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Abstract

The principal target of this work is to introduce and examine a novel kind of complex synchronization. This sort may be called complex anti-synchronization. There are surprising properties of complex anti-synchronization that do not exist in the writing, for example, (1) this sort of synchronization can dissect just for complex nonlinear frameworks. (2) The complex anti-synchronization contains or connects two sorts of synchronizations (anti-synchronization and complete synchronization). Anti-synchronization happens between the real part of main framework and the imaginary part of the slave framework, although complete synchronization accomplishes between the real part of slave framework and the imaginary part of the main framework. (3) In complex anti-synchronization, the attractors of the essential and slave structures are moving symmetrical to each other with a similar structure. (4) The state variable of the standard framework synchronizes with an other state variable of the slave structure. An explanation of complex anti-synchronization is presented for two indistinguishable chaotic complex nonlinear frameworks. In view of the Lyapunov function, a plan is intended to accomplish complex anti-synchronization of disordered or chaotic attractors of these frameworks. The effectiveness of the obtained results is outlined by a reenactment illustration. Numerical outcomes are plotted to show state variable, modulus errors, phase errors and the development of the attractors of these chaotic frameworks after synchronization to demonstrate that complex anti-synchronization is accomplished.

Keywords

Complex anti-synchronization chaotic Lyapunov function complex

Introduction

Nonlinear deterministic dynamical frameworks are pervasive in nature and are outstanding for their unusual property of delicate reliance on introductory conditions which offers ascend to their worldly unpredictability and obvious arbitrariness. This property for sure prompts a few intriguing qualities that have been examined for a long time in the fields of sciences, medicine and engineering,¹ electronics,² electric vehicles³ and the investigation of nonlinear frameworks in arithmetic.⁴ Specifically, clamorous elements, bifurcation structures, multistability of attractors, synchronization elements and numerous more have been thoroughly examined in endeavors to clarify and depict nonlinear practices and their potential applications. Synchronization which was initially proposed by Pecora and Carroll⁵ is one of the most critical of nonlinear frameworks and has been considered as an incredible leap forward in the confused flow,⁶ because of its potential applications in demonstrating cerebrum exercises, substance responses, and all the more significant in data handling and secure correspondence.⁷ These anticipated applications have set off the tremendous research consideration given to anti-synchronization (AS) for more than three decades. Expanding enthusiasm for the investigation of synchronization of chaotic frameworks has prompted the disclosure of different strategies and sorts of synchronization. In ordering AS, we particularly specify here the first thought of identical complete synchronization,^5,8 general chaos synchronization,⁹ stage synchronization,¹⁰ lag (or slack) synchronization,¹¹ generalized synchronization,^12,13 projective synchronization,^14–16 AS^17,18 and work projective synchronization.¹⁹ A vast number of past chips away at disarray synchronization have been gone for accomplishing synchronization between state factors of the principle frameworks and that of the comparable slave framework with real or complex factors.

There are some various types of synchronization which cannot be examined for authentic or genuine dynamical structures, for example, module-organize synchronization,²⁰ complex complete (or full) synchronization (CCS),⁸ complex lag (or slack) synchronization (CLS),²¹ complex anti-lag (or anti-slack) synchronization (CALS)²² and involved changed or modified projective synchronization (CMPS).²³ These different kinds of synchronizations are investigated for muddled or confused complicated nonlinear systems. In a complex state, there are two basic sums phase (or stage) and module. Thus, the acts of the stage and module are mulled over in.^20,22

Hold the turbulent (or chaotic) complex nonlinear system with explicit or certain parameters as pursues

{\begin{matrix} \overset{\cdot}{x} = Φ x + F (x, z) \\ \overset{\cdot}{z} = g (x, \bar{x}, z) \end{matrix}

(1)

where $x = [x_{1}, x_{2}, \dots, x_{n}]^{T_{r}}$ is a complex state vector; $x = x^{r} + j x^{i}$ , $x^{r} = [u_{1}, u_{3}, \dots, u_{2 n - 1}]^{T_{r}}$ , $x^{i} = [u_{2}, u_{4}, \dots, u_{2 n}]^{T_{r}}$ , $j = \sqrt{- 1}$ ; $T_{r}$ denotes a transpose, $Φ \in R^{n \times n}$ is complex or real matrix of framework parameters, $F = (f_{1}, f_{2}, \dots, f_{n})^{T_{r}}$ is a nonlinear vector capacity, $g : c^{k} \times R \to R$ is a real function of nonlinear and direct elements and $z$ is a real vector of variables. Spots speak to subordinates regarding time, an “overbar” signifies complex conjugate factors and superscripts $(r)$ and $(i)$ stay for the genuine and fanciful elements of the complex state vector $x$ .

