A hyperchaotic detuned laser model with an infinite number of equilibria existing on a plane and its modified complex phase synchronization with time lag

Chaos, Solitons and Fractals 130 (2020) 109442

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

A hyperchaotic detuned laser model with an infinite number of equilibria existing on a plane and its modified complex phase synchronization with time lag Emad E. Mahmoud a,b,∗, Bushra H. AL-Harthi c a b c

Department of Mathematics, Faculty of Science, Taif University, Taif 888, Saudi Arabia Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt Department of Mathematics, College of Science, Bisha University, Bisha, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 26 July 2019 Revised 28 August 2019 Accepted 11 September 2019

Keywords: Detuned laser system Hyperchaotic Attractors Synchronization Secure communication Complex

a b s t r a c t The objective of this research is to implement a contemporary hyperchaotic complex detuned laser system. Here, the hyperchaotic complex system is developed by combining a straight controller to the chaotic complex detuned laser system. The new system is a seven-dimensional real continuous autonomous hyperchaotic system. This system’s characteristics, including the Hamiltonian, dissipative, fixed points and its stability, Lyapunov dimension, Lyapunov exponents, and bifurcation diagrams are examined, as is the achievement of hyperchaos. Different forms of hyperchaotic complex detuned laser systems are constructed. Additionally, we present another type of synchronization for complex nonlinear systems only, termed modified complex phase synchronization with a time lag (MCPSTL). Given Lyapunov stability, the aim is to achieve MCPSTL of two indistinguishable hyperchaotic trajectories of these systems. A simulation is performed to demonstrate the viability of this approach. Numerical methods are used to calculate the variable and error states of these hyperchaotic trajectories after synchronization. The results provide a theoretical foundation for applications of the proposed approach in secure communication. Signal encryption and alteration are conducted numerically. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction A dynamical system is considered to be chaotic if it has sensitive reliance on initial conditions. In a structure, two adjacent attractors wander exponentially in time if one of the Lyapunov exponents is positive. Hyperchaotic trajectories are characterized at least two positive Lyapunov exponents and are viewed as substantially more perplexing than low-dimensional chaotic trajectories as regards topological system and dynamics. Over the past two decades, hyperchaotic systems have received increased attention from various scientific and engineering communities because of their many applications. These include nonlinear circuits, lasers, and secure communication [1–6]. Synchronization of chaotic (or hyperchaotic) systems involves synchronizing one dynamical system (usually called the slave system) with another (usually called the master system) by utilizing the chaotic control hypothesis. As chaotic systems are sen∗

Corresponding author. E-mail addresses: [email protected] (Emad E. Mahmoud), [email protected] (Bushra H. AL-Harthi). https://doi.org/10.1016/j.chaos.2019.109442 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

sitive to initial conditions, at that point, chaos synchronization can be utilized to create a particular desired signal. In 1990, Ott et al. [7] introduced a control technique, the OGY technique, that could be used to control a chaotic system to produce a particular required behavior. At about the same time, Pecora and Carroll [8] demonstrated that a chaotic system can be synchronized if the sub-Lyapunov exponents of the response system lie in the left half plane. Since then, chaotic (or hyperchaotic) synchronization has become an active field of research [9–11], in light of its applications in science, ecological systems, engineering, and secure communications [12–16]. Much research has focused on the synchronization event, owing to its importance. For example, there have been studies of complete (or full) synchronization [13,17], projective synchronization [18,19], modified projective synchronization [18,20], phase synchronization [17,21], and antiphase synchronization [21,22]. Phase and antiphase synchronization of real chaotic systems have attracted much attention because of their relevance to a wide range of applications, including lasers, electrical circuits, ecological systems, waves and turbulence in chemical oscillations and optical parametric oscillators, fluids, neuronal action of remote zones in the

2

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

human brain, multichannel nonlinear digital and secure communications, the heartbeat and the breath cycle, and neuroscience [23–26]. Recently, various types of synchronization with time lag have been examined, for instance, antilag synchronization (ALS) [27], lag (or slack) synchronization [28], and modified projective lag (or slack) synchronization (MPLS) [29]. In engineering contexts, a time lag always exists. For instance, in a telephone communication system, the sound heard by the collector party at time t is the sound produced by the transmitter party at time t − τ (τ ≥ 0 which is the time lag). Additionally, there is a time lag as the signal from the transmitter to the beneficiary ends in chaotic (or hyperchaotic) communication. The above synchronization methodologies can be applied to complex or real dynamic systems. Furthermore, some different types of synchronization have been used solely for complex dynamical systems, for example, complex complete (or full) synchronization [30], complex lag (or slack) synchronization [31], complicated (or complex) projective synchronization [32], complex modified projective synchronization [33], complex antilag synchronization (CALS) [34,35], complex phase synchronization [11], and complex modified projective lag synchronization [36]. These new types of synchronization will be explored in the context of chaotic (or hyperchaotic) complex nonlinear systems. The laser is an outgrowth of light amplification by stimulated emission of radiation from a device that amplifies microwaves. The first two lasers developed during 1960 were the ruby laser and the helium-neon gas laser. Soon, physicists realized that the laser action could be obtained using ions, atoms, liquids, flames, molecules in gases and solids, glasses, plastics, and semiconductors, at wavelengths ranging from the ultraviolet to radio-frequency regions and with power ranging from milliwatts to megawatts. The remarkable properties of lasers have led to many innovative applications. They have been utilized in meteorology, biology, telecommunication, metrology, computer circuits, cybernetic, medicine, and industry [37–40]. The complex nonlinear equations for detuned lasers as suggested by Zeghlache and Mandel are as follows [41]:

w˙ 1 (t ) = b1 w2 (t ) − b1 (1 − jb2 )w1 (t ), w˙ 2 (t ) = (b3 − z (t ) )w1 (t ) − (1 + jb2 )w2 (t ), 1 z˙ (t ) = −b4 z (t ) + (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ), 2

(1)

where w1 (t ) = ω1 (t ) + jω2 (t ), w2 (t ) = ω3 (t )√+ jω4 (t ) are the complex variables, z(t) is a real variable, j = −1, b1 , b2 , b3 , b4 are positive real parameters that shift the behavior of the system, dots denote derivatives with respect to time, and an overbar means a conjugate variable. The w1 (t), w2 (t), and z(t) variables of the problem are linked to electric field amplitude, polarization, and population inversion, respectively, in a ring laser system of twolevel atoms [42]. Eqs. (1) and specifically, their links to the popular Lorenz form, have been analyzed widely in the modern literature in the context of laser systems. The complex character of the state variables and the presence of an imagined detuning parameter jb2 result from solely physical events. These equations have chaotic behavior, as shown in [41]. In this work, we obtain hyperchaotic attractors by adding a linear controller to Eq. (1) as follows:

2. Dynamical features of system (2) In this section, we consider the essential features of our new system (2). The real form of system (2) is given by:

