A new memristive model with complex variables and its generalized complex synchronizations with time lag

Results in Physics 15 (2019) 102619

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A new memristive model with complex variables and its generalized complex synchronizations with time lag

T

Emad E. Mahmouda,b, , Ohood A. Althagafic ⁎

a

Department of Mathematics, Faculty of Science, Taif University, Taif 888, Saudi Arabia Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt c Department of Mathematics, Turabah College, Taif University, Turabah, Saudi Arabia b

ARTICLE INFO

ABSTRACT

Keywords: Memristive Synchronization Lyapunov function Complex Scaling function matrix

The point of this paper is to display and explore a memristive complex hyperchaotic model with a flux-controlled memristor. The model’s behavior is examined through numerical simulations by utilizing some tools of nonlinear theory, for example, generalized Hamiltonian, symmetry, dissipative, equilibria and their stability, bifurcation diagram, and Lyapunov exponents. Next, this paper centers around the complex function projective lag synchronization (CFPLS) for various-dimensional hyperchaotic models with complex variables and uncertain parameters. A simulation model displays the effectiveness of the obtained values. The effects of CFPLS were used to develop a precise application in secure correspondence. The digital outcomes were used to verify the adequacy and feasibility of the exhibited technique.

Introduction Chua [1] presented the missing fourth circuit component, the memristor, in 1971, as indicated by the theory of electrical circuits. He considered that component a memristor (or memory resistor). The memristor is recognized as the fourth circuit component, with the other standard ones consisting of the resistor, capacitor, and inductor. The memristor is characterized by a functional relationship between the charge across the device and the flux passing through the device. Furthermore, it is a fundamental circuit component portraying the relationship between the charge q and flux as d (q) = M (q) dq or dq ( ) = W ( ) d , where M (q) , and W ( ) are, memristance, and memductance, respectively [1]. Chua and Kang created a general concept of memristive models in 1976 [2]. In 2008, a group at HP Labs reported the first physical achievement of a memristor and a mathematical model representing its behavior [3]. In 2009, different nanoscale components with memory, the memcapacitor, and meminductor were presented [4]. Later on, in 2011, a group of multidisciplinary analysts from Harvard University presented an intriguing paper on programmable nanowire circuits for utilization in nano processors [5]. From that point forward, numerous potential applications of memristors have been discovered based on their properties. Nonvolatile memory applications [6], as well as memristor-based neural systems [7] and chaotic (or hyperchaotic) models [8,9], are some of them. ⁎

Chaos is an interesting complex nonlinear phenomenon, which has been examined seriously over the most recent four decades inside mathematics, physics, biology communities et cetera [8–11]. In [8], the memristive model can display standard chaotic behaviors with only one positive Lyapunov exponent. Itoh and Chua [12] proposed the primary memristor-based chaotic circuit. Likewise, the multi-wings [13] and multistability [14] of memristive chaos models have been hot research. After the primary hyperchaotic attractor was proposed by Rössler [15], then the hyperchaotic models are described with more than one positive Lyapunov exponent [9]. Memristive models have been presented by summing up the first meaning of a memristor. In general, the memristive form is described as follows:

x = X (x, u , t),

(1)

y = Y (x , u , t) u, such that u , y , and x denote the data, yield, and status of the memristive model, respectively. The function X denotes a continuous vector function with dimension (n ); moreover, Y is a continuous scalar function [2]. The Lorenz conditions initially included the complex variables, and further complex parameters were determined by Fowler et al. [16]. After that, Mahmoud et al. contemplated the essential characteristics, including chaotic synchronization for the Lü and Chen model with a

Corresponding author at: Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt. E-mail address: [email protected] (E.E. Mahmoud).

https://doi.org/10.1016/j.rinp.2019.102619 Received 22 June 2019; Received in revised form 16 July 2019; Accepted 27 August 2019 Available online 10 September 2019 2211-3797/ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

complex field [17]. Complex models have assumed a critical job in numerous areas of science, mathematics, engineering, particularly for hyperchaos-based safe correspondence, in which that complex increase in variables becomes secure for transferred data. A few model parameters cannot be known ahead of time. For instance, the recipient in secure correspondence is unquestionably experiencing a wide range of vulnerabilities, including unverifiable parameters and outer aggravations, which influence the accuracy of the communication. In this manner, it is an important issue to acknowledge chaos control with uncertain parameters [18]. The selection of complex variable hyperchaotic models has been broadly utilized in secure correspondence, and the complex variables (increase the number of elements) lead to incremental protection of the transferred data [18, and (references therein)]. Pecora and Carroll acknowledged the synchronization between dual chaos models [19]. At that point, the synchronization in chaos is vital in the field of nonlinear models, including secure correspondence [20], nonlinear circuits [21], secret signaling, and complex systems [22]. Thus, the synchronization of complex chaos models have attracted substantial consideration over the most recent two decades. Different sorts of synchronization have been proposed, for example, anti-synchronization (AS) [23], complete synchronization (CS) [24], projective synchronization (PS) [25], function-projective synchronization (FPS) [26], complex anti-synchronization (CAS) [27], complex complete synchronization (CCS) [28], complex-projective synchronization (CPS) [29], complex function-projective synchronization (CFPS) [30], lagsynchronization (LS) [31], anti-lag synchronization (ALS) [32], projective lag synchronization (PLS) [33], function projective lag synchronization (FPLS) [34], complex lag synchronization (CLS) [35], complex anti-lag synchronization (CALS) [36], modified projective synchronization (MPS) [37]. PS was first revealed by Mainieri and Rehacek [25]. In the previous years, Li [37] considered another type of projective synchronization strategy, called MPS, wherein each reaction of the dynamic synchronization state potentially synchronizes with a consistent scaling model. At that point, the idea of FPS was presented by a few specialists [26]. This means that responses in synchronized with dynamic phases can be joined to a scaling function. LS suggests that each phase variables in the coupled chaotic models become synchronized by a time lag concerning to each other. In engineering utilization, the time delay dependably exists. For example, within neural systems, also in the telephone-correspondence model, the sound one hears on the receiver side at time t is the sound of the transmitter side at interval t 0 [31]. At present, some synchronization plans in complex nonlinear models have been considered. For example, S. Liu and F. Zhang proposed CFPS of the complex chaotic model [30], Mahmoud et al. introduced many special cases of synchronization in time-delay models with non-diagonal complex scaling functions [38], and X. Tran and H. Kang proposed complex modified function projective lag synchronization of chaos complex models (CMFPLS) with known parameters [39]. Another four-dimensional four-wing memristive chaotic framework is acquired by including a memristor with the constitutive relation of W ( ) to a Lorenz-like model deformed by classical Lorenz model [26]. The scientific model can be depicted as:

x (t )=36y (t ) z (t ) Wz (t ) y (t )= dx (t ) z (t ) + ey (t ), z (t )=8y (t ) x (t ) w (t )=x (t ).