Comment 1

The most extraordinary of the turbulent complex systems can be depicted by equation (1), for instance, disordered complex Lorenz, Chen and Lü structures.^24–26

In this paper, we complete our investigation about complex anti-synchronization (CAS). In Mahmoud,²⁶ we study CAS with uncertain parameters. But in this manuscript, we study CAS with know parameters. We study the description of CAS of two same structures of the shape (1) with specific parameters. This kind of synchronization can be looked into for chaos and hyperchaos complex nonlinear structures figuratively speaking. The possibility of CAS can be viewed as syncretizing between AS¹⁷ and CS.⁸ AS occurs within the genuine part of the fundamental structure and the fanciful part of the slave model; however, CS achieves among the genuine part of the slave system and the nonexistent piece of the principal model. In CAS, the state variable of the main system synchronizes with other state factors of the slave structure. In this manner, the (CAS) gives progressively essential safety in secure communications. We might want to propose a global plan to study and to achieve the (CAS) of pair indistinct befuddled complicated nonlinear structures in the shape (1). By a particular end objective with display, the aftereffects about our plan for two undefined systems of the state (1), we choose, to the example, the disarranged mind boggling nonlinear model²⁷

\begin{matrix} \overset{\cdot}{x} = a (y - x) + yzw \\ \overset{\cdot}{y} = b (x + y) - xzw \\ \overset{\cdot}{z} = 1 / 2 (\bar{x} y + x \bar{y}) w - cz \\ \overset{\cdot}{w} = 1 / 2 (\bar{x} y + x \bar{y}) z - dw \end{matrix}

(2)

where $x = [x_{1}, x_{2}]^{T_{r}} = [x, y]^{T_{r}}, z = [z, w]^{T_{r}}$ , a, b, c and d are certain or individual parameters, $x = u_{1} + j u_{2}, y = u_{3} + j u_{4}$ are mind boggling capacities and $u_{l} (l = 1, \dots, 4)$ , $z = u_{5}, w = u_{7}$ are real capacities. Dabs name subordinates with regard to time and the overbar demonstrates conjugate complex variables. Our model in chaos action is a six-dimensional constant real self-confident framework. If the event $a = 30$ , $b = 14$ , $c = 1$ and $d = 2$ , framework (2) has turbulent trajectory. It is exciting to remark that the one positive Lyapunov example of the chaotic complex nonlinear model (2) is $λ_{1} = 4.47$ .²⁷

The course of action of this paper is as per the taking after: In section “A plan to accomplish CAS,” we layout the suggested plan to accomplish CAS of two vague befuddled complicated nonlinear structures in the shape (1). In Segment 3, we apply the results of our conspire to achieve CAS, for occasion, between two undefined disorganized complex nonlinear systems within the shape (2). Numerical results are used to demonstrate the authenticity of this survey. Finally, the essential fulfillment of our tests is consolidated in Segment 4.

A plan to accomplish CAS

We study two indistinguishable disorganized complicated nonlinear structures of the shape (1); one is the main structure (we act the main model including the index $(m)$ ) as

{\begin{matrix} {\dot{x}}_{m} = {\dot{x}}_{m}^{r} + j {\dot{x}}_{m}^{i m} = Φ x_{m} + F (x_{m}) \\ \dot{z} = g (x, \bar{x}, z) \end{matrix}

(3)

likewise, the other is the controlled slave structure (including index s) as

{\overset{\cdot}{x}}_{s} = {\overset{\cdot}{x}}_{s}^{r} + j {\overset{\cdot}{x}}_{s}^{im} = Φ x_{s} + F (x_{s}) + L

(4)

such that the added element complex controller $L = [L_{1}, L_{2}, \dots, L_{n}]^{T_{r}} = L^{r} + j L^{im}$ , $L^{r} = [l_{1}, l_{3}, \dots, l_{2 n - 1}]^{T_{r}}$ , $L^{im} = [l_{2}, l_{4}, \dots, l_{2 n}]^{T_{r}}$ .

