known impeccably [2]. Numerous chaotic (hyperchaotic) complex systems have been proposed and considered over the most recent couple of years by Mahmoud et al. [1, 3-8]. It is notable that the chaotic (hyperchaotic) complex nonlinear systems have significantly more extensive application. For instance, when the amplitudes of electromagnetic fields and nuclear polarization of laser systems are included. Another illustration is the point at which the chaotic (hyperchaotic) complex systems are utilized for communication, where multiplying the quantity of variable might be utilized to expand the substance and security of the transmitted data [1, 9, 10]. The synchronization issue of chaotic (hyperchaotic) systems implies two systems oscillate in a synchronized way. Given a chaotic (hyperchaotic) system, which is considered as the master system, and another indistinguishable (or nonidentical) system, which is considered as the slave system, the dynamical practices of these two systems might be indistinguishable after a transient when the slave system is driven by a control input. The synchronization of chaotic nonlinear systems is one of the basic issues in nonlinear science, for its different potential applications in material science, control hypothesis, secure communication, manufactured neural systems, synthetic reactors, and so forth [11-16]. Since the synchronization phenomena are extremely fascinating and imperative, a ton of work has been dedicated to studying it, for instance, (CS) complete (or full) synchronization [2,17], (GS) generalized synchronization [2,18], phase synchronization [2,19]. Furthermore, there are various sorts of synchronization with a time lag studied, for example, (ALS) anti-lag synchronization, (LS) Lag synchronization, and (MPLS) modified projective lag synchronization of two chaotic or hyperchaotic complex sys2

tems examined [20-23]. Lag synchronization infers that the state variables of the two coupled chaotic systems became synchronized with a time lag involving each other. In engineering applications, time delay dependably exists. For instance, in the phone communication system, the voice one hears on the beneficiary side at time t the voice from the transmitter side at time t − τ . Along these lines, entirely, it is not sensible to require the master system to synchronize the slave system at the very same time. Additionally, a time lag exists as the signal transmitted from the transmitter to the beneficiary end in chaos-communication [20-21]. Additionally, complex synchronization phenomena of chaotic (hyperchaotic) complex systems are imperative, a ton of work has been given to study it, for instance, complex complete synchronization (CCS) [24], complex lag synchronization (CLS) [25], complex projective synchronization (CPS) [26], and modified projective synchronization with complex scaling variables (CMPS) [27-28] are examined for coupled chaotic complex dynamical systems. Complex modified generalized projective synchronization (CMGPS) [29] and complex modified hybrid projective synchronization (CMHPS) [30] were actualized between the same or diverse dimensional partial request complex chaotic (hyperchaotic). Complex generalized synchronization concerning a complex vector map [31-32] was presented for two indistinguishable or nonidentical chaotic (hyperchaotic) complex-variable systems. A few outcomes on chaotic synchronization are determined by utilizing the known (certain) parameters of master and slave systems, and the controller is developed by those known parameters. To the best of our knowledge, a few system parameters cannot precisely be known ahead of time. The synchronization will be devastated with the impacts of these vulnerabilities. Then again, in real physical systems or test circumstances, chaotic systems may have some indeterminate parameters and may change every now and then [33-36]. Accordingly, it is a critical issue to acknowledge chaos synchronization for these chaotic systems with uncertain parameters. Numerous specialists had demonstrated the likelihood to accomplish synchronization between two distinctive chaotic systems with questionable parameters [37-43]. Subsequently, the inspiration of this paper is to study the idea of CALS and talk about the likelihood of accomplishing CALS between two indistinguishable complex chaotic systems with completely obscure parameters which have not been presented as of late in the literature. This kind of synchronization can be analyzed only for complex nonlinear systems. The idea of CALS can be considered as syncretizing between LS and ALS. In CALS the state variable of the master system synchronizes with an alternate state variable of the slave system. Along these lines, the CALS gives more important security in secure communications. So, the aftereffects of the CALS will give the hypothetical establishment to the secure communication applications. The masterminding of this paper is as per the following. A plan to achieve versatile CALS of complex chaotic nonlinear systems with completely questionable parameters is proposed 3

in section2. As an occasion for segment 2, we thought about CALS of two indistinguishable complex chaotic L¨ u systems with completely indeterminate parameters. Numerical simulation is utilized to show the legitimacy of this review. A basic application for the secure communication, in light of the aftereffects of the CALS, is demonstrated in section 3. Finally, the primary conclusions of our examinations are condensed in segment 4.

