In this work, an eight-term chaos system is presented. The effect of changing the initial conditions values on behavior Chen system was studied. The research provided a complete analysis of the system properties such as time series, attractor, FFT spectrum and bifurcation. Where the system appears steady state behavior at initial conditions x i, y i , z i that equal (0, 0, 0) respectively and it convert to quasi-chaotic at x i , y i , z i equal (-0.1, 0.5,-0.6). Finally, the system become hyper chaotic at x i , y i , z i equal (-1, 0.5,-0.6) that can used it in many applications like secure communication.
Research Article
Space phase inversions and initial conditions of Chen Model
https://doi.org/10.21203/rs.3.rs-1541117/v1
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Chaos
Chen system
time series
attractor
bifurcation
Chaotic dynamics are non-periodic, broadband systems that are similar to noise but are governed by a set of equations and are highly sensitive to initial conditions. Therefore, these systems have helped to understand many systems that show random behavior. There are many chaotic nonlinear systems which are becoming of great and increasing importance in many fields. The first scientist to study chaos was Edward Lorenz in 1963 [1]. Then great and rapid progress was made in the science of chaos with the help of a group of researchers such as Rossler and Chua [2-3] who presented a set of models to understand more about this strange system. Chaotic order appears in many fields, including physics and mathematics, and continues to grow in a variety of other areas of life, such as weather forecasting and wind movement, and in the sciences of population spread, and in the biological field such as the survival of single-celled organisms, the study of the nervous system and the signals sent from the brain to the organs of the organism, and also the study of fluid movement inside the organism [4-14]. Chaotic systems and their dynamic equations can be represented through electronic circuits, and this was actually done by the scientist Chua, who built the first electronic circuit that shows chaotic behavior in the simplest representation [15]. The simple Chua circuit has been used in various fields, mostly in the field of secure communications. Subsequently, many developed electronic circuits appeared, such as RLC circuits, which show complex chaotic behavior [16-19], electronic oscillators [20], energy storage circuits [21-22], digital filters [23-24] and capacitor circuits. Recently, new chaotic systems have emerged that show a different chaotic behavior than the previous systems that have been used in many fields [25-29]. Currently, the details of the chaotic Chen model system are presented and the effect of initial conditions on the behavior of this system is studied.
The Chen system is introduced by Guanrong Chen, where described by three first-order nonlinear differential equations, which resembles some familiar behavior from the Rossler and Lorenz attractors. The Chen system equation as follow [17]:
Where the typical parameters a, b, and c are real numbers that equal 40, 3 and 28 respectively at chaotic case. Likewise, for other parameter sets of {a, b, c}, system may not be chaotic. The system becomes not chaotically if this set of parameters is changed. The advantage of this system is that it has three ordinary differential equations and has two types of quadratic nonlinearities in (xy) and (xz). Until the mathematicians did not agree on a specific formula for defining chaos, but they agreed that the chaotic system is sensitive to the initial conditions and that the presence of double-period cycles leads to the emergence of chaos.
To analyze and study the effect of changing the initial conditions (xi , yi , zi) on the behavior of the chaotic system, two different conditions were taken, which are as follows:(-0.1 , 0.5 , -0.6) and (-0.5, 0.5 , -0.6). The MATLAB program was used to solve the differential equations using Rang- Kuta 4th, where the time scale taken is 25 a.u . The code results of Chen system as follow: The time series of all dynamics for Chen system at initial condition (-0.1 , 0.5 , -0.6) are given in Figure (1), where the Table (1) shows the amplitude values of all dynamics (peck to peck).
Table (1): The amplitude values of time series of Chen system
|
Dynamics |
xp.p (a.u) |
yp.p (a.u) |
zp.p (a.u) |
|
Amplitude |
24.8:-17.5 |
27:-18 |
39:-0.6 |
Through the Figure (1) notice, there is a similar in the behavior of the x and y dynamics approximately (but difference in amplitude) with a difference behavior in z dynamic. The strange attractant for the three dynamics x, y, and z are given in Figure (2). It is shaped like a butterfly (2-scrolls attractor) with complex topological structure, So Chen system is a specific case of Lorenz [18] according to this. Figure (2-d) shows the strange attractor in three dimensions (3D). To analysis the bandwidth of Chen system, Figure (3) shows the Fast Fourier Transformation (FFT) spectrum of it.
Chen system is showing exponential decay behavior and broadband system and Table (2) shows the values of maximum amplitudes, frequency bandwidth, and Full width at half maximum (FWHM). The Lyapunov exponent can quantitatively reflect the chaotic performance of a system. A Hyper- chaotic systems can be described as containing more than one positive Lyapunov exponent, which indicates that these systems expand in many directions and this leads to the emergence of a very complex attractor.
