Anti-control of chaos of two-degrees-of-freedom
loudspeaker system and chaos synchronization
of different order systems
Z.-M. Ge
*, W.-Y. Leu
Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC Accepted 23 July 2003
Abstract
The chaos anti-control and synchronization of a two-degrees-of-freedom loudspeaker system are studied in this paper. Anti-control term is added to change state from regular to chaos. The anti-control methods such as addition of a constant force, of a periodic square wave, of a periodic saw tooth wave, of a periodic triangle wave, of a periodic rectified sinusoidal wave and of the xjxj term are used. The results are illustrated by numerical results, i.e. bifurcation diagram and Lyapunov exponents. Next, chaos synchronization of different order system is studied. Two methods are presented to achieve the synchronization: the addition of the coupling terms, the linearization of the error dynamics. The results are illustrated by phase diagram and time response.
Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction
Lorenz studied the strange changes in the atmosphere which is the first example to study chaos in 1963. In the past four decades, a large number of studies have shown that chaotic phenomena are observed in many physical systems that possess non-linearity [1,2]. It was also reported that the chaotic motion occurred in many non-linear control systems [3]. Furthermore, the problem of anti-controlling chaos (from periodic to chaotic) is interesting, non-traditional, and indeed very challenging. More importantly, within the biological context, anti-control of chaos suggests great potential for future applications. Recently, there have been many successful papers towards the goal of anti-control, which are essentially experimental or semi-analytical [4].
In this paper, chaos anti-control and synchronization of a two-degrees-of-freedom loudspeaker system are re-searched by many methods. First, a two- degrees-of-freedom loudspeaker system model and states equations of motion for it are introduced. Next, the bifurcation diagram and the Lyapunov exponent are expressed by numerical analysis. Then, anti-control of chaos is applied by adding different kinds of external forces. The external forces are a constant force, a periodic square wave, a periodic saw tooth wave, a periodic triangle wave, a periodic rectified sin and xjxj term. The results are demonstrated by various numerical results.
Chaos synchronization of different order systems are studied in Section 4. First, synchronization of two degrees-of-freedom loudspeaker system and Chua system is achieved by application of unidirectional coupled term. Next, syn-chronization of two-degrees-of-freedom loudspeaker system and Duffing system is discussed by application of the linearization of the error dynamics.
Finally, the conclusion of the whole paper is briefly stated.
*Corresponding author. Tel.: +886-35712121; fax: +886-35720634. E-mail address:[email protected](Z.-M. Ge).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.07.001
Fig. 1. A schematic diagram of loudspeaker system.
3. Anti-control of chaos
Creating chaos is called anti-control of chaos at times [6]. The problem of anti-control of chaos is interesting, non-traditional, and indeed very challenging.
In this section, many methods of anti-control, such as addition of a constant force, of a periodic square wave, of a periodic saw tooth wave, of a periodic triangle wave, of a periodic rectified sinusoidal wave and of a xjxj term [7], are proposed, which can enhance the existing chaos of the originally chaotic system. The results are demonstrated by numerical results, i.e. bifurcation diagram and Lyapunov exponent.
3.1. Anti-control of chaos by addition of a constant force
One can add a constant term to control the system dynamics from periodic motion to chaotic motion in non-linear non-autonomous system. This process is called anti-control. It ensures effective controlling in a simple way by choosing the value of the force. The constant force Fcis applied on the plate of the capacitor. Thus Eq. (2.1) becomes
_xx1¼ x2 _xx2¼ a21x1 a22x2þ a23x3þ a24x23þ a25sin xX sþ Fc _xx3¼ x4 _xx4¼ a41x1þ a42x1x3 a43x3 a44x4 8 > > > < > > > : ð3:1:1Þ
Changing the force Fcfrom zero downwards, the chaotic behavior is increased when Fc¼ 0:35,)1. Bifurcation
diagram and corresponding Lyapunov exponent diagram are shown as Figs. 4–6. The spectral analysis of the Lyapunov exponent diagram has been proved to be the most useful dynamic diagnostic tool for checking chaotic motion.
