Synchronization is one of the most important characteristics of dynamic systems. For this paper, the authors obtained results for the nonlinear systems controller for the custom Synchronization of two 4D systems. The findings have allowed authors to develop two analytical approaches using the second Lyapunov (Lyp) method and the Gardano method. Since the Gardano method does not involve the development of special positive Lyp functions, it is very efficient and convenient to achieve excessive system SYCR phenomena. Error is overcome by using Gardano and overcoming some problems in Lyp. Thus we get a great investigation into the convergence of error dynamics, the authors in this paper are interested in giving numerical simulations of the proposed model to clarify the results and check them, an important aspect that will be studied is Synchronization Complete hybrid SYCR and anti-Synchronization, by making use of the Lyapunov expansion analysis, a proposed control method is developed to determine the actual. The basic idea in the proposed way is to receive the evolution of between two methods. Finally, the present model has been applied and showing in a new attractor, and the obtained results are compared with other approximate results, and the nearly good coincidence was obtained.

Chaotic systems with real state variables are boing found and studied with increased attention in several aspects of nonlinear dynamical systems, the first physical and mathematical model of a chaotic system is the system of Lorenz, which only includes real variables discovered in 1963 and opens the way to other chaos systems such as the system of Chen, Lu’s system (2002), Liu’s system (2004), and the system of the Pan system (2009). Each system has a 3-D of differential equations and just one positive Lyapunov exponent [

Exponent Lyapunov and nonlinear dynamic systems attractor play an important and actively involved role in classifying these systems and have attracted increasing interest in engineering application and different scientific research as, encrypting [

Rössler performed the first 4-D hyperchaotic system with real variables in 1979 with two positive exponents of Lyapunov and discovered a further 4-D and 5-D hyperchaotic system with three positive exponents of Lyapunov [

Several papers on the subject today are dedicated to studying the new hyperchaotic systems in higher dimensions (Dimension 5) [

The synchronization of similar 4-D hyperchaotic systems are studied and is then theoretically introduced as an Engineering application to detect error dynamics between each and its stable communication.

Non-linear stability-based control methods in Lyapunov, Gardano approach design the various controllers of synchronization phenomena.

By comparing the results of the Lyapunov method with the Gardano method, the best fitting controllers are found.

We use the second Lyapunov method and the methods of Gardano, where we infer that Lyapunov functions as a certain constructive tool as:

There

Be certainly negative, i.e.,

While in Gardano [

Let

This approach allows one to find the roots of the cubic equation (

If

When

When

Here, to construct all roots with negative real parts, not choosing an appropriate nonlinear controller

Briefly, this final point poses three fundamental questions. First, does the Lyapunov method always succeed? Second, is the Gardano method better? Thirdly, how can these two approaches be distinguished? This paper starts with two ways of answering these questions.

The Lorenz system was one of the most commonly studied 3-D chaotic systems. By adding a linear feedback controller, the original design was changed into a 4-D and 5-D hyperchaotic design. The new 4-D hyperchaotic system that contains is designed three positive Lyapunov Exponents _{1} = 0.94613, _{2} = 0.28714, _{3} = 0.0047625, and one negative Lyapunov Exponents _{4} = −12.4021. The 4-D system which is described by the following mathematical form:

where the real state variables are _{1}, _{2}, _{3}, _{4}, and

The numerical simulation of _{1} = 0.94613, _{2} = 0.28714, _{3} = 0.0047625.

The exponents of the plot of Lyapunov are shown in

Dimensions of Lyapunov are found as:

In this section one of the main applications of secure communication engineering is considered theoretical studies and numerical simulations. Therefore, the first system (

While the response system is as follows:

and let _{1}, _{2}, _{3}, _{4}]^{T} is the nonlinear controller to be designed.

The synchronization error dynamics between the 4-D hyperchaotic system _{i}_{i}_{i}

The dynamics of the error are defined as follows:

i.e.,

The matrix _{1} equal to _{1} =

Now, by designing several controllers based on Lyapunov and Gardano methods we will try to control the error system

The system

We are now building a positively defined Lyapunov candidate, based on the

where

where ^{4}. The nonlinear controller is suitable and the complete synchronization is achieved.

The characteristic equation between system

where

For simplified, we substitute the value of constants as

Therefore, we have

Of course, all roots with negative actual parts are successfully synced with system

The non-linear control approach, which uses two theoretical methods, offers an anti-synchronization between two related highly hyperchaotic systems. To stop collisions, it involves two systems; the first (called drive systems) reflects the first, and the second (called the responses system). This mechanism is the second train, which is used to ensure that there is no collision with the first train. The second is used to prevent collisions. The first train will have a second system. Suppose that the system

The 4D hyperchaotic system _{i}_{i}_{i}

The error dynamics is calculated as the following:

i.e.,

Now, based on the

The derivative of the Lyapunov function

where ^{4}. The nonlinear controller is suitable and the anti-synchronization is achieved.

In

After substituting the values of the constants (a,b,c,h), we get

Therefore, we have

Of course, all roots with negative actual parts are successfully synced with the system

Hybrid synchronization is a mixture of the previous two phenomena (

The hybrid synchronization error dynamics is defined as _{i}_{i}_{i}_{j}_{j}_{j}

The error dynamics is calculated as the following:

Now, based on the

The derivative of the Lyapunov function

where ^{4}. The nonlinear controller is suitable and the anti-synchronization is achieved.

In

After substituting the values of the constants (

S. No. | Second method of Lyapunov | Cardano’s method | |
---|---|---|---|

1. | Essentially, a quadratic function is created. | Based on the origins. | |

2. | Achieving conditions: A quadratically appositive function is a negative derivative. | Conditions: All roots with a very negative component. | |

3. | You need to often adjust this feature. | No modification required. | |

4. | Agreements with structures via co-factor systems (Lyapunov function) | Directly (no co-factor) processes structures. | |

5. | In a while, crashed. | Still effective. | |

6. | You did not have to find the solution. | The solution needs to be found. | |

7. | Dig in the late 19th century. | Discovered at the beginning of the 16th century |

The structure of the subject of synchronization phenomena in two methods is shown in

Therefore, we have

In addition, all roots with negative actual parts are successfully synced with system

Thus, all questions in this section are answered in these theorems, and the following table indicates that the Cardano method is stronger than the Lyapunova method.

The second method Lyapunov and the Cardano method are based on nonlinear models and two theoretical approaches. We have been trying to grasp the discrepancies in each process and how to achieve synchronization? Within this article, two identical 4D hyperchaotic systems deal with the synchronization phenomena. What is the best method? This paper, therefore, answers all these questions in the Cardano method and makes notice that a supporting function, like the Lyapunov method, should not be created or modified. The Cardano process is better than the Lyapunov procedure. The computational simulation was used to describe the same findings.

The following scheme shows the topics of research: