Dynamics, Control and Secure Transmission Electronic Circuit Implementation of a new 3D Chaotic System in Comparison with 50 Reported Systems

The very high demand for chaotic systems in the fields of sciences, particularly in secure communication, leads numerous researchers to build novel systems. This work announces a new easy to implement 3D chaotic system with five quadratic nonlinearities and three positive parameters, the proposed system is complex with larger bandwidth compared to at least 50 other systems that have been described. It contains eight terms and it can generate chaotic, periodic and quasi-periodic behaviors. The main dynamical properties of the proposed system are studied using Kaplan-Yorke dimension (KYD), Lyapunov exponents, bifurcation diagrams, multistability, equilibrium points stability and dissipativity. Then, the eight terms system feasibility is verified using Multisim software by designing its electronic circuit. In addition, active controllers are designed for leading the proposed system to achieve stability, tracking a desired dynamic and synchronizing with an identical model. Finally, using drive response synchronization, a novel secure communication electrical circuit design is built based on the suggested approach. The findings from numerical experiment demonstrated the new system’s success in completing the encryption/decryption process, as well as its secured transmission technique.


I. INTRODUCTION
After Lorenz first developed Lorenz chaotic system in 1963 [1], which proved that a three-dimensional system can generate a chaotic behavior with one positive Lyapunov exponent; many other chaotic systems was introduced in literature based on the Lorenz model or by constructing new models [2][3][4][5][6][7][8].
Chaotic systems are applicable in several fields, including secure communication schemes [9][10][11], physics [12], robotics [13], economy [14], lasers [15] and ecology [16]. Which make the construction of new chaotic systems an obligatory in line with the current demand for complex systems of this kind.
In addition, since the famous work of Pecora and Caroll [17], which proved the possibilities of synchronizing two identical chaotic systems having different initial guess, many of synchronization methods are employed for synchronizing chaotic systems, which include backstopping control [18], sliding mode control [19], adaptive control [20], active control [21], and so on.
In the current digital age, there has been a lot of interest on secure communication links due to the dramatic rise of online shopping, banking and trading transaction and this trend is set to increase exponentially in future [22]. In effect, it does not take much effort to realize that there will be a significant increase of users of digital communication Also, the existence of multistability phenomenon in certain chaotic systems is very interesting, it is means that the system not only exhibits chaotic behaviour with extreme sensitivity to the initial values; but also it can generate coexistence of different chaotic attractors depending only on its initial conditions. Multistability making a chaotic system more complex and more useful to use in many application that require complexity, especially in secure communication.
This study suggested the first new and easy to implement 3D chaotic system with five quadratic nonlinearities which is more complex and has larger bandwidth than at least 50 other systems that have been described, major properties of the announced system are investigated via theoretical and analytical methods. Also, Multisim software is used in designing and realizing an electronic circuit with the aim of validating the developed 3D model. In addition, active controllers are designed for leading the proposed system to achieve stability, tracking a desired dynamic and synchronizing.
Finally, using drive response synchronization, a novel secure communication electrical circuit design is built based on the new announced system. The rest part of the paper is designed as follows: In section 2, dynamical properties of the proposed system have been investigated using Kaplan-Yorke dimension, Lyapunov exponents, bifurcation diagrams, dissipativity, multistability and equilibrium points stability, also a comparison of the suggested system with 50 previously reported systems is introduced in section 3. The circuit schematic of the new model is realized using Multisim software in section 4. In section 5, active controllers are designed and applied to control the new system. Also, a new secure communication electronic circuit schematic is implemented based on the suggested three-dimensional chaotic system. Lastly, conclusions are dressed in section 6.

