Secure communications via modified complex phase synchronization of two hyperchaotic complex models with identical linear structure and adjusting in nonlinear terms

Abstract

We present a novel modified complex phase synchronization (MCPS) focusing on two hyperchaotic complex systems, including a comparative structure of direct terms with absolutely differing or not completely in nonlinear terms. We have used the active control theory for analytical control limits to achieve MCPS. It is shown that MCPS contains two sorts of synchronizations (phase and anti-phase synchronizations) and the state variables of the master system synchronize with substitute state components of the slave system. Numerical results are portrayed out to reveal the phases and modules errors of these hyperchaotic complex attractors, where the complex systems appear in various basic fields of material science and building. Also, we found that MCPS of the hyperchaotic complex structure synchronizes with another state variable of the slave structure is an empowering sort of synchronization as it contributes fabulous security in secure communication. In this secure communication, synchronization amongst transmitter and collector is closed and message signals are recovered. The encryption and recovery of the signals are imitated numerically.

Keywords

Complex synchronization hyperchaotic Stability theorem complex phase module

1. Introduction

In the previous two decades, synchronization of chaotic and hyperchaotic systems with real factors has turned into a dynamic research theme in nonlinear science since the fundamental work of Pecora and Carroll. Recently, it has potential applications in material science, biological science, neural systems and secure communications, and so on [1]. In 1990, Pecora and Carrol [2] proposed a methodology to synchronize two unclear systems with different introductory equations. Following this work, many research groups have been proposed and prescribed methods for the synchronization of chaotic and hyperchaotic system which hold complete synchronization [3], phase synchronization [4, 5], anti-phase synchronization, [4, 6] summed up synchronization [7], projective synchronization [8], modified projective synchronization [9], lag synchronization [10], changed projective phase synchronization [11] and so on. These sorts of synchronization are without question fitting real and complex nonlinear dynamical system [1 –13].

Phase synchronization (PS) demonstrates a sort of synchronization in which a chaotic oscillator modifies the frequencies of its inside movement to the rhythm of an external driving or to the flow of another chaotic or hyperchaotic oscillator while the amplitudes continue shifting in an erratic and uncorrelated plan. In PS, the state factors of the master and slave systems have a similar shape, however, there is an offset between them in view of the decision of introductory estimations of the master and slave system. Consequently, we trust that this component can be utilized as a part of secure communication. This sort of synchronization has been found in a wide variation of different physical, blend and natural systems, including coupled electronic oscillators [14], plasma discharge tubes remastered by a low recurrence wave generator [15], assortments of extensively coupled electrochemical oscillators [16], the phase secured visual response man [17], and the blood inferable from the neighboring practical units in the kidney [18]. It is moreover imperative to take note of that with a legitimate symmetry, other than in-phase synchronization, indistinct chaotic systems can in like manner work in anti-phase, and one can horizontally watch the occurrence of in-phase and anti-phase synchronization (APS) [5 , 20].

As of late, different new sorts of synchronization are presented, these sorts are fitting for complex (not real) nonlinear dynamical systems just, for example, complex anti- synchronization [21], complex complete synchronization [22], complex lag synchronization [23], the complex projective synchronization [24], complex modified projective synchronization [25] and complex against lag synchronization [26]. Certain new sorts of complex synchronizations are investigated for complex nonlinear dynamical systems with chaotic and hyper chaotic practices. In complex domain, there are two hazardous sums module and phase. Therefore, the exercises of the module and phase are considered and figured in [21 –26]. The complex dynamical system with nonlinear terms makes tremendous applications in outlining, detuned laser, and communications [27].

The objective of this paper is to achieve our examinations concerning complex synchronization. We propose a novel sort of complex synchronization of two hyperchaotic complex models with a comparative structure in coordinate terms and fluctuating absolutely or to some degree in nonlinear terms. We may call this kind of complex synchronization, modified complex phase synchronization (MCPS). The MCPS can be discussed and explored only for complex nonlinear system with chaotic and hyperchaotic practices. This new sort is depicted by that which consolidates or contains two sorts of synchronizations; these are the well-known phase synchronization (PS) and anti-phase synchronization (APS). The state variable of the master system synchronizes with another state variable of the slave structure. Along these lines, the MCPS presents all the more remarkable security in secure communications.

Whatever is left of this paper is as emerges. The depiction systems are clarified in Section 2. In Section 3, we present the importance of MCPS. The proposed plan to differentiate MCPS of two hyperchaotic complex models with moving completely or partially in nonlinear terms, while keeping the commence of straight parts is proficient. Section 4 is given to performing MCPS for instance of two hyper chaotic complex Chen models. A direct application for the secure communication, in light of the results of the MCPS, is presented in segment 5. The primary conclusions of our results are condensed inSection 6.

2. The hyperchaotic frameworks

Hyperchaotic system has extra complex direct and over owing flow than chaotic systems. A complex dynamical structure is depicted as hyperchaotic if certain necessities are satisfied: (i) to an incredible degree fragile to early on equations (ii) has no less than two positive Lyapunov sorts (iii) the hyperchaotic system is at most minimal 4 estimations. On account of the hyperchaotic complex system, including the benefit of high ability, high security, and world-class, it has a completely associated potential in nonlinear circuits, secure communications, lasers, neural systems, normal systems and so forth [28, 29]. Accordingly, the examination of complex nonlinear systems with hyperchaotic behavior is prominently basic for the present.

Assume the complex nonlinear framework with hyperchaotic attractors is as follows: $\dot{x} = Ax + f (x),$ (1) where x = (x₁, x₂, . . . , x_n) ^T is a state complex vector, x = x^r + jxⁱ, x^r = (u_1,u₃, . . . , u_2n-1) ^T, xⁱ = (u_2,u₄, . . . , u_2n) ^T, $j = \sqrt{- 1},$ T shows transpose, A ∈ R^n×n is real (or complex) matrix of framework parameters, f = (f₁, f₂, . . . , f_n) ^T is a vector of nonlinear complex capacities. The superscripts r and i denote the real and unbelievable individuals from the state complex vector x, separately. Comment 1. The most outrageous of the hyperchaotic complex structure can be qualified by (1), for instance, hyperchaotic complex Chen system. Remembering the true objective to present the aftereffects of our arrangement, we consider two hyperchaotic complex Chen systems with fluctuating most of the way in nonlinear terms which have been introduced and researched in [30]. The main hyperchaotic complex Chen framework is: $\dot{x} = α_{1} (y - x),$ (2) $\begin{matrix} \dot{y} & = & (γ_{1} - α_{1}) x - xz + γ_{1} y, \\ \dot{z} & = & 1 / 2 (\bar{x} y + x \bar{y}) + {jx}^{r} z^{r} - β_{1} z, \end{matrix}$ where x = (x₁, x₂, x₃) ^T = (x, y, z) ^T, α₁, β₁, γ₁ are certain parameters, x = u₁ + ju₂, y = u₃ + ju₄, z = u₅ + ju₆ are intricate capacities. Dabs disclose subsidiaries with respect to time and an overbar implies complex conjugate factors.

The hyperchaotic complex Chen is 6-dimensional consistent real self-sufficient framework. In the case α₁ = 22, β₁ = 2, γ₁ = 15.8 framework (2) has hyperchaotic attractor [30]. The second hyperchaotic complex Chen framework is: $\dot{x} = α_{2} (y - x),$ (3) $\begin{matrix} \dot{y} & = & (γ_{2} - α_{2}) x - xz + γ_{2} y, \\ \dot{z} & = & 1 / 2 (\bar{x} y + x \bar{y}) + {jx}^{r} y^{i} - β_{2} z . \end{matrix}$ At the point when α₂ = 22, β₂ = 2, γ₂ = 13 framework (3) has hyperchaotic conduct [30]. Frameworks (2) and (3) has the same straight sections in their conditions while there are fractional contrast in the nonlinear terms.

3. A scheme to perfect MCPS

We consider two hyperchaotic complex nonlinear models of the frame (1) with a comparative structure in straight terms and differing absolutely or to some degree in nonlinear terms. The first is the master system (we model the master system with the subscript m) as: ${\dot{x}}_{m} = {\dot{x}}_{m}^{r} + j {\dot{x}}_{m}^{i} = A_{1} x_{m} + {f (x}_{m}) .$ (4) What’s more, the second is the controlled slave framework (with subscript s) as: ${\dot{x}}_{s} = {\dot{x}}_{s}^{r} + j {\dot{x}}_{s}^{i} = A_{2} x_{s} + {g (x}_{s}) + L,$ (5) with the added substance complex controller L = (L₁, L₂, . . . , L_n) ^T = L^r + jLⁱ, L^r = (v₁, v₃, . . . , v_2n-1) ^T, Lⁱ = (v₂, v₄, . . . , v_2n) ^T.

Definition. Two complex dynamical models with hyperchaotic conduct in master-slave setup can model MCPS if there is a vector of the complex error work δ know, for example: $δ = δ^{r} + j δ^{i} = x_{s} + j x_{m} = H, as, t \to \infty,$ (6)where δ = (δ₁, δ₂, . . . , δ_n) ^T,x_mandx_sare the state complex vectors of the master and slave frameworks, $lim_{t ⟶ \infty} δ^{r} = lim_{t ⟶ \infty} ‖ x_{s}^{r} - x_{m}^{i} ‖ = H, lim_{t ⟶ \infty} δ^{i} = lim_{t ⟶ \infty} ‖ x_{s}^{i} + x_{m}^{r} ‖ = 0$ , δ ^r = (δ_{u
₁}, δ_{u
₃}, . . . , δ_{u
_2n-1}) ^T, δ ⁱ = (δ_{u
₂}, δ_{u
₄}, . . . , δ_{u
_2n}) ^TandHis vector of real constants.

