Complexity

Volume 2017, Article ID 3848953, 13 pages

https://doi.org/10.1155/2017/3848953

## A New Nonlinear Chaotic Complex Model and Its Complex Antilag Synchronization

^{1}Department of Mathematics, College of Science, Sohag University, Sohag 82524, Egypt^{2}Department of Mathematics, College of Science, Taif University, Taif, Saudi Arabia^{3}Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia

Correspondence should be addressed to Emad E. Mahmoud; moc.oohay@naule_dame

Received 4 April 2017; Revised 23 May 2017; Accepted 8 June 2017; Published 3 August 2017

Academic Editor: Olfa Boubaker

Copyright © 2017 Emad E. Mahmoud and Fatimah S. Abood. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Another chaotic nonlinear Lü model with complex factors is covered here. We can build this riotous complex system when we add a complex nonlinear term to the third condition of the complex Lü system and think of it as if every one of the factors is mind boggling or complex. This system in real adaptation is a 6-dimensional continuous autonomous chaotic system. Different types of chaotic complex Lü system are developed. Also, another sort of synchronization is presented by us which is simple for anybody to ponder for the chaotic complex nonlinear system. This sort might be called a complex antilag synchronization (CALS). There are irregular properties for CALS and they do not exist in the literature; for example, (i) the CALS contains or fused two sorts of synchronizations (antilag synchronization ALS and lag synchronization LS); (ii) in CALS the attractors of the main and slave systems are moving opposite or similar to each other with time lag; (iii) the state variable of the main system synchronizes with a different state variable of the slave system. A scheme is intended to accomplish CALS of chaotic complex systems in light of Lyapunov function. The acquired outcomes and effectiveness can be represented by a simulation case for our new model.

#### 1. Introduction

The chaotic system is an extremely specific nonlinear dynamical system. This chaotic system has numerous properties like the sensibility to starting conditions and in addition a sporadic, unusual conduct. The “butterfly effect” is the well-known name of the sensibility of the introductory states of the chaotic systems [1]. Since the development of the first chaotic attractor in a three-dimensional (3D) self-governing chaotic system in 1963 by Lorenz [2], over the most recent 30 years chaos has been seriously explored [3–5]. Other chaotic systems were ordered in progression relying upon the arrangement of Lorenz, for example, Lü and Chen chaotic systems [6] and Liu chaotic system [7]. Each of the past chaotic systems has one positive Lyapunov exponent, and every one of them is three-dimensional chaotic system. Be that as it may, there are likewise many fascinating cases including complex factors which have not been effectively investigated. For instance, we say here the mind-boggling Lorenz conditions which are utilized to depict and reproduce the physics of detuned laser and thermal convection of fluid flows [8] and some of its dynamical properties are contemplated in [9]. The electric field abundance and the nuclear polarization sufficiency are both complex; for details see [10] and reference therein. In secured communications, utilizing complex variable expands the substance security of the transmitted data [11]. Complex Lü and Chen systems are presented and contemplated as of late in [12].

In 2007, Mahmoud et al. [12] introduced the complex Lü system as follows:where , and are positive parameters, and are complex components, is real variable, , , dots refer to derivatives according to time, and the meaning of overbar is complex conjugate variables.

Complex synchronization of chaotic (hyperchaotic) complex systems is a critical nonlinear occurrence [13–25]. Complex synchronization of chaotic complex systems considers a couple of complex chaotic systems called main and slave systems, and it means to accomplish asymptotic tracking of the conditions of the slave system to the conditions of the main system. In the quest for the higher ability for enhancing the security of communication systems, some efforts have been dedicated to synchronization with complex-esteemed scaling components between chaotic (hyperchaotic) complex-variable systems. Lately, complex complete synchronization (CCS) [13], complex lag synchronization (CLS) [14], complex projective synchronization (CPS) [15], and modified projective synchronization with complex scaling components (CMPS) [16, 17] are explored for coupled chaotic complex dynamical systems. Complex modified generalized projective synchronization (CMGPS) [18] and complex modified hybrid projective synchronization (CMHPS) [19] were executed between the same or different dimensional partial request complex chaotic (hyperchaotic). Complex generalized synchronization as for a mind-boggling vector map [20, 21] was presented for two indistinguishable or nonidentical chaotic (hyperchaotic) complex-variable systems. The complex modified projective synchronization of complex chaotic (hyperchaotic) systems with uncertain complex parameters was studied in [22, 23] while the complex modified function projective synchronization of complex chaotic systems is investigated in [24, 25].