In this work, we generalize the chaotic Lorenz-like model deformed in (2) to the hyperchaotic complex Lorenz deformation model and consider model (2) under the complex design. This model (2) in a complex structure is described as:

x (t )=36y (t ) z (t ) Wz (t ) y (t )= dx (t ) z (t ) + ey (t ),

cx (t ),

z (t )=4[y¯ (t ) x (t ) + y (t ) x¯ (t )] w (t )=x (t ),

(3)

fz (t ),

such that x , y , z , and w are complex variables and z (t ) is a real variable, i.e., (x (t ), y (t ), z (t ), w (t )) = (x r (t ), y r (t ), z (t ), w r (t )) + j (x i (t ) , W r = a + b ((w r (t ))2 (w i (t ))2) , W i = 2b y i (t ), 0, wi (t )) . Also (w r (t ) wi (t )) such that W = W r + jW i is a complex memductance, j= 1 and “overbar” denotes complex conjugation. This paper is intended to develop a complex Lorenz-like model deformed by the classical Lorenz model with a flux-controlled memristor and to acknowledge CFPLS of the same two memristors found on complex Lorenz distorted models. For a four-dimensional (4D) model, the memristor-based real Lorenz deformed model by replacing real variables to complex variables, which can be described as the seventh dimension (7D) real ordinary differential equations (ODEs). Similarly, the symmetry, fixed points, and stability of the model are investigated theoretically. The dynamic forms are numerically explored utilizing the bifurcation diagram, Lyapunov exponent, and Lyapunov dimension. The equilibrium points and hyperchaos behaviors are shown in the model by different parameters and basic conditions. Moreover, CFPLS is intended to synchronize the memristive-based Lorenz deformed models quickly and efficiently, which is verified by mathematical simulations. Then, a method to produce proper masking and more secure encoding is used to increase the nonlinear feedback condition. In Section “Features of dynamic model”, this paper pursues a complex hyperchaotic memristive model, introduced by presenting a smooth flux-controlled memristor to this model and its properties, such as the generalized Hamiltonian, symmetry, invariance, and equilibria. The extent of the parameter estimations by the numerical results from the system (3) at hyperchaos attractors is found to be dependent on the calculations of the Lyapunov exponents. The Lyapunov dimension is computed to hyperchaos trajectories of (3). The bifurcation diagram is used to indicate the chaos and hyperchaos behaviors of (3). In Section “Synchronization of hyperchaotic model (4)”, a procedure is expected to achieve CFPLS and its model. We used the results of CFPLS are in secure communications. In the final section, the conclusions of our studies are outlined. Features of dynamic model (3) In the following section, we present the complex behaviors of the new complex hyperchaotic model, or, in other words, the complex Lorenz deformed model for simplicity. Dividing the real and imaginary elements of the complex Lorenz deformed the model (3), we obtain the real form:

x r (t )=36y r (t ) z (t ) W rz (t ) cx r (t ), x i (t )=36y i (t ) z (t ) W iz (t ) cx i (t ), y r (t )= dx r (t ) z (t ) + ey r (t ), y i (t )= dx i (t ) z (t ) + ey i (t ),

cx (t ), (2)

fz (t ),

z (t )=8[y r (t ) x r (t ) + y i (t ) x i (t )] wr (t )=x r (t ), w i (t )=x i (t ).

The cubic flux-controlled memristor is described as b q ( ) = a + 3 3 , where w = is the magnetic flux and W (w ) = a + b w 2 is the memductance. Both a and b are memristor parameters deciding its qualities; and c , d , e , and f are model parameters that determine the movement states [8].

(4)

fz (t ),

In this paper, we examine the Eqs. (4) and the extent to which complex variables affect dynamic model behaviors. These equations represent one form of equations in the higher dimension. High dimension and complex variables have an important role in secure 2

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

communication and make transferring information safer. The fundamental dynamic properties of the model (4) are described in the accompanying subsection.

Dissipative property

Generalized Hamiltonian of model (4)

x (t ) =

Our study of a continuous nonlinear model (4), which can be portrayed as follows:

such that (t ) = [ 1, 2 , 3, 4 , 5, 6 , 7 ]T , the set of parameters are , and the indicates […]T is the transpose for matrix or vector. So, x (t ) can be formed as:

x = [ (x) +

(x)]

H , x

In vector notation, the 7D model (4) can be considered as follows:

(5)

x=

Let (t ) be any area in R7 and V (t ) denote the volume. By Liouville’s theorem [41], we have

[ (x)]T = 0.

dV (t ) = dt

H x

That vector field (x) shows the conservative section of the model; it is likewise alluded to as the workless section of the model. H Interestingly, (x) x is the working section of the model [40]. We characterize H (x) for model (4) as follows: H (x ) =

1 2

2 x r (t ) 2 x i (t ) + + 8 8

y r (t ) 2 + d

2

y i (t ) d

7

·

(6)

4W r 4W i 8dx r 8dx i 0 0 0

4 0 0 0 0 0 0

4 0 0 0 0 0 0

0 4 0 0 0 0 0

x i (t ) 8 y r (t ) d yi (t ) d z (t ) 8 w r (t )

w r (t ),

f ) V (t ).

(10)

f ) t ] V (0),

0=36y r (t ) z (t ) W rz (t ) cx r (t ), 0=36y i (t ) z (t ) W iz (t ) cx i (t ), 0= dx r (t ) z (t ) + ey r (t ), 0= dx i (t ) z (t ) + ey i (t ), 0=8[y r (t ) x r (t ) + y i (t ) x i (t )] 0=x r (t ), 0=x i (t ).

t

0.

(11)

(12)

fz (t ),

After a few estimations of model (12), we acquire the fixed points:

Generally, the symmetry property exists in many dynamic models with many trajectories. The memristive model (4) keeps the symmetry of the model (4), and it is invariant under the transformation:

z (t ),

(9)

The fixed points of the model (4) are constructed by obtaining the accompanying nonlinear equations:

.

Symmetry and invariant

(t ),

f.

Fixed points and their stability

The result of the Hamiltonian in the model (4) demonstrates its ability to measure physical quantities; furthermore, it indicates that the model (4) is dynamic and has complex behavior.