Table (2): The important values of FFT spectrum of x, y, and z-dynamic
|
Dynamics |
x-dynamic |
y-dynamic |
z-dynamic |
|
Max amplitude (a.u.) |
0.8 |
0.8 |
0.8 |
|
Freq. Bandwidth (a.u.) |
8 |
10 |
8 |
|
FWHM (a.u.) |
1.8 |
1.8 |
1.4 |
One of the important tools for studying chaotic behavior and studying the effect of changing system parameters is the bifurcation diagram shown in Figure (4). When changing parameter (c), we notice that the system shows different behaviors over time, starting from fixed points and ending with excessive chaos and passing through different periodic states. The system can return to the periodic state at certain values of parameter (c) at range [6.5-10] and range [20.5-21], and the system shows hyper chaotic state at range [25-29].
To verify that the behavior of the Chen system changes by changing its initial conditions, the new value of the initial conditions in x-dynamic is taken as (-1) as shown in Figure (5). Table (3) shows the amplitude values that occurred as a result of this changing, where noticed that when changing the initial condition of xi, the values of amplitudes of x and y have increased compared to the first case with remain the amplitude value in z-dynamic, as well as the general behavior of the Chen system has remained the same but different in the positions of pecks in time series of three dynamics.
By comparing between two Figures (6) and (2) of strange attractor, where it seems the Figure (6) is lower dynes (trajectories) than Figure (2), when changing the initial condition in x-dynamic. So not all initial conditions values will be investigate the hyper chaotic system. Sometimes, it converts the system to quasi-chaotic that is not advantage in our applications especially in secure communication. Indeed, this hyper chaotic (figure (3)) system has wide range of frequencies.
Table (3): The amplitude values with changing in initial condition
|
Dynamics |
xp.p (a.u) |
yp.p (a.u) |
zp.p (a.u) |
|
Amplitude |
24.9:-19 |
28:-20 |
39:-0.6 |
To certain the effect of the initial condition values for changing the system behaver Figure (7) shows fast Fourier transformation after changing the initial condition in x, so the bandwidth frequency have less than the first condition, where Table (4) shows these changes.
The bifurcation diagram in Figure (8) shows less chaotic behavior in the [25-28] range, with the transition states of the system remaining unchanged. This means that this value chosen from the initial conditions does not affect the behavior and states of the system.
For comprehensive study, Figure (9) shows the time series in x-dynamic when changing a set of initial conditions as the Chen system appears a change from hyper-chaotic to steady state and Table (5) shows some changes in the initial state of the system along with its behavior. Through this study, it was noted that the selection of initial conditions is important in obtaining an excessively chaotic system, especially when applied in secure communications.
Table(4): The important values of FFT spectrum of x, y, and z-dynamic
|
Dynamics |
x-dynamic |
y-dynamic |
z-dynamic |
|
Max amplitude |
0.65 |
0.65 |
0.45 |
|
Freq. Bandwidth |
8.5 |
8.2 |
8 |
|
FWHM |
1.6 |
1.78 |
1.45 |
Table (5): Shows some changing in initial condition of system with its behavior
|
Dynamics state |
x-dynamic |
y-dynamic |
z-dynamic |
State dynamic |
|
(a) |
-0.1 |
0 |
-0.6 |
Chaotic |
|
(b) |
-0.1 |
0.5 |
0 |
Chaotic |
|
(c) |
0 |
0 |
-0.6 |
Steady state |
|
(d) |
0 |
0 |
0 |
Steady state |
In this paper, and from obtained results, conclude the Chen system displayed sensitive dependence on the initial conditions, where at specific value of initial conditions made the behavior system hyper chaotic (increased the complexity) of the Chen system that will be used in secure communication with quickly transmit information and high security.
Declaration of interest statement
Authors would like to declare that they do not have any conflict of interest.
- Edward N Lorenz, Deterministic nonperiodic flow, Journal of the atmospheric sciences, 20(2):130–141, 1963.
- Otto E Rossler, An equation for continuous chaos, Physics Letters A, 57(5):397–398, 1976.
- Muthusamy Lakshmanan and Kallummal Murali, Chaos in nonlinear oscillators: controlling and synchronization, 13, World scientific, 1996.
- Kejeen M. Ibrahim, Raied K. Jamal, Kais A. Al-Naimee, Complex Dynamics in incoherent source with ac-coupled optoelectronic Feedback, Iraqi Journal of Science, Special Issue, Part B, 328-340, 2016.
- Kejeen M. Ibrahim and Raied K. Jamal, Full Synchronization of 2×2 Optocouplers Network Using LEDs, Australian Journal of Basic and Applied Sciences, 10(16): 8-13, 2016.