Comment 2. The combination of the nonexistent piece of slave framework $x_{s}^{i}$ and the real piece of master framework $x_{m}^{r}$ is vanishing when t⟶ ∞. This demonstrates the significance of APS [4].

Comment 3. The error between the real piece of slave framework $x_{s}^{r}$ and the whimsical piece of master framework $x_{m}^{i}$ ascends to real constants as t ⟶ ∞ . This appears in the importance of PS [4].

Comment 4. MCPS fuses amongst APS and PS as shown from comments 2,3.

Theorem 1. On the off chance that the nonlinear controller is created as: $\begin{matrix} L & = & L^{r} + j L^{i} = - A_{2} x_{s} - {g (x}_{s}) - j [A_{1} x_{m} \\ + {f (x}_{m})] + Ω δ^{r} + j Ψ δ^{i} \\ = & - A_{2} x_{s}^{r} - g^{r} {(x}_{s} {) + A}_{1} x_{m}^{i} + f^{i} {(x}_{m}) + Ω δ^{r} \end{matrix}$ (7) $+ j [- A_{2} x_{s}^{i} - g^{i} {(x}_{s} {) - A}_{1} x_{m}^{r} - f^{r} {(x}_{m})] + Ψ δ^{i}],$ where $Ω$ and $Ψ$ are an askew lattice of the real corner to corner componentsζ_l and ψ_l, l = 1, 2, . . . , n, respectively.Then the slave framework (5) and the master framework (4) can be accomplished MCPS whenζ_l = 0, ψ_l < 0 .

Verification. From the meaning of MCPS: $δ = δ^{r} + j δ^{i} = x_{s} + j x_{m} .$ (8) In this way, $\begin{matrix} \dot{δ} & = & {\dot{δ}}^{r} + j {\dot{δ}}^{i} = {\dot{x}}_{s} + j {\dot{x}}_{m} = A_{2} x_{s} \\ + {g (x}_{s}) + j [A_{1} x_{m} + {f (x}_{m})] + L \end{matrix}$ (9) $\begin{matrix} = & {\dot{x}}_{s}^{r} + j {\dot{x}}_{s}^{i} + j [{\dot{x}}_{m}^{r} + j {\dot{x}}_{m}^{i}] \\ = & {\dot{x}}_{s}^{r} - {\dot{x}}_{m}^{i} + j [{\dot{x}}_{s}^{i} + {\dot{x}}_{m}^{r}] . \end{matrix}$ By utilizing the hyperchaotic complex frameworks (4) and (5), we get the error complex dynamical framework as: $\begin{matrix} \dot{δ} & = & {\dot{δ}}^{r} + j {\dot{δ}}^{i} = A_{2} x_{s}^{r} + g^{r} {(x}_{s} {) - A}_{1} x_{m}^{i} \\ - f^{i} {(x}_{m} {) + L}^{r} + j [A_{2} x_{s}^{i} + g^{i} {(x}_{s}) \\ + A_{1} x_{m}^{r} + f^{r} {(x}_{m}) + L^{i}] . \end{matrix}$ (10) Via isolating the real and the unbelievable parts in Equation (10), the error complex framework is composed as: ${\begin{matrix} {\dot{δ}}^{r} = A_{2} x_{s}^{r} + g^{r} {(x}_{s} {) - A}_{1} x_{m}^{i} - f^{i} {(x}_{m} {) + L}^{r}, \\ {\dot{δ}}^{i} = A_{2} x_{s}^{i} + g^{i} {(x}_{s} {) + A}_{1} x_{m}^{r} {+ f}^{r} {(x}_{m})] + L^{i} . \end{matrix}$ (11) By substituting from Equation (7) about L^r, Lⁱ in Equation (11) we get: ${\begin{matrix} {\dot{δ}}^{r} = Ω δ^{r}, \\ {\dot{δ}}^{i} = Ψ δ^{i}, \end{matrix} \Rightarrow \dot{δ} = Ω δ^{r} + j Ψ δ^{i}$ (12) In the event that we select ζ_l = 0, ψ_l < 0 (e . g . ψ_l = -1) , l = 1, 2, . . . , n, then Ω = diag (0, 0, . . . , 0) ,Ψ = diag (-1, - 1, . . . , -1) and Equation (12) takes the shape: $\begin{matrix} {\dot{δ}}^{r} = 0, \\ {\dot{δ}}^{i} = - δ^{i}, δ^{i} \to 0. \end{matrix} \Rightarrow δ^{r} \to c o n s \tan t s,$ (13) Utilizing Lyapunov soundness we get, δ ⁱ ⟶ 0 and δ ^r ⟶ steady vector as t⟶ ∞ and the MCPS is accomplished for all state factors. This completes the confirmation.