( x r (t ),

2c + 2e

From (11), we find that V (t ) contracts to zero exponentially when t . Consequently, the 7D complex hyperchaotic Lorenz deformed model (4) is dissipative and the asymptotic movement of the 7D complex hyperchaotic Lorenz deformed model (4) settles exponentially to a strange attractor of measure zero.

w i (t )

(x r (t ), x i (t ), y r (t ), y i (t ), z (t ), w r (t ), wi (t ))

=

V (t ) = exp[( 2c + 2e

yi (t ) d z (t ) 8 w r (t )

x r (t ) 8

4W r 4W i 0 0 8f 0 0

(8)

Integrating the first ODE (10), we get the single answer:

w i (t )

8c 0 18dz 0 0 8c 0 18dz 18dz 0 de 0 + 0 18dz 0 de 4W r 4W i 0 0 4 0 0 0 0 4 0 0

l

xl

dV (t ) = ( 2c + 2e dt

x i (t ) 8 y r (t ) d

0 4 0 0 0 0 0

( · ) dx rdx idy r dy idzdw r dwi.

Substituting (9) into (8), we obtain

x r (t ) 8

0 18dz 0 0 0 18dz 0 0 0 18dz 0 0 4W i 8dx r 8dx i 0 0 0 4 0 0

= l =1

In this way, model (4) can be acquired based on Eq. (5) as

x r (t ) 0 x i (t ) 0 y r (t ) 18dz y i (t ) = 0 z (t ) 4W r 4 w r (t ) 0 wi (t )

(t )

The divergence of the 7D model (4) is effectively acquired as

z (t ) 2 + (w r (t ))2 + (w i (t ))2 . 8

+

36y r (t ) z (t ) W rz (t ) cx r (t ) 36y i (t ) z (t ) W iz (t ) cx i (t ) 2 (t ) dx r (t ) z (t ) + ey r (t ) 3 (t ) . dx i (t ) z (t ) + ey i (t ) 4 (t ) = r r 8[y (t ) x (t ) + y i (t ) x i (t )] fz (t ) 5 (t ) x r (t ) 6 (t ) 7 (t ) x i (t ) 1 (t )

such that x(t ) = [x r , x i , y r , y i , z, w r , wi]T . In addition, to indicate a smooth energy function, H (x) is globally positive definite in Rn . Furthermore, the column derivative vector of H indicated by H is expected x to exist over the entire domain. We utilize the quadratic energy func1 tion H (x) = 2 xT x with as a constant symmetric positive definite matrix. The square matrices, (x) (anti-symmetric matrix) and (x) (symmetric matrix), fulfill the accompanying properties, which portray the overall energy structure of the previously mentioned model:

[ (x)] + [ (x)]T = 0, [ (x)]

(7)

(x (t ); ),

E0=(0, 0, 0, 0, 0, 0, 0) E1=(0, 0, 0, 0, 0, w r , wi ).

x i (t ), y r (t ), y i

For a helpful depiction, we contemplate the stability of E0 and E1. Interestingly, the equilibrium points have the equivalent Jacobian matrix and the characteristics of polynomials. At that point, we can ascertain the Jacobian matrix as follows:

wi (t ))

Also, attractors in state space must be symmetric, as for the y r (t ) , axis.

y i (t )

3

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

0 0 0 0 1 0

JE0 =

0

0 c 0 0 e 0 0 0 0 0 0 1 0

c

0 0 0 e 0 0 0

0 0 0 0 0 0 0

a 0 0 0 f 0 0

0 0 0 0 . 0 0 0

The characteristic polynomials relating to the equilibria E0 and E1 utilizing the Jacobian JE0 are composed as

( + c )2 ( + f )(e

(13)

) 2 ( ) 2 = 0.

The eigenvalues of Eq. (13) are 1

=

2

=

c,

=

3

f,

4

=

5

= e,

6

=

7

(14)

= 0.

The trivial fixed points E0 and E1 are stable if and only if c > 0 , f > 0 , e < 0 . Otherwise, they are unstable. Lyapunov exponents The Lyapunov exponent is an essential segment of hyperchaotic dynamics, and likewise implies the rate of assembly and divergence of adjacent directions in the stage space of the model. It is noticed that for a 7D model to be hyperchaotic, there ought to be at least two positive Lyapunov exponents. The equations of the small deviations x(t ) from the path x(t ) are as follows:

x (t ) = Jlk (x (t ); ) x such that Jlk (x (t ); ) = shape:

c 0 dz 0 8y r 1 0

Jlk =

(15)

l, k = 1, 2, 3, 4, 5, 6, 7, l

xk

refers to the matrix of Jacobian that takes the

36z 0 36y r W r c 0 36z 36y i W i 0 e 0 dx r dz 0 e dx i 8y i 8x r 8x i f 0 0 0 0 1 0 0 0 0

2bw rz 2bw iz 0 0 0 0 0

Fig. 1. For b = 3, c = 15, d = 8, e = 3, f = 20 and vary a with the initial conditions (x r (0), x i (0), y r (0), y i (0), z (0), w r (0), w i (0)) = (1, 2, 3, 4, 5, 6, 7) . (a) Lyapunov exponents of model (4): L1, L 2 , L3 , L4 , L5 , L6 and L7 , (b) Bifurcation diagrams in (a, x r ) , (c) Bifurcation diagrams in (a, y i ) .

2bwiz 2bw r z 0 . 0 0 0 0

Assume that the Lyapunov exponents of model (4) are Lk for k = 1, 2, 3, 4, 5, 6, 7 satisfying L1 > … > L7 . At that point, the dynamic practices of the model (4) can be classified as follows:

Therefore, the Lyapunov exponents Lk of the model are described as

Lk = lim t

1 log t

x k (t ) , x k (0)

k = 1, 2, 3, 4, 5, 6, 7,

7

L < 0, i) For 0 < L 2 < L1, L4 > L5 > L6 > L 7 , L3 = 0 , and k=1 k model (4) is hyperchaos. ii) For L7 < L6 < L5 < L4 < L3 < L2 < L1, model (4) is an equilibrium point.

(16)

such that the differential of x(t ) is x(t ) , and the distance is equivalent to zero, as touted by x(0) [42]. To obtain Lk models, (7) and (15) must be numerically decided at the same time. The fourth-order Runge–Kutta technique is used to calculate Lk . For the new model (4), we set the parameter values and initial conditions as a = 10, b = 3, c = 15, d = 8, e = 3, f = 20 and (x r , x i , y r , y i , z , wr , wi ) = (1, 2, 3, 4, 5, 6, 7) and t = 0 ; then, the related Lyapunov exponents appear in Figs. 1 and 2. The numerical estimations of the Lyapunov exponents are L1 = 6.73, L2 = 3.80 , L3 = 0 , L4 = 1.22, L5 = 7.96 , , L7 = 9.30 . This implies that our model (4) under the specified option of (a, b , c, d, e , f )T is a hyperchaotic system because two of the Lyapunov exponents are greater than zero. The dimension of the Lyapunov of the attractors from model (4), per the Kaplan–Yorke measurement, is resolved as

In the two sections below, some properties of the new 7D model are additionally examined with varying a and e . Fix b, c , d , e , f and vary a [1, 50]. For the new hyperchaotic model, Fig. 1(a) shows the seven Lyapunov exponents of the model (4) for parameter a . The comparative bifurcation diagram is shown in Fig. 1(b and c). Fig. 1(b and c) contains (a, x r ) and (a, y i ) and demonstrates that the dynamic practices of model (4) can be unmistakably watched. Whenever a [1, 50], the first and the second largest Lyapunov exponents are positive, inferring that model (4) is hyperchaotic. With expansion, the model (4) can develop into hyperchaos. The Lyapunov exponent range and bifurcation diagram show that model (4) has complex behaviors as a shifts. Fig. 1 demonstrates that when a varies in a broad range, the model is still hyperchaotic. Some unique attractors are quickly delineated in Fig. 2.

