SOUND AND VIBRATION
www.elsevier.com/locate/jsvi
Journal of Sound and Vibration 270 (2004) 685–712
Chaos synchronization and chaos anticontrol of
a suspended track with moving loads
Zheng-Ming Ge*, Hong-Wen Wu
Department of Mechanical Engineering, Nation Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC
Received 6 August 2002; accepted 4 February 2003
Abstract
In this paper, we first study the chaotic synchronization phenomenon of the suspended track with moving load system and then the transient response of chaos synchronization. The Lyapunov exponent is utilized to prove the chaos synchronization, but under certain conditions it is more complicated. Next, we study the synchronization of autonomous and non-autonomous systems. We find that slave systems cannot be synchronized with master system for certain time excitations no matter how large A is. Next, the phase synchronization between them will be studied. An application of chaos synchronization and secure communication is presented.
Finally, in order to increase the chaos phenomena, we use anticontrol. Constant torque, periodic torque, periodic impulse signal, time delay function, and adaptive control are used successfully to control the state from order to chaos.
r2003 Elsevier Science Ltd. All rights reserved.
1. Introduction
Synchronization is a basic phenomenon in physics, engineering and many other scientific disciplines. In the classical sense, synchronization means frequency and phase locking of periodic oscillators. However, even chaotic systems may be linked in such a way that their chaotic oscillations are synchronized, so that the difference of the state vectors of both chaotic systems converges to zero. Recently, chaos synchronization has been studied extensively. Identical chaotic systems can be successfully synchronized by linearly and non-linearly coupled terms discussed in this paper.
*Corresponding author. Tel.: +886-3-5712121; fax: +886-3-5720634. E-mail address:zmg@cc.nctu.edu.tw (Z.-M. Ge).
0022-460X/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0022-460X(03)00187-1
The suspended track with moving load system [1], which is excited by a harmonic torque (M sin ot) and a periodic force (F sin ot), is explored. The chaos synchronization phenomena of master and slaver systems are studied[2–6]. These phenomena and the transient response of chaos synchronization are described. The Lyapunov exponent is utilized to prove the chaos synchronization , but under certain conditions it is more complicated.
When a usual Runge–Kutta numerical scheme is used, the full state variables may be integrated. The Euclidean distance d ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx1 x2Þ2þ ðy1 y2Þ2þ ðz1 z2Þ2 q
between the two trajectories is monitored for various choices of coupling parameter A: The distance between the trajectories of the subsystems, i.e., the stability of the chaotic attractor in the invariant subset is monotonic if the distance decreases to zero monotonically in time. Euclidean distance d between the drive and the response trajectories will converge to zero if systems synchronize.
The synchronization of the autonomous and the non-autonomous systems is studied. It is found that slave systems cannot be synchronized with master systems, no matter how large A is. Next, the phase synchronization between them can be realized [7,8].
An application of synchronization and secure communication is presented [9,10]. This paper presents a way to transmit and retrieve a signal via chaotic systems. In contrast to existing schemes with one transmission line, a two-channel transmission method is adopted for the purpose of faster synchronization and higher security. Basically, an output of the chaotic transmitter is sent for synchronization, only with no connection to the information signal. The other channel transmits a signal generated from a highly non-linear function of the chaotic states, while the first complicated encryption and improves privacy. Simulation results validate the new chaotic-based secure communication method.
Finally, in order to increase the chaos phenomena, anticontrol is used[11]. Constant torque, periodic torque, periodic impulse signal, time delay function[12,13,14], and adaptive control are used successfully to control the state from order to chaos.
2. Synchronization phenomena of coupled chaotic systems
2.1. Description of the system model and differential equations of motion
The suspended track system is depicted in Fig. 1. The beam which can rotate freely about a vertical axis is suspended by a string. There are two heavy loads linked with beam by a spring moving on the track of the beam with viscous damping. Neglecting the dry friction referring to
Fig. 1, one can write the expression for kinetic energy T and potential energy V as T ¼12J’j þ mð’r þ r2’j2Þ;
V ¼ Kðr r0Þ2;
where J is the moment of inertia of the beam about the vertical axis, m the mass of each load, K the spring coefficient, r the distant between vertical axis and the center of the load, r0the original length of the spring, j the rotating angle of the beam.