ω˙ 1 (t ) = b1 (ω3 (t ) − ω1 (t ) − b2 ω2 (t ) ), ω˙ 2 (t ) = b1 (ω4 (t ) − ω2 (t ) + b2 ω1 (t ) ), ω˙ 3 (t ) = (b3 − z(t ) )ω1 (t ) − ω3 (t ) + b2 ω4 (t ), ω˙ 4 (t ) = (b3 − z(t ) )ω2 (t ) − ω4 (t ) − b2 ω3 (t ), z˙ (t ) = −b4 z (t ) + ω1 (t )ω3 (t ) + ω2 (t )ω4 (t ) + ω5 (t ) − ω6 (t ), ω˙ 5 (t ) = k1 ω1 (t ) + k2 ω3 (t ), ω˙ 6 (t ) = k1 ω2 (t ) + k2 ω4 (t ). (3) system (3) has the following accompanying essential dynamical attribute. 2.1. Extrapolated Hamiltonian of system (3) We think about a nonlinear continuous system (3), given by the equation:

w˙ = α (w )

∂h ∂h + σ (w ) , ∂w ∂w

(4)

where w= [ω1 (t ), ω2 (t), ω3 (t), ω4 (t), z(t), ω5 (t), ω6 (t)]T , h(w) is h smooth function of energy and positive globally definite, ∂∂w is a vector of gradient column of h(w) [43]. By using quadratic energy [43]:

h (w ) =

1 T w γ w, 2

(5)

anywhere γ is a steady diagonal matrix, being positive definite and h symmetric, with respect to ∂∂w = γ w, α (w) is antisymmetric matrix, and it is the vector field workless part and, σ (w) a matrix symmetric, relating to the non-conservative part of the system or the working and a negative definite [43].

w˙ 1 (t ) = b1 w2 (t ) − b1 (1 − jb2 )w1 (t ), w˙ 2 (t ) = (b3 − z (t ) )w1 (t ) − (1 + jb2 )w2 (t ), 1 z˙ (t ) = −b4 z (t ) + (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ) 2 1 j + (w3 (t ) + w¯ 3 (t ) ) + (w3 (t ) − w¯ 3 (t ) ), 2 2 w˙ 3 (t ) = k1 w1 (t ) + k2 w2 (t ),

where b1 , b2 , b3 , and b4 are real positive parameters that modify the conduct of the system, w1 (t ) = ω1 (t ) + jω2 (t ), w2 (t ) = ω3 (t ) + jω4 (t ), w3 (t ) = ω5 (t ) + jω6 (t ) are the complex variables, z(t) is a real variable, and k1 and k2 are parameters of the control. In this article, we describe and present another type of complex synchronization that we call modified complex phase synchronization with time lag (MCPSTL). This is based on syncretizing between phase synchronization with time lag (PSTL) and anti lag synchroniztion (ALS) [27]. PSTL occurs between both the actual or real part of the slave system and the imaginary part of the master system with a time lag, whereas ALS happens between both the imaginary part of the slave system and the real part of the master system with a time lag. In MCPSTL, the variable state of the master system synchronizes with a different variable state of the slave system. Thus, MCPSTL ensures greater safety in secure communication. The following is the organization of this document. In the following section, we explore the basic features of system (2). We construct different forms of hyperchaotic complex detuned laser systems in Section 3. In Section 4, a definition of MCPSTL is introduced, and a plan to accomplish MCPSTL of hyperchaotic complex nonlinear systems is proposed. We study MCPSTL of two hyperchaotic complex detuned laser systems (2). The primary intended use, in secure communication, is demonstrated with reference to the results of the MCPSTL. Finally, we determine the critical findings of our studies in Section 5.

(2)

α (w ) = [−α (w )]T , σ ( w ) = [ σ ( w )] T .

(6)

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

The system (3) has one isolated fixed point E0 = (0, 0, 0, 0, 0, 0, 0), as well as a plane of equilibria represented by:

Let we determine h(w) of system (3):



1 (ω1 (t ) ) (ω2 (t ) ) (ω3 (t ) ) (ω4 (t ) ) + + + 2 b1 b1 b3 b3

h (w ) =

+ ⎤ ω˙ 1 (t ) ω ˙ ( t ) ⎢ 2 ⎥ ⎢ω˙ 3 (t )⎥ ⎢ ⎥ ⎢ω˙ 4 (t )⎥ ⎢ ⎥ ⎢ z˙ (t ) ⎥ ⎣ ⎦ ω˙ 5 (t ) ω˙ 6 (t ) ⎡

2

(z(t ) )2 b3

2

2

2

ω5 (t ) − ω6 (t ) − b4 z(t ) = 0.



+ (ω5 (t ) ) + (ω6 (t ) ) 2

2

Thus, system (3) has non-trivial fixed points given by:

,

(7)

⎡

Eg = (0, 0, 0, 0, z (t ),

ω5 (t ), ω5 (t ) − b4 z(t ) ).

To study the stability of E0 , we require the Jacobian matrix of system (3) at E0 :

⎛

0

⎢ 2 ⎢ b1 b2 ⎢ ⎢ ⎢ 0 ⎢ =⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ b1 k1 ⎣ 2 ⎡

3

0 −b21

−b21 b2

0

0

0

0

0

0

0

b3 b2

0

0

−b3 b2

0

0 b3 k2 2

0

−b3 ω2 (t )

0

b3 ω1 (t )

b3 ω2 (t )

0

1 2

0

b3 k2 2

0

− 12

0

b1 k1 2

0

b3 k2 2

1 2

0

0

b1 k1 2

b1 b3

0

0

b1 k1 2

0

b1 b3

0

0

−b3

0

0

b3 k2 2

0

−b3

0

0

0

0

−b3 b4

1 2

b3 k2 2

0

1 2

0

0

b3 k2 2

− 12

0

0

JE0

⎤

⎡ ω1 (t ) ⎤ ⎥ b1 ⎥⎢ ω2 (t ) ⎥ ⎥⎢ b 1 ⎥ ⎥⎢ ⎥ 0 ⎥⎢ ω3 (t ) ⎥ ⎥⎢ b 3 ⎥ b k ⎥⎢ ⎥ − 32 2 ⎥⎢ ω4 (t ) ⎥ ⎥⎢ b 3 ⎥ 1 ⎥⎢ ⎥ −2 ⎥⎢ zb(t ) ⎥ ⎥⎣ 3 ⎦ 0 ⎦ ω5 (t ) ω6 (t ) 0 0

b k − 12 1

−b3 ω1 (t ) −

⎢ ⎢ 0 −b21 ⎢ ⎢ 0 ⎢b 1 b 3 ⎢ +⎢ b1 b3 ⎢ 0 ⎢ ⎢ 0 0 ⎢ ⎢ b1 k1 0 ⎣ 2 0

0

b k − 12 1

⎤

⎡ ω1 (t ) ⎤

−b1 ⎜b1 b2 ⎜ b ⎜ 3 =⎜ ⎜ 0 ⎜ 0 ⎝ k 1 0

0 b1 b2 −1 0 0 k2

0 0 0 0 −b4 0 0

⎞

0 0 0 0 1 0 0

0 0⎟ 0⎟ ⎟ 0⎟ ⎟ −1⎟ ⎠ 0 0

and the characteristic polynomial:

  (−b4 − λ )λ2 λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0,

such that:

a1 = 2 b1 + 2 , 2

a2 = b21 + (b1 b2 ) + b1 (4 − 2b3 ) + b2 + 1, 2





a4 = b21 (1 + b2 − 2b3 ) + (b1 b2 ) 3 + b22 − 2b3 . At this stage, the polynomial’s own values are defined as:

λ1 = 0, λ2 = 0, λ3 =

The presence of the Hamilton in system (3) implies its capacity to gauge physical amounts.