Lk D=

+

k=1

|L

such that

+1 |

,

is the largest whole number, where

(17) k=1

Fix a, b, c, d , f and vary e [ 20, 8.5]. At the point when e shifts, the seven Lyapunov exponents of the model (4) are shown in Fig. 3(a) with the similar initial condition as that in Fig. 1. At the point when e [ 20, 0], the maximum Lyapunov exponent quickly becomes negative and the bifurcation of x r and y i changes into a line (see Fig. 3(b and c)). Both indicate that the

Lk > 0 and

Lk < 0 [43]. The Lyapunov dimension of this hyperchaotic trajectory using (17) is D = 11.6311. This indicates that if the Lyapunov dimension is high, the geometric structure of model (4) is more difficult. +1 k=1

4

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

Fig. 2. Solutions of model (4) for b = 3, c = 15, d = 8, e = 3, f = 20 and vary a with the same initial conditions in Fig. 1. (a) Hyperchaotic trajectories, a = 10 in (z, y i , x i ) , (b) Hyperchaotic trajectories, a = 30 in (x r , z , y r ) , (c) Hyperchaotic trajectories, a = 40 in (z, y r , x r ) , (d) Hyperchaotic trajectories, a = 50 in (y r , x r , z ) .

trajectory of model (4) reaches equilibrium (shown in Fig. 3). At the point when e (0, 8.5], there appears a hyperchaotic attractor, and the hyperchaotic trajectory advances as e expands. Fig. 3 demonstrates that model (4) is hyperchaotic for an extensive variety of e , and the model can likewise advance into fixed trajectories and hyperchaotic trajectories. Fig. 4(a) and (b) demonstrate that when e varies in a wide range, the model is still in equilibrium. For e = 5 and e = 7 , model (4) is hyperchaotic and the related stage pictures appear in Fig. 4c and d, respectively. When comparing the results of the complex model (3) and the real model (2), we conclude the effect of complex variables on the dynamic.

where x = xr + j x i = [x1, x2, …, xn]T Cn is a complex state vector, (x) Cn × m is a complex matrix and the elements of this matrix are Rn is a real vector of the parameters state complex variables, R is a real model, G (x, z) is a linear or nonlinear part and Z: Cn × R nonlinear function, z R . Then, the master and the salve models take the form:

Synchronization of hyperchaotic model (4)

{x s = x rs + j x is =

Proposition to achieve CFPLS

= where a complex control vector is i i T r r T i = [ i, r = [ r, 1 2, …, n] . 1 2, …, n] ,

We study the hyperchaos memristive complex nonlinear model in the form:

Definition. With regard to two identical systems by a complex dynamic, we define the complicated error between slave and master systems to produce CFPLS as

x = (x) + G (x, z), z = Z (x , x¯ , z),

xm = x rm + j x im = .

(18)

(xm) + G (xm, z), (19)

z = Z (xm, x¯ m, z),

5

(x s) + G (xs, z) +

(20)

, r

+j

i

= [ 1,

T 2, …, n] ,

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

e = er + jei = limt

x s (t )

Ax m (t

such x rs (t )

0)

(21)

,

+ = [e x1r , e x2r , …, e xnr ]T = that e = [e x1, e x2, …, e xn] = r i i r i e = [e x1i, e x2i , …, e xni]T = xis (t ) A x m (t 0 ) + Ax m (t 0) , i r r i A x m (t Ax m (t 0) 0 ) , x m (t ) and x s (t ) are the state complex vectors of the master and slave models, respectively, the complex scaling function matrix is known A , such that the A = Ar + j Ai = diagonal ( (t ), (t ), …, (t )) , (t ) is a bounded complex function and the time lag definite as 0 T

j ei ,

er

er

Remark 1. The matrix A is known as the complex scaling matrix. Some of the synchronizations are extraordinary instances of our definition in Table 1. Consequently, this new kind of complex synchronization is in consideration of CAS [27], CCS [28], CPS [29], CFPS [30] of complex hyperchaotic models when 0 = 0 , which have appeared up in recent literature, as shown in Table 1. Remark 2. Additionally, the CFPLS of time lag complex models is a generalization of most sorts of synchronization of dynamic models, where the change matrix is complex and 0 0 , e.g., CLS [35], CALS [36], and so forth, as shown in Table 2. Theorem 1. For the provided continuous differential scaling matrix function, (A), the CFPLS between models (19) and (20) will obtain and the uncertain parameters, and assessed evaluated if the controller and the parameter update laws are produced as follows:

=

r

+j

i

=

[ (xs (t ))

+ G (x m (t r

= [ +

0 ),

m (t

Ai [ i (x

0)

0 ),

m (t

r

(x m (t

0 ))

Gi (x

+

m (t

0 ),

z)] +

A r x rm (t

0)

0)

er + j { [ i (x s (t )) + Gi (xs (t ), z)] + Ar [ i (xm (t i r (x (t + Gi (x m (t + G r (x m (t 0 ), z)] + A [ m 0 )) + A r xim (t

Fig. 3. For a = 10 , b = 3, c = 15, d = 8, f = 20 and vary e with the initial conditions in Fig.1. (a) Lyapunov exponents of model (4): L1, L 2 , L3 , L4 , L5 , L6 and L7 , (b) Bifurcation diagrams in (e, x r ) , (c) Bifurcation diagrams in (e, y i ) .

z)]

0 ))

Aixim (t

0 ))

e,

+ Gr (xs (t ), z)] + Ar [

(x s (t ))

G r (x

+ G (xs (t ), z)] + A [ (x m (t

z)] + Axm (t

0)

+ A ix rm (t

0 )) 0 ),

e = er + jei = [

z)]

+

ei},

0)

Ar [

and r

Ar [

(x s (t ))]

+ ([ i (x s (t ))]

r

(x m ( t

Ar [

i (x

0 ))]

m (t

+ Ai [

0 ))]

i (x

Ai [

r

m (t

(x m (t

+

T r 0 ))]) (e ) T ( e i) ))]) 0

(23)

Axm (t

e=er + jei = [x rs (t )

