Ain Shams Engineering Journal - Hashemite UniversityAin Shams Engineering Journal xxx (2017)...
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A new four-dimensional chaotic attractor M. Ababneh Department of Mechatronics Engineering, The Hashemite University, Zarqa, Jordan article info Article history: Received 10 June 2016 Revised 26 July 2016 Accepted 14 August 2016 Available online xxxx Keywords: Chaos attractor Lyapunov exponents Bifurcation diagram Poincaré map Riccati equation Stability abstract A new four-dimensional chaotic system is introduced in this paper, it is mainly consist of four multiplier terms and four simple terms. The system has different structure and topology of existing four- dimensional systems and produces two equilibrium points with one at the origin. The fundamental char- acteristics of the new system are investigated by means of equilibrium points, their stabilities, power spectrum, bifurcation diagram, and Poincaré map. Furthermore, an optimal controller using Riccati equa- tion is established to run system trajectories to the zero equilibrium. The dynamics of the new system is simulated using Matlab and Simulink. Ó 2017 Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction Chaotic systems are nonlinear systems which are sensitive to initial conditions and exhibit rich dynamic behavior [1]. Further- more, a chaotic attractor is defined as a chaotic set toward which a dynamic system tends to evolve [2]. Chaotic systems have wide range of applications in many Engineering and non-Engineering fields, some of their applications are found in intelligent controls [3], power systems [4], secure communication [5], biology [6], and mathematics [7]. Furthermore, chaotic attractors were discovered by Lorenz while he was studying atmospheric convection, then, he intro- duced the first three-dimensional chaotic system in 1963 [8]. Afterwards, Rossler continued this work of dissipative dynamical system and proposed a new chaotic system in 1976 [9]. More work has been conducted since then, for example Chen introduced a new three-dimensional attractor in 1999 which is not topologically equivalent to Lorenz system [10]. In addition, recently there has been great interest in investigating hidden attractors where basin of attraction does not intersect with small neighborhood of equi- libria [11–15]. Proposing of new chaotic attractors with new structures and dynamics is very useful to the field of chaos theory and its applica- tions [3–7]. In this paper, a new four-dimensional chaotic system is introduced; it is mainly consist of four multiplier terms and four simple terms. The contribution of this work is that it proposes a novel system which has different structure and topology of exist- ing four-dimensional systems [16–18]. The fundamental character- istics of the new system are investigated by means of equilibrium points, their stabilities, bifurcation diagrams, Poincare maps, and power spectrum. Further investigations of the system are con- ducted in next section. However, the system is completely new and does not belong to a known family of known systems. The rest of paper is organized as follows. In Section 2, the new four-dimensional chaotic system is presented. In Section 3, the numerical analysis and simulation of system dynamics are shown. In Section 4, an optimal controller design based on Riccati equation is derived and implemented. Finally, conclusions are presented in Section 5. 2. The new four dimensional chaotic system The new chaotic system has the following set of four dynamic equations: _ w ¼ aðx wÞ _ x ¼ bw wy _ y ¼ wx xz _ z ¼ xy y ð1Þ where x ¼½w; x; y; z T is the state vector of the system. The signifi- cance of the system in (1) is that it presents a new 4D chaotic sys- tem with a different dynamics than existing systems. To study the system we first look at the equilibrium points which are found by http://dx.doi.org/10.1016/j.asej.2016.08.020 2090-4479/Ó 2017 Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer review under responsibility of Ain Shams University. E-mail address: [email protected] Ain Shams Engineering Journal xxx (2017) xxx–xxx Contents lists available at ScienceDirect Ain Shams Engineering Journal journal homepage: www.sciencedirect.com Please cite this article in press as: Ababneh M. A new four-dimensional chaotic attractor. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j. asej.2016.08.020
Transcript of Ain Shams Engineering Journal - Hashemite UniversityAin Shams Engineering Journal xxx (2017)...