Thus, the Lagrangian is given by L ¼ T þ V ¼1
2Jj
2þ mð’r2þ r2’j2Þ Kðr r 0Þ2:
Since the system includes a non-conservative damping force, their energy is lost. Rayleigh’s dissipation function of the system is
R ¼ B’r2;
where B is the damping coefficient and Lagrange’s equations are: d dt @L @’j @L @j¼ 0; d dt @L @’r @L @r ¼ Qr ¼ @R @’r: The dynamics equations of the system are:
.j þ 4mr’r ’j J þ 2mr2 ¼ 0; m.r mr ’j2þ Kðr r0Þ ¼ B’r:
It is assumed that the beam is subjected to a harmonic torque M sin ot along the direction of j; and each load is subjected to a periodic force F sin ot along the direction of r: Then the equations become
.j þ 4mr’r ’j
J þ 2mr2¼ M sin ot; m.r mr ’j2þ Kðr r0Þ ¼ B’r F sin ot;
where o is the frequency of the external torque and external force. To show our system in dimensionless form is a better way for research. Use the dimensionless time t ¼ Ot; where O is a
normalized frequency. Substituting t ¼ Ot; the following dimensionless equations are obtained: j00 rr 0j0 Jrþ12r2 ¼ Mjsin ott; r00 rj02þ Kmðr 1Þ ¼ Bmr0 Frsin ott; where j0¼dj dt; j 00¼d2j dt2; r 0¼dr dt; r 00¼d2r dr2; ot¼ o O; r ¼ r r0 ; Jr ¼ J 4mr20; Km¼ K mO2; Bm¼ B mO; Mj¼ M 4mr2 0O 2; Fr ¼ F mr0O2 :
The phase portrait is the evolution of a set of trajectories emanating from various initial conditions in the state space. During the investigations of the dynamical system we are particularly interested in the periodic and chaotic behaviors of the phase trajectories. Using the dimensionless equation and letting a ¼ j; x ¼ j0; y ¼ r; z ¼ r0; the system equations become ’x ¼ xyz Jrþ1 2y2 Mjsin ott; ’y ¼ z; ’z ¼ x2y K mðy 1Þ Bmz Frsin ott: ð1Þ
It is noted that a ¼ j is cyclic. Since a does not appear on the right side of the last three equations of the system, it produces no effect on the dynamics of the last three equations.
Typical graphs of the three computed Lyapunov exponents for the non-linear dynamical system (Eq. (1)) are plotted in Fig. 2 as Mj ranges from 4 to 6 and Fr¼ 1: The system has the chaos phenomena when we choose Mj¼ 5 and Fr¼ 1:
2.2. Synchronization of mutual coupled chaotic systems 2.2.1. Synchronization by linear coupling term
In this subsection, we consider two identical mutually coupled systems[15,16]. These are more complex than uni-directional systems. The master and slave system can be expressed as follows:
Master system: ’x1¼ x1y1z1 Jrþ12y21 Mjsin ott Aðx1 x2Þ; ’y1¼ z1; ’z1¼ x21y1 Kmðy1 1Þ Bmz1 Frsin ott: ð2Þ
Slave system: ’x2¼ x2y2z2 Jrþ12y22 Mjsin ott þ Aðx1 x2Þ; ’y2¼ z2; ’z2¼ x22y2 Kmðy2 1Þ Bmz2 Frsin ott; ð3Þ
where A is the coupling strength and Aðx1 x2Þ is the coupling term. These two systems have different initial conditions: ðx10; y10; z10Þ ¼ ð0:1; 0:2; 0:3Þ and ðx20; y20; z20Þ ¼ ð0:1; 0:2; 0:3Þ: When Ao0:0824; the systems are not synchronized and the results are shown inFig. 3. The phase portrait y versus z gives the relation between the displacement r and velocity ’r; the phase portrait x versus z gives the relation between angular velocity ’j and linear velocity ’r; the phase portrait x versus y gives the relation between angular velocity ’j and linear displacement r for completeness, since these three state variables are in equality mathematically. When AX0:0825; the systems are synchronized and the results are shown in Fig. 4. These results can also be proved from the Lyapunov exponent diagram inFig. 5, from which the critical value of A for synchronization can be found. At synchronization, one of the Lyapunov exponent transverses the zero value from positive to negative. This indicates that the transversality means synchronization and the transverse value is the critical value A ¼ 0:0825: Although the critical value from phase portraits and that from the Lyapunov exponent diagram are not identical, they are very close and the difference is because of only the computational error.
2.2.2. Synchronization by non-linear coupled term
In this subsection, two systems beginning with two different initial conditions will be synchronized by non-linear coupled term. The coupling term is A sinðx1 x2Þ: Using Lyapunov exponent as a criterion for analyzing whether synchronization occurs or not is more complex.
