Realization of a novel chaotic system using coupling dual chaotic system

This paper establishes coupling between two various chaotic systems for Lorenz and Rössler circuits. The x-dynamics of Lorenz circuit was coupled numerically with the x dynamics of Rössler circuit. As a result of the optical coupling between these two chaotic systems, it has been observed an exceptional variation in the time series and attractors, which exhibits a novel behavior, leading to a promises method for controlling the chaotic systems. However, performing fast Fourier transforms of chaotic dynamics before and after coupling showed an increase in the bandwidth of the Rössler system after its coupling with the Lorenz system, which, in turn increases the possibility of using this system for secure and confidential optical communications.


Introduction
The nonlinear dynamics is an important branch in science in which inclusive researchers have been performed in the last decades (Chen 2003). Basically, this branch is an interdisciplinary of science that deals with the systems that can be represented using mathematical nonlinear equations. The nonlinear dynamics study has substantial importance in technology and science due to nonlinearity in the most engineering and natural systems (Mehrotra 2005).The nonlinear systems exhibit a wide variety of phenomena such as pattern formation, self-sustained oscillations and chaos (Ott 2000).
The study of nonlinear dynamics has improved after the invention of the deterministic chaos phenomenon by Lorenz (1963). Chaos may be defined as the phenomenon of the irregular fluctuations of the system outputs obtained from models represented by deterministic equations. The dynamical system is said to be chaotic when it has three important properties: infinite recurrence, boundedness and sensitive dependence on the initial conditions (Vaidyanathan and Volos 2015). Chaos phenomenon can be found in a various disciplines, such as engineering, physics, economics, biology, chemistry (Jamal and Kafi 2016), steganography (Banupriya et al. 2019) and encryption (Hua et al. 2019). Recently, the concept of dual synchronization of two different pairs of chaotic dynamical systems has been investigated and used experimentally in communication applications (Uchida et al. 2003).
In this work, a novel chaotic system scheme is presented by coupling two kinds of hyper chaotic systems (Lorenz and Rössler circuit) and studying the effect of the x-dynamics of Lorenz circuit on the x-dynamics of Rössler circuit. The data are analyzed numerically using MATLAB (9.2) to investigate the effect of the coupling on the Rössler circuit using the time series, attractor, and the fast Fourier transformation.

Lorenz and Rössler circuits
The Lorenz model is an essential computational approach in the nonlinear dynamics studies (Jamal and Kafi 2016). This model consists of a system of three ordinary differential equations which have two nonlinear terms (x 1 y 1 ) and (x 1 z 1 ) (Jamal and Kafi 2016): The parameters σ, r, and β > 0, and their values are 10, 28, and 8/3 respectively. This model is explicit in the sense that both of attractor size and time scale depends on some parameters each of which has inverse time dimension.
In order to have the simplest attractor with a chaotic behavior without a property of symmetry, a Rössler model was suggested also. The Rössler model is a system includes three non-linear ordinary differential equations, which represent a continuous-time nonlinear system that shows a chaotic behavior associated with the fractal characteristics of the attractor (Rössler 1976).
where a, b, and c are constants. The Rössler model is simpler than the Lorenz system because it only exhibits a single spiral. In this model there is only a single nonlinear term (z 2 x 2 ), and it is chaotic when a = b = 0.2, c = 5.7 (Rössler 1976).
In order to attain a novel chaotic scheme, two chaotic systems are connected. The two systems; Lorenz and Rössler are coupled where the output of the variable x-dynamics for Lorenz circuit is the input of variable of x-dynamics of Rössler circuit. The main components of both Rössler system and Lorenz system are not identical. Consequently the output signal of Lorenz system will enter to Rössler system which in turn will be affected by any variation that occurs in Lorenz circuit. The result of the coupling of Lorenz and Rössler system will be as follows: (1)

Results and discussion
Fourth-order Runge-Kutta method was used to solve the systems of differential equations in all numerical simulations with step of Δ = 0.0078. Figure 1 represents the variation of the x1-dynamic with time at the initial conditions are taken as (x1 = y1 = z1 = 0.01) according to x-dynamics of time series of Lorenz system. The range of x1-dynamics are [− 20 to 22].
The attractors of Rössler system are a strange attractors as shown in Fig. 3 which takes a pair of state variables (x 2 − y 2 ), (x 2 − z 2 ), and (y 2 − z 2 ) respectively.
When the variable x 1 of Lorenz system is coupled with the variable x 2 of Rössler system, the time series of Rössler system will take different values while the dynamics remain chaotic as shown in Fig. 4 which illustrate the dynamics of Rössler at (x1 + x2)for three state variables (x 2 , y 2 , z 2 ) respectively.
The attractors after coupling between the variable x 1 of Lorenz system and the variable x 2 of Rössler system remain strange (chaotic behavior), but the attractor shapes vary, as shown in Fig. 5 which explain the attractor of Rössler at (x1 + x2) which take a pair of state variables (x 2 − y 2 ), (x 2 − z 2 ), and (y 2 − z 2 ) respectively. It is observed that there is an increase in the value of chaotic dynamic systems ranges as shown in Table 1, this positively causes an increase in bandwidth, which plays an important role in communication applications especially confidential and secure ones. This is confirmed during the process of performing fast Fourier transforms (FFT) of chaotic dynamics before and after coupling. It is believed that the chaotic systems with higher dimensional attractors have much wider applications. The spectrum of FFT of the Lorenz and Rössler systems, is illustrated in Fig. 6,where it was observed that when comparing the two spectra, the Lorenz system has a wider spectrum of the Rössler system with a distinct frequency at the low frequency of about 1.7(a.u). As for Figs. 7, 8 and 9, it is noticed that the bandwidth of the x2, y2, and z2 dynamics of Rössler system has increased and became wider after the mixing process, and this observation is important and the basis for the work of secure communications. The distinct unwanted frequency of Rössler system has (3) x 2 = −y 2 −z 2 + σ y 1 − x 1 ̇y 2 = x 2 + ay 2 z 2 = b + z 2 x 2 − c   [− 28,13] been noticed to disappear within the new frequency range. The best bandwidth that can be used in communications is within z-dynamics due to its distinct exponential dicey distribution.

Conclusions
In this paper the coupling between two chaotic systems (Lorenz and Rössler) is investigated. The study revealed that there are variations in time series and attractor of Rössler system. This means that the behavior of Rössler system experienced a significant change after coupling, which can be considered as a good method for controlling the chaotic systems. This new system can be used in secure communications due to its chaotic behavior and its wide bandwidth, by performing fast Fourier transforms (FFT) of chaotic dynamics before and after coupling. It is possible to build an electronic circuit using the new chaotic system.Over all, the results indicate that the theoretical model has practical feasibility for implementation.
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