Definition

Pair joined complex dynamical structures in a main-slave arrangement may display CAS if the vector of the unpredictable mistake or error function ∇ define as²⁶

\nabla = \nabla^{r} + j \nabla^{im} = lim_{t \to \infty} ‖ x_{s} + j x_{m} ‖ = 0

(5)

where $\nabla = (\nabla_{1}, \nabla_{2}, \dots, \nabla_{n})^{T}$ , $x_{m}$ and $x_{s}$ are the state complex vectors of the main and slave structures where they are pair indistinguishable, respectively

\nabla^{r} = {(\nabla_{u_{1}}, \nabla_{u_{3}}, \dots, \nabla_{u_{2 n - 1}})}^{T} = x_{s}^{r} - x_{m}^{im} = 0

(6)

and

\nabla^{im} = {(\nabla_{u_{2}}, \nabla_{u_{4}}, \dots, \nabla_{u_{2 n}})}^{T} = x_{s}^{im} + x_{m}^{r} = 0

(7)

Comment 2

In equation (6), the wrong or error among the genuine part of slave model or framework $x_{s}^{r}$ and the imaginary part of main framework $x_{m}^{im}$ perform to zero essentially $t \to \infty$ (CS).^7,8

Comment 3

From equation (7), the total of the fanciful part of slave structure $x_{s}^{im}$ and the genuine ( or real) part of main structure $x_{m}^{r}$ is vanished during $t \to \infty$ (AS).¹⁷

Comment 4

1. The difference among CAS and AS¹⁷ can be delineated from

\begin{matrix} Complex anti - synchronization (CAS) Anti - synchronization (AS) \\ \begin{matrix} In CAS, we define the error in simple case : \\ x_{s} + j x_{m} = 0, as t \to \infty, where x = u_{1} + j u_{2} \\ [u_{1 s} + j u_{2 s}] + j [u_{1 m} + j u_{2 m}] = 0 \\ [u_{1 s} + j u_{2 s}] + [- u_{2 m} + j u_{1 m}] = 0 \\ [u_{1 s} - u_{2 m}] + j [u_{2 s} + u_{1 m}] = 0 \\ \begin{matrix} \Rightarrow u_{1 s} - u_{2 m} = 0 \Rightarrow (CS) \\ \Rightarrow u_{2 s} + u_{1 m} = 0 \Rightarrow (AS) \end{matrix}} \Rightarrow (CAS) \end{matrix} \begin{matrix} In AS, the error in simple case : \\ x_{s} + x_{m} = 0, as t \to \infty, where x = u_{1} + j u_{2} \\ [u_{1 s} + j u_{2 s}] + [u_{1 m} + j u_{2 m}] = 0 \\ [u_{1 s} + j u_{2 s}] + [u_{1 m} + j u_{2 m}] = 0 \\ [u_{1 s} + u_{1 m}] + j [u_{2 s} + u_{2 m}] = 0 \\ \begin{matrix} \Rightarrow u_{1 s} + u_{1 m} = 0 \Rightarrow (AS) \\ \Rightarrow u_{2 s} + u_{2 m} = 0 \Rightarrow (AS) \end{matrix}} \Rightarrow (AS) \end{matrix} \end{matrix}