4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 M L y a p u nov ex pon ent
-1 0 1 2 3 -10 -5 0 5 10 15 x1 -1 0 1 2 3 -20 -10 0 10 20 x1 -10 -5 0 5 10 15 -20 -10 0 10 20 y1 -1 0 1 2 3 -10 -5 0 5 10 15 a = 0.08 x2 -1 0 1 2 3 -20 -10 0 10 20 x2 -10 -5 0 5 10 15 -20 -10 0 10 20 y2 4000 4200 4400 4600 4800 5000 -0.05 0 0.05 t (sec) 4000 4200 4400 4600 4800 5000 -0.05 0 0.05 t (sec) 4000 4200 4400 4600 4800 5000 -0.05 0 0.05 t (sec) x1-x2 y1-y2 z1-z2 y1 z1 z1 z2 y2 y2
Fig. 3. Phase portrait and time–response error of mutual coupled systems with coupling terms Aðx2 x1Þ and Aðx1 x2Þ for A ¼ 0:08: -1 0 1 2 3 -10 -5 0 5 10 15 x1 -1 0 1 2 3 -20 -10 0 10 20 x1 -10 -5 0 5 10 15 -20 -10 0 10 20 y1 -1 0 1 2 3 -10 -5 0 5 10 15 A = 0.0825 x2 -1 0 1 2 3 -20 -10 0 10 20 x2 -10 -5 0 5 10 15 -20 -10 0 10 20 y2 0 1000 2000 3000 4000 5000 -2 -1 0 1 2 t (sec) 0 1000 2000 3000 4000 5000 -10 -5 0 5 10 t (sec) 0 1000 2000 3000 4000 5000 -10 -5 0 5 10 t (sec) y1 z1 z1 y2 z2 z2 z1-z2 y1-y2 x1-x2
Fig. 4. Phase portrait and time–response error of mutual coupled systems with coupling terms Aðx2 x1Þ and Aðx1 x2Þ for A ¼ 0:0825:
These two systems have different initial conditions: ðx10; y10; z10Þ ¼ ð0:1; 0:2; 0:3Þ and ðx20; y20; z20Þ ¼ ð0:1; 0:2; 0:3Þ: When Ap0:02 and near 0:06 and 0:083; the systems are not synchronized and the result is shown inFig. 6. When AX0:084; the systems are synchronized and the result is shown in Fig. 7. These results can also be proved from the Lyapunov exponent diagram inFig. 8from which the critical value of A for synchronization can be found, but this is not accurate. At synchronization, one of the Lyapunov exponent transverses the zero value from positive to negative. This indicates that the transversality means synchronization and the transverse value is the critical value A ¼ 0:083: Although two critical values from phase portraits and that from Lyapunov exponent diagram are not identical, they are very close because of computational error.
2.3. Synchronization via adaptive feedback
In this section, we study the adaptive control of the system. The adaptive control directs a chaotic trajectory to stable trajectory.
The master and slave system can be described as follows: Master system: ’x1¼ x1y1z1 Jrþ12y21 Mjsin ott; ’y1¼ z1; ’z1¼ x21y1 Kmðy1 1Þ Bmz1 Frsin ott: ð4Þ 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 A Lyapunov exponent
Fig. 5. The Lyapunov exponent of mutual coupled systems with coupling terms Aðx2 x1Þ and Aðx1 x2Þ for A between 0.02 and 0.2.