Ar x rm (t

0)

i i (t + Ax m

+ + j [x is (t )

A ixim (t

+ Ar [ +

0)

0)

A r xim (t

0)

Ax m (t

0)

+

0 ))

+ Gi (x m (t

z)]

A r x im (t

+

r

0 ),

(x m (t

G i (x

r ,+ j {[ i (x

0)

+

i (x

Aix im (t

0)

[

r

(xs (t ))

0)

Ar [

0 ), 0) m (t

z)] +

z)] Ai [ 0 ))

m (t

0 ),

r (x

Aix rm (t

m (t

0)

[

z)] +

i (x

and

+

Gr (x

m (t

s (t ))

(25)

i

in (25), we

0 )) 0 ), s (t ),

i (x

z)]

z)]

m (t

0 ))

er ,

0) 0 ))

0 ),

z)]

+ Gi (xs (t ), z)]

0 ),

z)] + Ai [

r

+

Aix rm (t

0)

0)

0 )) i},

+

+ Gr (x m (t

0 ))

z)]

(x m (t

(x m (t

A ixim (t

i (x

+ G i (x m ( t A r x im (t

r

r

r

0)

i 0 ), z)] A [

i s (t )) + G (x s (t ), z)]

0 ),

Ar [

0 ),

i 0 )) + G (x m (t

m (t

+ G r (x m (t

0 ))

m (t

Aix rm (t

(x s (t )) + Gr (xs (t ), z)]

0 ))

i s (t )) + G (x s (t ), z)]

Ai [

z)]

0)

Ar xrm (t

i (x

Gr (x

e = er + jei = [ [

r

+j {[ + G ( x m (t

0 ))

Ar [

(x m (t

0 ))

(26)

ei},

By exchange or enhancement of (26), we obtain:

e = er + jei 0 ))

m (t

0) +

i 0 ), z)] +A [

m (t

i (x

+ Ar [

0)

From the Eqs. (19) and (20) we find the error derivative as

A [ (x m (t

r

(x m (t

A r x im (t

0 )],

= [ (xs (t )) + G (xs (t ), z)]

0 ),

+ G i (x m ( t

+

Ar x im (t

m (t

r

G i (x

+j {[

0 )]

i r (t Ax m

Aix rm (t

A r x rm (t

0)

m (t

A r x rm (t

(24)

0 ),

i (x

Gr (x

+ Gr (x m (t

Proof. The differentiation for the time of complex error dynamic model is 0)

0 ),

z)]+Ai [ i (x

i i 0 ) + A x m (t

e = er + jei = [

where = diag ( , , …, ) is the control strength matrix, and every one of its segments is composed of positive real numbers. Furthermore, represents the estimated vectors of real uncertain parameters ; = and = diag ( , , …, ).

Axm (t

m (t

(x s (t )) + Gr (xs (t ), z)]

Consequently, substituting from Eq. (22) for find:

.

e = e r + j e i = x s (t )

r

A r x rm (t

(22)

=([

Gr (x

0 ),

z)]

[

r

(xs (t )) ] + i (x i (x

s (t ))

s (t ))

Ar [

(x s (t )) ] ]

]+

Ar [ r (x Ar [

m (t

i (x

Ar [ i (x

m (t

m (t

r

] + Ai [

(x m (t

0 ))

0 ))

Ai [ i (x

0 )) 0 ))

]

m (t

i (x

m (t

0 ))

0 ))

]

]

Ai [

r

( x m (t

0 ))

]

]+

Ai [ r

(x m (t

0 ))

]

]

er , ei}. (27)

, We can form (27) as

Isolating Eq. (24) into real and imaginary parts, we obtain: 6

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

Fig. 4. Solutions of model (4) for a = 10 , b = 3, c = 15, d = 8, f = 20 and vary e with the same initial conditions in Fig.1. (a) Fixed trajectory, e = (b) Fixed trajectory, e = 3 in (z, x r , y r ) , (c) Hyperchaotic trajectories, e = 5 in (y r , x r , z ) , (d) Hyperchaotic trajectories, e = 7 in (y r , x r , z ) . Table 1 Special cases of (CFPLS) when

0

Setting the Matrix Type of A

A = diagonal (1, 1, …, 1) . A = diagonal ( 1, 1, …, 1) . A = diagonal ( , , …, ) Rn × n where is a constant. A = diagonal ( (t ), (t ), …, (t )) Rn × n where (t ) is a function. A = diagonal (j, j, …, j ) Cn × n . A = diagonal ( j, j, …, j) Cn× n .

A = diagonal ( , A = diagonal ( (t ),

Cn × n where is a constant. (t ), …, (t )) C n× n where (t ) is a function.

, …, )

Table 2 Special cases of (CFPLS) when

0

e = er + jei = [

= 0.

+Ai [ Synchronization types

+ Ar [

A = diagonal (1, 1, …, 1) . A = diagonal ( 1, 1, …, 1) . A = diagonal ( , , …, ) Rn × n where is a constant. A = diagonal ( (t ), (t ), …, (t )) Rn × n where (t ) is a function. A = diagonal (j, j, …, j ) Cn × n . A = diagonal ( j, j, …, j) Cn× n .

m (t

i (x

)] + Ar [

(x s (t ))( 0 ))(

m (t

)] + Ai [

+

(x m (t

er , + j {[

)]

0 ))(

r

r

i (x

0 ))(

+

s (t ))(

(x m (t

)]

)]

0 ))(

+

)] (28)

ei}.

CS [24] AS [23] PS [25]

For the parameter vector, errors are acknowledged as accordingly,

FPS [26]

;

=

e = er + jei

CCS [28]

CAS [27]

r

=[

CPS [29]

(xs (t ))( )] + Ar [

er ,

CFPS [30]

+ Ai [

r

i (x

+ j {[ (x m (t

r

s (t ))(

0 ))(

(x m (t )] +

0 ))(

Ar [ i (x

)]

m (t

Ai [

i (x

0 ))(

m (t

0 ))(

(29)

ei}.

)]

)]

)]

By segregating the real and the imaginary sections in Eq. (29), the differentiation for the time of complex error model is composed in real form as

0.

Setting the Matrix Type of A

i (x

r

20 in (x r , y i , y r ) ,

Synchronization types

er =

[

r

(x s (t ))( )] + Ar [

er ,ei

LS [31] ALS [32] PLS [33]

+ Ai [

FPLS [34]

r

=

(x m (t

[

i (x

r (x

s (t ))(

0 ))(

)

m (t

)] +

0 ))(

Ar [ i (x

)] m (t

Ai [ i (xm (t 0 ))(

0 ))(

)]

)] (30)

e i].