2. The difference among CAS and CCS⁸ can be delineated from

\begin{matrix} Complex anti - synchronization (CAS) & Complex complete snchronization (CCS) \\ \begin{matrix} In CAS, we define the error in simple case : \\ x_{s} + j x_{m} = 0, as t \to \infty, where x = u_{1} + j u_{2} \\ [u_{1 s} + j u_{2 s}] + j [u_{1 m} + j u_{2 m}] = 0 \\ [u_{1 s} + j u_{2 s}] + [- u_{2 m} + j u_{1 m}] = 0 \\ [u_{1 s} - u_{2 m}] + j [u_{2 s} + u_{1 m}] = 0 \\ \begin{matrix} \Rightarrow u_{1 s} - u_{2 m} = 0 \Rightarrow (CS) \\ \Rightarrow u_{2 s} + u_{1 m} = 0 \Rightarrow (AS) \end{matrix}} \Rightarrow (CAS) \end{matrix} & \begin{matrix} In CCS, the error in simple case : \\ x_{s} - j x_{m} = 0, as t \to \infty, where x = u_{1} + j u_{2} \\ [u_{1 s} + j u_{2 s}] - j [u_{1 m} + j u_{2 m}] = 0 \\ [u_{1 s} + j u_{2 s}] - [j u_{1 m} - u_{2 m}] = 0 \\ [u_{1 s} + u_{2 m}] + j [u_{2 s} - u_{1 m}] = 0 \\ \begin{matrix} \Rightarrow u_{1 s} + u_{2 m} = 0 \Rightarrow (AS) \\ \Rightarrow u_{2 s} - u_{1 m} = 0 \Rightarrow (CS) \end{matrix}} \Rightarrow (CCS) \end{matrix} \end{matrix}

Comment 5

From Comments 2–4, CAS combines between CS and AS.

Theorem 1

In the event that a nonlinear controller is originated essentially

\begin{matrix} L & = L^{r} + j L^{im} = - Φ x_{s} - F (x_{s}) - j [Φ x_{m} + F (x_{m})] - ξ \nabla \\ = - Φ x_{s}^{r} - F^{r} (x_{s}) + Φ x_{m}^{im} + F^{im} (x_{m}) - ξ \nabla^{r} \\ + j [- Φ x_{s}^{im} - F^{im} (x_{s}) - Φ x_{m}^{r} - F^{r} (x_{m}) - ξ \nabla^{im}] \end{matrix}

(8)

at that point the slave framework (4) synchronizes the main or master framework (3) asymptotically under the requirements of CAS, where $ξ > 0$ .

Proof

Of the meaning of CAS

\nabla = \nabla^{r} + j \nabla^{im} = x_{s} + j x_{m}

(9)

\begin{matrix} \overset{\cdot}{\nabla} = {\overset{\cdot}{\nabla}}^{r} + j {\overset{\cdot}{\nabla}}^{im} = {\overset{\cdot}{x}}_{s} + j {\overset{\cdot}{x}}_{m} \\ \overset{\cdot}{\nabla} = {\overset{\cdot}{\nabla}}^{r} + j {\overset{\cdot}{\nabla}}^{im} = [{\overset{\cdot}{x}}_{s}^{r} - {\overset{\cdot}{x}}_{m}^{im}] + j [{\overset{\cdot}{x}}_{s}^{im} + {\overset{\cdot}{x}}_{m}^{r}] \end{matrix}

(10)

From disorganized complex frameworks (3) and (4), the blunder or error complex dynamical framework was identified as

\begin{matrix} \overset{\cdot}{\nabla} = & {\overset{\cdot}{\nabla}}^{r} + j {\overset{\cdot}{\nabla}}^{im} = Φ x_{s}^{r} + F^{r} (x_{s}) - F x_{m}^{im} - F^{im} (x_{m}) + L^{r} \\ + j [Φ x_{s}^{im} + F^{im} (x_{s}) + Φ x_{m}^{r} + F^{r} (x_{m}) + L^{im}] \end{matrix}

(11)

By dividing the genuine and the fanciful elements in equation (11), the blunder complex model is formulated essentially as

{\begin{matrix} {\overset{\cdot}{\nabla}}^{r} = Φ x_{s}^{r} + F^{r} (x_{s}) - Φ x_{m}^{im} - F^{im} (x_{m}) + L^{r} \\ {\overset{\cdot}{\nabla}}^{i} = Φ x_{s}^{im} + F^{im} (x_{s}) + Φ x_{m}^{r} + F^{r} (x_{m}) + L^{im} \end{matrix}

(12)

For the positive parameters in the structure (12), the unmistakable positive measure of the Lyapunov work is⁸

\begin{matrix} V (t) & = \frac{1}{2} [{(\nabla^{r})}^{T} \nabla^{r} + {(\nabla^{i})}^{T} \nabla^{im}] \\ = \frac{1}{2} (\sum_{l = 1}^{n} \nabla_{u_{2 l - 1}}^{2} + \sum_{l = 1}^{n} \nabla_{u_{2 l}}^{2}) \end{matrix}