-1 0 1 2 3 -10 -5 0 5 10 15 x1 -1 0 1 2 3 -20 -10 0 10 20 x1 -10 -5 0 5 10 15 -20 -10 0 10 20 y1 -1 0 1 2 3 -10 -5 0 5 10 15 A = 0.06 x2 -1 0 1 2 3 -20 -10 0 10 20 x2 -10 -5 0 5 10 15 -20 -10 0 10 20 y2 0 1000 2000 3000 4000 5000 -2 -1 0 1 2 t (sec) 0 1000 2000 3000 4000 5000 -10 -5 0 5 10 t (sec) 0 1000 2000 3000 4000 5000 -20 -10 0 10 20 t (sec) y1 z1 z1 y2 z2 z2 z1-z2 y1-y2 x1-x2
Fig. 6. Phase portrait and time–response error of mutual coupled systems with coupling terms A sinðx2 x1Þ and A sinðx1 x2Þ for A ¼ 0:06: -2 0 2 4 -10 -5 0 5 10 15 x1 -2 0 2 4 -20 -10 0 10 20 x1 -10 -5 0 5 10 15 -20 -10 0 10 20 y1 -2 0 2 4 -10 -5 0 5 10 15 A = 0.088 x2 -2 0 2 4 -20 -10 0 10 20 x2 -10 -5 0 5 10 15 -20 -10 0 10 20 y2 0 50 100 150 200 -0.1 0 0.1 0.2 0.3 t (sec) 0 50 100 150 200 -0.5 0 0.5 1 t (sec) 0 50 100 150 200 -1 -0.5 0 0.5 1 t (sec) y1 z1 z1 y2 z2 z2 z1-z2 y1-y2 x1-x2
Fig. 7. Phase portrait and time–response error of mutual coupled systems with coupling term A sinðx2 x1Þ and A sinðx1 x2Þ for A ¼ 0:088:
Slave system: ’x2¼ x2y2z2 Jrþ12y22 Mjsin ott Axsinðx1 x2Þ; ’y2¼ z2; ’z2¼ x22y2 Kmðy2 1Þ Bmz2 Frsin ott; ð5Þ
and increase in the linear feedback is given by
’Bm¼ Ayðy1 y2Þsgnðx2Þ; ð6Þ
where the system parameter ’Bmis an adjustable function, Ayis a constant adaptive control gain, and Ax is a coupling strength. In Fig. 9(a), Ax ¼ 0:01 and Ay¼ 0:011 are shown. In Fig. 9(b), Ax ¼ 0:02 and Ay¼ 0:0008 are shown.
2.4. Transient time for uni-directional chaotic synchronization 2.4.1. Transient time of uni-directional linear coupled system
In this subsection we consider uni-directional coupled chaotic systems by linear coupling term [17]. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 A Lyapunov exponent
-1 -0.5 0 0.5 1 1.5 -2 0 2 4 6 x1 -1 -0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 x1 -2 0 2 4 6 -6 -4 -2 0 2 4 y1 -1 -0.5 0 0.5 1 1.5 -2 0 2 4 6 x2 Ax=0.01, Ay=0.011 -1 -0.5 0 0.5 1 1.5 -6 -4 -2 0 2 4 x2 -2 0 2 4 6 -6 -4 -2 0 2 4 y2 0 20 40 60 80 -20 0 20 40 60 t (sec) 0 20 40 60 80 -100 -50 0 50 100 150 t (sec) 0 20 40 60 80 -500 0 500 1000 t (sec) -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 x1 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 x1 0 0.5 1 1.5 2 2.5 -1.5 -1 -0.5 0 0.5 1 y1 1.105 1.1055 1.10 6 2.3794 2.3796 2.3798 2.38 2.3802 x2 Ax=0.02, Ay=0.0008 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 x2 2.3794 2.3796 2.3798 2.38 2.3802 0.1984 0.1986 0.1988 0.199 0.1992 0.1994 y2 0 20 40 60 80 100 120 -50 0 50 t (sec) 0 20 40 60 80 100 120 -1000 -500 0 500 t (sec) 0 20 40 60 80 100 120 -1000 -500 0 500 1000 1500 t (sec) y1 z1 z1 y1 z1 z1 z2 z2 y2 z2 z2 y2 x1-x2 y1-y2 z1-z2 x1-x2 y1-y2 z1-z2 (a) (b)
Master system: ’x1¼ x1y1z1 Jrþ12y21 Mjsin ott; ’y1¼ z1; ’z1¼ x21y1 Kmðy1 1Þ Bmz1 Frsin ott: ð7Þ Slave system: ’x2¼ x2y2z2 Jrþ12y22 Mjsin ott Aðx1 x2Þ; ’y2¼ z2; ’z2¼ x22y2 Kmðy2 1Þ Bmz2 Frsin ott: ð8Þ
The Euclidean distance d ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx1 x2Þ2þ ðy1 y2Þ2þ ðz1 z2Þ2 q
between the two trajectories is monitored for various choices of the coupling strength A; as shown inFig. 10(a). By increasing the value of the coupling strength, the distance d approaches zero when AthrD0:145; and the two subsystems display the same output. For values of A greater than Athr; the synchronized state is stable. This scenario is specifically for the given initial conditions and different initial conditions would qualitatively produce the same response shown in Fig. 10(b).
In Fig. 11, we show curves representing the evolution of dðtÞ; again on linear-log scale, with A ¼ 0:19 fixed, for other initial conditions. More precisely, y1ð0Þ ¼ 0:2; z1ð0Þ ¼ 0:3; x2ð0Þ ¼ 0:1; y2ð0Þ ¼ 0:2; z2ð0Þ ¼ 0:3 are kept fixed, and we use different values of x1ð0Þ; as shown in
Fig. 11. The slope of the linearly decaying part or the decaying transient for each of the three curves is almost the same, corresponding to the intuitive conjecture that the convergence is governed by the strength of the dissipation transverse to the attractor.