Comment 5. Another definition of complex phase synchronization has been examined in [31] for chaotic complex nonlinear frameworks. The difference between the two definitions can be explained in the following table:

Modified Complex Phase Synchronization (MCPS) $\begin{matrix} (MCPS) \end{matrix}$ In MCPS, we define the error in simple case: xs + jxm = h, as t⟶ ∞, where x = u₁ + ju₂, h is real constant [u_1s + ju_2s] + j[u_1m + ju_2m] = h, [u_1s + ju_2s] + [-u_2m + ju_1m] = h, [u_1s - u_2m] + j[u_2s + u_1m] = h, ⇒ u_1s - u_2m = h ⇒ (PS) ⇒ u_2s + u_1m = 0 ⇒ (APS)} ⇒ (MCPS) $\begin{matrix} Complex Phase Synchronization in [31] \end{matrix}$ In [31], we defined the error in simple case: x_s - jx_m = jh, as t ⟶ ∞,

where x = u₁ + ju₂, h is real constant [u_1s + ju_2s] + j[u_1m + ju_2m] = jh, [u_1s + ju_2s] + [-u_2m + ju_1m] = jh, [u_1s - u_2m] + j[u_2s + u_1m] = jh, ⇒ u_1s - u_2m = h ⇒ (APS) ⇒ u_2s + u_1m = 0 ⇒ (PS)} ⇒ (CPS)

Comment 6. In numerical propagation, one can figure the phases and modules errors of the master and slave systems. For any complex number ξ = ξ^r + jξⁱ, the phase θ_ξ and modulus P_ξ are registered as follows [21 –26]: $θ_{ξ} = {\begin{matrix} arctan (ξ^{i} / ξ^{r}) ξ^{r} > 0, ξ^{i} \geq 0, \\ 2 π + arctan (ξ^{i} / ξ^{r}) ξ^{r} > 0, ξ^{i} < 0, \\ π + arctan (ξ^{i} / ξ^{r}) ξ^{r} < 0 . \end{matrix}$ (14) and $P_{ξ} = \sqrt{(ξ^{r})^{2} + (ξ^{i})^{2}} .$ (15) The MCPS occurs if the phase errors are constrained and can run chaotically or hyperchaotically inside a little range [4, 11]. While our system differentiates MCPS in the equations where these modulus errors are chaotic or hyperchaotic, and uncorrelated (straightly free). In the extensive variation of complex synchronizations, the phases and the modulus errors go to zero or specific characteristics. In any case, in MCPS, these errors have another property, move chaotically or hyperchaotically.

Finally, our arrangement is clarified by using it for two hyperchaotic edifices Chen models while fluctuating for the most part in nonlinear terms inSection 4.

4. Example

To prove the aftereffects of the recommended design, a trademark case is resolved. Two of the general interminable hyperchaotic systems of the shape (1), two hyperchaotic complex Chen frameworks (2) and (3), are brought for example with numerical reproductions simulations.

Assume the master framework as: $\begin{matrix} {\dot{x}}_{m} & = & α_{1} (y_{m} - x_{m}), \\ {\dot{y}}_{m} & = & (γ_{1} - α_{1}) x_{m} + γ_{1} y_{m} - x_{m} z_{m}, \end{matrix}$ (16) ${\dot{z}}_{m} = 1 / 2 ({\bar{x}}_{m} y_{m} + x_{m} {\bar{y}}_{m}) + {jx}_{m}^{r} z_{m}^{r} - β_{1} z_{m},$ which can be revised as A₁x_m + f (x_m ), where: $\begin{matrix} A_{1} & = & (\begin{matrix} - α_{1} & α_{1} & 0 \\ (γ_{1} - α_{1}) & γ_{1} & 0 \\ 0 & 0 & - β_{1} \end{matrix}), \\ {f (x}_{m}) & = & (\begin{matrix} 0 \\ - x_{m} z_{m} \\ 1 l 1 / 2 ({\bar{x}}_{m} y_{m} + x_{m} {\bar{y}}_{m}) + {jx}_{m}^{r} z_{m}^{r} \end{matrix}) . \end{matrix}$ Let, in the simulation, the parameters of (16) are picked as α₁ = 22, β₁ = 2, γ₁ = 15.8 and the initial estimations of the master framework state vector are taken as (x_m (0) , y_m (0) , z_m (0)) ^T = (1 + 2j, 3 + 4j, 5 + j6) ^T .