3.2. Anti-control of chaos by addition of a periodic force
Another way to anti-control chaos is using a periodic force as a control force. For this purpose, the added periodic force Fpis applied on the plate of the capacitor. Eq. (2.1) becomes
_xx1¼ x2 _xx2¼ a21x1 a22x2þ a23x3þ a24x23þ a25sin xX sþ Fp _xx3¼ x4 _xx4¼ a41x1þ a42x1x3 a43x3 a44x4 8 > > > < > > > : ð3:2:1Þ
Fig. 4. Bifurcation diagram for A between 38 and 44, Fc¼ 0:35.
Fig. 5. The bifurcation diagram for A between 38 and 44, Fc¼ 1.
37 38 39 40 41 42 43 44 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 A L
3.2.1. Adding a periodic force of square wave
In Eq. (3.2.1) periodic force Fpis a square wave. The square wave form is described by
FpðtÞ ¼
X1 k¼0
½ð1Þk A uðt sÞ ð3:2:2Þ
where A is the amplitude of the square wave, s¼ kP
2, P is the period of the square wave, uðtÞ is a unit step function.
The parameters of the periodic force are chosen such that amplitude A varies for fixed period P or the period varies with fixed amplitude. The chaotic behavior is effectively increased. Bifurcation diagram is shown as Figs. 7–9. 3.2.2. Adding a periodic force of saw tooth wave
In Eq. (3.2.1) periodic force Fpis a saw tooth wave. The saw tooth wave form is described by
FpðtÞ ¼ A Pt A X1 k¼1 ½uðt sÞ ð3:2:3Þ
where A is the amplitude of the saw tooth wave, s¼ kP , P is the period of the saw tooth wave, uðtÞ is a unit step function.
Fig. 7. Bifurcation diagram for A between 38 and 44, the parameter of square wave chosen as P¼ 0:5, A ¼ 0:4.
The parameters of the periodic force are chosen such that amplitude A varies for fixed period P . The chaotic be-havior is effectively increased. Bifurcation diagram is shown as Figs. 10 and 11.
3.2.3. Adding a periodic force of triangle wave
In Eq. (3.2.1) periodic force Fpis a triangle wave. The triangle wave form is described by
FpðtÞ ¼ A 2 16A p2 X1 k¼0 1 ð2 þ 4kÞ2cos 2p P t ! ð3:2:4Þ
where A is the amplitude of the triangle wave, P is the period of the triangle wave.
The parameters of the periodic force are chosen such that amplitude A varies for fixed period P or the period varies with fixed amplitude. The chaotic behavior is effectively increased. Bifurcation diagram is shown as Figs. 12–14. 3.2.4. Adding a periodic force of rectified sinusoidal wave
In Eq. (3.2.1) periodic force Fpis a rectified sinusoidal wave. The rectified sinusoidal wave form is described by
Fp¼ Aj sinðxtÞj ð3:2:5Þ
where A is the amplitude of the rectified sinusoidal wave, P ¼p
x, P is the period of the rectified sinusoidal wave, Fig. 9. Bifurcation diagram for A between 38 and 44, the parameter of square wave chosen as P¼ 1, A ¼ 4.
Fig. 11. Bifurcation diagram for A between 38 and 44, the parameter of saw tooth wave chosen as P¼ 0:5, A ¼ 2.
Fig. 12. Bifurcation diagram for A between 38 and 44, the parameter of triangle wave chosen as P¼ 0:5, A ¼ 1.
The parameters of the periodic force are chosen such that amplitude A varies for fixed frequency P or the fre-quency varies with fixed amplitude. The chaotic behavior is effectively increased. Bifurcation diagram is shown as Figs. 15–17.
3.3. Anti-control of chaos by addition of a xjxj term
We add k2x2jx2j to the second equation of Eq. (2.1) where k2is the strength. Bifurcation diagrams are shown as Fig.
18. We add k4x4jx4j to the fourth equation of Eq. (2.1) where k4is the strength. Bifurcation diagrams are shown as Fig.
19. We add k1x1jx1j, k2x2jx2j and k4x4jx4j to the first equation, second equation and fourth equation of Eq. (2.1),
re-spectively, where k1, k2and k4 are the strengths. Bifurcation diagrams are shown as Fig. 20.
4. Chaos synchronization of different order systems
The chaos synchronization of different order systems is discussed in this section. Two kinds of methods to achieve the synchronization are presented: the addition of the coupling terms and the linearization of the error dynamics.
Fig. 14. Bifurcation diagram for A between 38 and 44, the parameter of triangle wave chosen as P¼ 1, A ¼ 3.
Fig. 16. Bifurcation diagram for A between 38 and 44, the parameter of rectified sinusoidal wave chosen as P¼ 0:5, A ¼ 0:4.
Fig. 17. Bifurcation diagram for A between 38 and 44, the parameter of rectified sinusoidal wave chosen as P¼ 0:5, A ¼ 1.
4.1. Chaos synchronization of different order coupled chaotic systems
The chaotic synchronization of fourth-order response systems and third-order drive oscillator is studied in this section. Eq. (2.1) is as follows _xx1¼ x2 _xx2¼ a21x1 a22x2þ a23x3þ a24x23þ a25sin xX s _xx3¼ x4 _xx4¼ a41x1þ a42x1x3 a43x3 a44x4 8 > > < > > : ð4:1:1Þ
The parameters are chosen as follows: a21¼ 1, a22¼ 0:05, a23¼ 2, a24¼ 0:0847, a25¼ 5:5652, a41¼ 0:0694,
a42¼ 0:0694, a43¼ 1:27, a44¼ 0:5.
Chua system is an electronic circuit with one non-linear resistive element. The circuit equations can be written as a third-order system that is given by the following dimensionless form [8]
_yy1¼ c1½y1þ y2 f ðy1Þ
_yy2¼ y1 y2þ y3
_yy3¼ c2y2
ð4:1:2Þ
Fig. 19. Bifurcation diagram for A between 38 and 44, where k4¼ 0:0015.
where fðy1Þ ¼ c3y1þ 0:5ðc4 c3Þ½jy1þ 1j jy1 1j, ci, i¼ 1; 2; 3; 4; are positive constants. Let us take system Eq.
(4.1.2) as the drive system. The parameters are chosen as follows: r¼ 10:0, r2¼ 14:87, r3¼ 0:68, r4¼ 1:27. Initial
condition is arbitrarily located at the point yð0Þ ¼ ð0:1; 0:5; 0:2Þ.
The coupled chaotic systems is presented by adding linear coupling term Dðx2 y2Þ on the second equation of Eq.
(4.1.2) as
_yy1¼ c1½y1þ y2 f ðy1Þ
_yy2¼ y1 y2þ y3þ Dðx2 y2Þ
_yy3¼ c2y2
ð4:1:3Þ
where D is coupling strength. When D¼ 10; 000, the system will be synchronized and the result is shown in Fig. 21. 4.2. Chaos synchronization of different order systems by linearization of error dynamics
The chaotic synchronization of a fourth-order loudspeaker drive system and a second-order Duffing response os-cillator is studied in this section.
It is shown that dynamical evolution of the second-order response oscillators can be synchronized with the canonical projection of the fourth-order chaotic system. In this sense, it is said that synchronization is achieved in reduced order. Duffing equation is chosen as response system whereas loudspeaker system equation is defined as drive system. The synchronization scheme has non-linear feedback structure. The loudspeaker system Eq. (2.1) is
_xx1¼ x2 _xx2¼ a21x1 a22x2þ a23x3þ a24x23þ a25sin xX s _xx3¼ x4 _xx4¼ a41x1þ a42x1x3 a43x3 a44x4 8 > > < > > : ð4:2:1Þ
The parameters are chosen as follows: a21¼ 1, a22¼ 0:05, a23¼ 2, a24¼ 0:0847, a25¼ 5:5652, a41¼ 0:0694, a42¼
0:0694, a43¼ 1:27, a44¼ 0:5. The chaotic phase portrait is shown as Fig. 22.