2.4.2. Transient time of uni-directional non-linear coupled system
In this section, we consider uni-directional coupled chaotic systems by non-linear couplingffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiterm. The coupling term is A sinðx1 x2Þ: The Euclidean distance d ¼
ðx1 x2Þ2þ ðy1 y2Þ2þ ðz1 z2Þ2 q
between the two trajectories is monitored for various choices of the coupling strength A; as shown inFig. 12(a). By increasing the value of the coupling strength, the distance d approaches zero when AthrD0:15; and the two subsystems display the same output. For values of A greater than Athr the synchronized state is stable. This scenario is specifically for the given initial conditions, and different initial conditions would qualitatively produce the same response shown inFig. 12(b).
In Fig. 13, we show curves representing the evolution of dðtÞ; again on linear-log scale, with A ¼ 0:19 fixed, for other initial conditions. More precisely, y1ð0Þ ¼ 0:2; z1ð0Þ ¼ 0:3; x2ð0Þ ¼ 0:1; y2ð0Þ ¼ 0:2; z2ð0Þ ¼ 0:3 are kept fixed and we use different values of x1ð0Þ; as given in
Fig. 13. The slopes of the linearly decaying part or the decaying transient for each of the three curves are almost the same, corresponding to the intuitive conjecture that the convergence is governed by the strength of dissipation transverse to the attractor.
2.5. Synchronization of coupled chaotic different systems
Consider that the slave system is a R.ossler system and the master system is still the suspended track system. Their chaos synchronization will be studied.
Fig. 10. Plot of several values of the Euclidean distance dðtÞ for different values of coupling strength A: The transition to a stable synchronized state is located approximately at Athr¼ 0:145: (a) I.C.: ðx1ð0Þ; y1ð0Þ; z1ð0ÞÞ ¼ ð0; 0:2; 0:3Þ and ðx21ð0Þ; y2ð0Þ; z2ð0ÞÞ ¼ ð0:1; 0:2; 0:3Þ: (b) I.C.: ðx1ð0Þ; y1ð0Þ; z1ð0ÞÞ ¼ ð0:2; 0:2; 0:3Þ and ðx21ð0Þ; y2ð0Þ; z2ð0ÞÞ ¼ ð0:1; 0:2; 0:3Þ:
Master system: ’x1¼ x1y1z1 Jrþ12y21 Mjsin ott Aðx1 x2Þ; ’y1¼ z1; ’z1¼ x21y1 Kmðy1 1Þ Bmz1 Frsin ott: ð9Þ
Slave system (R.ossler system):
’x2¼ 0:65y2 z2þ Aðx1 x2Þ; ’y2¼ 0:65x2þ 0:15y2; ’z2 ¼ 0:2 þ z2ðx2 10:0Þ:
ð10Þ
We can find it very hard to carry out synchronization for Mj¼ 5; Fr ¼ 1 even if we increase the coupling strength A (Figs. 14(a) and (b)). When the coupling strength A is increased, the error values ðx1 x2; y1 y2; z1 z2Þ of the system continues to exist. We consider the two systems (Eqs. (9) and (10)) with the master system forced by a periodic signal. The slave system is coupled to the master system, but remains autonomous. We find that slave systems cannot be synchronized with master systems no matter how large A is. Next, the phase synchronization between them will be studied.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 A = 0.19 x (0) =0.3 x (0) = 0.1 x( 0) = -0.2
Fig. 11. The behavior of dðtÞ for three cases of convergence onto the synchronized subset for other different initial conditions.