2

A scheme for design a complex controller to achieve CALS

Consider the chaotic complex nonlinear system as follows: x˙ = Φ(x)A + G(x),

(1)

where x = (x1 , x2 , ..., xk )T is a state complex vector, x = xRe + jxIm , xRe = (w1, w3 , ..., w2k−1 )T , √ xIm = (w2, w4 , ..., w2k )T , j = −1, T signifies transpose, Φ(x) is n × n complex matrix and

the elements of this matrix are state complex variables, A =(a1 , a2 , ..., an )T is n × 1 vector of system parameters, G = (g1 , g2 , ..., gk )T is a vector of direct or nonlinear complex functions,

and coordinating Re and Im planning and symbolize for the genuine and fanciful parts of the state complex vector x. The versatile CALS phenomenon of two indistinguishable systems of the form (1) with a completely obscure parameter by building a control plan, will be talked about in this paper. We tried its rightness numerically. Two indistinguishable chaotic complex nonlinear systems of the form (1) are viewed, one is the master system (we mean the master system with the subscript m ) as: x˙ m = x˙ Re ˙ Im m + jx m = Φ(xm )A + G(xm ),

(2)

and secondly the controlled slave system (with subscript s ) as x˙ s = x˙ Re ˙ Im s + jx s = Φ(xs )A + G(xs ) + L,

(3)

where the additive complex controller L=(L1 , L2 , ..., Ln )T =LRe +jLIm , LRe =(ζ 1 , ζ 3 , ..., ζ 2n−1 )T , LIm =(ζ 2 , ζ 4 , ..., ζ 2n )T . Definition [44] Two indistinguishable complex dynamical systems coupled in a master-slave design can display CALS if there exists a vector of the complex error function e defined as: e = eRe + jeIm = lim kxs (t) + jxm (t − τ )k = 0, t−→∞

4

(4)

where e = (e1 , e2 , ..., en )T , xm and xs are the state complex vectors of the master and slave sys

Im Re Im

tems individually, eRe = lim xRe = lim xIm s (t) + xm (t − τ ) = s (t) − xm (t − τ ) = 0, e t−→∞

t−→∞

0, eRe = (ew1 , ew3 , ..., ew2n−1 )T , eIm = (ew2 , ew4 , ..., ew2n )T and τ is the positive time lag. Remark (1)

At the point when τ = 0 in Eq (4) we characterize CAS between (2) and (3) [45]. Remark (2) In the event that we characterize e = lim kxs (t) − jxm (t − τ )k we got CLS of system (2) t−→∞

and (3) [25]. While on the off chance that τ = 0 we get CCS of similar systems. Theorem 1. On the off chance that a nonlinear controller is composed as:

ˆ − G(x (t)) − j[Φ(xm (t − τ ))A ˆ + G(x (t − τ ))] − ξe, L = LRe + jLIm = −Φ(xs (t))A s m ˆ − GRe (x (t))+ΦIm (x (t − τ ))A ˆ + GIm (x (t − τ ))−ξeRe = −ΦRe (xs (t))A s m m

(5)

ˆ − GIm (x (t))−ΦRe (x (t − τ )) A ˆ −GRe (xm (t − τ ))−ξeIm ], +j[−ΦIm (xs (t))A s m

and the adaptive law of parameter is selected as . Re ˆ ˜ A=[Φ (xs (t)) − ΦIm (xm (t − τ ))]T eRe + [(ΦIm xs (t)) + ΦRe (xm (t − τ ))]T eIm − ΛA,

(6)

then the slave system (3) complex anti-lag synchronization the master system (2) asymptotically, where ξ = diag(ξ, ξ, ....ξ), Λ = diag(k, k, ....k), ξ, k are certain constants, and the parameter of ˆ is the parameter estimation of vector A, A ˜ =A ˆ − A. vector A Verification. From the meaning of CALS:

e = eRe +jeIm = xs (t) + jxm (t − τ ).

(7)

In this way, e˙ = e˙ Re +j e˙ Im = x˙ s (t) + j x˙ m (t − τ ),

e˙ = e˙ Re +j e˙ Im = x˙ Re ˙ Im ˙ Im ˙ Re s (t) − x m (t − τ )+j[x s (t)+x m (t − τ )].

(8)

What’s more, from chaotic complex systems (2) and (3), we get the error complex dynamical system as follows: e˙ = e˙ Re +j e˙ Im = ΦRe (xs (t))A + GRe (xs (t)) − ΦIm (xm (t − τ ))A −GIm (xm (t − τ ))+LRe

+j[ΦIm (xs (t))A + GIm (xs (t))+ΦRe (xm (t − τ ))A +GRe (xm (t − τ )) + LIm ].