λ4 =

2.2. Dissipative nature of the system It is clear that:

λ5 =

∂ ω˙ 1 (t ) ∂ ω˙ 2 (t ) ∂ ω˙ 3 (t ) ∂ ω˙ 4 (t ) ∂ z˙ (t ) ∇ ·V = + + + + ∂ω1 (t ) ∂ω2 (t ) ∂ω3 (t ) ∂ω4 (t ) ∂ z(t ) ∂ ω˙ 5 (t ) ∂ ω˙ 6 (t ) + + = − ( 2b1 + b4 + 2 ) < 0. ∂ω5 (t ) ∂ω6 (t )

λ6 = dV dt

=

e−(2b1 +b4 +2 )t . This means that the product volume V0 corresponds to the sample quantity V0 e−(2b1 +b4 +2 )t at time t. At the point where t → ∞, each volume segment that contains the system direction tends to 0, including the exponent comparison form 2b1 + b4 + 2. In this way, the vast majority of the system headings can eventually be confined to a zero volume subset, and the dynamic enhancement follows a trajectory.

λ7 =

1 1 (−1 − b1 ) − μ1 2 2    1 2 − 1 − b2 + 2μ1 − μ2 + 2b1 −1 − b22 + μ1 + 2b3 , 2 1 1 (−1 − b1 ) − μ1 2 2    1 2 + 1 − b2 + 2μ1 − μ2 + 2b1 −1 − b22 + μ1 + 2b3 , 2 1 1 (−1 − b1 ) + μ1 2 2    1 2 − 1 − b2 − 2μ1 − μ2 − 2b1 1 + b22 + μ1 − 2b3 , 2 1 1 (−1 − b1 ) + μ1 2 2    1 2 + 1 − b2 − 2μ1 − μ2 − 2b1 1 + b22 + μ1 − 2b3 , 2 −b4 , (9)



where μ1 =





−(−1 + b1 ) b22 and μ2 = b21 −1 + b22 . The trivial 2

fixed point E0 is a stable point if:

b22 + 1 < b3 , b4 > 0.

(10)

Anything else, the points fixed are unstable. To consider the stability of Eg , we require the eigenvalues of the Jacobian matrix of system (3) at Eg :

2.3. Fixed point and its stability The equilibria of system (3) can be determined using the accompanying system of conditions:

0 = b1 (ω3 (t ) − ω1 (t ) − b2 ω2 (t ) ), 0 = b1 (ω4 (t ) − ω2 (t ) + b2 ω1 (t ) ), 0 = (b3 − z (t ) )ω1 (t ) − ω3 (t ) + b2 ω4 (t ), 0 = (b3 − z (t ) )ω2 (t ) − ω4 (t ) − b2 ω3 (t ), 0 = −b4 z (t ) + ω1 (t )ω3 (t ) + ω2 (t )ω4 (t ) + ω5 (t ) − ω6 (t ),

⎛

−b1 ⎜ b1 b2 ⎜b − z(t ) ⎜ 3 JEg = ⎜ ⎜ 0 ⎜ 0 ⎝ k 1 0

−b1 b2 −b1 0 b3 − z (t ) 0 0 k1

b1 0 −1 −b2 0 k2 0

0 b1 b2 −1 0 0 k2

0 0 0 0 −b4 0 0

0 0 0 0 1 0 0

⎞

0 0⎟ 0⎟ ⎟ 0⎟ ⎟. −1⎟ ⎠ 0 0

We use the characteristic polynomial situation to acquire these own values, which takes the form of the above matrix:

0 = k1 ω1 (t ) + k2 ω3 (t ), 0 = k1 ω2 (t ) + k2 ω4 (t ).

b1 0 −1 −b2 0 k2 0

a3 = b21 (2 − 2b3 ) + b1 (2 + 2b2 − 2b3 ),

⎥ b1 ⎥⎢ ω2 (t ) ⎥ ⎥⎢ b 1 ⎥ ⎥⎢ ⎥ 0 ⎥⎢ ω3 (t ) ⎥ ⎥⎢ b 3 ⎥ b 3 k 2 ⎥⎢ ω (t ) ⎥ ⎥. 2 ⎥⎢ 4 ⎥⎢ b 3 ⎥ 1 ⎥⎢ ⎥ −2 ⎥⎢ zb(t ) ⎥ ⎥⎣ 3 ⎦ 0 ⎦ ω5 (t ) ω6 (t ) 0 b1 k1 2

When 2b1 + b4 + 2 > 0, the system is dissipative, and

−b1 b2 −b1 0 b3 0 0 k1

(8)



−λ2 (b4 + λ )

 λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0.

(11)

4

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442 Table 1 The solutions of system (3).

This yields:

λ1 = −b4 , λ2 = 0, λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0, where:

a1 = 2 b1 + 2 , a2 = b21 + (b1 b2 )2 + b1 (4 − 2b3 ) + b2 + 1,









a3 = b21 2 − 2b3 + 2b22 + b1 2 + 2b22 − 2b3 , a4 =

b21

(

b42

+ ( 1 − b3 ) + 2 ( 1 − 2

) ).

b3 b22

In line with the Routh–Hurwitz theorem [44], the necessary and adequate circumstances for the stability of Eg is stable are that all the roots of the above equation (11) have negative real components. This occurs when:

ai > 0, i = 1 , 2 , 3 , 4 , a1 a2 > a3 , a1 a2 a3 > a23 + a21 a4 . Otherwise, it is unstable. 2.4. Lyapunov exponents system (3) can be written in vector form as:

T

∂ w˙ (t ) = Jsi (w(t ); )∂ w;

s, i = 1, 2, 3, 4, 5, 6, 7,

(13)

where Jsi = ∂∂wfs is the Jacobian matrix of the form: i

−b1 b2 −b1 0 b3 − z (t ) ω4 (t ) 0 k1

b1 0 −1 −b2 ω1 (t ) k2 0

0 b1 b2 −1 ω2 (t ) 0 k2

0 0

−ω1 (t ) −ω2 (t ) −b4 0 0

0 0 0 0 1 0 0

⎞

0 0⎟ 0⎟ ⎟ 0⎟ ⎟. −1⎟ ⎠ 0 0

The Lyapunov exponents [45] Li of the system are defined by:

Li = lim

1

t→∞ t

log

∂ w(t ) , i = 1, 2, 3, 4, 5, 6, 7, ∂ w ( 0 ) 

(14)

where ∂ w(t) is a differential of w(t), ∂ w(0) represents the original form when the distance is equal to zeroare. Conditions (12) and (13) need be numerically determined by the Runge–Kutta technique of order 4 to calculate Li . For the case b1 = 30.7, b2 = 0.001, b3 = 30, b4 = 5, k1 = −7, and k2 = −6, with initial conditions ω1 (0 ) = 1, ω2 (0 ) = 2, ω3 (0 ) = 3, ω4 (0 ) = 4, z(0 ) = 5, ω5 (0 ) = 6, and ω6 (0 ) = 7, we computed the Lyapunov exponents as L1 = 5.12,L2 = 0.25, L3 = 0, L4 = 0, L5 = −0.42, L6 = −45.50, and L7 = −57.98. This shows that our system (3) under this choice of b1 , b2 , b3 , b4 , k1 , and k2 is a hyperchaotic system with two positive Lyapunov exponents. The Lyapunov dimension of the trajectories of (3) as indicated by the Kaplan–Yorke conjecture is defined as [46]:

D=ε+

ε

i=1 Li

|Lε+1 |

L3

L4

L5

L6

L7

Type of solution

+ −

0 −

− −

− −

− −

− −

Hyperchaotic attractors. Solutions approach fixed points.