Now let us consider the Duffing equation, which is given by _yy1¼ y2
_yy2¼ y1 y13 dy2þ seðsÞ þ u
ð4:2:2Þ
where d is a positive parameter that represents damping coefficient, seðsÞ ¼ a cosðxsÞ denotes driving force and u is the
coupling force (controller), parameters are chosen as d¼ 0:15, a ¼ 0:3, and x ¼ 1:0. Initial condition is arbitrarily located at the point yð0Þ ¼ ð0; 0Þ. The phase portrait s is shown as Fig. 23.
Fig. 21. (a) Time history of error, (b) time history of x2(solid line) and y2(dotted line) and (c, d) phase portraits of the synchronization system for D¼ 10; 000.
The differences between the states of the drive system and response system are e1¼ y1 x1, e2¼ y2 x2. The figures
are shown as Figs. 24 and 25. Then the error dynamics is
_ee1¼ e2
_ee2¼ e1 de2þ s0eþ u
ð4:2:3Þ
where
s0e¼ ½ða23þ a24x3Þx3þ a25sinð-=XÞs a cosðwsÞ ð1 þ a21 y12Þy1 ða22 dÞy2
Let u¼ s0
eþ k1e1þ k2e2.
Linearization of Eq. (4.2.3) becomes _ee1¼ e2
_ee2¼ ða21 k1Þe1 ða22 k2Þe2
ð4:2:4Þ i.e. _ee1 _ee2 ¼ 0 1 ða21 k1Þ ða22 k2Þ e1 e2 ð4:2:5Þ
Fig. 22. Phase portrait of the loudspeaker system without control term.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 y1 y2
We can rewrite Eq. (4.2.5) as
_ee¼ Ae ð4:2:6Þ
The characteristic equation of the system isjA kIj ¼ 0 and k are the eigenvalues of the system. And so
k2þ ða22 k2Þk þ ða21 k1Þ ¼ 0 ð4:2:7Þ
By the theory of linear system, if the eigenvalues are all negative, eðtÞ ¼ eð0Þ expðAtÞ will converge. So the eigenvalues are chosen as follows, k1;2¼ 32; 32, such that k1¼ 1023, k2¼ 63:05.
Then the form of controller is u¼ ða23þ a24x3Þx3þ a25sin
x X
s a cosð-sÞ ð1 þ a21 y12Þy1þ ða22 dÞy2 1023e1 63:05e2
The phase portraits and errors in the presence of control term are shown in Figs. 26–29.
Another example, Duffing equation is chosen as drive system, whereas loudspeaker system equation is defined as response system. The synchronization scheme has non-linear feedback structure.
Fig. 24. Time history of error e1without control term.
Fig. 26. Phase portrait of the loudspeaker system with control term.
Fig. 27. Phase portrait of the Duffing equation with control term.
Loudspeaker system Eq. (2.1) becomes _xx1¼ x2 _xx2¼ a21x1 a22x2þ a23x3þ a24x23þ a25sin xX sþ u _xx3¼ x4 _xx4¼ a41x1þ a42x1x3 a43x3 a44x4 8 > > < > > : ð4:2:8Þ
The parameters are chosen as follows: a21¼ 1, a22¼ 0:05, a23¼ 2, a24¼ 0:0847, a25¼ 5:5652, a41¼ 0:0694, a42¼
0:0694, a43¼ 1:27, a44¼ 0:5 and u is the coupling force. The phase portrait is shown as Fig. 30.
Now let us consider the Duffing equation, which is given by _yy1¼ y2
_yy2¼ y1 y13 dy2þ seðsÞ
ð4:2:9Þ
where d is a positive parameter which represents damping coefficient, seðsÞ ¼ a cosðxsÞ denotes driving force.
Parameters are chosen as d¼ 0:15, a ¼ 0:3, and x ¼ 1:0. Initial condition is arbitrarily located at the point yð0Þ ¼ ð0; 0Þ. The phase portrait is shown as Fig. 31.
Consider the differences between the states of the drive system and response system are e1¼ y1 x1, e2¼ y2 x2.
Their time histories are shown in Figs. 32 and 33.
Fig. 29. Time history of error e2with control term.