Choose the following Lyapunov function:

CLS [35]

L f (t ) =

CALS [36]

7

1 [(er )T (er ) + (ei)T (ei) + ( 2

)T (

)],

(31)

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

(19) and (20), separately, we find:

1

=2 [(er )T (er ) + (ei)T (ei) + ( )T ( )], 1 =2 [[(e x1r )2 + (e x2r ) 2 +…+ (e xnr )2] + + [( 1 ) 2 + ( 2 ) 2 +…+ ( n )2]] n n n 1 =2 [ (e xlr ) 2 + e xli 2 + l=1 l=1 l=1

i )2

[(e x1

( )

(e x2i ) 2+…+ (e xni )2]

+

( l )2]

r

(x s (t )) + Gr (xs (t ), z)]

i (x

m (t

0 ),

r (x

m (t

0 ))

Ar x rm (t

z)]

+ G r ( x m (t

0 ),

+ (ei)T ([ Gi (x

Ai [

i (x

s (t ))

m (t

r (x

0 ),

m (t

+ Gi (x s (t ), z)]

Ar [

i (x

m (t

+ A ix im (t

0)

0)

A ix rm (t

0)

0 ))

Ar x im (t

+ G r ( x m (t

0 ),

Ar [

r (x

m (t

0 ))]

+ Ai [

i (x

m (t

T r 0 ))]) (e )

Ar [

i (x

m (t

0 ))]

Ai [

r (x

m (t

T i 0 ))]) (e )

z)]

+ ( )T ([ (

r

(xs (t ))]

i (x

)T

s (t ))]

(ei)T (ei)

( )T ( ),

1=[a z s (t )

xm (t )=36ym (t ) z m (t ) Wm z m (t ) ym (t )= dxm (t ) z m (t ) + eym (t ),

1,

xs 0 0

0 0 xs z s ys 0 0

+ 36ym (t

0)

0 ) z m (t

0 )]

0 ) z m (t

0)

0 )]

0 )]

ey ,

xs (t ) + (3 + jsin(t ))[xm (t

+ b z s (t )((wsr (t ))2

(33)

0 )]

+ j cos(t )[wm (t

a 36ym zm b c + , 0 xm d e

a 36ys z s b c + + 0 xs d e

0 )]

e w.

r 0 )((wm (t

0

36ysr (t ) z s (t )]

))2

(wmi (t

0 ))

2)

0) 0 ) z m (t

0 )]

r 0 )(wm (t

+ sin(t )[2b z m (t

i 0 ) wm (t

0 ))

+ c x mi (t ex

36ymi (t

0)

36ymi (t

0 ) z m (t

+ sin(t )[ az m (t +

(34)

c xmr (t 36ymr (t

+

sin(t )[d xmi (t ey

3

cos(t )[xmi (t

0 )]

36ysi (t ) z s (t )]

i 0 ) wm (t

c xmi (t

0 ))

0)

0 )]

0)

r 0 )((wm (t

b z m (t

0 ))

2

(wmi (t

0 ))

2)

0 )]

+ cos(t )[xmr (t

e ysr (t )] + 3[ d xmr (t 0 ) z m (t

0)

0 )]

0 ) z m (t

e ymi (t

0 )]

e xi, r 0 ) + e ym (t

0 )]

cos(t )[ymi (t

0 )]

r,

i i 2=[d xs (t ) z s (t )

(35)

0 )]

0) 0 ) z m (t

r r 2=[d xs (t ) z s (t )

,

+ c xsi (t )

r 0 )(wm (t

+ 3[ 2b zm (t +

0 ) z m (t

r,

i r i 1=[2b z s (t )(ws (t ) ws (t ))

1 2

(wsi (t ))2) + c xsr (t )

b z m (t

0)

c xmr (t + 36ymr (t

z s ws2 0 0

0)

c x m (t ex ,

0)

e ys (t )] + (3 + jsin(t ))[ d x m (t

+ 3[ a zm (t

where Wm = Wmr + jWmi , Ws = Wsr + jWsi such that Wm = a + b (wm ) 2 and Ws = a + b (ws )2 are memductances, x m = xmr + jxmi , ym = ymr + jymi , wm = wmr + jwmi , xs = xsr + jxsi , ys = ysr + jysi , ws = wsr + jwsi are complex variables, z m (t ) and z s (t ) are a real variables. Then, 1 = 1r + j 1i , r i r i 2 = 3 + j 3 are complex functions of control. 2 + j 2, 3 = Complex systems (32) and (33) can be molded, independently, according to the following:

zs 0 0

36ys (t ) z s (t )]

In this way, the controller in (36) can be composed as:

fzm (t ),

0 0 xm z m ym 0 0

0 ))

z)]

e,

+ jcos(t )[ym (t 3=

3,

xm 0 0

0 0 0 . 0 j sin(t ))

(36)

r 1 =[az s (t )

zm wm2 0 0

0 ),

+ b z s (t ) ws2 (t ) + c xs (t )

+ e ym (t

wm (t )=xm (t ),

xs ys = ws

0)

2=[d x s (t ) z s (t )

(32)

xs (t )=36ys (t ) z s (t ) Ws z s (t ) cxs (t ) + ys (t )= dxs (t ) z s (t ) + eys (t ) + 2,

a b = c , d e

a b = c , d e

+ G (x s, z)] + A [ (x m (t

+ (3 + jsin(t ))[ az m (t

cxm (t ),

z m (t )=4[y¯m (t ) xm (t ) + ym (t ) x¯m (t )]

[ (x s (t ))

2 b zm (t 0 ) wm (t + j cos(t )[xm (t 0 )]

In the present subsection, we can show the likelihood and adequacy of the advanced synchronization outlined in this section. We examine the CFPLS of an identical hyperchaos complex memristive system (4). The master and slave models are hence decided, separately, as follows:

zm 0 0

=

+ Axm (t

CFPLS between two identical hyperchaotic complex memristive models with uncertain parameters

xm ym = wm

i

+j

+ G (x m (t

Along these lines, according to the Lyapunov stability theorem, since Lf (t ) is positive definite and Lf (t ) is negative definite. So, the CFPLS between models (19) and (20) is achieve. □

ws (t ) = xs (t ) +

r

=

( ),

= (er )T (er )

0 0 xm z m ym , 0 0

According to Theorem 1, the controller is calculated as:

i)

+ + ( )T ([

xm 0 0

(3 + j sin(t )) 0 0 0 0 (3 + j sin(t )) 0 0 A= 0 0 (3 + jsin(t )) 0 0 0 0 (3 + jsin(t )) 0 0 0 0 (3 +

z)]

0 ))

z m wm2 0 0

z)]

0)

r)

+ +

Ar [

+ Gi (x m (t

0 ))

0 0 xs z s ys , 0 0

36ys z s 36ym z m G (xs , z)= 0 , G (x m, z) = , 0 xs xm

Lf (t )=(er )T (er ) + (ei)T (e i) + ( )T ( ), + Ai [

xs 0 0

zm 0 0

(xm) =

where = and with the controller (22) and the parameter update laws (23), the time derivation of Lf (t ) along the trajectory of error model (25) is =(er )T ([

z s ws2 0 0

zs 0 0

(x s ) =

e ysi (t )] + 3[ d xmi (t

0 ) z m (t

0)

+ e ymi (t

0 )]

(37)

Thus, by comparing models (34) and (35) with the form of models 8

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

Fig. 5. Time evolutions of CFPLS errors between models (32) and (33). (a) (e x r , t ) diagram, (b) (e x i, t ) diagram, (c) (e yr , t ) diagram, (d) (e yi , t ) diagram, (e) (e wr , t ) diagram, (f) (e wi, t ) diagram.