Ain Shams Engineering Journal xxx (2017) xxx–xxx
Contents lists available at ScienceDirect
Ain Shams Engineering Journal
journal homepage: www.sciencedirect .com
A new four-dimensional chaotic attractor
http://dx.doi.org/10.1016/j.asej.2016.08.0202090-4479/� 2017 Ain Shams University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer review under responsibility of Ain Shams University.E-mail address: [email protected]
Please cite this article in press as: Ababneh M. A new four-dimensional chaotic attractor. Ain Shams Eng J (2017), http://dx.doi.org/10asej.2016.08.020
M. AbabnehDepartment of Mechatronics Engineering, The Hashemite University, Zarqa, Jordan
a r t i c l e i n f o
Article history:Received 10 June 2016Revised 26 July 2016Accepted 14 August 2016Available online xxxx
Keywords:Chaos attractorLyapunov exponentsBifurcation diagramPoincaré mapRiccati equationStability
a b s t r a c t
A new four-dimensional chaotic system is introduced in this paper, it is mainly consist of four multiplierterms and four simple terms. The system has different structure and topology of existing four-dimensional systems and produces two equilibrium points with one at the origin. The fundamental char-acteristics of the new system are investigated by means of equilibrium points, their stabilities, powerspectrum, bifurcation diagram, and Poincaré map. Furthermore, an optimal controller using Riccati equa-tion is established to run system trajectories to the zero equilibrium. The dynamics of the new system issimulated using Matlab and Simulink.� 2017 Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under
the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Chaotic systems are nonlinear systems which are sensitive toinitial conditions and exhibit rich dynamic behavior [1]. Further-more, a chaotic attractor is defined as a chaotic set toward whicha dynamic system tends to evolve [2]. Chaotic systems have widerange of applications in many Engineering and non-Engineeringfields, some of their applications are found in intelligent controls[3], power systems [4], secure communication [5], biology [6],and mathematics [7].
Furthermore, chaotic attractors were discovered by Lorenzwhile he was studying atmospheric convection, then, he intro-duced the first three-dimensional chaotic system in 1963 [8].Afterwards, Rossler continued this work of dissipative dynamicalsystem and proposed a new chaotic system in 1976 [9]. More workhas been conducted since then, for example Chen introduced a newthree-dimensional attractor in 1999 which is not topologicallyequivalent to Lorenz system [10]. In addition, recently there hasbeen great interest in investigating hidden attractors where basinof attraction does not intersect with small neighborhood of equi-libria [11–15].
Proposing of new chaotic attractors with new structures anddynamics is very useful to the field of chaos theory and its applica-tions [3–7]. In this paper, a new four-dimensional chaotic system isintroduced; it is mainly consist of four multiplier terms and four
simple terms. The contribution of this work is that it proposes anovel system which has different structure and topology of exist-ing four-dimensional systems [16–18]. The fundamental character-istics of the new system are investigated by means of equilibriumpoints, their stabilities, bifurcation diagrams, Poincare maps, andpower spectrum. Further investigations of the system are con-ducted in next section. However, the system is completely newand does not belong to a known family of known systems.
The rest of paper is organized as follows. In Section 2, the newfour-dimensional chaotic system is presented. In Section 3, thenumerical analysis and simulation of system dynamics are shown.In Section 4, an optimal controller design based on Riccati equationis derived and implemented. Finally, conclusions are presented inSection 5.
2. The new four dimensional chaotic system
The new chaotic system has the following set of four dynamicequations:
_w ¼ aðx�wÞ_x ¼ bw�wy_y ¼ wx� xz_z ¼ xy� y
ð1Þ
where x ¼ ½w; x; y; z�T is the state vector of the system. The signifi-cance of the system in (1) is that it presents a new 4D chaotic sys-tem with a different dynamics than existing systems. To study thesystem we first look at the equilibrium points which are found by
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w
X
Fig. 1. A new chaotic attractor x-w phase plane.
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Fig. 2. A new chaotic attractor y-z phase plane.
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y
Fig. 3. Three-dimensional z-x-y portrait.
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setting Eq. (1) to zero. This produces two equilibrium points: (0, 0,0, 0) and (1, 1, b, 1). In order to study their stabilities the Jacobian of(1) is found as follows:
J ¼
�a a 0 0b� y 0 �w 0x w� z 0 �x0 y x� 1 0
26664
37775 ð2Þ
The linearized system at first equilibrium point (0, 0, 0, 0) isgiven by the following Jacobian matrix:
J ¼
�a a 0 0b 0 0 00 0 0 00 0 �1 0
26664
37775 ð3Þ
The characteristics equation for (0, 0, 0, 0) is k2ðk2 þ ak� abÞand the roots are given as double poles at origin and the other
poles at �a�ffiffiffiffiffiffiffiffiffiffiffia2þ4ab
p2 which makes the point is unstable point.