(13)

Remark presently that the total time subordinate of $V (t)$ along the heading of the mistake model (12) is according to the taking after

\begin{matrix} \overset{\cdot}{V} (t) & = {({\overset{\cdot}{\nabla}}^{r})}^{T} \nabla^{r} + {({\overset{\cdot}{\nabla}}^{im})}^{T} \nabla^{im} \\ {(F x_{s}^{r} + F^{r} (x_{s}) - F x_{m}^{im} - F^{im} (x_{m}) + L^{r})}^{T} \nabla^{r} \\ + {(F x_{s}^{im} + F^{im} (x_{s}) + F x_{m}^{r} + F^{r} (x_{m}) + L^{im})}^{T} \nabla^{im} \end{matrix}

(14)

With replacing from equation (8) regarding $L^{r}$ , $L^{im}$ in equation (14), we obtain

\begin{matrix} \overset{\cdot}{V} (t) = - ξ [{(\nabla^{r})}^{T} \nabla^{r} + {(\nabla^{im})}^{T} \nabla^{im}] \\ = - ξ (\sum_{l = 1}^{n} \nabla_{u_{2 l - 1}}^{2} + \sum_{l = 1}^{n} \nabla_{u_{2 l}}^{2}) \end{matrix}

(15)

For $V (t)$ is a certain positive clear function and its subordinate holds negatively distinct, thusly as shown by the outstanding Lyapunov speculation, the perplexing error model (11) is asymptotically stable, which implies that $\nabla_{u_{2 l}}$ and $\nabla_{u_{2 l - 1}}$ will in general zero as $t \to \infty$ , $l = 1, 2, \dots, n$ . Therefore, the conditions of the slave framework and the main framework will be all inclusive complicated synchronized asymptotically. That performs the confirmation.

Comment 6

Our scheme is suitable also for hyperchaotic frameworks. At long last, our scheme is outlined in section “For example” by applying it, for instance, for two indistinguishable clamorous complex frameworks (eqaution (2)) with specific parameters.

For example

Description of the controller

In this section, we think about CAS within the disordered complex (2). Along these lines, the master framework is depicted by

\begin{matrix} {\begin{matrix} {\overset{\cdot}{x}}_{m} = a (y_{m} - x_{m}) + y_{m} z_{m} w_{m} \\ {\overset{\cdot}{y}}_{m} = b (x_{m} + y_{m}) - x_{m} z_{m} w_{m} \end{matrix} \\ {\begin{matrix} {\overset{\cdot}{z}}_{m} = 1 / 2 ({\bar{x}}_{m} y_{m} + x_{m} {\bar{y}}_{m}) w_{m} - c z_{m} \\ {\overset{\cdot}{w}}_{m} = 1 / 2 ({\bar{x}}_{m} y_{m} + x_{m} {\bar{y}}_{m}) z_{m} - d w_{m} \end{matrix} \end{matrix}

(16)

The equations of the slave framework are

\begin{matrix} {\overset{\cdot}{x}}_{s} = a (y_{s} - x_{s}) + y_{s} z_{s} w_{s} + L_{1} \\ {\overset{\cdot}{y}}_{s} = b (y_{s} + x_{s}) - x_{s} z_{s} w_{s} + L_{2} \end{matrix}

(17)

The master and slave frameworks with complicated variables can be modified separately as

(\begin{matrix} {\overset{\cdot}{x}}_{m} \\ {\overset{\cdot}{y}}_{m} \end{matrix}) = (\begin{matrix} - a & a \\ b & b \end{matrix}) (\begin{matrix} x_{m} \\ y_{m} \end{matrix}) + (\begin{matrix} y_{m} z_{m} w_{m} \\ - x_{m} z_{m} w_{m} \end{matrix})

(18)

and

(\begin{matrix} {\overset{\cdot}{x}}_{s} \\ {\overset{\cdot}{y}}_{s} \end{matrix}) = (\begin{matrix} - a & a \\ b & b \end{matrix}) (\begin{matrix} x_{s} \\ y_{s} \end{matrix}) + (\begin{matrix} y_{s} z_{s} w_{s} \\ - x_{s} z_{s} w_{s} \end{matrix}) + (\begin{matrix} L_{1} \\ L_{2} \end{matrix})