A. New chaotic system
The new 3D chaotic system contains eight terms with five quadratic nonlinearities and three positive constant parameters defined by the algebraic equations that follows: Where the state variables denoted by 1 2 3 , and x x x while , aband c are the positive constant coefficients. By choosing (1, 1, 1) as the initial conditions and 1.1, 13 and 11 a b c = = = as the parameters values; the proposed system (1) exhibit a complex chaotic behavior as depicted in Figure 1.
For the previous parameters values, the Lyapunov exponents of system (1) are obtained using Wolf's algorithm, results are shown in Figure 2 as: 1 2 3 1.051, 0.000, 3.745 As depicted in Figure 2 there are one negative exponent, one zero exponent, and one positive Lyapunov exponents. So, the suggested system is chaotic, and the corresponding KYD is computed as follow: The KYD is fractional, implying that the new eight terms system (1) demonstrate a complicated chaotic behavior. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.

B. Equilibrium points stability
The following equations are used to determine the points of equilibrium of the suggested system (1): By considering the previous values of parameters, three equilibrium points is obtained as the following:  Table 1 demonstrate the chaos in the dissipative system due to the instability of all the equilibrium points.

C. Dissipativity
The sum of Lyapunov exponents (2) is negative, so it is clear that the proposed eight terms system defined by (1) is dissipative. Therefore, orbits of the whole of system (1) are ultimately constricted to a particular subset of zero volume, and the asymptotic motion relaxes on a chaotic attractor.

D. Bifurcation analysis
The bifurcation parameter a versus system (1) dynamical behaviors is analyzed using Lyapunov exponents spectrum and bifurcation diagram presented in Figures 3  and 4, respectively, as parameter a varies. It can be seen from Figure 3 and 4 that the proposed model has a rich dynamical behavior, as it can show chaotic behavior, periodic and quasi-periodic behaviors when parameter a increase as summarized in Table 2.

E. Frequency spectrum and bandwidth
The bandwidth of a chaotic system should be large in order to completely cover the masked signal in chaos-based secure transmission applications. This is not the case in many of reported chaotic systems, which the frequency spectrum of its signals is not broad enough. Therefore, it is of great significance to construct new chaotic systems with larger bandwidth than existing ones. Figure 8 shows the normalised average frequency spectrum of the signal generated by the second state variable 2 x of system (1), The novel chaotic system has a bandwidth (BWT) of about 15, that is greater when compared it to more than 50 chaotic systems documented in the study, indicated in Table II. (Red color) Fix b=13, c=11 and vary a, it can be proved from the numerical simulations that the new 3-D system (1) is multistable. When a=0.9, we can see from Figure 9 that system (1) has coexistence of two different chaotic attractors starting from 0 X and 0 Y for the same values of parameters. Coexistence of one chaotic attractor and one periodic attractor starting from 0 Y and 0 X respectively are determined when a=9.6 as depicted in Figure 10. When a=6.6, we can see from Figure 11 that system (1) has coexistence of two different periodic attractors starting from 0 X and 0 Y for the same values of parameters. The obtained results enable us to observe the phenomenon of multistability, this strange phenomenon proves the high complexity of the new system (1), which make it very suitable to use in engineering applications that require complexity; especially, chaos communication.

III. COMPARISON BETWEEN THE NEW CHAOTIC SYSTEM AND 50 REPORTED SYSTEMS
Based on many studies: 1-The Kaplan-York dimension (KYD) is a measure of a chaotic system's unpredictability and complexity. The higher the Kaplan-York dimension, the more complex the chaotic behaviour [23]. 2-A chaotic system's carrier signal's bandwidth (BWT) should be big enough to cover the disguised information. In chaotic-based secure communications, large bandwidth is essential for achieving high data rate transmission [24]. In this section, a comparison between the Kaplan-York dimension and the bandwidth of the proposed chaotic system (1) with those of 50 reported chaotic systems is introduced. This comparison proved that the proposed model has a larger bandwidth and it is more complex when compare it to more than 50 considered systems, making it more helpful in a wide range of applications; especially in secure communication (see Table II).
Rossler |27] Chen et al. [28]   This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.