The linearized system at second equilibrium points (1, 1, b, 1) isgiven by the following Jacobian matrix:
J ¼
�a a 0 00 0 �1 01 0 0 �10 b 0 0
26664
37775 ð4Þ
Moreover, The characteristics equation for the point isk4 þ ak3 þ ða� bÞk� ab and applying Routh-Hurwitz criterionyields that this point is VðtÞ ¼ Vð0Þe�at stable only for a > 0 andab < 0.
The new system is proved to be dissipative system for all posi-tive a since the divergence of flow of the system is:
1V
dVdt
¼ divV ¼ @ _w@w
þ @ _x@x
þ @ _y@y
þ @ _z@z
¼ �a ð5Þ
And VðtÞ ¼ Vð0Þe�at with rate of contraction dVdt ¼ Vð0Þe�a. Eq. (5)
proves the existence of a bounded and attracting chaotic set thatform the attractor.
3. Numerical analysis
The new chaotic system has been tested for a wide range ofparameter values and proved to yield chaotic behavior for manyselections of parameters a and b. By choosing one pair values asa ¼ 23 and b ¼ 9 the chaotic system in (1) is dissipative and thetwo equilibrium points are unstable. The portrait for two dimen-sions and three dimensions, for initial conditions (1, 1, 1, 1), areshown below. First, the phase-plane for x-w is shown in Fig. 1,and the phase-plane for y-z is shown in Fig. 2.
Then, the 3D portraits are shown in Figs. 3–5, for differentarrangements of the axes.
The power spectrum of signal describes how the variance of thedata is distributed over the frequency domain [19]. Fig. 6 showsthe power spectrum of the signal w(t) of the system (1). It is obvi-ous from the figure that the system has a bandwidth of roughly 0–50 Hz.
Based on the algorithm presented [20] by Alan Wolf, the Lya-punov exponents are calculated using Matlab with step-size 0.05and after 10,000 iterations as follows:
Please cite this article in press as: Ababneh M. A new four-dimensionalasej.2016.08.020
LE1 = 1.86137, LE2 = �2.95652, LE3 = �3.88644, LE4 = �18.0247and Lyapunov dimension of 1.6296.
chaotic attractor. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.
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wx
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Fig. 4. Three-dimensional w-x-y portrait.
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w
Fig. 5. Three-dimensional z-x-w portrait.
0 50 100 150 200 250 300 350 400 450 500-40
-30
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-10
0
10
20
30
Frequency (Hz)
Pow
er/fr
eque
ncy
(dB
/Hz)
Welch Power Spectral Density Estimate
Fig. 6. Power spectral density.
0 5 10 15 20 25 30-500
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a
Xm
ax
Fig. 7. Bifurcation diagram as a function of a.
0 2 4 6 8 10 12 14 16-500
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b
Xm
ax
Fig. 8. Bifurcation diagram as a function of b.
M. Ababneh / Ain Shams Engineering Journal xxx (2017) xxx–xxx 3
Moreover, bifurcation diagram displays the maxima and peaksof numerically computed solutions of a system as a function of abifurcation parameter in the system while other parameters arefixed [21]. The chaotic dynamic of the system as a function of
Please cite this article in press as: Ababneh M. A new four-dimensionalasej.2016.08.020
parameter a, and b = 9, is displayed in Fig. 7 and shows a stablesystem when parameter a is less than 8 and chaotic system whenit is greater than 8.
Furthermore, the chaotic dynamic of the system as a function ofparameter b, and a = 23, is displayed in Fig. 8, and shows a stablesystemwhen parameter b is less than 7.5 and chaotic system whenit is greater than 7.5. Furthermore, a closer look at the system tra-jectories would place it as hidden attactor.
Another way to look at the system dynamics is through Poin-caré maps of the system, which is a technique that representsthe phase space of the system by isolating some trajectories [22].Figs. 9–14 show the Poincaré maps in different 2D planes whileother two variables are set to zeros. It is noticed that several planesof the system are symmetrical. For example, both planes in Fig. 10shows even symmetry around the y-axis, while Figs. 11(a) and 13(a) show odd function curves. In addition, Fig. 14 shows even sym-metry function curves around the z-axis. Note that the range ofcoordinates is in magnitude of hundreds, therefore the other vari-ables are set to |variable| < 5 rather = 0 to compensate for numeri-cal approximations.
chaotic attractor. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.