2.4.1. Fix b2 , b3 , b4 k1 , k2 and vary b1 where b1 ∈ [10, 43] By equation (14) we can calculate Lyapunov exponents L1 , L2 , . . . , L7 . Fig. 1 shows the behavior of the Lyapunov exponent values. The components of system (3) change between the hyperchaotic trajectories and fixed point solutions. It is clear from Fig. 1 that when b1 ∈ [10, 38.95], system (3) has hyperchaotic trajectories with (2) positive Lyapunov exponents (++ 0 − − − − ), whereas it has fixed point solutions (− − − − − − − ) when b1 ∈ [39, 43]. We performed an examination of the other parameters, in the same manner as for parameter b1 , and obtained hyperchaotic arrangements with two positive Lyapunov exponent for every solution. As such, we were satisfied with the eventual outcomes for parameter b1 . The solutions of (3) can be classified as shown in Table 1.

(12)

where w(t ) = [ω1 (t ), ω2 (t ), ω3 (t ), ω4 (t ), z (t ), ω5 (t ), ω6 (t )] is the state space vector, f (t ) = [ f1 (t ), f2 (t ), f3 (t ), f4 (t ), f5 (t ), f6 (t ), f7 (t )]T , ϰ is the course of action of the arrangement of parameters, and [. . .]T is a matrix transpose. The equalization for small deviations ∂ w(t) from the attractor w(t) is:

−b1 ⎜ b1 b2 ⎜b − z(t ) ⎜ 3 Jsi = ⎜ ⎜ 0 ⎜ ω3 (t ) ⎝ k 1 0

L2

+ −

2.5. Bifurcation diagram of system (3)

w˙ (t ) = f (w (t ); ),

⎛

L1

,

(15)

ε ε+1 where ε is the largest integer for which i=1 Li > 0 and i=1 Li < 0. The Lyapunov dimension of this hyperchaotic attractor using L +L +L +L +L (15) is D = 5 + 1 2 |L3 | 4 5 ≈ 5.1088. 6

A bifurcation diagram provides a summary of the changes that can occur in various types of movement as one parameter of the system varies. A bifurcation diagram layout has a system parameter on the horizontal axis and a description of the trajectory’s movement on the perpendicular axis. Thus, bifurcation diagrams represent a useful way to picture how a system’s behavior varies, as demonstrated by the estimation of a parameter [47]. In this subsection, we assume the other device to determine the dynamical behavior of system (3) using bifurcation diagrams. Fig. 1c–f show the bifurcation diagrams of system (3) when the parameter b1 is varied, to ensure that our system is hyperchaotic. Fig. 1c–f show the bifurcation diagrams of system (3) for b1 ∈ [10, 43], and Fig. 1c shows the (b1 , ω1 (t)) bifurcation diagrams for b1 ∈ [10, 43]. Note that when b1 ∈ [10, 38.95], system (3) has solutions that are hyperchaotic attractors. However, when b1 ∈ ]38.95, 43] our system (3) has solutions that approach fixed points. The results shown in Fig. 1c correspond to those in Fig. 1d–f. Fig. 1d– f show the (b1 , ω2 (t)), (b1 , ω3 (t)) and (b1 , ω4 (t)) bifurcation diagrams when b1 ∈ [10, 43], respectively. The results are the same as those obtained previously in the case of the plane (b1 , ω1 (t)). To examine this, we solved (3) numerically (using the Mathematica 7 program) in some situations, showing good communication with the previous results. For example, choosing b2 = 0.001, b3 = 30, b4 = 5, k1 = −7, and k2 = −6, with the initial conditions ω1 ( 0 ) = 1, ω2 ( 0 ) = 2, ω3 ( 0 ) = 3, ω4 ( 0 ) = 4, z ( 0 ) = 5, ω5 ( 0 ) = 6, and ω6 (0 ) = 7, for b1 = 25, the solution of system (3) was hyperchaotic with two positive Lyapunov examples (see Fig. 2(a)). The hyperchaotic trajectory with two positive Lyapunov examples is illustrated in Fig. 2(b) for b1 = 30. In Fig. 2(c) and (d), the solutions are fixed points for b1 = 40 and b1 = 42, respectively. 3. Different forms of hyperchaotic complex detuned laser systems In this section we point out that the complex detuned laser system (2) is not just the one with hyperchaotic attractors. So, we’re looking for other structures that produce hyperchaotic behavior. We can construct these systems via system (1). We can construct eight different forms of hyperchaotic complex detuned laser systems. These systems with their Lyapunov exponents are:

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

5

Fig. 1. For b2 = 0.001, b3 = 30, b4 = 5, k1 = −7, k2 = −6 and vary b1 with the initial conditions ω1 (0 ) = 1, ω2 (0 ) = 2, ω3 (0 ) = 3, ω4 (0 ) = 4, z(0 ) = 5, ω5 (0 ) = 6, ω6 (0 ) = 7. (a) The Lyapunov examples of system (3): L1 , L2 , L3 , (b) The Lyapunov examples of system (3): L4 , L5 , L6 , L7 , (c) The bifurcation diagrams in (b1 , ω1 (t )) plane, (d) The bifurcation diagrams in (b1 , ω2 (t )) plane, (e) The bifurcation diagrams in (b1 , ω3 (t )) plane, (f) The bifurcation diagrams in (b1 , ω4 (t )) plane.

1.

2.

w˙ 1 (t ) = w˙ 2 (t ) =

z˙ (t ) = w˙ 3 (t ) =

j b1 w2 (t ) − b1 (1 − jb2 )w1 (t ) + (w3 (t ) + w¯ 3 (t ) ), 2 (b3 − z(t ) )w1 (t ) − (1 + jb2 )w2 (t ) 1 + (w3 (t ) − w¯ 3 (t ) ), 2 1 −b4 z (t ) + (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ), 2 k1 w1 (t ) + k2 w2 (t ), (16)

for b1 = 50, b2 = 0.001, b3 = 42, b4 = 5, k1 = −7, and k2 = −6 system (16) has L1 = 3.05, L2 = 1.20, L3 = 0, L4 = 0, L5 = 0, L6 = −12.23, and L7 = −30.57.

w˙ 1 (t ) = b1 w2 (t ) − b1 (1 − jb2 )w1 (t ) 1 + (w3 (t ) − w¯ 3 (t ) ), 2 w˙ 2 (t ) = (b3 − z (t ) )w1 (t ) − (1 + jb2 )w2 (t ) j + (w3 (t ) + w¯ 3 (t ) ), 2 1 z˙ (t ) = −b4 z (t ) + (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ), 2 w˙ 3 (t ) = k1 w1 (t ) + k2 w2 (t ),

(17)

6

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

Fig. 2. Solutions of system (3) for b2 = 0.001, b3 = 30, b4 = 5, k1 = −7, k2 = −6 and change b1 with the similar fundamental conditions in Fig. 1. (a) Hyperchaos trajectory, b1 = 25 in (z(t ), ω3 (t ), ω1 (t )), (b) Hyperchaos trajectories, b1 = 30 in (z(t ), ω3 (t ), ω1 (t )), (c) Solutions approach fixed points, b1 = 40 in (ω4 (t ), ω3 (t ), ω2 (t )), (d) Solutions approach fixed points, b1 = 42 in (z(t ), ω3 (t ), ω1 (t )).