IV. CIRCUIT IMPLEMENTATION OF THE NEW SYSTEM
This section present an equivalent electronic circuit for our defined easy to use chaotic system (1), the electronic circuit is designed using Multisim software as depicted in Figure 12. By affixing the law of Kirchhoff to Figure 12 circuit, it will generate nonlinear equations described as the following: x The circuital components values are selected as:

A. Design of active controllers for controlling the new chaotic system to equilibrium
In this subsection, an active control law is presented for control of the proposed chaotic system to one of its three points of equilibrium.
In order to control the proposed system to 1 (0, 0, 0) E = , active control functions are added to the new chaotic system. Then, the controlled system is considered as follows: (1 ) where 1 2 3 , and ,, with ( ) By considering system (10) and the desired state (12), we obtained the following state errors dynamics: Then, the state errors' dynamics is said to converge asymptotically to zero. Therefore, the controlled system (10) converges to the point of equilibrium 1 (0, 0, 0) E = .
Proof: The state errors' dynamics (13) is re-written as follows, using the active control functions provided in (14): Since the eigenvalues for the states matrix are negatives, then, based on Routh-Hurwitz condition [75]; the errors dynamics are stables, ensuring the convergence of the controlled new system (10) to equilibrium.
• Simulation results: System (10) initial conditions and parameters are chosen as So, results of simulation proving the efficiency of the new active controllers (14) for controlling the new chaotic system (1) to equilibrium.

B. Design of active controllers for tracking control of the new chaotic system
In this subsection, an active tracking control law is calculated for the proposed system for tracking any desired function of time () ft. For that, the controlled system is considered as follows: , and x x x P P P is the functions for the active trailing control to be obtained, and the state errors are described by :   1  2  3   1  1  2  2  3  3 ,, By considering system (17) and the desired state (19), we obtained the following state errors dynamics: Theorem 2. Suppose the functions for the active tracking control is chosen as: The dynamical of state errors will be converge asymptotically to zero. Therefore, system (17) is controlled to track any desired function of time () ft.
Proof: The dynamics of state errors (20) can be recast using the active control tracking functions provided in (21), as follows: Because all of the state matrix's eigenvalues are negative, the errors dynamics are stable, ensuring that the controlled new system (17) converges to the function () ftaccording to the Routh-Hurwitz criterion [75].  Figures  19, 20 and 21.
The results shows that all the three new system state variables generate a complex chaotic behaviour when the active controllers are deactivated (when 150 ts  ). After that (when 150 ts  ), the controllers are activated and it can be seen that all state variables switch quickly to exhibit a sine wave behaviour and to track the sine function ( ) 4sin(0.5 ) So, simulation findings proving the success of the suggested active tracking controllers (21) for controlling the new chaotic system (1) in tracking any desired function of time including ( ) 4sin(0.5 ) f t t = .

C. Design of active controllers for synchronizing the proposed chaotic system
In this subsection, we use active control to synchronise two identical new chaotic systems (1) with differing initial values.
The master system is demonstrated as the following: The, the slave system is described as: x ax where functions of active control represented by 1 2 3 , and x x x Q Q Q are to obtained, and the state errors are described by: ,, By considering systems (24), (25) and (26), the dynamic state errors is obtained: Then, the dynamical state errors will asymptotically converge to zero. As a result, the master (24) and slave (25) systems will be synced.

Proof:
The dynamics of state errors (27) can be simplified using the active control functions given in (28).
Because all of the states matrix's eigenvalues are negative, the error dynamics are stable, ensuring synchronisation between the master and slave systems (24) according to the Routh-Hurwitz criterion [75].
• Simulation results: The master and slave systems' initial conditions defined in (24) and (25)  The results shows that all the three state synchronization errors evolve chaotically with time by deactivating the active controllers (when 150 ts  ). After that (when 150 ts  ), the controllers are activated, then, the convergence all the state synchronization errors to zero is very obvious.
So, simulation findings showing the feat of the new active controllers (28) to synchronize two new identical chaotic systems (1) starting from various initial guesses.