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Fig. 9. Poincaré map in wx plane when (a) |y| < 5, (b) |z| < 5.
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w
y
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Fig. 10. Poincaré map in wy plane when (a) |x| < 5, (b) |z| < 5.
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1000(a) |x|<5
w
z
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w
z
Fig. 11. Poincaré map in wz plane when (a) x < 5, (b) y < 5.
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1400(a) |w|<5
x
y
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x
y
Fig. 12. Poincaré map in xy plane when (a) w < 5, (b) z < 5.
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2000(a) |w|<5
x
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x
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Fig. 13. Poincaré map in xz plane when (a) w < 5, (b) y < 5.
-500 0 500 1000 1500-2000
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2000(a) |w|<5
y
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y
z
Fig. 14. Poincaré map in yz plane when (a) w < 5, (b) x < 5.
4 M. Ababneh / Ain Shams Engineering Journal xxx (2017) xxx–xxx
Please cite this article in press as: Ababneh M. A new four-dimensional chaotic attractor. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.asej.2016.08.020
M. Ababneh / Ain Shams Engineering Journal xxx (2017) xxx–xxx 5
4. Optimal control design
Linear Quadratic Regulator (LQR) is a popular case of optimalcontrol where a measure of the quadratic continuous time costfunction
J ¼ 12
Z 1
0XTðtÞQðtÞXðtÞ þ uTðtÞRðtÞuðtÞh i
dt ð6Þ
is minimized. And it is subjected to linear dynamic constraints asgiven in the linearzed Jacobian system discussed in Section 2. BothQ and R matrices are positive definite matrices to ensure the cost
Fig. 15. System block diagram with optimal controller.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-200
-100
0
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w
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-200
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seconds
Fig. 16. Trajectories w and x with controller after 3 s.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-500
0
500
y
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1000
-500
0
500
1000
z
seconds
Fig. 17. Trajectories y and z with controller after 3 s.
Please cite this article in press as: Ababneh M. A new four-dimensionalasej.2016.08.020
measure remain positive. Furthermore, the negative feedback con-troller, shown in Fig. 15, is in the form
uðtÞ ¼ �KxðtÞ ð7ÞIt has been shown in optimal control theory that the feedback
controller K is given by K ¼ R�1BTS where S is the solution of thewell known Algebraic Riccati equation (6)
ATSþ SA� SBR�1BTSþ Q ¼ 0 ð8Þwhere S is symmetrical solution matrix [23]. This controller is cal-culated around every point of the trajectory. Figs. 16 and 17 showthe four trajectories with the controller is applied after 3 s to drivethe trajectories to the origin.
5. Conclusions
A new chaotic system, with different structure and topology ofexisting four-dimensional systems, is introduced in this paper, ithas a simple structure since it has only four multiplier terms andfour simple terms. The system produced two equilibrium pointsat (0, 0, 0, 0) and (1, 1, b, 1). The new chaotic system was testedfor a wide range of parameters values and proved to yield chaoticbehavior for many selections of parameters a and b. The new sys-tem was proved to be dissipative system for all positive a and forall values of b. Therefore, by choosing a ¼ 23; b ¼ 9 makes thechaotic system (1) dissipative and the two equilibrium pointsunstable. Optimal controller, based on Riccati equation, wasdesigned and utilized to control system trajectories to the zeroequilibrium. Future work is to study the prospect of system self-synchronization in order to apply it to cryptography and securecommunication. In addition, it is suggested to investigate the pos-sibility of using the system in chaotic fluid mixing.
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Please cite this article in press as: Ababneh M. A new four-dimensionalasej.2016.08.020
Dr. Ababneh received his Ph.D. in 2004 from theDepartment of Electrical and Computer Engineeringfrom University of Houston, Texas. Before that heworked for three major US industries, where he workedas Project Engineer with FMC Energy Systems in Hous-ton, System Engineer with Compaq Computers Corpo-ration in Houston, and Maintenance Engineer withInteplast Corporation in Lolita, Texas. He has been withthe Department of Mechatronics Engineering of theHashemite University in Jordan Since 2004. Where heserved as department chair for two years in 2005 and2006. His main research interest has been the in the
areas of control systems, system synchronization, and energy systems.
chaotic attractor. Ain Shams Eng J (2017), http://dx.doi.org/10.1016/j.