(19)

Considering frameworks (18) and (19) are equal to frameworks (3) and (4) separately so

\begin{matrix} Φ = (\begin{matrix} - a & a \\ b & b \end{matrix}) \\ F (x_{m}) = (\begin{matrix} y_{m} z_{m} w_{m} \\ - x_{m} z_{m} w_{m} \end{matrix}), F (x_{s}) = (\begin{matrix} y_{s} z_{s} w_{s} \\ - x_{s} z_{s} w_{s} \end{matrix}) \end{matrix}

As per Theorem 1, the controller is planned as

\begin{matrix} (\begin{matrix} L_{1} \\ L_{2} \end{matrix}) = (\begin{matrix} - a (y_{s} - x_{s}) - y_{s} z_{s} w_{s} - j (a (y_{m} - x_{m}) + y_{m} z_{m} w_{m}) - ξ \nabla_{1} \\ - b (y_{s} + x_{s}) + x_{s} z_{s} w_{s} - j (b (x_{m} + y_{m}) - x_{m} z_{m} w_{m}) - ξ \nabla_{2} \end{matrix}) \\ (\begin{matrix} l_{1} \\ l_{3} \end{matrix}) + j (\begin{matrix} l_{2} \\ l_{4} \end{matrix}) = (\begin{matrix} a (- u_{3 s} + u_{1 s} + u_{4 m} - u_{2 m}) - u_{3 s} u_{5 s} u_{7 s} + u_{4 m} u_{5 m} u_{7 m} - ξ \nabla_{u_{1}} \\ b (- u_{3 s} - u_{1 s} + u_{2 m} + u_{4 m}) + u_{1 s} u_{5 s} u_{7 s} - u_{2 m} u_{5 m} u_{7 m} - ξ \nabla_{u_{3}} \end{matrix}) \\ + j (\begin{matrix} a (- u_{4 s} + u_{2 s} + u_{1 m} - u_{3 m}) - u_{4 s} u_{5 s} u_{7 s} - u_{3 m} u_{5 m} u_{7 m} - ξ \nabla_{u_{2}} \\ b (- u_{4 s} - u_{2 s} - u_{1 m} - u_{3 m}) + u_{2 s} u_{5 s} u_{7 s} + u_{1 m} u_{5 m} u_{7 m} - ξ \nabla_{u_{4}} \end{matrix}) \end{matrix}

(20)

Numerical simulation

To manifest and assert the handiness of the suggested design, we light up the reproduction aftereffects of the CAS amid two same chaos complex systems (16) and (17). Structures (16) and (17) with the controller (20) are lighted numerically. The parameters of the framework (16) are picked as $a = 30, b = 14, c = 1$ and $d = 2$ , where the chaotic behavior is obtained.²⁷ The initial requirement of the main framework, the initial state of the slave display and $ξ$ are basically considered $(x_{m} (0), y_{m} (0), z (0), w (0))^{T} = (1 + 2 j, 3 + 4 j, 5, 6)^{T}, (x_{s} (0), y_{s} (0))^{T} = (- 1 - 2 j, - 3 - 4 j)^{T}$ and $ξ = 10$ .

In Figure 1, the arrangements of equations (16) and (17) are proposed subject to different starting states and show that CAS is achieved after a by no time t. We can see that each $u_{1 s}, u_{3 s}$ have a similar indication of $u_{2 m}, u_{4 m}$ while $u_{2 s}, u_{4 s}$ have an inverse indication of $u_{1 m}, u_{3 m}$ . This implies CS accomplishes between the imaginary element of model (16) and the real element of model (17) while AS happens between the real part of slave structure (16) and the imaginary part of the main structure (17). It is evident from Figure 1 that the state variable of the main system synchronizes with a different state variable of the slave system. Like this, the CAS provides more prominent safety in secure communications. Figure 1 demonstrates CAS is achieved after a short time period. CAS errors are proposed in Figure 2. Of course from the above exschemeatory purposes of the CAS errors, $\nabla_{u_{2 l - 1}}, \nabla_{u_{2 l}}$ converge to (0) as $t \to \infty, l = 1, 2$ . In Figure 2, it can be observed that the errors will tend to (0) after a small evaluation of t. The development of the trajectories of main and slave systems subsequent to accomplishing the CAS is shown in Figure 3. Another wonder is described in Figure 3 and does not appear in a broad scope of synchronizations in the literature. The trajectories of the main and slave structures in CAS are moving inverse or symmetrical to one another with the same form from appeared in Figure 3(a) and (b).