+ sin(t )[ d x mr (t

0 ) z m (t

0)

+ e ymr (t

+ cos(t )[ymr (t

0 )]

+ { xsi (t ) + 3[xmi (t

0 )]

d ={ xsr (t ) z s (t ) + 3[xmr (t

e yi, r 3=

[xsr (t )] ew

i 3=

+

3[x mr (t

sin(t )[x mi (t

0 )]

cos(t )[wmi (t

0 )]

+

0 )]

+

sin(t )[xmr (t

0 )]

+

cos(t )[wmr (t

0 )]

+

e w i. Because lizing (23):

0 ))

0 )]} e x

r

(wsi (t )) 2)

+ {sin(t )[zm (t + 3[z m (t

0 )]} e x i

r 0 )((wm (t

a, 0

e x r =xsr (t )

))2

sin(t )[x mi (t

0 )]

0 ) z m (t

c, 0 ) z m (t

0 ) z m (t

3[ymr (t 3[ymi (t

0 )]

+

0 )]

0 )]

0 )]} e yi

d,

sin(t )[ymi (t + sin(t )[

0 )]} e y r

ymr (t

0 )]} e yi

r 0 )(wm (t

i 0 ) wm (t

+ { 2z s (t )(wsr (t ) wsi (t )) + 3[2zm (t r 0 )((wm (t

c ={[ xsr (t )] + 3[xmr (t

0 )]

0 ))

2

(wmi (t

+ sin(t )[ xmi (t

e yr =ysr (t )

0 ))]} e x r

r 0 )(wm (t

i 0 ) wm (t 0 ))

2)]} e i x

3xmr (t

0)

+ sin(t )[xmi (t

0 )],

sin(t )[xmr (t

2)]

+ sin(t )[ 2z m (t + sin(t )[z m (t

0 )]} e x i

e,

where

z s (t )((wsr (t ))2 (wmi (t

sin(t )[xmr (t

e ={[ysr (t )] + {[ysi (t )]

= [abcde]T , we can determine the parameter law by uti-

a ={[ z s (t )] + 3[z m (t b ={[

0 ) z m (t

+ { xsi (t ) z s (t ) + 3[xmi (t

3[xmi (t

+ sin(t )[xmr (t

0 )]} e yr

0 )]

r,

[xsi (t )]

0 )]

3ymr (t

0)

+ sin(t )[ymi (t

0 )],

e wr =wsr (t )

b,

3wmr (t

i 0 ) + sin(t )[wm (t

(38) 9

0)

3ymi (t

0)

0 )],

i 0 )], e w i = ws (t )

sin(t )[wmr (t

0 )]} e x r

3xmi (t

0 )],

e yi = ysi (t )

sin(t )[ymr (t

0 ))]

e x i = xsi (t )

0 )].

3wmi (t

0)

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

Fig. 6. Approximate values of uncertain parameters a , b , c , d and e of the slave model versus (t ) . (a) (a , t ) diagram, (b) (b , t ) diagram, (c) ( c , t ) diagram, (d) (d , t ) diagram, (e) ( e , t ) diagram..

Numerical simulations

and xi

In this section, the numerical simulations are exhibited to verify the adequacy of the advanced synchronization controller. Models (32) and (33) with (36) and (38) are determined numerically for a = 10 , b = 3, c = 15, d = 8, e = 3, f = 20 with initial conditions (x r (0) , x i (0), y r (0) , y i (0) , z (0) , wr (0)) = (1, 2, 3, 4, 5, 6, 7) and 0 = 15. The dynamics of the CFPLS errors are plotted in Fig. 5, which shows that the paths of e x r , e x i , e yr , e yi , e wr and e wi of the error model tend to zero. In Fig. 5 it is shown that after a small value of t the errors converge to zero. The complex function matrix A(t ) is chosen as diag ((3 + jsin(t )) , (3 + jsin(t )) , (3 + jsin(t )) , (3 + jsin(t )) , (3 + j sin(t ))) to decrease the complexity of the simulation. Choose the uncertain parameters of a memristor complex hyperchaotic model as (a , b , c , d , e )T = ( 2, 3, 4, 6, 7)T . The positive = diag (100, 100, 100, 100, 100) . Fig. 6 deconstants are selected as monstrates that the assessed estimates of the uncertain parameters a , b , . c , d , and e converge to 10, 3, 15, 8, and 3, respectively, as t As a result of our numerical model, we ascertained the phase errors with the master and slave frameworks. For any complex value, a module x and phase x are computed as x

=

(x r ) 2 + (x i ) 2

tan( x r ), x

x r > 0, x i

( ), x + tan ( ) ,

= 2 + tan

xi

xr

xi xr

r

0

> 0, x i < 0 x r < 0.

(40)

We calculated a module error and phase error with the master and slave frameworks (32) and (33) by utilizing relation (39) and (40). It is obvious from Fig. 7(a–c) that the module errors xs ym , and xm , ys ws wm vary in the hyperchaotic state. The hyperchaotic trajectory of these errors is shown inFig. 7(d–f). The phase errors xs xm , ys ym , and ws when t tends wm vary with time, ranging between and to , as seen in Fig. 7(g–i). Secure communications based on the results of CFPLS synchronization Over the previous decades, there has been massive enthusiasm for studying the behavior of a hyperchaotic model. Numerous highlights portray, for example, the sensitive dependence of hyperchaotic models on initial conditions, periodicity, and random-like behavior. These properties concur with the requirements for signals connected in secure

(39) 10

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

Fig. 7. The modules errors and phases errors of models (32) and (33).