-20 -15 -10 -5 0 5 10 15 20 25 -20 -15 -10 -5 0 5 10 15 20 x1 x2
Fig. 31. Phase portrait of the Duffing equation without control term.
Fig. 32. Time history of error e1without control term.
Then the error dynamics is _ee1¼ e2 _ee2¼ e1 de2 s0e u ð4:2:10Þ where s0e¼ ½y3
1 a25sinð-=XÞs þ a cosðxsÞ þ ða21þ 1Þx1þ ða22 dÞx2 ða23þ a24x3Þx3
Let u¼ s0
eþ k1e1þ k2e2. Linearization of Eq. (4.2.13) becomes
_ee1¼ e2
_ee2¼ ð1 k1Þe1 ðd þ k2Þe2
ð4:2:11Þ as _ee1 _ee2 ¼ 0 1 ð1 k1Þ ðd þ k2Þ e1 e2 ð4:2:12Þ
We can rewrite Eq. (4.2.12) as
_ee¼ Ae ð4:2:13Þ
The characteristic equation of the system isjA kIj ¼ 0 and k are the eigenvalues of the system, and so k2þ ðd þ k
2Þk ð1 k1Þ ¼ 0 ð4:2:14Þ
By the theory of linear system, if the eigenvalues are all negative, eðtÞ ¼ eð0Þ expðAtÞ will converge. So the eigenvalues are chosen as follow k1;2¼ 29,)29, such that k1¼ 842, k2¼ 57:85.
Then the form of controller is u¼ y3
1 ða23þ a24x3Þx3 a25sin
x X
sþ a cosð-sÞ þ ða21þ 1Þx1þ ða22 dÞx2þ 842e1þ 57:85e2
The phase portraits and errors in presence of the control term are shown in Figs. 34–37.
5. Conclusions
In this paper, anti-control and synchronization of a two-degrees-of-freedom loudspeaker system are studied. In Section 2, a two-degrees-of-freedom loudspeaker system model and states equations of motion are introduced. Next, the bifurcation diagram and the Lyapunov exponent are expressed by numerical analysis.
Fig. 35. Phase portrait of the Duffing equation with control term.
Fig. 36. Time history of error e1with control term.
In Section 3, anti-control of chaos is accomplished by adding a constant force, a periodic square wave, a periodic saw tooth wave, a periodic triangle wave, a periodic rectified sinusoidal wave or a xjxj term. The originally existing chaos is enhanced which is illustrated by the bifurcation diagrams and Lyapunov exponents.
The chaos synchronization of different order systems is discussed in the Section 4. First, synchronization of two degrees-of-freedom loudspeaker system and Chua system is achieved by application of unidirectional coupling term, while coupling strength is rather large. Finally, synchronization of two-degrees-of-freedom loudspeaker system and Duffing system is accomplished by application of the linearization of the error dynamics. The results are demonstrated by applying various numerical results.
Acknowledgements
This research was supported by the National Science Council, Republic of China, under grant number NSC 91-2212-E-009-025.
References
[1] Ott E. Chaos in dynamical systems. 2nd ed. England: Cambridge; 2002. [2] Khailil HK. Nonlinear systems. New Jersey: Prentice-Hall; 2002.
[3] Holmes P. Bifurcation and chaos in a simple feedback control system. In: Proceedings of the IEEE 22nd Conference on Decision and Control, 1983, p. 365–70.
[4] Lai D, Chen G. Distribution of controlled Lyapunov exponents a statistical simulation study. Comput Stat Data Anal 2000;33:69– 77.
[5] Ge ZM, Bao TS. Chaos, chaos control and synchronization of electromechanical loudspeaker system. To be published. [6] Chen G. Control and anticontrol of chaos. In: Proceedings of the 1st International Conference on Control of Oscillations and
Chaos, 1997, p. 181–86.
[7] Tang KS, Man KF, Zhong GQ, Chen G. Generating chaos via xjx. IEEE Trans Circ Syst 2001;48(5):636–41. [8] Ricardo F, Gualberto SP. Synchronization of chaotic system with different order. Phys Rev E 2002;65(5):036226-1-7.