Figure 1.

CAS between two indistinguishable frameworks (16) and (17) with the controller (19): (a) $u_{1 m}$ and $u_{2 s}$ vary t, (b) $u_{2 m}$ and $u_{1 s}$ vary t, (c) $u_{3 m}$ and $u_{4 s}$ vary t and (d) $u_{4 m}$ and $u_{3 s}$ vary t.

Figure 2.

CAS errors between frameworks (16) and (17): (a) $u_{1 s} - u_{2 m}$ vary t, (b) $u_{2 s} + u_{1 m}$ vary t, (c) $u_{3 s} - u_{4 m}$ vary t and (d) $u_{4 s} + u_{3 m}$ vary t.

Figure 3.

The orthogonal trajectories of frameworks (16) and (17).

In the numerical amusements, we figure the module blunders and stages mistakes of primary and slave frameworks, freely. For each complex number, the module and stage are resolved as pursues

ρ_{x} = \sqrt{{(x^{r})}^{2} + {(x^{im})}^{2}}

(21)

and

θ_{x} = {\begin{matrix} \arctan (x^{im} / x^{r}), x^{r} > 0, x^{im} \geq 0 \\ 2 π + \arctan (x^{im} / x^{r}), x^{r} > 0, x^{im} < 0 \\ π + \arctan (x^{im} / x^{r}), x^{r} < 0 \end{matrix}

(22)

Figure 4 shows the modules errors and phase errors of the main systems (16) and slave frameworks (17). It is evident from Figure 4(a) and (b) that the modules errors $ρ_{x_{m}} - ρ_{x_{s}}$ , $ρ_{y_{m}} - ρ_{y_{s}}$ meet to (0) as $t \to \infty$ . While the phase errors $θ_{x_{m}} - θ_{x_{s}}, θ_{y_{m}} - θ_{y_{s}}$ tend to $π / 2$ as $t \to \infty$ , see Figure 4(c) and (d).

Figure 4.

The phases errors and modules errors of frameworks (16) and (17).

Conclusion

In this work, we present another kind of complicated synchronization which is named CAS. We analyze and concentrate the (CAS) concerning two same disordered complicated nonlinear structures. The CAS may be gathered only in complicated nonlinear systems. The CAS may be seen as syncretizing among AS and CS (see Figure 1). In CAS, the state variable of the primary structure synchronizes with different state variables of the slave model (see Figure 1). This way, CAS provides increasingly unmistakable safety in secure exchanges. The common striking characteristic for the CAS is that the trajectories of the master (or principle) and slave systems are leading converse or symmetrical to one another including a comparative shape (see Figure 3(a) and (b)). These wonders did not result and showed up for any sorts of synchronization in the writing.

A design is spread out to recognize CAS of two indistinct complicated nonlinear systems in light of Lyapunov limits. In the midst of this design, we shut the control complex abilities demonstratively to achieve CAS. It is straightforward and accommodating to use this plan for chaos and hyperchaos complex structures. We apply our method, for example, for two vague wild, complicated systems with different beginning characteristics the fundamental structure (16) and slave frameworks (17). Numerical entertainments of our case affirm all the theoretical results. A splendid understanding is built up as appeared in Figures 1 –3. In Figure 4, we procedure the modules mistake and stages blunders because, in the complicated nonlinear dynamical model, the recognizable or quantifiable physical sums generally are phase and module.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Emad E Mahmoud

References

Strogatz

. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. Reading, MA: Perseus Books Publishing, LLC, 2000, pp. 1–20.

Kissaoui

Abouloifa

Chaoui

, et al. Output-feedback nonlinear adaptive control of single-phase half-bridge ups systems. Asian J Control 2016; 18: 1995–2009.