- Raied Kamel Jamal, Dina Ahmed Kafi, Secure communications by chaotic carrier signal using Lorenz model, Iraqi Journal of Physics, 14(30):51-63, 2016.
- Dina Ahmed Kafi, Raied Kamel Jamal, K. A. Al- Naimee. Lorenz model and chaos masking /addition technique. Iraqi Journal of Physics, 14(31):51-60, 2016.
- K M Ibrahim1, R K Jamal and F H Ali, Chaotic behaviour of the Rossler model and its analysis by using bifurcations of limit cycles and chaotic attractors, IOP Conf. Series: Journal of Physics: Conf. Series, 1003, 012099, 2018.
- Raied K. Jamal and Dina A. Kafi, Secure Communication Coupled semiconductor Laser Based on Rössler Chaotic Circuits. IOP Conf. Series: Materials Science and Engineering, 571, 012119, 2019.
- Raied K. Jamal and Dina A. Kafi, Secure Communication Coupled Laser Based on Chaotic Rössler Circuits, Nonlinear Optics, Quantum Optics, 51:79–91, 2019.
- Raied K. Jamal, Falah H. Ali, Falah A-H. Mutlak, Studying the Chaotic Dynamics Using Rossler-Chua Systems Combined with A Semiconductor Laser, Iraqi Journal of Science, 62(7):2213-2221, 2021.
- Ryam Salam Abdulaali, Raied K. Jamal. Salam K. Mousa, Generating a new chaotic system using two chaotic Rossler‑Chua coupling systems. Optical and Quantum Electronics, 53(667), 2021.
- Salam K. Mousa and Raied K. Jamal, Realization of a novel chaotic system using coupling dual chaotic system, Optical and Quantum Electronics, 53(188), 2021.
- Ibrahim A. Hamadi, Raied K. Jamal and Salam K. Mousa, Image encryption based on computer generated hologram and Rossler chaotic system. Optical and Quantum Electronics, 54(33), 2022.
- Leon O Chua, Chai Wah Wu, Anshan Huang, and G-Q Zhong, A universal circuit for studying and generating chaos, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40(10):745–761, 1993.
- Ryam Salam Abdulaali, Raied K. Jamal, Comprehensive Study and Analysis of the Chaotic Chua Circuit, Iraqi Journal of Science, 63(2)556-570, 2022.
- S Nakagawa and T Saito, Circuits and Systems Connecting the World, ISCAS 96, 3:245–248. IEEE, 1996.
- A Tamasevicius, A Namajunas, and A Cenys. Simple 4d chaotic oscillator. Electronics Letters, 32(11):957–958, 1996.
- Maciej J Ogorzalek, Order and chaos in a third-order rc ladder network with nonlinear feedback, IEEE Transactions on circuits and systems, 36(9):1221–1230, 1989.
- Hiroshi Kawakami, Bifurcation of periodic responses in forced dynamic nonlinear circuits: Computation of bifurcation values of the system parameters, IEEE Transactions on circuits and systems, 31(3):248–260, 1984.
- G Poddar, K Chakrabarty, and S Banerjee, Control of chaos in the boost converter, Electronics letters, 31(11):841–842, 1995.
- Chi K Tse, Flip bifurcation and chaos in three-state boost switching regulators, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 41(1):16–23, 1994.
- David F Delchamps, Some chaotic consequences of quantization in digital filters and digital control systems, In IEEE International Symposium on Circuits and Systems, 602–605. IEEE, 1989.
- Leon O Chua and Tao Lin, Chaos and fractals from third-order digital filters. International journal of circuit theory and applications, 18(3):241–255, 1990.
- Leon O Chua and T Lin, Fractal pattern of second-order non-linear digital filters: a new symbolic analysis, International Journal of Circuit Theory and Applications, 18(6):541–550, 1990.
- Chunbiao Li, Ihsan Pehlivan, Julien Clinton Sprott, and Akif Akgul, A novel four-wing strange attractor born in bistability, IEICE Electronics Express, 12(4):20141116–20141116, 2015.
- Akif Akgul, Irene Moroz, Ihsan Pehlivan, and Sundarapandian Vaidyanathan, A new four scroll chaotic attractor and its engineering applications, Optik, 127(13):5491–5499, 2016.
- Z. Hou, N. Kang, X. Kong, G. Chen and G. Yan, On the nonequivalence of Lorenz system and Chen system, Int. J. Bifurcation Chaos, 20:557-560, 2010.
- Azhaar Akram abdallua, Alaa kadhim Farahan,"A new Image encryption algorithem based on multi chaotic system", Iraqi journal of science, 36:1, 324-337,2022.