A new hyper-chaotic system with equilibrium lines: dynamic analysis and synchronization.

. Based on our previous work investigating chaos theory, in this article we suggest a hyperchaotic system with an equilibrium line and its synchronization. The paper introduces a new hyperchaotic system with highly complex dynamic behavior, so it is interesting to note that this system has three fixed point lines. Finally, we propose a method for synchronizing the proposed hyperchaotic system.


Introduction
Chaotic systems are non-linear dynamic systems governed by deterministic laws, dependent on several parameters, whose evolution over time is unpredictable.Since E. Lorenz's discovery [2], chaos theory has always been one of the most appealing disciplines of mathematics.Several systems have been presented in recent years exploiting chaotic behavior in various disciplines [3][4][5].
Continuous chaotic systems give rise to hyper-chaotic systems, which exhibit more complicated dynamic behavior than regular chaotic systems.The hyper-chaotic system is characterized by a chaos attractor with more than one positive Lyapunov exponent [7].In general, hyper-chaotic systems exists for systems of order 4 and higher.
The use of chaos theory in communication systems was initiated by the work of Pecora and Carroll [14].They observed that if two similar chaotic systems with differing start conditions are correctly coupled, they can synchronize.Then, chaos synchronization phenomena have piqued the interest of researchers studying chaotic systems because they may be used in a wide range of engineering and information science domains, most notably secure communication and cryptology.
Motivated by the numerous and diverse uses of chaos theory, the goal of our contribution is to present a novel chaotic system competes in terms of complexity and sensitivity to initial conditions as well as simple to apply in a variety of fields.
Our research is based on chaos theory, which proposes several techniques for chaos detection [1], including Lyapunov exponents, attractor shapes, phase portraits, equilibrium points, and time series analysis.We propose a novel four-dimensional hyperchaotic system in the first part of our study.We suggest a strong synchronization of this system in the second half.Finally, we validate the theoretical synchronization results obtained via circuit simulations.
We conclude that system (1) is chaotic with Kaplan-Yorke dimension D = 3.09 for the parameters(a, b, c) = (0.5, 1, 1), while for the parameters (a, b, c) = (0.5, 1, −1) we find a hyperchaotic behavior of system (1) with a Kaplan-York dimension, D = 3.375 which shows the highest degree of complexity in this case.Based on research by Eric Campos-Canton et.al [8], we show that system (1) belongs to the family of unstable dissipative systems (UDS).UDS type I for c=-1, and UDS type II for c=1.We also prove that this is a system with three lines of equilibrium, and no other equilibrium.This is a very interesting case, as it belongs to the class of systems with hidden attractors, compared to other new works [15][16].
We will choose different values of a, b, c and plot the attractor in 3D and in phase plane in each case, summarizing the results in the following table : Table 1.Attractor forms.We've drawn the phase portraits of system (1) for different parameter values a, b and c.As can be seen in Fig. 1, the system's behavior is chaotic; Fig. 2 represents the periodic behavior of the system, while Fig. 3 shows an asymptotically stable orbit.Fig. 4 shows hyperchaotic behavior, with the attractor taking the form of two shells.3 Synchronizing the new hyper-chaotic system Chaotic system synchronization is a concept based on two chaotic systems named master and slave, so that the slave asymptotically follows the master.Several methods for synchronizing chaotic systems have been presented, such as linear and nonlinear feedback control [9], backstepping nonlinear control approach [11], adaptive control [12], PID control [13], sliding mode control [10], and so on.

Initial Conditions
Motivated by the work of Jun-Juh Yan et, al [10], in this paper we design a synchronization of the chaotic system (1) using a robust adaptive sliding mode controller (RASMC).Thus, the slave system is of the form: We confirm the synchronization result by numerical simulation using Matlab.To carry out a simulation, we will choose the parameter values(a, b, c ) = (0.5, 1, −1) and the master's initial condition  = (0.5, 0, 0,0)and that of the slave  = (−0.2,1.5, −0.3,0) (see fig. 5 and Fig. 6) and represent the master/slave system state variables and the synchronization error of the state variables between the master and slave (see Figures 5 and 6).

CIRCUIT DESIGN AND SIMULATION
In the following part, Based on the circuits built by YU Simin [6], we present a circuit corresponding to system (1).
We explain the circuit seen in fig.7: The circuit consists of 20 resistors, four capacitors, four integrating amplifiers (U1A-U2A-U3A-U9A), three invertors (U4A-U8A-U11A-), two summers (U7A-U10A), a multiplier A1, and a circuit of two amplifiers for the realization of the sign(x) function (U5A-U6).We display the results of the circuit simulation using tree oscilloscopes, which are the phase planes (x − x ), (x − x ) and (x − x ) respectively (see fig. 8).We infer that these simulations are equivalent to Matlab numerical simulations in fig.We obtained system master synchronization using the slave system.In four oscilloscopes, we display the result of synchronization in mode X-Y of the four variables' states.In Fig. 10

RESULTS & DISCUSSION
These results confirm that the error states of the system are asymptotically regulated to zero, as seen in Fig. 5 and Fig. 6 show how the slave states converge to the master states.i.e. synchronization between master and slave is assured.
The proposed synchronization method is robust, it has a number of intriguing characteristics, including low sensitivity to external perturbations, as well as resistance to plant uncertainty caused by structural differences.It should also be noted that the circuit structures of master/slave system is robust, and very simple, and easy to implement.

Conclusion
In conclusion, in our paper we have presented a new hyper-chaotic Jerk system and performed an investigation on this system using Kaplen York dimension, phase portrait, and equilibrium point studies.
The proposed system is among the most interesting and competitive new systems in the literature.Because it's not only a hyper-chaotic system, but it is also a highly complex system with dynamic behavior and a very interesting value of Kaplan York dimension D = 3.375, and relates to the class of hidden attractor systems.
We propose a robust synchronization for this system, and we demonstrate that the theory fits with the application by running numerous simulations of the system and its associated electrical circuit.
We offer the system 4D and its synchronization (master/slave) for technical applications in robotics, arbitrary bit generator, picture encryption, secure communication, and many other engineering applications.

Fig. 5 .Fig. 6 .
Fig. 5.The time evolution of the errors Figure 6 depicts the error of the different state variables, and we infer that the error goes to zero, allowing the master and slave systems to be synchronized.

Fig. 8 .
Fig. 8. Plot of the new system observed on the oscilloscope in different phase planes The realization of the synchronization of chaotic systems has solved the problem of secure chaotic communication technology.We validate the theoretical result of synchronization, by implementing a slave circuit shown in fig.9.We obtained system master synchronization using the slave system.In four oscilloscopes, we display the result of synchronization in mode X-Y of the four variables' states.In Fig.10depicts appropriate synchronization.