Consider the slave framework as:

$\begin{matrix} \dot{x_{s}} & = & α_{2} (y_{s} - x_{s}) + L_{1}, \\ \dot{y_{s}} & = & (γ_{2} - α_{2}) x_{s} + γ_{2} y_{s} - x_{s} z_{s} + L_{2}, \end{matrix}$ (17) $\dot{z_{s}} = 1 / 2 ({\bar{x}}_{s} y_{s} + x_{s} {\bar{y}}_{s}) + {jx}_{s}^{r} y_{s}^{i} - β_{2} z_{s} + L_{3},$

where: $\begin{matrix} A_{2} & = & (\begin{matrix} - α_{2} & α_{2} & 0 \\ (γ_{2} - α_{2}) & γ_{2} & 0 \\ 0 & 0 & - β_{2} \end{matrix}), \\ {g (x}_{s}) & = & (\begin{matrix} 0 \\ - x_{s} z_{s} \\ 1 l 1 / 2 ({\bar{x}}_{s} y_{s} + x_{s} {\bar{y}}_{s}) + {jx}_{s}^{r} y_{s}^{i} \end{matrix}) . \end{matrix}$

The parameters of (17) and the initial estimations of the master framework state vector are chosen as α₂ = 22, β₂ = 2, γ₂ = 13 and (x_s (0) , y_s (0) , z_s (0)) ^T = (14 - j, 18 - 4j, 28 - 6j) ^T .

As per the theorem 1, the complex control capacities L₁, L₂ and L₃ are figured as follows: $(\begin{matrix} L_{1} \\ L_{2} \\ L_{3} \end{matrix}) = (\begin{matrix} - α_{2} (y_{s} - x_{s}) - j α_{1} (y_{m} - x_{m}) \\ - (γ_{2} - α_{2}) x_{s} - {γ_{2} y}_{s} + x_{s} z_{s} - j (γ_{1} - α_{1}) x_{m} - j γ_{1} y_{m} + {jx}_{m} z_{m} \\ β_{2} z_{s} - Γ_{1} - x_{m}^{r} z_{m}^{r} - j (Γ_{2} - β_{1} z_{m} + x_{s}^{r} y_{s}^{i}) \end{matrix}) + Ω δ^{r} + j Ψ δ^{i},$ (18)

where $Γ_{1} = \frac{1}{2} ({\bar{x}}_{s} y_{s} + x_{s} {\bar{y}}_{s}), Γ_{2} = \frac{1}{2} ({\bar{x}}_{m} y_{m} + x_{m}$ ${\bar{y}}_{m}), Ψ = diag (- 1, - 1, - 1), Ω = diag (0, 0, 0)$ , δ ^r = (δ_{u
₁}, δ_{u
₃}, δ_{u
₅}) ^T, δ ⁱ = (δ_{u
₂}, δ_{u
₄}, δ_{u
₆}) ^T.

Fig. 1.

MCPS between two systems (16) and (17). (a) u_1m and u_2s versus t. (b) u_2m and u_1s versus t. (c) u_3m and u_4s versus t. (d) u_4m and u_3s versus t. (e) u_5m and u_6s versus t: (f) u_6m and u_5s versus t.

Fig. 2.

MCPS error between two systems (16) and (17). (a) δ_{u
₁} versus t. (b) δ_{u
₂} versus t. (c) δ_{u
₃} versus t. (d) δ_{u
₄} versus t. (e) δ_{u
₅} versus t. (f) δ_{u
₆} versus t.

Fig. 3.

The phases errors of systems (16) and (17) and the relations between these errors.

Frameworks (16) and (17) with the controllers (refCONN) are settled numerically to prove the authenticity of our examinations. In Fig. 1 the MCPS is addressed. Clearly, the MCPS is joined between the PS and the APS. The APS shows up in Figs. 1a, 1c, 1e while the PS occurs in Figs. 1b, 1d, 1f. Its unmistakable from Fig. 1, the state variables of the master system synchronizes with substitute state elements of the slave structure and this phenomenon does not appear in the composition examined. Fig. 2 demonstrates MCPS errors. These errors approach constant values in Figs. 2a, 2c, 2e which infers the PS is refined as required by the scientific course of action. While these errors go to zero in Figs. 2b, 2d, 2f, obviously from the investigative idea of our arrangement. The PS appears, since ${\dot{δ}}^{r} = 0$ and δ ^r = (δ_{u
₁}, δ_{u
₃}, $δ_{u_{5}})^{T} = x_{s}^{r} (0) - x_{m}^{i} (0) = (12,$ 14, 22) .

We figure the phases distinction θ_{x
_m} - θ_{x
_s}, θ_{y
_m} - θ_{y
_s}, θ_{z
_m} - θ_{z
_s} and plot these in Figs. 3a, 3b, 3c. It is clear the phase contrasts θ_{x
_m} - θ_{x
_s}, θ_{y
_m} - θ_{y
_s}, θ_{z
_m} - θ_{z
_s} are constrained and flounder about the starting point in the chaotic or hyperchaotic way which infers that MCPS is proficient. Figs. 3d, 3e, 3f presents the relations between θ_{x
_m} - θ_{x
_s}, θ_{y
_m} - θ_{y
_s}, θ_{z
_m} - θ_{z
_s}. We found that these relations are chaotic or hyperchaotic attractors and this disclosure shows that θ_{x
_m} - θ_{x
_s}, θ_{y
_m} - θ_{y
_s}, θ_{z
_m} - θ_{z
_s} move chaotically or hyperchaotically and MCPS is capable.

Figs. 4a, 4b, 4c demonstrate the modules errors p_{x
_m} - p_{x
_s}, p_{y
_m} - p_{y
_s}, p_{z
_m} - p_{z
_s}, individually. While p_{x
_m} - p_{x
_s} versus p_{y
_m} - p_{y
_s}, p_{x
_m} - p_{x
_s} versus p_{z
_m} - p_{z
_s}, p_{y
_m} - p_{y
_s} versus p_{z
_m} - p_{z
_s} are presented in Figs. 4d, 4e, 4f. Plainly, from Figure 4 the modules errors are moving chaotically or hyperchaotically and uncorrelated (Straightly independent) clearly from our arrangement.

Fig. 4.

The modules errors of systems (16) and (17) and the relations between these errors.

5. Secure communications based on the MCPS results

The fundamental thought of utilization of hyperchaos in secure communication depends on utilizing hyperchaotic nonlinear oscillator as a broadband signal age. The signal is joined with a message to deliver a complex signal which is transmitted from the transmitter to the collector. At the receiver, the pseudorandom is created through the reverse task and unique message is recouped. In order for this plan to appropriately work, the collector must synchronize powerfully enough in order to concede the little irritation in the drive motion because of the expansion of the message [29, 30].

The energy of data signal must be lower than that of the hyperchaotic signal to effectively cover the data signal. The signal from the master serves two needs: to control the slave structure in order to synchronize it with the master and to convey the data signal, much the same as some other communication scheme, the reason for hyperchaos secure communication is to conceal message amid transmission. The reasonableness of hyperchaotic systems for application in secure communication depends on the element of hyperchaotic bearers, for example, broadband or wide range (which decreases the blurring of the signal and increases the transmission limit); orthogonality (which lessens signal mutilation); affectability to slight changes in the initial equations and system parameter and intricacy and noise similarity elements which prompt unconventionality, accordingly making extraction of shrouded message difficult.

The MCPS of hyperchaotic complex systems in which a state variable of the master system synchronizes with an alternate state variable of the slave structure is an enabling sort of synchronization as it contributes wonderful security in secure communication. We study a system (16) as transmitter structure and the system (17) as receiver structure. For a certain reason, we pick subjectively the information motion as r (t) =10 sin t cos t . Take $\hat{r} (t) = r (t) + u_{5 m}$ and accept that $\hat{r} (t)$ is added to the variable u_6m $\Rightarrow \bar{r} (t) = \hat{r} (t) + u_{2 m} = r (t) + u_{5 m} + u_{2 m} .$ Figure 5demonstrates the hyperchaotic system to cover and recoup the data bearing signal r (t) . It delineates a draw intended for our communication scheme utilizing MCPS, in which the transmitter structure and the collector system are the hyperchaotic system (16) and (17), individually. Numerical results of employment to secure communication are presented in Figure 6. The information signal r (t) and the transmitted signal $\bar{r} (t)$ are presented in Figures 6a,b, independently. It is clear that the information signal r (t) is conceal by adding it to the hyperchaotic signals u_5m, u_2m . If we utilize the past plans in [12 , 32–34] the information signal is recouped by subtracting the hyperchaotic signals u_5s, u_2s, which is created by the slave structure, from the received signal. Be that as it may, in our plan, the information signal is recouped by subtracting u_1s - [u_1s (0) - u_2m (0)] and adding u_6s to the received signal (See Figure 5). According to the outcomes of past section u_1s (0) - u_2m (0) =12 and this value can change contingent upon the determination of the initial equations. Along these lines, we find that our plan to actuate the association security relies upon the decision of the initial values and this has not occurred before in secure communications. Self-evident, we subtract and include the hyperchaotic signals with a constant value to recuperate the information signal which is an uncommon property in secure communication and does not exist in the composition. This uncommon phenomenon occurs as a result of the MCPS contains or incorporates two sorts of synchronizations (PS and APS) and the state variable of the fundamental system synchronizes with a different state variable of the slave structure. These highlights can upgrade and enhance the security of communication. This clarifies the upsides of our plan over the current one. In this way, the recovered data signal, whatever is communicated by r^∗ (t) = r (t) + u_6s - u_1s + u_1s (0) - u_2m (0), is presented in Figure 6c (in light of the way that resulting to achieving MCPS u_5m + u_6s = 0, u_1s - u_2m = 12). Figure 6d models the error motion between the first data signal and the recuperated one and demonstrates the oversight movement between the first information signal and the recovered one. From Figure 6d, it is not difficult to find that the information signal r (t) is recovered definitely after a short transient.