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

for b1 = 20, b2 = 0.005, b3 = 33, b4 = 15, k1 = −15, and k2 = −16 system (17) has L1 = 4.15, L2 = 0.72, L3 = 0, L4 = 0, L5 = 0, L6 = −18.36, and L7 = −28.15.

z˙ (t ) = −b4 z (t ) +

+

j (w3 (t ) − w¯ 3 (t ) ), 2 w˙ 3 (t ) = k1 w1 (t ) + k2 w2 (t ),

1 (w3 (t ) + w¯ 3 (t ) ) 2

8.

w˙ 1 (t ) = b1 w2 (t ) − b1 (1 − jb2 )w1 (t ), w˙ 2 (t ) = (b3 − z (t ) )w1 (t ) − (1 + jb2 )w2 (t ) j + (w3 (t ) − w¯ 3 (t ) ), 2 1 z˙ (t ) = −b4 z (t ) + (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ) 2 1 + (w3 (t ) + w¯ 3 (t ) ), 2 w˙ 3 (t ) = k1 w1 (t ) + k2 w2 (t ),

(18)

for b1 = 20, b2 = 0.005, b3 = 33, b4 = 3, k1 = −5, and k2 = −6 system (18) has L1 = 3.69, L2 = 1.32, L3 = 0, L4 = 0, L5 = 0, L6 = −8.56, and L7 = −21.30. 4.

w˙ 1 (t ) = b1 w2 (t ) − b1 (1 − jb2 )w1 (t ), w˙ 2 (t ) = (b3 − z (t ) )w1 (t ) − (1 + jb2 )w2 (t ) 1 j (w3 (t ) + w¯ 3 (t ) ) + (w3 (t ) − w¯ 3 (t ) ), 2 2 1 z˙ (t ) = −b4 z (t ) + (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ), 2 w˙ 3 (t ) = k1 w1 (t ) + k2 w2 (t ),

(22)

for b1 = 21, b2 = 0.002, b3 = 30, b4 = 4, k1 = −7, and k2 = −5 system (22) has L1 = 4.26, L2 = 0.79, L3 = 0, L4 = 0, L5 = 0, L6 = −16.09, and L7 = −36.12.

j (w3 (t ) − w¯ 3 (t ) ), 2

w˙ 2 (t ) = (b3 − z (t ) )w1 (t ) − (1 + jb2 )w2 (t ), 1 z˙ (t ) = −b4 z (t ) + (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ), 2 w˙ 3 (t ) = k1 w1 (t ) + k2 w2 (t ),

1 (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ) 2

+

3.

w˙ 1 (t ) = b1 w2 (t ) − b1 (1 − jb2 )w1 (t ) +

7

(23)

for b1 = 21, b2 = 0.001, b3 = 30, b4 = 4, k1 = −7, and k2 = −5 system (23) has L1 = 3.52, L2 = 1.25, L3 = 0, L4 = 0, L5 = 0, L6 = −11.60, and L7 = −27.20.

+

(19)

for b1 = 22, b2 = 0.004, b3 = 35, b4 = 3, k1 = −5, and k2 = −6 system (19) has L1 = 3.90, L2 = 0.98, L3 = 0, L4 = 0, L5 = 0, L6 = −12.14, and L7 = −35.80. 5.

All the systems mentioned above have two positive Lyapunov exponents. We calculate these exponents as we did for system (2) with the same initial conditions as in Fig. 1. The fundamental characteristics of these systems can be explored in the same way as we did in Section 2. 4. Hyperchaos synchronization for system (3) 4.1. Definition of MCPSTL

w˙ 1 (t ) = b1 w2 (t ) − b1 (1 − jb2 )w1 (t ) +

1 (w3 (t ) + w¯ 3 (t ) ), 2

w˙ 2 (t ) = (b3 − z (t ) )w1 (t ) − (1 + jb2 )w2 (t ), 1 z˙ (t ) = −b4 z (t ) + (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ) 2 j + (w3 (t ) − w¯ 3 (t ) ), 2 w˙ 3 (t ) = k1 w1 (t ) + k2 w2 (t ),

(20)

for b1 = 21, b2 = 0.001, b3 = 30, b4 = 4, k1 = −5, and k2 = −6 system (20) has L1 = 4.23, L2 = 0.39, L3 = 0, L4 = 0, L5 = 0, L6 = −23.13, and L7 = −41.15. 6.

w˙ 1 (t ) = b1 w2 (t ) − b1 (1 − jb2 )w1 (t ) +

j (w3 (t ) − w¯ 3 (t ) ), 2

w˙ 2 (t ) = (b3 − z (t ) )w1 (t ) − (1 + jb2 )w2 (t ), 1 z˙ (t ) = −b4 z (t ) + (w¯ 1 (t )w2 (t ) + w1 (t )w¯ 2 (t ) ) 2 1 + (w3 (t ) + w¯ 3 (t ) ), 2 w˙ 3 (t ) = k1 w1 (t ) + k2 w2 (t ),

(21)

w˙ 1 (t ) = b1 w2 (t ) − b1 (1 − jb2 )w1 (t ), w˙ 2 (t ) = (b3 − z (t ) )w1 (t ) − (1 + jb2 )w2 (t ) +

1 (w3 (t ) + w¯ 3 (t ) ), 2

w˙ = φw + H(w, z ) , ¯ , z) z˙ = g(w, w

(24)

where w = (w1 (t ), w2 (t ), . . . , wη (t ) )T is a complex state vector, w = wRe + jwIm , wRe = (ω1 (t ), ω3 (t ), . . . , ω2η−1 (t ) )T , wIm = √ (ω2 (t ), ω4 (t ), . . . , ω2η (t ) )T , j = −1, η = 1, 2, . . . , n, T denotes the n × n transpose, φ ∈ R is a real matrix of system parameters, H = (h1 , h2 , . . . , hη )T is a vector of nonlinear functions, g:Cn × R → R is real nonlinear function, z ∈ R, and superscripts Re and Im represent the real and imaginary components, respectively, of the complex state vector w. In this work, we present and study the definition of MCPSTL of two indistinguishable systems of the form (24) with known parameters by developing a control plan, and attempt to confirm its correctness numerically. We study two indistinguishable hyperchaotic complex nonlinear systems of the form (24) in a master-slave configuration:



for b1 = 21, b2 = 0.001, b3 = 30, b4 = 5, k1 = −5, and k2 = −6 system (21) has L1 = 3.49, L2 = 1.17, L3 = 0, L4 = 0, L5 = 0, L6 = −9.61, and L7 = −28.36. 7.