5

(9)

Subsequently, substituting from Eq. (5) about LRe , LIm in Eq. (9) we get: e˙ =

Re

e ˙

Im

+j e ˙

=

ΦRe (xs (t))A + GRe (xs (t)) − ΦIm (xm (t − τ ))A − GIm (xm (t − τ ))

(10)

ˆ − GRe (x (t))+ΦIm (x (t − τ ))A ˆ + GIm (x (t − τ ))−ξeRe −ΦRe (xs (t))A s m m +j[ΦIm (xs (t))A + GIm (xs (t))+ΦRe (xm (t − τ ))A + GRe (xm (t − τ ))

ˆ − GIm (x (t))−ΦRe (x (t − τ )) A ˆ −GRe (xm (t − τ ))−ξeIm ], −ΦIm (xs (t))A s m

by alternate way or improve (10), we obtain: ˆ e˙ = e˙ Re +j e˙ Im = ΦRe (xs (t))A − ΦIm (xm (t − τ ))A − ΦRe (xs (t))A

(11)

ˆ − ξeRe +ΦIm (xm (t − τ ))A

ˆ +j[ΦIm (xs (t))A + ΦRe (xm (t − τ )) A − ΦIm (xs (t))A

ˆ −ξeIm ]. −ΦRe (xm (t − τ )) A We can form (11) as:

ˆ + ΦIm (xm (t − τ ))(A ˆ − A)−ξeRe e˙ = e˙ Re +j e˙ Im = ΦRe (xs (t))(A − A)

(12)

Re ˆ ˆ − ξeIm ]. +j[ΦIm (xs (t))(A − A)+Φ (xm (t − τ ))(A − A)

˜ =A ˆ − A, therefore: Since the vector of the parameter errors is known as A Re ˜ + ΦIm (xm (t − τ ))(A)−ξe ˜ e˙ = e˙ Re +j e˙ Im = ΦRe (xs (t))(−A)

(13)

Re ˜ ˜ − ξeIm ]. (xm (t − τ ))(−A) +j[ΦIm (xs (t))(−A)+Φ

By segregating the nonexistent and the genuine segments in Eq. (13), the error complex system is composed in genuine form as: ( Re ˜ + ΦIm (xm (t − τ ))(A)−ξe ˜ e˙ Re = ΦRe (xs (t))(−A) , ˜ + ΦRe (xm (t − τ ))(−A) ˜ − ξeIm , e˙ Im = ΦIm (xs (t))(−A)

(14)

For positive parameters, we may now characterize as a Lyapunov function for this system by the accompanying positive unequivocal amount: ˆ − A)T (A ˆ − A)], + (eIm )T eIm + (A

V (t) =

1 [(eRe )T eRe 2

=

1 [(eRe )T eRe 2

=

1 [(e2w1 + ... + 2 n X 1 e2w2h−1 2 h=1

=

˜ T A], ˜ + (eIm )T eIm + A

e2w2n−1 ) + (e2w2 + ... + e2w2n ) + (˜ a21 + a ˜22 + ... + a ˜2n )] ! n n X X 2 2 + ew2h + a ˜h . h=1

h=1

6

(15)

Note now that the aggregate time subordinate of V (t) along the direction of the error system (14) is as per the following: .

˜ T A, ˜ V˙ (t) = [e˙ Re ]T eRe + [e˙ Im ]T eIm +A Re T Re ˜ + ΦIm (xm (t − τ ))(A)−ξe ˜ = [ΦRe (xs (t))(−A) ] e

.

(16)

Re ˜ ˜ − ξeIm ]T eIm +A ˜ T A. ˆ (xm (t − τ ))(−A) +[ΦIm (xs (t))(−A)+Φ

.

.

.