As of late, a few sorts of synchronization with time lag were concentrated; for example, antilag synchronization (ALS), lag synchronization (LS), and modified projective lag synchronization (MPLS) of two riotous or hyperchaotic complex systems are investigated in [26–29]. In designing the applications, time delay always exists. For example, in the arrangement of telephone communication, the receiver hears the speaker voice at time . This is the speaker voice or transmitter at time ( and it is the lag time). In chaos communication, the time lag is the transmit flag that transmits to the receiver’s end [26, 27].

In this research, we introduce a modern chaotic model with complex components by embedding a complex nonlinear expression to the third equation of the complex Lü system (1) aswhere , and are positive parameters, , , and are complex functions, is imaginary part of equal to , and is the real part of equal to .

In addition, we present a novel sort of synchronization which we can name as complex antilag synchronization (CALS). The term CALS can be dealt with as synchronizing among ALS and LS. ALS occurs between the real part of the main system and the imaginary part of a slave system, while LS occurs between a real part of the slave system and an imaginary part of the main system.

This paper is organized as follows: in the tracking section invariance, dissipation, fixed points, and their stability analysis of some points are contemplated. The complex comportment of system (2) can without much of a stretch be studied. As indicated by estimations of Lyapunov exponents, we can figure the scope of parameters qualities at the chaotic attractors numerically. We get the best classification of the progression of (2) by the signs of Lyapunov types. We get the Lyapunov dimension of (2). We develop different types of chaotic complex Lü systems in Section 3. In Section 4 a definition of CALS is acquainted and a scheme to achieve CALS of chaotic complex nonlinear systems is proposed. In Section 5, we study CALS of two indistinguishable chaotic complexes Lü systems (2) as a case of Section 4. Finally, we will find that the fundamental conclusions of our investigations are totalized in Section 6.

#### 2. Basic Properties of System (2)

We study the basic dynamical analysis of our new system (2).

The real version of system (2) reads

System (3) has many main dynamical qualities as tracking.

##### 2.1. Symmetry and Invariance

In system (3), we notice that this system is invariant transformation: ; therefore if is the solution of system (3), then is known as the solution of the similar system.

##### 2.2. Dissipation

It is clear to find . When , the system is dissipative and meets with type shape. It implies that the volume component contracts to the volume component at the time . At the point when , each volume component which contains the system direction congregates to with exponent rate form . Thus, the majority of the system directions will finally be limited to zero volume subset, and the dynamic development is fixed on an attractor.

##### 2.3. Equilibria and Their Stability

The equilibria of system (3) can be found by solving the tracking system of equations:Obviously, is trivial fixed point. Other nonzero equilibria are given by

###### 2.3.1. Stability of

To study the stability of the Jacobian matrix of system (3) at is The characteristic polynomial equation isThen the eigenvalues are , , and The trivial fixed point is a stable point if has negative values and and are positive and else, it is not stable fixed point.

###### 2.3.2. Stability of , , , and

, , , and have the same characteristic polynomial, which isAccording to the Routh-Hurwitz theorem [30] the necessary and sufficient conditions for all roots to have negative real parts are if and only if (stable fixed points) otherwise they are unstable fixed points.

Likewise we can study the stability of , , , and .

##### 2.4. Lyapunov Exponents

System (3) in vector notation can be written aswhere is the state space vector, , is the set of parameters, and indicates transpose. The equations for small deviations from the trajectory arewhere is the Jacobian matrix of the formThe Lyapunov exponents of the system are defined by [31]To find , (10) and (11) must be numerically determined simultaneously. Runge-Kutta method of order 4 is used to compute .

For the case of , , and , the initial conditions are as follows: , , , , , , and . We compute the Lyapunov exponents as , , , , , and

This implies that our system (3) under this selection of is a chaotic system due to the fact that one of the Lyapunov exponents is positive.

The Lyapunov dimension of the attractors of (3) according to Kaplan-Yorke conjecture is defined as [32]such that is the largest integer that and The Lyapunov dimension of this chaotic attractor using (14) with , , and is

###### 2.4.1. Fix and and Vary

In light of Lyapunov exponents (13) the system’ parameter qualities were computed (3) at which chaotic attractors, periodic, quasiperiodic attractors and fixed points exist. The review modifies and differs one parameter and fixes the extra parameters as indicated by their states which fulfill the dissipative condition and the dependability state of the trifling fixed focuses.

Utilizing (13) we ascertain , , and the estimations of versus are plotted in Figure 1.