Fig. 5.

A scheme to achieve secure communication between the hyperchaotic frameworks (16) and (17).

Fig. 6.

Simulation results of secure communication using MCPS of two identical hyperchaotic complex Chen frameworks.

From the above, we can state, the secure communication in light of MCPS has the accompanying points of interest:

1- High security in data exchange, on the grounds that the state variable of the master system synchronizes with another state variable of the slave structure. This component has not been presented in past examinations in secure communications.

2- Previous security communications plans rely upon one kind of synchronization. Yet, in our plan, we utilized two sorts of synchronizations (PS and APS). This would prompt expanded security in the transmission of data.

3- We see that our plan to activate the association security relies upon the choice of the initial values and this has not happened beforehand in secure communications.

6. Conclusions

We have invented a new sort of modified complex synchronization of hyperchaotic complex nonlinear structures. It is shown that the visibility of this sort of synchronization can be seen for complex (not real) nonlinear structures as well as the state variable of the master system synchronizes with another state variable of the slave system which does not appear in the composition outline. Our definition based on the connection between two sorts of synchronizations in the written work, phase synchronization PS and anti-phase synchronization APS. Another arrangement is obtained in light of the theory’s strength to acknowledge MCPS of two hyperchaotic complex structures, including the same direct terms with moving absolutely or to some degree in non-straight terms. In this arrangement, we have considered scientifically the complex control capacities which gave rise to MCPS and tested to contemplate, MCPS of two hyper chaotic complex Chen structures with changing most of the way in nonlinear terms. Finally, in the perspective of the state variable of the transmitter structure (16) synchronizes with another state variable of the collector system (17), a fundamental secure communication project is planned by methods of hyperchaotic-veiling. It is found that the hyperchaotic-veiling technique which includes the transmitted signal specifically to a similarly strong hyperchaotic signal to outline the information carrier wave has strong security and additionally recover the information signal enough. Our secure communication scheme in light of MCPS has a few highlights, and does not appear in the writing, for example, (i) great insurance and security in information exchange, because the state variable of the transmitter structure synchronizes with another state variable of the collector or receiver system, (ii) new arrangement has been given, where we have adopted two surely understood sorts of synchronizations (PS and APS), (iii) our plan was to initiate the association security relies upon the decision of the initial values which (as far as we are aware) has not been considered before in secure communications plans.

References

Boccaletti ,

Kurths ,

Osipov ,

D.L.

Valladares and

C.S.

Zhou , The synchronization of chaotic systems, Phys Rep366 (2002), 1–101.

L.M.

Pecora and

T.L.

Carroll , Synchronization in chaotic frameworks, Phys Rev Lett64 (1990), 821–830.

G.M.

Mahmoud and

E.E.

Mahmoud , Complete synchronization of chaotic complex nonlinear frameworks with uncertain parameters, Nonlinear Dyn62 (2010), 875–882.

G.M.

Mahmoud and

E.E.

Mahmoud , Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear frameworks, Nonlinear Dyn61 (2010), 141–152.

Juan and

Xing-yuan , Nonlinear observer based phase synchronization of chaotic frameworks, Phys Lett A369 (2007), 294–298.

Liu , Anti-phase synchronization in coupled chaotic oscillators, Phys Rev E73 (2006), 057203.

A.E.

Hramov and

A.A.

Koronovskii , Generalized synchronization onset, Europhys Lett6 (2005), 01–907.

G.M.

Mahmoud and

E.E.

Mahmoud , Synchronization and control of hyperchaotic complex Lorenz system, Mathematics and Computers in Simulation80 (2010), 2286–2296.

G.H.

Li , Modified projective synchronization of chaotic framework, Chaos Solitons Fractals32 (2007), 1786–1790.

10.

Li ,

Liao and

Wong , Lag synchronization of hyperchaos with application to secure communications, Chaos, Solitons & Fractals23 (2005), 183–193.

11.

E.E.