Consider the following hyperchaotic complex nonlinear system:



w˙ m = w˙ m + jw˙ m = φwm + H(wm , z ) , ¯ m, z) z˙ = g(wm , w

(25)

w˙ s = w˙ s + jw˙ s = φws + H(ws , z ) + ,

(26)

Re

Re

Im

Im

where  is the additive complex controller and  = ( 1 , 2 , . . . , n )T = Re + jIm , Re = (ζ1 , ζ3 , . . . , ζ2n−1 )T , Im = (ζ2 , ζ4 , . . . , ζ2n )T . We indicate the master system with the subscript m and the controlled slave system with the subscript s. Definition 1. Two coupled dynamical complex systems in a master-slave configuration can show MCPSTL if there exists a vector of the complicated (or complex) error function δ defined as:

8

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

δ = δRe + jδIm = ws (t ) + jwm (t − τ ) = C, Re

(27)

Im

where δ = δ + jδ = (δ1 , δ2 , . . . , δn )T , wm and ws are the complex state vectors of the master and slave systems, respectively, τ is a positive time lag, C = [c1 , c2 , . . . , cn ]T is a vector of real constants, Im δRe = (δω1 , δω3 , . . . , δω2n−1 )T = wRe and s (t ) − wm (t − τ ) = C,

δ

Im

= ( δ ω 2 , δω 4 , . . . , δ ω 2 n ) = T

wIm s (t )

+

wRe m (t

− τ ) = 0.

Remark 1. The errors between the real part of the slave system wRe s (t ) and the imaginary part of the master system with time lag wIm m (t − τ ) equal real constants as t → ∞, as in the definition of PSTL [48]. Remark 2. The sum of the imaginary part of the slave system wIm s (t ) and the real part of the master system with time lag wRe m (t − τ ) vanishes when t → ∞. This is consistent with the definition of ALS [27]. Remark 3. MCPSTL is incorporated between ALS and PSTL, as shown by Remarks 1 and 2. Remark 4. When C = 0 in Eq. (27), we define CALS [34,35] between systems (25) and (26). Remark 5. When τ = 0 in Eq. (27), we define MCPS [49] between systems (25) and (26). Theorem 1. If the nonlinear controller is given by:

 = Re + jIm = −φws (t ) − H(ws (t ), z(t ) )   Re Im − j φwm (t − τ ) + H(wm (t − τ ), z (t − τ ) ) + δ + jδ , Re Im = −φwRe s (t ) − H (ws (t ), z (t ) ) + φ wm (t − τ ) Re

Im Re + j −φwIm s (t ) − H (ws (t ), z (t ) ) − φ wm (t − τ )

− HRe (wm (t − τ ), z (t − τ ) ) + δ

Im



Re   δ˙ = δRe Re Im → δ˙ = δ + jδ . Im δ˙ = δIm

(33)

If we select υl = 0, ν l < 0 (e.g.νl = −1), l = 1, 2, . . . , n, then  = diag(0, 0, . . . , 0 ) and  = diag(−1, −1, . . . , −1 ), and Eq. (33) takes the form:

 Re   δ˙ = 0, Im → δ˙ = −δ , Im δ˙ = −δ Im ,

(34)

using the Lyapunov stability [50], we get δRe → constant vector and δIm → 0 as t → ∞, and the MCPSTL is achieved for all state variables. This completes the proof.  Remark 6. If we choose υl < 0 (e.g. υl = −1 ) (e.g. νl = −1 ), then Eq. (33), becomes:

νl < 0

and



Re   δ˙ = −δRe ˙ = −δ , → δ Im δ˙ = −δIm

and CALS [34,35] is achieved for all state variables. 4.2. Example In this subsection, we demonstrate the adequacy of the recommended synchronization scheme by calculating the MCPSTL of two identical hyperchaotic complex detuned laser systems. The master and the slave systems are described, respectively, as follows:

w˙ 1m (t ) = b1 w2m (t ) − b1 (1 − jb2 )w1m (t ),

+ HIm (wm (t − τ ), z (t − τ ) ) + δ



By substituting Re and Im from Eq. (28), into Eq. (32), we obtain:



,

(28)

where  and  are diagonal matrices of real diagonal elements υl and ν l , l = 1, 2, . . . , n, respectively, then the slave system (26) and the master system (25) can achieve MCPSTL when υl = 0, ν l < 0. Proof. From the definition of MCPSTL:

w˙ 2m (t ) = (b3 − z (t ) )w1m (t ) − (1 + jb2 )w2m (t ), 1 z˙ (t ) = −b4 z (t ) + (w¯ 1m (t )w2m (t ) + w1m (t )w¯ 2m (t ) ) 2 1 j + (w3m (t ) + w¯ 3m (t ) ) + (w3m (t ) − w¯ 3m (t ) ), 2 2 w˙ 3m (t ) = k1 w1m (t ) + k2 w2m (t ) and

w˙ 1s (t ) = b1 w2s (t ) − b1 (1 − jb2 )w1s (t ) + 1,

δ = δRe + jδIm = ws (t ) + jwm (t − τ ).

(29)

w˙ 2s (t ) = (b3 − z (t ) )w1s (t ) − (1 + jb2 )w2s (t ) + 2 , w˙ 3s (t ) = k1 w1s (t ) + k2 w2s (t ) + 3,

So, Re

δ˙ = δ˙ + jδ˙

Im

= w˙ s (t ) + jw˙ m (t − τ ),

Re

Im

= w˙ s (t ) − w˙ m (t − τ )

δ˙ = δ˙ + jδ˙ 

Re

Im



+ j w˙ s (t ) + w˙ m (t − τ ) , Im

Re

(30)

and, based on hyperchaotic complex systems (25) and (26), we have the following error complex dynamical system: Re

δ˙ = δ˙

+ jδ˙

Im

=φ

Re ws

(t ) + H (ws (t ), z (t ) ) − φ Re

− HIm (wm (t − τ ), z (t − τ ) ) + 



Im wm



(t − τ )

⎧ Re Im Re ⎪ δ˙ = φwRe s (t ) + H (ws (t ), z (t ) ) − φ wm (t − τ ) ⎪ ⎨ Re − HIm (wm (t − τ ), z (t − τ ) ) +  , Im Re Im ⎪δ˙ = φwIm ⎪ s (t ) + H (ws (t ), z (t ) ) + φ wm (t − τ ) ⎩ Im Re + H (wm (t − τ ), z (t − τ ) ) +  .



w˙ 1m (t ) w˙ 2m (t ) w˙ 3m (t )



=

−b1 (1 − jb2 ) b3 k1



+ (31)

By isolating the real and the imaginary components of (31), the error complex system can be written as:

(36)

where w1m (t ) = ω1m (t ) + jω2m (t ), w2m (t ) = ω3m (t ) + jω4m (t ), w3m (t ) = ω5m (t ) + jω6m (t ), w1s (t ) = ω1s (t ) + jω2s (t ), w2s (t ) = ω3s (t ) + jω4s (t ), w3s (t ) = ω5s (t ) + jω6s (t ), and 1 = ζ1 + jζ2 , 2 = ζ3 + jζ4 , 3 = ζ5 + jζ6 are the unknown control capacities to be resolved. The complex systems (35) and (36) can be written, respectively, as

Re

Re Im φwIm s (t ) + H (ws (t ), z (t ) ) + φ wm (t − τ )  Im + HRe (wm (t − τ ), z (t − τ ) ) +  .