ˆ in Eq. (16) we acquire: ˜ By substituting from Eq. (6) about A ˆ A. where A= ˜ + ΦIm (xm (t − τ ))(A) ˜ − ξeRe ]T eRe V˙ (t) = [ΦRe (xs (t))(−A)

˜ + ΦRe (xm (t − τ ))(−A) ˜ − ξeIm ]T eIm +[ΦIm (xs (t))(−A)

(17)

˜ T ([ΦRe (xs (t)) − ΦIm (x (t − τ ))]T eRe +A m

˜ T (ΛA), ˜ +[ΦIm (xs (t)) + ΦRe (xm (t − τ ))]T eIm ) − A T

˜ (ΛA), ˜ = −[ξeRe ]T eRe − [ξeIm ]T eIm −A ! n n X X ˜ ˜ T (ΛA). ξe2w2h −A = − ξe2w2h−1 + h=1

h=1

Unmistakably V (t) is a positive distinct function and its subordinate is negative unequivocal, subsequently, as per the Lyapunov hypothesis, the complex error system (9) is asymptotically steady, which implies that ew2h → 0, ew2h−1 → 0 as t → ∞, h = 1, 2, ..., n. In this way, the

CALS between systems (2) and (3) is accomplished. This supplements the confirmation.

At last, by applying it to two indistinguishable complex chaotic L¨ u systems in segment 3, our diagram is expressed.

3

An example

We can see the impact of our plan of two indistinguishable systems of the form (1) we select, as an example, confused complex L¨ u system. The riotous complex L¨ u system is presented in [44] as: x˙ = ρ(y − x), y˙ = υy − xz, 1 z˙ = (¯ xy + x¯ y ) − µz + jw2 w3 , 2

(18)

where x = (x1 , x2 , x3 )T = (x, y, z)T , ρ, υ, µ, positive parameters, and x = w1 + jw2 , y = w3 + jw4 , z = w5 + jw6 , are complex function. Specks allude to subordinates as per time and the importance of overbar is unpredictable conjugate factors. 7

For the case ρ = 21, υ = 10, µ = 6 and the underlying conditions t0 = 0, w1 (0) = 1, w2 (0) = 2, w3 (0) = 3, w4 (0) = 4, w5 (0) = 5 and w6 (0) = 6. We figured the Lyapunov exponents as: l1 = 1.75, l2 = 0, l3 = −1.22, l4 = −3.86, l5 = −23.89, l6 = −24.9 [44].

This infers our structure (18) under this choice of ρ, υ, µ is a chaotic system due to one of the

Lyapunov exponents is positive.

3.1

Formula of the controller

Give us now a chance to explore the CALS of two indistinguishable complex confused L¨ u systems with questionable parameter as an occurrence for area 2. The master and slave systems are in this manner characterized, separately, as takes after:

x˙ m = ρ (ym − xm ) , y˙ m = υym − xm zm ,

(19)

z˙m = 1/2 (¯ xm ym + xm y¯m ) − µzm + jw2m w3m , and x˙ s = ρ (ys − xs ) + L1 , y˙ s = υys − xs zs + L2 ,

(20)

z˙s = 1/2 (¯ xs ys + xs y¯s ) − µzs + jw2s w3s + L3 , where xm = w1m +jw2m , ym = w3m +jw4m , zm = w5 m +jw6m , xs = w1s +jw2s , ys = w3s +jw4s , zs = w5s + jw6s , L1 = ζ 1 + jζ 2 , L2 = ζ 3 + jζ 4 , L3 = ζ 5 + jζ 6 ,are perplexing control functions, individually, which are to be resolved. The complex systems (19) and (20) can be shaped, separately, as:        x˙ m (ym − xm ) 0 0 ρ o         y˙ m  =   υ  +  −xm zm  , (21) 0 y 0 m        z˙m 0 0 −zm 1/2 (¯ xm ym + xm y¯m ) + jw2m w3m µ          x˙ s 0 (ys − xs ) 0 o ρ L1           y˙ s  =      + L2  . (22) 0 ys 0      υ  +  −xs zs    z˙s 0 0 −zs µ 1/2 (¯ xs ys + xs y¯s ) + jw2s w3s L3

So, by comparing the complex systems (21) and (22) with the form of systems (2) and (3) respectively, we find: 

 Φ(xm ) =  

(ym − xm )

0

0

ym

0

0

0





   0   , Φ(xs ) =  −zm 8

(ys − xs )

0

0

ys

0

0

0



 0  , −zs





0   , G(xm ) =   −xm zm  1/2 (¯ xm ym + xm y¯m ) + jw2m w3m     ρ 0         G(xs ) =  −xs zs  , A = υ  µ 1/2 (¯ xs ys + xs y¯s ) + jw2s w3s

According to Theorem 1, the controller is computed as:

ˆ − G(x (t))] − j[Φ(xm (t − τ ))A ˆ + G(x (t − τ ))] − ξe. L = [−Φ(xs (t)) A s m     L1 −ˆρ(ys (t) − xs (t)) − jˆρ(ym (t − τ ) − xm (t − τ )) − ξe1     L2  =   −ˆ υ y (t) + ϕ − j(ˆ υ y (t − τ ) − ϕ ) − ξe s m 2 1 2     L3 −ϕ3 + µ ˆ zs (t) − jw2s w3s − j(ϕ4 − µ ˆ zm (t − τ ) + jw2m w3m )−ξe3

(23)

where ϕ1 = xs (t)zs (t), ϕ2 = xm (t − τ )zm (t − τ ), ϕ3 = 1/2 (¯ xs (t)ys (t) + xs (t)¯ ys (t)) , ϕ4 =

1/2 (¯ xm (t − τ )ym (t − τ ) + xm (t − τ )¯ ym (t − τ )) . So, the controller in (23)   ζ 1 + jζ 2   ζ + jζ  = 3 4   ζ 5 + jζ 6

can be written as:   −ˆρ(w3s (t) − w1s (t) − w4m (t − τ ) + w2m (t − τ )) − ξew1     − ξe −ˆ υ (w (t) − w (t − τ )) + ϕ w3 5 3s 4m   −ϕ7 + µ ˆ (w5s (t) − w6m (t − τ )) + w2m w3m − ξew5   −ˆρ(w4s (t) − w2s (t) + w3m (t − τ ) − w1m (t − τ )) − ξew2    +j  −ˆ υ (w (t) + w (t − τ )) + ϕ − ξe w4 6 4s 3m   −ϕ8 + µ ˆ (w6s (t) + w5m (t − τ )) − w2s w3s − ξew6

(24)

where ϕ5 = w1s (t)w5s (t) −w2s (t)w6s (t) − w1m (t − τ )w6m (t − τ ) − w2m (t − τ )w5m (t − τ ), ϕ6 =

w1s (t)w6s (t) + w2s (t)w5s (t) + w1m (t − τ )w5m (t − τ ) − w2m (t − τ )w6m (t − τ ), ϕ7 = w1s (t)w3s (t) + w2s (t)w4s (t), ϕ8 = w1m (t − τ )w3m (t − τ ) + w2m (t − τ )w4m (t − τ ),

ew1 = w1s (t)−w2m (t − τ ), ew2 = w2s (t)+w1m (t − τ ), ew3 = w3s (t)− w4m (t − τ ), ew4 = w4s (t)+w3m (t − τ )

ew5 = w5s (t)−w6m (t − τ ), ew6 = w6s (t)+ w5m (t − τ ).

Since A =(ρ, υ, µ)T we can calculate the adaptive law of parameter by using Eq. (6) as: .   ew ew + ew4 ew2 − k˜ρ ρˆ . .  3 1     ˆ A = υˆ  =  e2w3 + e2w4 − k˜ (25) υ   . µ ˆ −e2w5 − e2w6 − k˜ µ

The formula of the slave system after adding the controller functions as: x˙ s = (ρ − ρˆ) (ys − xs ) − jˆρ(ym − xm ) − ξe1 , y˙ s = (υ − υˆ )ys + j(ˆ υ ym + xm zm ) − ξe2 ,

z˙s = (ˆ µ − µ)zs + w2m w3m + j(ˆ µzm + 1/2 (¯ xm ym + xm y¯m )) − ξe3 , 9

(26)

where e1 = ew1 + jew2 , e2 = ew3 + jew4 , e3 = ew5 + jew6 .

3.2

Numerical simulation

In order to explain and prove the feasibility of the proposed scheme, we researched the simulation effect of the CALS between two identical chaotic complex L¨ u systems (19) and (20). The systems (19) and (20) are solved in a numerical way, and also the parameters will be chosen as ρ = 21, υ = 10, µ = 6. The initial requirement of the master model, the initial condition of the slave model, and the diagonal constant matrices are essentially considered (xm (0), ym (0), zm (0))T = (1 + 2j, 3 + 4j, 5 + 6j)T , (xs (0), ys (0), zs (0))T = (−1 − 2j, −3 − 4j, −5 − 6j)T and