Mahmoud , Modified projective phase synchronization of chaotic complex nonlinear systems, Math Comput Simul89 (2013), 69–85.

12.

G.M.

Mahmoud ,

E.E.

Mahmoud and

A.A.

Arafa , Projective synchronization for coupled partially linear complex-variable frameworks with known parameters, Math Meth Appl Sci40 (2017), 1214–1222.

13.

E.E.

Mahmoud and

M.A.

AL-Adwani , Dynamical behaviors, control and synchronization of a new chaotic model with complex variables and cubic nonlinear terms, Results in Physics7 (2017), 1346–1356.

14.

Anischenko ,

Vadivasova ,

Postnov and

Safanova , Synchronization of chaos, Int J Bifur Chaos2 (1992), 633–644.

15.

Rosa Jr ,

W.B.

Pardo ,

C.M.

Ticos ,

J.A.

Walkenstein and

Monti , Phase synchronization of chaos in a plasma discharge tube, Int J Bifur Chaos10 (2000), 2551–2563.

16.

Tallon-Baudry ,

Bertrand ,

Delpuech and

Pernier , Stimulus specificity of phase-locked and non-phase-locked 40 Hz visual responses in human, Journal of Neuroscience16 (1996), 4240–4249.

17.

Wang ,

I.Z.

Kiss and

J.L.

Hudson , Experiments on arrays of globally coupled chaotic electrochemical oscillators: Synchronization and clustering, Chaos10 (2000), 248–256.

18.

Dabrowski ,

Galias and

Ogorzalek , Observations of phase synchronization phenomena in one-dimensional arrays of coupled chaotic electronic circuits, Int J Bifurc and Chaos10 (2000), 2391–2398.

19.

L.Y.

Cao and

Y.C.

Lai , Antiphase synchronism in chaotic frameworks, Phys Rev E58 (1998), 382–386.

20.

Jiang ,

Liu ,

Zhang and

Yu , Anti-phase synchronization and symmetry-breaking bifurcation of impulsively coupled oscillators, Communications in Nonlinear Science and Numerical Simulation39 (2016), 199–208.

21.

E.E.

Mahmoud , An unusual kind of complex synchronizations and its applications in secure, Eur Phys J Plus132 (2017), 466.

22.

E.E.

Mahmoud , Complex complete synchronization of two nonidentical hyperchaotic complex nonlinear frameworks, Math Methods App Sci37 (2014), 321–328.

23.

E.E.

Mahmoud and

K.M.

Abualnaja , Complex lag synchronization of two identical chaotic complex nonlinear frameworks, Cent Eur J Phys12 (2014), 63–69.

24.

Wu ,

Duan and

Fu , Complex projective synchronization in coupled chaotic complex dynamical frameworks, Nonlinear Dyn69 (2012), 771–779.

25.

G.M.

Mahmoud and

E.E.

Mahmoud , Complex modified projective synchronization of two chaotic complex nonlinear frameworks, Nonlinear Dyn73 (2013), 2231–2240.

26.

E.E.

Mahmoud and

F.S.

Abood , A new nonlinear chaotic complex model and it's complex antilag synchronization, Complexity2017, ID 3848953.

27.

G.M.

Mahmoud ,

Bountis and

E.E.

Mahmoud , Active control and global synchronization of the complex Chen and Lü frame works, Int J Bifurcat Chaos17 (2007), 4295–4308.

28.

E.E

Mahmoud , Dynamics and synchronization of new hyper-chaotic complex Lorenz framework, Mathematical and Computer Modelling55 (2012), 1951–1962.

29.

E.E

Mahmoud , Adaptive anti-lag synchronization of two identical or non-identical hyperchaotic complex nonlinear frame-works with uncertain parameters, J Franklin Inst349 (2012), 1247–1266.

30.

E.E.

Mahmoud , Generation and suppression of a new hyperchaotic nonlinear model with complex variables, Applied Mathematical Modelling38 (2014), 4445–4459.

31.

E.E.

Mahmoud , A novel sort of complex synchronizations, Acta Physica Polonica B48 (2017), 1441–1454.

32.

E.E.

Mahmoud and

S.M.

Abo-Dahab , Dynamical properties and complex anti synchronization with applications to secure communications complex anti synchronization with applications to secure communications for a novel chaotic complex nonlinear model, Chaos, Solitons and Fractals106 (2018), 273–284.

33.

Wua ,

Wanga and

Lu , Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication, Nonlinear Analysis: Real World Applications13 (2012), 1441–1450.

34.

E.E.

Mahmoud and

F.S.

Abood , A novel sort of adaptive complex synchronizations of two indistinguishable chaotic complex nonlinear models with uncertain parameters and its applications in secure communications, Results in Physics7 (2017), 4174–4182.