We define the average frequency Oi by Oi¼ dyiðtÞ dt ¼ lim T-N 1 T Z T 0 ’yiðtÞ dt: ð11Þ
Fig. 12. Plot of several values of the Euclidean distance dðtÞ for different values of coupling strength A: The transition to a stable synchronized state is located approximately at Athr¼ 0:15: (a) I.C.: ðx1ð Þ; y0 1ð0Þ; z1ð0ÞÞ ¼ ð0; 0:2; 0:3Þ and ðx21ð0Þ; y2ð0Þ; z2ð0ÞÞ ¼ ð0:1; 0:2; 0:3Þ: (b) I.C.: ðx1ð0Þ; y1ð0Þ; z1ð0ÞÞ ¼ ð0:2; 0:2; 0:3Þ and ðx21ð0Þ; y2ð0Þ; z2ð0ÞÞ ¼ ð0:1; 0:2; 0:3Þ:
The phase is defined by riðtÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xiðtÞ2þ yiðtÞ2 q ; yiðtÞ ¼ tan1 yiðtÞ xiðtÞ ; i ¼ 1; 2: ð12Þ
InFig. 15, we take A ¼ 0:2; change the frequency ot from 0.9 to 1.4, and plot the O1=ot and O2=ot versus ot: In Fig. 15, it appears clearly that the slave system is locked to the forcing frequency ot at ot¼ 1:13: In Fig. 16Oi=ot is plotted versus A; and the same frequency locking O2=ot¼ 1 occurs for A ¼ 0:2 while the two systems still stay at the chaotic state.
InFig. 16, it must be emphasized that when A ¼ 0:2; ot¼ 1:13; the average frequency O2of the slave system (Rossler system) without excited periodic term equals the forcing periodic signal ot: However, for the master system, the defined average frequency O1 is never equal to ot:
InFig. 16, at A ¼ 0:04 the average frequencies of the master and slave system are equal. We call this phenomena phase synchronization.
2.6. Application of synchronization
The topic of synchronization of chaotic oscillators has attracted increased attention in recent years because of possible relevance to secure communication and biological systems. In this section, our point is the application of synchronization. InFig. 17, an explicit analytic condition
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 t (sec) A = 0.19 x (0) = 0.3 x( 0) = -0.2 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 0 2 10 10 log (d)
Fig. 13. The behavior of dðtÞ for three cases of convergence onto the synchronized subset for other different initial conditions.
-4 -2 0 2 4 -20 -10 0 10 20 30 x1 -4 -2 0 2 4 -40 -20 0 20 40 x1 -20 -10 0 10 20 30 -40 -20 0 20 40 y1 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 A = 0.1 x2 -20 -10 0 10 20 30 0 20 40 60 80 100 x2 -30 -20 -10 0 10 20 y2 -30 -20 -10 0 10 20 -40 -20 0 20 40 -100 -50 0 50 -50 0 50 100 150 200 0 2 4 6x 10 50 x1 -50 0 50 100 150 200 -1 -0.5 0 0.5 1x 10 51 x1 0 2 4 6 x 1050 -1 -0.5 0 0.5 1x 10 51 y1 -40 -20 0 20 40 -30 -20 -10 0 10 20 A = 0.2 x2 -40 -20 0 20 40 0 50 100 150 200 x2 -30 -20 -10 0 10 20 y2 -600 -400 -200 0 200 -2 0 2 4 6x 10 50 0 1000 2000 3000 4000 5000 -1 -0.5 0 0.5 1x 10 51 t (sec) 0 1000 2000 3000 4000 5000 t (sec) 0 1000 2000 3000 4000 5000 t (sec) 0 1000 2000 3000 4000 5000 t (sec) 0 1000 2000 3000 4000 5000 t (sec) 0 1000 2000 3000 4000 5000 t (sec) y1 z1 y1 z1 z1 y2 z2 0 20 40 60 80 100 z2 y2 z2 0 50 100 150 200 z2 x1-x2 y1-y2 z1-z2 x1-x2 y1-y2 z1-z2 (a) (b) z1
Fig. 14. Phase portrait and time–response error of mutual coupled systems with coupling terms Aðx2 x1Þ and Aðx1 x2Þ for (a) A ¼ 0:1 and (b) A ¼ 0:2:
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 wt = 1.13 A master system slave system frequency-locking phase synchronization (d(phase)/dt)/wt
Fig. 16. O1=otand O2=otplotted versus A; ot¼ 1:13:
0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 A = 0.2 wt master system slave system frequency-locking (d(phase)/dt)/w
of communication of encryption and decryption is shown. The communication is composed of three steps:
1. Encrypt the signal.
2. Synchronize the master and slave system. 3. Decrypt the signal.
Encryption Function φ Decryption Function ψ Private Information-bearing Signal S Chaotic Slave system Decrypted SignalSd Chaotic Master system Encrypted Signal Se Coupling term Transmission Chaotic Signal 1 X Chaotic Signal 2 X
Fig. 17. The structural figure of base encryption and decryption.
