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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.

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 [

In 2007, Mahmoud et al. [

Complex synchronization of chaotic (hyperchaotic) complex systems is a critical nonlinear occurrence [

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 [

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 (

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 (

We study the basic dynamical analysis of our new system (

The real version of system (

System (

In system (

It is clear to find

The equilibria of system (

To study the stability of

Likewise we can study the stability of

System (

For the case of

This implies that our system (

The Lyapunov dimension of the attractors of (

In light of Lyapunov exponents

Utilizing (

The Lyapunov exponents

Obviously from Figure

In Figure

The bifurcation diagram of system (

Testing this numerically (utilizing, e.g., Mathematica 7 programming) in different sections and best notes is found and is agreeable with our outcomes. For instance, picking

For

In this section, we demonstrate that the complex Lü system (

All the above systems have one positive Lyapunov exponent. We ascertain these Lyapunov types as we accomplished in system (

Consider the chaotic complex nonlinear system as follows:

In this work we present and study the definition of CALS of two systems of shape (

We consider two indistinguishable chaotic complex nonlinear systems of shape (

Two indistinguishable complex dynamical systems coupled in a main-slave configuration can exhibit CALS if there exists a vector of the complex error function

When

If we define

The complex state is as follows:

The sum of the imaginary part of slave system

The error between the real part of slave system

The difference between CALS and ALS [

In CALS we define the error in simple case:

In ALS the error in simple case:

If nonlinear controller is designed as

From the definition of CALS,

At last, by applying it for two comparable riotous complexes Lü systems in Section

In the current subsection, we can demonstrate the likelihood and effectiveness of the proposed synchronization plot in the forward segment. We review the CALS of two indistinguishable chaotic complex Lü systems. The main and the slave systems are consequently defined, individually, as follows:

Complex systems (

So, the controller in (

To demonstrate and confirm the value of the suggested plot, we clarify the recreation consequences of the CALS among two indistinguishable chaotic complex systems (

In Figure

CALS between two identical systems (

CALS errors between systems (

The attractors of systems (

In the numerical reenactments, we register the module errors and phase errors of main and slave models, respectively. For every unpredictable number, the module and phase are resolved as follows:

The modules errors and phases errors of systems (

In this paper, we have presented a cutting edge chaotic complex Lü system (

We present another kind of complex synchronization which is called complex antilag synchronization (CALS). A definition of CALS of two indistinguishable chaotic complex systems is given. We dissect and concentrate the CALS concerning two indistinguishable chaotic complex nonlinear systems. The CALS can be concentrated just in complex nonlinear systems. The CALS can be considered as syncretizing among ALS and LS (see Figure

A scheme is laid out to acknowledge CALS of two indistinguishable chaotic complex nonlinear systems in view of Lyapunov capacities. Amid this scheme, we analytically concluded the control complex capacities to accomplish CALS. It is simple and advantageous to utilize this scheme for chaotic and hyperchaotic complex systems. We apply our scheme, for instance, for two indistinguishable chaotic complex systems with different introductory qualities, main system (

The authors declare that there are no conflicts of interest regarding the publication of this paper.