VI. SECURE COMMUNICATION CIRCUIT DESIGN BASED ON THE NEW CHAOTIC SYSTEM
To demonstrate the applicability of the proposed system (1) in securing communications, we create a new secure communication circuit schematic centered around the proposed system and employing drive response synchronization, with the use of a single state variable to attain strong self-synchronization.

A. Drive Response Synchronization
In this subsection, we present the synchronization electronic circuit schematic of two identical new chaotic systems (1) starting from different initial values using drive response method. We use the proposed system (1) as a master system, also as a slave system. The driving variable generated by the master system ensures the synchronization of states of the master system and the states of the slave system.
The slave system presented below is described by considering the proposed eight terms chaotic system (1) as a master system: (1 ) y ay y by y y cy y y y y y y By selecting the master system's second state variable of the 2 x as the driving variable, the slave system become as the following: Electronic circuit schematic of the drive response synchronization between master system and slave system is depicted in Figure 25.
The initial condition of system (1) and (32) are chosen as (1,1,1) and ( 5,10) −− respectively. Time evolutions of the first states ( ) 11 and xy are depicted in Figure 26; Figure 27 shows the time evolution of the third states ( ) 33 and xy .
The drive response approach was successful in synchronizing the two identical master and slave chaotic systems starting from differing initial circumstances, as shown in simulation results in Figures 26 and 27.  Figure 28 shows the electronic circuit design for the secure communication strategy created for the new chaotic system. We can see that two signals are transmitted from the transmitter to the receiver, the first one is the driving variable 2 x so as to complete synchronization between the master system and the slave system, the second signal transmitted is the encrypted message m .
The encrypted message m is obtained using the electronic circuit schematic depicted in Figure 31  m k m k w k x k x = −  +  +  +  (33) where 13 , xx represent the master system's state variables, m is the clear message, w is a wrong message assumed to be known which is included to increase the complexity of the encrypted message; 1 2 3 4 , , , k k k k are parameters (constant) used as an additional transmission secret key.
When the synchronization is achieved, the decrypted message m is reconstructed in the receiver using the electronic circuit schematic depicted in Figure 32  ,0 = − +  +  +   m m k w k y k y k k (34) where 13 , yy representing the slave system's state variables, m is the encrypted message and w is the same wrong message as in the transmitter.
For computational simulation: We used the following constant parameters for the encryption and decryption functions: Case 1: the sine wave is chosen as the clear message to be communicated as shown in Figure 33 and the triangle wave as the wrong message to be included in the encryption and decryption functions as shown in Figure 34. Figure 35 depicts the encrypted message's complicated chaotic behaviour, demonstrating the effectiveness of the encrypting process in concealing the clear information. The decrypted message is shown in Figure 36, and it is identical to the clear message. Figure 37 depicts the convergence of the message reconstruction error to zero after synchronisation, indicating that the decryption procedure was successful in reconstructing the original message.  The proposed secure communication electronic circuit schematic based on the novel eight terms chaotic system (1) was shown to be successful in completing the secure transmission/reception procedure.

VII. CONCLUSION
This paper suggested a new three-dimensional eight terms chaotic system with five quadratic nonlinearities for the first time.
The new system is more complicated and has a wider bandwidth than at least 50 other chaotic systems studied recently. Dynamical analysis of the new system are widely studied via many tools which include: phase portraits, equilibrium points, dissipativity, multistability, bifurcation diagram and Lyapunov exponents spectrum. The physical feasibility of our proposed theoretical model is proved by using Multisim software to design the equivalent electronic circuit. After that, control of the proposed chaotic system for achieving stability, tracking and synchronization is studied and applied using active controllers, the obtained results illustrate the effectiveness of the suggested control laws. Finally, employing drive response synchronization, a new secure communication electronic circuit schematic is derived for the new chaotic system. Multisim simulation findings show that the proposed circuit schematic completed the secure transmission/reception process successfully.
We are convinced that the new 3D chaotic system with its easy to implement secure communication electronic circuit schematic, multistability and large bandwidth would be useful in the near future for real-world secure transmission systems.