0 20 40 60 80 100 -0.1 0 0.1 0.2 0.3 t (sec) 0 20 40 60 80 100 -0.5 0 0.5 1 t (sec) 99.5 99.6 99.7 99.8 99.9 100 -0.05 0 0.05 t (sec) S 0 20 40 60 80 100 -40 -30 -20 -10 0 10 t (sec) 99.5 99.6 99.7 99.8 99.9 100 -0.05 0 0.05 t (sec) 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 t (sec) x1-x2 Sd y1-y2 Se S-Sd (a)
We consider the master and slave system of Eqs. (7) and (8). When A ¼ 1; synchronization occurs.
In step1, use the function f to encrypt the information signal sðtÞ and the chaotic signal x1ðtÞ: We take the encryption function fðx1; sÞ ¼ se¼ f1ðx1Þ þ f2ðx1Þs; where f1 and f2 are the continuous functions and f2 is non-zero everywhere. In step2, the master and slave system are in synchronization via coupling term Aðx2 x1Þ: A chaotic signal x1ðtÞ is transmitted from master to slave via the coupling term. In step 3, we used the synchronization signal x2ðtÞ and the decryption function c to re-produce an approximate estimate sdðtÞ of the masked confidential signal. We take the decryption function cðx2; fðx1; sÞÞ ¼ cðx2; seÞ ¼ f1ðx2Þ=f2ðx2Þ þ se=f2ðx2Þ ¼ s: A schematic description of the entire process is depicted in Fig. 17.
For example, we take s ¼ 0:05sinð60ptÞ; f1ðxiÞ ¼ x2i; f2ðxiÞ ¼ ð1 þ x2iÞ; so fðx1; sÞ ¼ se¼ x21þ ð1 þ x2
1Þs; cðx2; fðx1; sÞÞ ¼ x32=ð1 þ x22Þ þ se=ð1 þ x22Þ: Fig. 18(a) shows the result of the encrypted and decrypted signal. We can prove that the exchange of f1 and f2 give the same result. Next, if we take s ¼ 0:05sinð60ptÞ; f1ðxiÞ ¼ sin xiexpðxiÞð1 x2iÞ and f2ðxiÞ ¼ 2cos xisin xið1 xiÞ: The result is shown in Fig. 18(b). If we take A ¼ 0:04 synchronization disappears, s sd becomes large as shown in Fig. 18(c). In this case, secure communication fails. 0 20 40 60 80 100 -0.1 0 0.1 0.2 0.3 t (sec) 0 20 40 60 80 100 -0.5 0 0.5 1 t (sec) 99.5 99.6 99.7 99.8 99.9 100 -0.05 0 0.05 t (sec) 99.5 99.6 99.7 99.8 99.9 100 t (sec) S 0 20 40 60 80 100 -100 0 100 200 300 t (sec) -0.05 0 0.05 0 10 20 30 40 50 -0.1 0 0.1 0.2 0.3 0.4 t (sec) x1-x2 Sd S-Sd y1-y2 Se (b) Fig. 18 (continued).
3. Chaos anticontrol
To improve the chaotic phenomena of the dynamic system, we must anticontrol a periodic motion to a chaotic system. For this purpose, the addition of constant torque, periodic torque, periodic impulse input, delay feedback control and adaptive control are used to control periodic to chaotic phenomena.
3.1. Chaos anticontrol by the addition of constant torque
One can add an external input torque f1 in the system. Eq. (1) can be rewritten as ’x ¼ xyz
Jrþ12y2
Mjsin ott þ f1: ð13Þ
ot (from 1.0 to 2.0) and f1 (from 0 to 5) are changed to improve the chaotic phenomena of the dynamic system. The result is shown inFig. 19(a). It is clear that when otand f1change, the chaos region T also changes. It is a very simple way to improve the chaos region of the dynamic system. We cannot predict the chaos region T ; because inFig. 19(a)there exists no explicit clue.Fig. 19(b)
shows Lyapunov exponent diagram, when f1¼ 0; ot¼ 1:8:
99.5 99.6 99.7 99.8 99.9 100 t (sec) 99.5 99.6 99.7 99.8 99.9 100 t (sec) -0.05 0 0.05 S 0 20 40 60 80 100 -2 -1 0 1 2 3 t (sec) 0 20 40 60 80 100 -10 -5 0 5 10 t (sec) 0 20 40 60 80 100 -40 -30 -20 -10 0 10 t (sec) -50 0 50 100 0 20 40 60 80 100 -1 0 1 2 3 4 x 10 4 t (sec) x1-x2 Sd S-Sd Se y1-y2 (c) Fig. 18 (continued).