ξ =diag(10, 10, 10), Λ = diag(12, 12, 12) and τ = 0.2. The initial values of estimate for unknown parameters vector are considered as (ˆρ(0), υˆ (0), µ ˆ (0))T = (4, 5, 6)T . The results are outlined in Figures 1,2,3,4,5. In Figure 1 the solutions of (19) and (20) are plotted subject to different initial conditions and show that CALS is indeed achieved after a very little time t. We can see that each w1m (t − τ ),

w3m (t−τ ), w5m (t−τ ) have the opposite sign of w2s (t), w4s (t), w6s (t) while w2m (t−τ ), w4m (t−τ ),

w6m (t − τ ) have the same sign of w1s (t), w3s (t), w5s (t). This implies that ALS accomplishes be-

tween the real part of the system (19) and the fanciful part of the system (20) while LS happens between the real part of the slave system (20) and the nonexistent part of the master system (19). Figure 1 demonstrates that CALS is accomplished after little time interim and demonstrates CALS has sporadic properties. The CALS errors are plotted in Figure 2. Of course from the above diagnostic contemplations the CALS errors ew2h−1 , ew2h meet to zero as t −→ ∞,

h = 1, 2, 3. In Figure 2 it can be seen that the errors will approach zero after little estimation of t. Figure 3 demonstrating that the assessed estimations of the obscure parameters ρˆ(t), υˆ (t),ˆ µ(t) meet too 22, 10, 6 respectively. Another phenomenon is delineated in Figure 4 and does not show up in a wide range of synchronizations in the writing. The attractors of the master and slave structures in CALS are moving the inverse or comparable shape to each other with various state factors and time slack as found in Figure 4. In Figure 4.a, the attractor of the fundamental structure in (w1m (t − τ ),w3m (t − τ ),w5m (t − τ )) has the inverse type of the slave system in (w2s (t),w4s (t),w6 (t)).

While the attractor of the slave structure in (w1s (t),w3s (t),w5 (t)) has the similar shape of the main system with time lag in (w2m (t − τ ),w4m (t − τ ),w6m (t − τ )) as shown in Figure 4.b.

In the numerical simulations, we figure the module errors and stage errors of master and slave models, individually. For every perplexing number, the module and stage are resolved as takes after: ρx =

p (xRe )2 + (xIm )2 , 10

(27)

+a/

+b/ w1m+tW/___ w2s +t/

14

w2m+tW/___ w1s +t/

10

7 5 0

0

-7

-5 -10

-14 0

3

6

9

12

0

3

6

t +c/

12

+d/ w3m+tW/___ w4s +t/

14

9

t w4m+tW/___ w3s +t/

10

7

5

0

0

-7

-5 -10

-14 0

3

6

9

12

0

3

6

t +e/

12

+f/ w5m+tW/___ w6s +t/

24

9

t

w6m+tW/___ w5s +t/

10

12

5

0 0 -12 -5 -24 0

3

6

9

12

0

3

t

6

9

12

t

Figure 1: CALS between two identical systems (19) and (20) with the controller (24). (a) w1m (t − τ ) and w2s (t). (b) w2m (t − τ ) and w1s (t). (c) w3m (t − τ ) and w4s (t). (d) w4m (t − τ )

and w3s (t). (e) w5m (t − τ ) and w6s (t). (f) w6m (t − τ ) and w5s (t).

11

+b/

0.02

0.02

0.01

0.01

ew2

ew1

+a/

0 -0.01

-0.01

-0.02

-0.02 0

10

20

0

10

t

+c/

+d/

0.02

0.02

0.01

0.01

0 -0.01

0 -0.01

-0.02

-0.02 0

10

20

0

10

t

+f/ 0.02

0.01

0.01

ew6

0.02

0

0

-0.01

-0.01

-0.02

-0.02 0

10

20

t

+e/

ew5

20

t

ew4

ew3

0

20

0

t

10

20

t

Figure 2: CALS errors between systems (19) and (20): (a) w1s (t) − w2m (t − τ ). (b) w2s (t) + w1m (t − τ ). (c) w3s (t) − w4m (t − τ ). (d) w4s (t) + w3m (t − τ ). (e) w5s (t) − w6m (t − τ ). (f) w6s (t) + w5m (t − τ ).

12

+a/

Ò

U

22

21

20

0

10

20

30

20

30

t +b/

Q

Ò

11

10

9

0

10

t +c/

Ò

P

7

6

5

0

10

20

30

t Figure 3: Adaptive parameters estimation laws versus t. 13

-10

w1m +t  W/ +a/ w2s +t/

+b/

0 10

w3m +t  W/ w4s +t/

10

10

0

10 0

-10 0

w5m +t  W/ w6s +t/

5 0

-10

-5

-5

w6m +t  W/ w5s +t/

w2m +t  W/ w1s +t/

0 5 10

5

0

-5

-10 -10

w4m +t  W/ w3s +t/

Figure 4: The attractors of systems (19) and (20).(a) Attractor of the main framework has the opposite shape of the slave framework. (b) Attractor of the main framework has the similar shape of the slave framework. and θx =

 Im Re    arctan(x /x ),

xRe > 0, xIm > 0,

2π + arctan(xIm /xRe ),    π + arctan(xIm /xRe ),

xRe > 0, xIm < 0, x

Re

(28)

< 0.