Abdelouahad

Hassan

Fouad

, et al. Nonlinear adaptive control of hybrid fuel cell power system for electric vehicles—a Lyapunov stability based approach. Asian J Control 2016; 18: 166–177.

Martynyuk

Khusainov

Chernienko

. Integral estimates of solutions to nonlinear systems and their applications. Nonlinear Dynam Syst Theory 2016; 16: 1–11.

Pecora

Carroll

. Synchronization in chaotic systems. Phys Rev Lett 1990; 64: 821–824.

Vincent

Saseyi

McClintock

PVE

. Multi-switching combination synchronization of chaotic systems. Nonlinear Dynam 2015; 80(1): 845–854.

Mahmoud

Abualnaja

Althagafi

. High dimensional, four positive Lyapunov exponents and attractors with four scroll during a new hyperchaotic complex nonlinear model. AIP Adv 2018; 8: 065018.

Mahmoud

. Complex complete synchronization of two nonidentical hyperchaotic complex nonlinear systems. Math Method Appl Sci 2014; 37: 321–328.

Adel

. A new synchronization scheme for general 3D quadratic chaotic systems in discrete-time. Nonlinear Dynam Syst Theory 2015; 15: 163–170

10.

Mahmoud

. Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems. Nonlinear Dynam 2010; 61: 141–152.

11.

Mahmoud

. Lag synchronization of hyperchaotic complex nonlinear systems. Nonlinear Dynam 2012; 67: 1613–1622.

12.

Rulkov

Sushchik

Tsimring

, et al. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys Rev E 1995; 51(2): 980–994.

13.

Roy

Hens

Grosu

, et al. Engineering generalized synchronization in chaotic oscillators. Chaos 2011; 21: 0131061.

14.

. Adaptive projective synchronization of unified chaotic systems and its application to secure communication. Chinese Phys 2007; 16(11): 3231–3237.

15.

. Stability criterion for projective synchronization in three-dimensional chaotic systems. Phys Lett A 2001; 282(3): 175–179.

16.

Mahmoud

Arafa

. Projective synchronization for coupled partially linear complex-variable systems with known parameters. Math Method Appl Sci 2017; 40: 1214–1222.

17.

Liu

. Anti-synchronization between different chaotic complex systems. Phys Scripta 2011; 83: 065006.

18.

Liu

. Adaptive anti-synchronization of chaotic complex nonlinear systems with unknown parameters. Nonlinear Anal-Real 2010; 12: 3046–3055.

19.

Kareem

Ojo

Njah

. Function projective synchronization of identical and nonidentical modified finance and Shimizu-Morioka systems. Pramana-J Phys 2012; 79(1): 71–79.

20.

Mahmoud

. A novel sort of complex synchronization. Acta Phys Pol B 2017; 48: 1441–1454.

21.

Mahmoud

Abualnaja

. Complex lag synchronization of two identical chaotic complex nonlinear systems. Cent Eur J Phys 2014; 12: 63–69.

22.

Mahmoud

Abood

. A new nonlinear chaotic complex model and it’s complex anti lag synchronization. Complexity 2017; 2017: 3848953.

23.

Mahmoud

. Complex modified projective synchronization of two chaotic complex nonlinear systems. Nonlinear Dynam 2013; 73: 2231–2240.

24.

Mahmoud

. Generation and suppression of a new hyperchaotic nonlinear model with complex variables. Appl Math Model 2014; 38: 4445–4459.

25.

Mahmoud

. Dynamics and synchronization of new hyperchaotic complex Lorenz system. Math Comput Model 2012; 55: 1951–1962.

26.

Mahmoud

. An unusual kind of complex synchronizations and its applications in secure communications. Eur Phys J Plus 2017; 132: 466.

27.

Mahmoud

Al-Adwani

. Dynamical behaviors, control and synchronization of a new chaotic model with complex variables and cubic nonlinear terms. Results Phys 2017; 7: 1346–1356.

Complex anti-synchronization of two indistinguishable chaotic complex nonlinear models

Abstract

Keywords

Introduction

Comment 1

A plan to accomplish CAS

Definition

Comment 2

Comment 3

Comment 4

Comment 5

Theorem 1

Proof

Comment 6

For example

Description of the controller

Numerical simulation

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References