+ j

(35)

and





w˙ 1s (t ) w˙ 2s (t ) w˙ 3s (t )

 =



+





0 0 0



w1m (t ) w2m (t ) w3m (t )

0 −z (t )w1m (t ) , 0

−b1 (1 − jb2 ) b3 k1

(32)

b1 −(1 + jb2 ) k2

b1 −(1 + jb2 ) k2



0 −z (t )w1s (t ) 0

(37)



 1 + 2 . 3

0 0 0





w1s (t ) w2s (t ) w3s (t )

(38)

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

9

Fig. 3. MCPSTL between two systems (35) and (36). (a) ω1m (t − τ ) and ω2s (t ) vary t. (b) ω2m (t − τ ) and ω1s (t ) vary t. (c) ω3m (t − τ ) and ω4s (t ) vary t. (d) ω4m (t − τ ) and ω3s (t ) vary t. (e) ω5m (t − τ ) and ω6s (t ) vary t. (f) ω6m (t − τ ) and ω5s (t ) vary t.

Thus, by examining the complex systems (37) and (38) and the design of systems (25) and (26), respectively, we find

 φ=  H(wm (t ), z (t ) ) =

 =

−b1 (1 − jb2 ) b3 k1



b1 −(1 + jb2 ) k2

0 −z (t )w1m (t ) , H(ws (t ), z (t ) ) 0



0 −z (t )w1s (t ) . 0



0 0 , 0

According to Theorem 1, the complex control functions 1 , 2 , and 3 can be computed as follows:



1 2 3 ⎛

 =

⎞

b1 (1 − jb2 )w1s (t ) − b1 w2s (t ) − j (−b1 (1 − jb2 )w1m (t − τ ) ⎜ + b1 w2m (t − τ ) ) − b3 w1s (t ) + (1 + jb2 )w2s (t ) + β1 ⎟ ⎝ − j (b w (t − τ ) − (1 + jb )w (t − τ ) − β ) − k w (t ) ⎠ 3 1m 2 2m 2 1 1s −k2 w2s (t ) − j (k1 w1m (t − τ ) + k2 w2m (t − τ ) ) + δ

Re

+ jδ , Im

(39)

10

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

Fig. 4. MCPSTL errors between two systems (35) and (36). (a) δω1 vary t. (b) δω2 vary t. (c) δω3 vary t. (d) δω4 vary t. (e) δω5 vary t. (f) δω6 vary t.

where β1 = z (t )w1s (t ), β2 = z (t − τ )w1m (t − τ ),  = diag(0, 0, 0 ),  = diag(−1, −1, −1 ), δRe = (δω1 , δω3 , δω5 )T , and δIm = ( δω 2 , δ ω 4 , δ ω 6 ) T . Therefore, the complex controllers in (39) can be transcribed as:



 ζ1 + jζ2 ζ3 + jζ4 = ζ5 + jζ6 ⎛ ⎞ b1 (ω1s (t ) − ω2m (t − τ ) ) − b1 (ω3s (t ) − ω4m (t − τ ) ) ⎜ + b1 b2 (ω2s (t ) + ω1m (t − τ ) ) − b3 (ω1s (t ) − ω2m (t − τ ) ) ⎟ ⎝ + (ω (t ) − ω (t − τ ) ) + β − b (ω (t ) + ω (t − τ ) ) ⎠ 3s 4m 3 2 4s 3m − β4 − k1 (ω1s (t ) − ω2m (t − τ ) ) − k2 (ω3s (t ) − ω4m (t − τ ) ) ⎛b ω (t ) + ω (t − τ ) − b ω (t ) + ω (t − τ ) ⎞ ) ) 1 ( 2s 1m 1 ( 4s 3m ⎜ −b1 b2 (ω1s (t ) − ω2m (t − τ ) ) + ν1 δω2 − b3 (ω2s (t ) ⎟ + j⎜ + ω1m (t − τ ) ) + (ω4s (t ) + ω3m (t − τ ) ) + β5 + β6 ⎟, (40) ⎝ ⎠ + ν2 δω4 − k1 (ω2s (t ) + ω1m (t − τ ) ) − k2 (ω4s (t ) + ω3m (t − τ ) ) + ν3 δω6

where β3 = z (t )ω1s (t ), β4 = z (t − τ )ω2m (t − τ ), β5 = z (t )ω2s (t ), and β6 = z (t − τ )ω1m (t − τ ) + b2 (ω3s (t ) − ω4m (t − τ ) ). 4.3. Numerical outcomes To confirm our analysis of the scheme, we show results for the MCPSTL between two identical hyperchaotic complex systems (35) and (36). systems (35) and (36), including the controllers (40), are shown numerically, and the parameters are chosen as b1 = 30, b2 = 0, 001, b3 = 30, b4 = 5, k1 = −7, and k2 = −6. The basic conditions of the master system state vector and the basic estimation of the slave system state vector are (w1m (0 ), w2m (0 ), z (0 ), w3m (0 ) )T = (1 + 2 j, 3 + 4 j, 5, 6 + 7 j )T , (w1s (0 ), w2s (0 ), w3s (0 ) )T = (52 − 1 j, 50 − 3 j, −06 − 6 j )T , and τ = 0, 2. The MCPSTL is depicted in Fig. 3. It is clear that the MCPSTL is incorporated or combined between the PSTL and the ALS. The ALS appears in Fig. 3(a), (c), and (e), while the PSTL is shown in

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

11

Fig. 5. The phases errors of systems (35) and (36) and the relations between these errors.

Fig. 3(b), (d), and (f). It is also clear from Fig. 3 that the variable state of the master system synchronizes with a different variable state of the slave system. In this way, the MCPSTL provides progressively greater safety in secure communications. The MCPSTL errors are plotted in Fig. 4. These errors approach constant values, as shown in Fig. 4(a), (c), and (e), consistent with our Re Re analytical results, and PSTL is implied since δ˙ = 0 and δ =

(δω1 , δω3 , δω5 )T = (50, 46, −67 )T . However, these errors go to zero in Fig. 4(b), (d), and (f), indicating that ALS was achieved as in the analytical results. In the numerical simulations, we determine the modulus and phase errors of the master and slave systems, respectively. For each complex value w1 = wRe + wIm , the modulus and phase can 1 1 be written as [30–36]:

ρw 1 =

(wRe )2 + (wIm )2 , 1 1

(41)

and

θw 1 =



Re arctan(wIm 1 /w1 ), Re 2π + arctan(wIm 1 /w1 ), Im Re π + arctan(w1 /w1 ),

Im wRe 1 > 0, w1  0, Im wRe > 0 , w 1 < 0, 1 wRe 1 < 0.