Step 2. The data signal r (t ) is added to the inner state ymi (t the hyperchaotic masking can be depicted as:

correspondence models [44]. Various kinds of communication protocol have been planned, including the utilization of logical and scientific strategies, signal-handling controls, chaotic dynamic models, and quantum data approaches [45]. To date, numerous chaos-based secure communication models have been proposed, which can be classified into the accompanying classes: hyperchaotic masking [25], chaotic modulation, digital communication [46], and so on. In this work, we care about the hyperchaotic masking technique, which is characterized as a shared-secret data strategy. An autonomous model creates the data signal superimposed on the hyperchaotic signal. The blended signal is transmitted to the collecting end. An equal hyperchaotic model at the receiving end is used to synchronize the hyperchaotic model at the transmitting end by the hyperchaotic driving signal. Finally, the inverse operation is done to decode the data signal [47]. The outline of the hyperchaotic masking communication application for the new hyperchaotic model utilizing CFPLS is delineated in Fig. 8. At the transmitter side, the data signal r (t ) is added to the driving hyperchaotic signal ymi (t 0 ) , and a modulated signal r¯ (t ) is created. The modulated signal r¯ (t ) is sent to the receiver. The other r r i yield signal of the model x mr (t 0 ) , x m (t 0), 0 ) , wm (t 0 ) , ym (t wmi (t 0 ) is additionally transmitted to the receiver. At the receiver side, with the CFPLS technique utilizing hyperchaotic signals r (t ) , ymi (t Subtracting signal r¯ (t ) from hyperchaotic signals 0).

[

ysi (t )

r (t sin(t )[ym

3

0 )]

] or [

i (t ysr (t ) + 3ym

sin(t )

0)

r¯ (t ) = r (t ) + ymi (t

0),

and

0 ).

Step 3. The modulated signal r¯ (t ) is transmitted to the receiving end. Step 4. By the controls ( ) composed with Theorem 1, a CFPLS synchronization is accomplished following synchronization at time t. Step 5. The data signal can be obtained by the reverse process, which is achieved in one of two directions as follows: I) This relationship describes the first method:

r (t ) = r¯ (t )

1 i [y (t ) 3 s

sin(t )(ymr (t

0 ))].

(41)

II) This relationship describes the second method:

r (t ) = r¯ (t )

ysr (t ) + 3ymi (t sin(t )

0)

.

(42)

Fig. 9 demonstrates that the data signal can be recovered at the receiving end with the related parameters and initial state of Fig. 1. The message r (t ) to be encoded in the transmitter model is r (t ) = 5sin(3 t ). cos(4 t )], as shown in Fig. 9a, and the consolidated signal r¯ (t ) of the transmitted model appears in Fig. 9b. As can be seen, after a transient state, the recovered messages can be recuperated in two ways by utilizing Eqs. (41) and (42), and is delineated in Fig. 9(c and d). Fig. 9(e and f) shows the signal of the error between the first data signal in two ways and the recuperated signal, and demonstrates the misstep between the primary data signal and the recovered one. From Fig. 9(e and f), we find the signal of data r (t ) is recovered following the short-time transient. Furthermore, we can conclude the most important results for

], the data signal r (t ) is obtained.

Calculation of the secret communication This subsection follows two methods by which the message is extracted as follows: Step 1. Develop the transmitting model by the current hyperchaotic masking, and build the receiving model by the relating hyperchaotic masking, see Fig. 8. 11

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

Fig. 8. Masking diagram hyperchaotic secure communication model based on CFPLS.

Fig. 9. For the hyperchaotic masking communication model: (a) The Data signal, (b) The encrypted message, (c) The received message by the first method, (d) The received message by the second method, (e) The error between the original message and recovered message by the first method, (f) The error between the original message and recovered message by the second method.

12

Results in Physics 15 (2019) 102619

E.E. Mahmoud and O.A. Althagafi

the application of secure communications in this type of synchronization (CFPLS), which has not been mentioned in any of the previous studies.

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1. The method of extracting the message in this paper is done in more than one way, although the previous studies obtained the signal in it only once. 2. Previously, the extraction of the message depended on the slave variables, but here, the command evolves to becomes the extraction of the signal based on slave variables and master variables. 3. Using this type of synchronization (CFPLS) allows us to rely on the complex function where the method of extracting the message changes as the complex function changes. Conclusion In this paper, a novel complex Lorenz deformed model with a complex flux-controlled memristive has been developed and researched. The dynamical behaviors of the memristive model are found through computing equilibrium points, Lyapunov exponents, bifurcation diagram and Lyapunov dimension. Furthermore, this work is focused on another kind of synchronization, called complex function projective lag synchronization (CFPLS). This sort of synchronization can be examined in complex nonlinear models and can be considered as generalization of many types of synchronization (see Tables 1 and 2). A plan is intended to accomplish the (CFPLS) of two identical hyperchaotic complex memristive models based on Lyapunov functions. The module and phase errors were studied. The most important features are that the module error close to that of the hyperchaotic solutions, whereas the phase error is limited to period [ , ] when time passes to infinity (see Figs. 6 and 7). Usually, the approximation of both errors leads to a constant value, so these results did not appear in any previous paper. We have executed a secure connection based on the complex hyperchaotic memristive model. Furthermore, we list the most important results achieved when applying secure communications, where we noted that: i) The method of extracting the message varies according to the complex function that is assumed, and consequently, the process of obtaining the data changes. ii) The extraction of data based on master and slave variables, unlike that in previous research, was based on the analysis of slave variables (see Figs. 8 and 9). This phenomenon has not been shown in any of the previous research. This result has emerged because of the complex function and the definition of synchronization, making the behavior of the model more complicated. Further, the necessary research has recently started, and the memristive-based complex models and their synchronization should be explored from the perspective of uses later on. References [1] Chua LO. Memristors-the missing circuit element. IEEE Trans Circuit Theory 1971;18:507–19. [2] Chua LO, Kang SM. Memristive devices and systems. Proc IEEE 1976;64:209–23. [3] Strukov DB, Snider GS, Stewart DR, Williams RS. The missing memristor found. Nature 2008;453:80–3. [4] Ventra MD, Pershin YV, Chua LO. Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc IEEE 2009;97:1717–24. [5] Yan H, Choe HS, Nam S, Hu Y, Das S, Klemic JF, Ellenbogen JC, Lieber CM. Programmable nanowire circuits for nanoprocessors. Nature 2011;470:240–4. [6] Valov I, Kozicki M. Non-volatile memories: organic memristors come of age. Nat Mater 2017;16:1170–2. [7] Cai Z, Lihong H, Zhu M, Wang D. Finite-time stabilization control of memristor-

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M, Stankovski T. Experimental realization of the coupling function secure communications protocol and analysis of its noise robustness. IEEE Trans Inf Forensics Secur 2018;13:2591–601. [46] Abd Elzaher MF, Shalaby M, Kamal Y, El Ramly S. Securing digital voice communication using non-autonomous modulated chaotic signal. J Inf Secur Appl 2017;34:243–50. [47] Xu G, Xu J, Xiu C, Liu F, Zang Y. Secure communication based on the synchronous control of hysteretic chaotic neuron. Neurocomputing 2017;227:108–12.

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