Figure 5 shows the modules errors and phases errors of the master system (19) and slave systems (20). It is clear from Figure 5a,b,c the modules errors ρxm − ρxs , ρym − ρys , ρzm − ρzs converge to zero as t −→ ∞. While the phases errors θxm − θxs , θym − θys , θzm − θzs go to

t −→ ∞, see Figure 5d,e,f.

3.3

π 2

or − π2 as

Application to secure communication

The chaotic (or hyperchaos) signals generated by chaotic (or hyperchaos) systems have some properties such as randomness, complexity and sensitive dependence on initial conditions, which make them particularly suitable for secure communications. The chaotic (or hyperchaos) synchronous control and its application have turned into the problem section in nonlinear fields, specifically, the applications to secure communication [46-49]. Secure communication is conveying a message from the transmitter to the recipient inside chaotic (or hyperchaotic) systems. As such, the message is immunized or included into chaotic (or hyperchaotic) systems, transmitted, and after that found and recouped by the beneficiary. Different sorts of secure communication plans have been presented like chaotic (or hyperchaotic) covering [46,47,49]. In chaotic (or hyperchaotic) veiling, the message which we have to send it is added to one of chaotic (or 14

+d/ S

0.2

S cccc 2

Txm Txs

Uxm Uxs

+a/ 0.4

0 -0.2

0 S  cccc 2 S

-0.4 0

10

20

0

10

t

+e/

0.4

S

0.2

S cccc 2

Tym Tys

Uym Uys

+b/

0 -0.2

0 S  cccc 2 S

-0.4 0

10

20

0

10

t

20

t

+c/

+f/

0.4

S

0.2

S cccc 2

Tzm Tzs

Uzm Uzs

20

t

0 -0.2

0 S  cccc 2 S

-0.4 0

10

20

0

t

10

20

t

Figure 5: The modules errors and phases errors of systems (19) and (20).

15

hyperchaotic) motion to conceal it, then the signal is transmitted to the beneficiary. Under specific conditions, the message might be recuperated at the collector. CALS of complex chaotic systems in which a state variable of the master system synchronizes with an alternate state variable of the slave system is an encouraging kind of synchronization as it contributes excellent security in secure communication. We consider the system (19) as transmitter system and the system (20) as a beneficiary system. For a certain something, we pick self-assertively the data motion as r(t) = cos 5t + sin 2t. Take rˆ(t) = r(t) + w3m (t − τ ) and

assume that rˆ(t) is added to the variable w2m =⇒ r¯(t) = rˆ(t) + w2m = r(t) + w3m (t − τ ) + w2m (t − τ ).

The numerical after effects of utilizations to secure communication are shown in Figure 6 with similar parameters and beginning states of figure 1 The data signal r(t) and the transmitted signal r¯(t) are shown in Figures 6.a and 6.b, separately. The recuperated information signal, whatever is communicated by r∗ (t) = r¯(t)+w4s (t)−w1s (t), is shown in Figure 6.c (in light of the fact that in the wake of accomplishing CALS w3m (t − τ ) + w4s (t) = 0, w2m (t − τ ) − w1s (t) = 0).

Figure 6.d presentations the blunder motion between the first data signal and the recouped one. From Figure 6, it is anything but difficult to find that the data signal r(t) is recuperated precisely after a short transient.

4

Conclusions

Synchronization of chaotic nonlinear systems with obscure parameters is a vital issue on the grounds that there are numerous chaotic systems having questionable parameters and may change every once in a while. This paper researches the meaning of CALS of chaotic systems in which we can see the complex primary variables take an interest in the dynamics. Consequently, when the real and imaginary parts are isolated anybody acquires a higher dimensional real system. In communications, multiplying the quantity of variables might be utilized to build the substance and security of the transmitted data [1,3]. In this manner, we can see that the exploration of CALS of complex chaotic nonlinear systems with uncertain parameters is so difficult. There are several sorts of synchronization considered as uncommon instances of this definition, which are CCS, CAS, CLS. The CALS can be considered as syncretizing between ALS and LS (see Figure 1). Depending on Lyapunov functions, a plan is intended to accomplish CALS of two indistinguishable complex chaotic nonlinear systems with unverifiable parameters of the form (1). Through this plan, we decided systematically the control complex functions and versatile laws of parameters to accomplish CALS. An illustrative case is given to confirm the rightness of our plan. We connected this plan, for instance, to contemplate CALS 16