(42)

Figs. 5 and 6 show the phase and modulus errors of the master system (35) and slave system (36). It is obvious from Fig. 5(a) and (b) that the phase differences θw1m − θw1s and θw2m − θw2s are swayed in a chaotic or hyperchaotic way, implying that MCPSTL is accomplished, while θw3m − θw3s goes to π2 (see Fig. 5(c)). Fig. 5(d)– (f) show the relations between θw1m − θw1s , θw2m − θw2s , and θw3m − θw3s . We have shown these relations are chaotic or hyperchaotic attractors, which implies that θw1m − θw1s , θw2m − θw2s , and θw3m − θw3s move chaotically or hyperchaotically and that MCPSTL is efficacious. The modulus errors pw1m − pw1s , pw2m − pw2s , and pw3m − pw3s are shown versus t in Fig. 6(a)–(c), while pw1m − pw1s versus pw2m − pw2s , pw1m − pw1s versus pw3m − pw3s , and pw2m − pw2s versus pw3m − pw3s are illustrated in Fig. 6(d), (e),

12

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

Fig. 6. The modules errors of systems (35) and (36) and the relations between these errors.

and (f), respectively. It is clear from Fig. 6 that the modulus errors move in a chaotic or hyperchaotic manner and are uncorrelated (linearly independent). 4.4. Applications in secure communications In recent years, there has been significant focus on the use of chaos and hyperchaos in secure communication. The principle of this approach is that a chaotic (or hyperchaotic) signal can be used as a bearer and transferred simultaneously to the receiver via an open channel that could be disturbed by noise. On the collector hand, the synchronization of chaos (or hyperchaos) is used to retrieve the data signal [12,17]. Here the consequences of MCPSTL were used to demonstrate its potential applications in secure connections, using a straightforward design with the master system (35) being the transmitter system and system (36) being the receiver system. The signal for the information r(t) was concealed by the hyperchaotic signals ω2m (t − τ ), ω3m (t − τ ), and we characterized r¯ (t ) = r (t ) +

ω2m (t − τ ) + ω3m (t − τ ). The data signals were retrieved by subtracting ω1s (t) plus a constant value, and adding ω4s (t) to the received signal r¯ (t ). This constant value depends on the values in the selected underlying estimation of the master and slave systems (i.e. ω1s (0 ) − ω2m (0 ) = 50). Thus, the recovered information was r ∗ (t ) = r¯ (t ) − ω1s (t ) + 50 + ω4s (t ) (see Fig. 7). actually Fig. 7 illustrates the hyperchaotic construct that used to mask and retrieve the signal r(t) data. This illustrates an application of MCPSTL in communication, where the systems (35) and (36) are the transmitter and the collector systems, respectively. So we can say that there is an advantage in using MCPSTL in secure communications. High data transfer security since this type of synchronization is based on constant values resulting from the selection of the initial values of the transmitter and receiver models. In previous investigations, this segment was not presented in secure or safe communications. Digital simulations assign the same values as those used in Section 4.3 to the system parameters and the underlying assessment of the receiver and transmitter systems. For a

Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

13

Fig. 7. A plan to accomplish secure communication between two systems (35) and (36) based on MCPSTL.

Fig. 8. Simulation outcomes of secure communication using MCPSTL between two indistinguishable hyperchaotic complex detuned lasers systems. (a) The message of original ∗ r(t). (b) The signal of transmitted r¯ (t ). (c) The message of recovered r∗ (t). (d) The signal of error r (t ) − r (t ).

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Emad E. Mahmoud and Bushra H. AL-Harthi / Chaos, Solitons and Fractals 130 (2020) 109442

specific something, we pike the information signal as r (t ) = sin(5t ) cos(9t ). The numerical simulations in secure communication on the use of hyperchaos synchronization can be seen in Fig. 8. The information signal r(t) and the transmitted signal r¯ (t ) are shown in Fig. 8(a) and (b), respectively. Fig. 8(c) shows the system for the recovered data signal r ∗ (t ) = r¯ (t ) − ω1s (t ) + 50 + ω4s (t ) (in light of the results achieving MCPSTL ω4s (t ) + ω3m (t − τ ) = 0, ω1s (0 ) − ω2m (0 ) = 50). The errors between the master data signal and the recovered signal are outlined in Fig. 8(d). From Fig. 8(d), it is clear that the information signal r(t) is recovered precisely following a short transition. 5. Conclusions In this work, we have presented a new complex nonlinear hyperchaotic system (2), which is seven-dimensional and contains the new parameters k1 and k2 . Dynamical features including dissipative, fixed points and their stability, and hyperchaos trajectories were investigated. Notably, the non-trivial fixed points moved in a way that could be represented as follows: ω5 (t ) − ω6 (t ) − b4 z (t ) = 0. The new system exhibited hyperchaotic behaviors and arrangements that approximate fixed points, as shown in Figs. 1 and 2. We introduced, in Section 3, different forms of hyperchaotic complex detuned laser systems and can be similarly investigated as we did for system (2) in Sections 2 and 4. These new systems leave room for further investigations. We also presented another kind of complex synchronization, named MCPSTL, which has irregular properties not previously reported. Namely, (i) MCPSTL includes two (fused) types of synchronization, PSTL and ALS (see Fig. 3); (ii) the variable state of the master system synchronizes with a different variable state of the slave system (see Fig. 3), thus providing progressively improved safety in secure communications; (iii) MCPSTL can be found only in complex nonlinear systems. In summary, a novel scheme was proposed to realize MCPSTL of two identical complex nonlinear systems with hyperchaotic behavior based on the stability theorem. We determined scientifically the complex control functions that produce MCPSTL, and implemented the scheme specifically in the case of MCPSTL of two indistinguishable hyperchaotic complex detuned laser systems. The resulting assertions are shown in Figs. 3–6; these were based on calculations of phase errors and modulus errors because, in nonlinear complex dynamical systems, the recognizable or quantifiable physical amounts are generally the phase and module. Finally, the aftereffects of hyperchaos synchronization were used to demonstrate a straightforward application in secure communication. Declaration of Competing Interest The authors declare that there is no conflict of interest regarding this paper. References [1] Mahmoud EE. An unusual kind of complex synchronizations and its applications in secure communications. Eur Phys J Plus 2017;132:1–14. [2] He J, Cai J, Lin J. Synchronization of hyperchaotic systems with multiple unknown parameters and its application in secure communication. Optik (Stuttg) 2016;127:2502–8. [3] Mohammadzadeh A, Ghaemi S. Synchronization of uncertain fractional-order hyperchaotic systems by using a new self-evolving non-singleton type-2 fuzzy neural network and its application to secure communication. Nonlinear Dyn 2017;88:1–19. [4] Vafamand N, Khorshidi S, Khayatian A. Secure communication for non-ideal channel via robust TS fuzzy observer-based hyperchaotic synchronization. Chaos Solut Fract 2018;112:116–24. [5] Smaoui N, Zribi M, Elmokadem T. A novel secure communication scheme based on the karhunen–loéve decomposition and the synchronization of hyperchaotic lü systems. Nonlinear Dyn 2017;90:271–85.

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