+b/ 16

0.5

8

rr +t/

r Sin#2t' Cos#5t'

+a/ 0.9

0

0 -8

-0.5

-16

-0.9 0

5

10

15

0

5

t +c/

15

+d/ 0.9

0.5

0.5

r+t/r+t/

0.9

0

r+t/

10

t

-0.5

0

-0.5

-0.9

-0.9 0

5

10

15

0

t

5

10

15

t

Figure 6: Simulation results of secure communication using CALS of two identical chaotic complex L¨u systems. (a) The original message r(t). (b) The transmitted signal r¯(t). (c) The recovered message

r∗ (t). (d) The error signal r(t) − r∗ (t).

17

of two indistinguishable complex chaotic L¨ u systems with completely obscure parameters. In numerous critical sections of engineering and physics, we can see the presence of these intricate systems. The intricate control functions (24) and versatile laws of parameters (25) are determined systematically to accomplish CALS with indeterminate parameters. All the hypothetical outcomes are confirmed by numerical simulation of our case. These outcomes demonstrate the adequacy and practicality of the proposed plan to study versatile CALS of systems in segment 2. Our discoveries demonstrate that the proposed control plan is especially effective and of wide appropriateness to systems of ODEs with polynomial nonlinearity having attractors which indicate chaotic dynamics. At long last, in light of the state variable of the transmitter system (19) synchronizes with an alternate state variable of the recipient system (20), a basic secure communication venture is planned by means of chaotic covering. The outcomes demonstrate that utilizing the chaotic covering strategy which adds the transmitted signal straightforwardly to a similarly solid chaotic signal to frame the data bearer wave has solid security as well as recoup the data signal successfully as appeared in Figure 6.

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[37] H. Zhang, W. Huang, Z. Wang, T. Chai, Adaptive synchronization between two different chaotic systems with unknown parameters, Phys. Lett. A 350, 363-366 (2009) [38] S. Li, W. Xu, R. Li, synchronization of two different chaotic systems with unknown parameters, Phys. Lett. A 361, 98-102 (2007) [39] X. Chen, J. Lu, Adaptive synchronization of different chaotic systems with unknown parameters, Phys. Lett. A 364, 123-128 (2007) [40] M. Pourmahmood, S. Khanmohammadi, G. Alizadeh, synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller, Commun. Nonlinear Sci. Numer. Simulat. 16, 2853-2868 (2011) [41] J. Huang, Adaptive synchronization between different hyperchaotic systems with fully unknown parameters, Phys. Lett. A 372, 4799-4804 (2008) [42] S. Chen, J. L¨ u, Parameters identification and synchronization of chaotic systems based upon adaptive control, Phys. Lett. A 299 353–358 (2002). [43] G.M. Mahmoud, E.E. Mahmoud, Complete synchronization of chaotic complex nonlinear systems with unknown parameters, Dyn. 62 875-882 (2010) [44] E.E. Mahmoud, F.S. Abood, A new nonlinear chaotic complex model and it’s complex anti-lag synchronization, Complexity (2017), Accepted. [45] E.E. Mahmoud, M.A. AL-Adwani, Complex anti synchronization of two indistinguishable chaotic complex nonlinear models. Journal of Franklin Institute, Submitted for publication. [46] K. M. Abualnaja, E.E. Mahmoud, Analytical and Numerical Study of the Projective Synchronization of the Chaotic Complex Nonlinear Systems with Uncertain Parameters and Its Applications in Secure Communication, Mathematical Problems in Engineering. (2014) Article ID 808375. [47] E.W. Bai, K.E. Lonngren, A. Ucar, Secure communication via multiple parameter modulation in a delayed chaotic system, Chaos Solitons Fractals 23 1071–1076 (2005) [48] G.M. Mahmoud, E.E. Mahmoud, A.A. Arafa, Projective synchronization for coupled partially linear complex-variable systems with known parameters, Math. Meth. Appl. Sci. 40 1214–1222 (2017)

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[49] E.E. Mahmoud, M. A. AL-Adwani, Dynamical behaviors, control and synchronization of a new chaotic model with complex variables and cubic nonlinear terms, Results in Physics, (2017) Accepted. http://dx.doi.org/10.1016/j.rinp.2017.02.039.

22