3.2. Chaos anticontrol by the addition of periodic torque
One can add an external input periodic torque f2sinðottÞ in the system. Eq. (1) can be rewritten as ’x ¼ xyz Jrþ12y2 Mjsin ott þ f2sin o2t: ð14Þ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.2 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 f1 wt T 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 F 1.4 (a) Lyapunov exponent (b)
Fig. 19. (a) T is the chaos region, f1 is the external force region, ot is the frequency region. (b) Lyapunov exponent diagram when f1¼ 0; ot¼ 1:8:
When ot and f2 change, the chaos region also changes. For a pair of ot; f2; the chaos region is increased as shown inFigs. 20(a) and (b), o2¼ ot:Fig. 20(a)shows the bifurcation andFig. 20(b) shows the Lyapunov exponent.
4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 -0.5 -0.4 0 0.5 1.5 0 2 F 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 F Lyapunov exponent (a) (b)
Fig. 20. (a) Bifurcation diagram of x for Frbetween 1 and 6, ot¼ 1:8; f2¼ 2: (b) Lyapunov exponent diagram for Fr between 4.1 and 6, ot¼ 1:8; f2¼ 2:
3.3. Chaos anticontrol by the addition of periodic impulse input
One can add an external input periodic impulse in the system. Eq. (1) can be rewritten as ’x ¼ xyz
Jrþ12y2
Mjsin ott þ u: ð15Þ
The periodic impulse input
u ¼ f3 XN
i¼0
dðt iTIÞ; ð16Þ
where f3is a constant impulse intensity, TI is the period between two consecutive impulses, and d is the standard delta function.
With different values of f3 and TI the chaos region also changes; the chaos region is increased, as shown in Fig. 21.
3.4. Chaos anticontrol by the addition of delay feedback term
One can add a delay feedback term in the system. Eq. (1) can be rewritten as ’x ¼ xyz
Jrþ12y2
Mjsin ott: ð17Þ
The delay feedback function u is as the following: u ¼ f4u
n
ðyðt tdÞÞ; ð18Þ
where un
is a linear (or non-linear) function, and td is the delay time. With different un
ð¼ sinðyðt 5ÞÞÞ the chaos region also changes and the chaos region is increased as shown inFig. 22.
3.5. Chaos anticontrol by adaptive control
Adaptive control [11] is one of the main approaches in control engineering that deals with uncertain systems. An adaptive system is capable of adapting to a changing environment as well as varying internal parameters. We successfully choose that when parameter Fr is perturbed as ’Fr ¼ e½ðx xsÞ ðy ysÞ ðz zsÞ; the following system can be controlled from order to chaos: ’x ¼ xyz Jrþ12y2 Mjsin ott; ’y ¼ z; ’z ¼ x2y K mðy 1Þ Bmz Frsin ott; ’Fr ¼ e½ðx xsÞ ðy ysÞ ðz zsÞ: ð19Þ
Fig. 23(a) shows that the system is controlled from period-1 to chaos with e ¼ 0:019; and
Fig. 23(b) shows the system is controlled from period-2 to chaos with e ¼ 0:02:
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 -2 0 2 4 6 8 t (sec) F -1 0 1 2 3 -10 -5 0 5 10 15 Xs -10 -5 0 5 10 15 -20 -10 0 10 20 Ys 0.4 0.6 0.8 1 1.2 1.4 -5 0 5 10 15 X Y -5 0 5 10 15 -10 -5 0 5 10 Y Z -5 0 5 10 F -0.5 0 0.5 1 1.5 2 2.5 -10 -5 0 5 10 15 Xs -10 -5 0 5 10 15 -20 -10 0 10 20 Ys 0 0.5 1 1.5 2 -5 0 5 10 15 X Y -5 0 5 10 15 -10 -5 0 5 10 Y Z Ys Zs 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 t (sec) Zs Ys (a) (b)
3.6. Chaos anticontrol by another style of adaptive control
One can add an adaptive control term in the system instead of control of Fr: Eq. (1) can be rewritten as ’x ¼ xyz Jrþ12y2 Mjsin ott; ’y ¼ z; ’z ¼ x2y K mðy 1Þ Bmz Frsin ott þ f5; ð20Þ
where the adaptive control term is f5:
’f5 ¼ eððx xsÞ ðy ysÞ ðz zsÞÞ; ð21Þ where xs; ys; zs is the desired steady state and e indicates the stiffness of control.
We can change e to change the chaotic region. Fig. 24(a) shows the bifurcation diagram of Eq. (1).Fig. 24(b)shows the bifurcation diagram of Eq. (20), e ¼ 0:1: InFig. 24(a), when Fr ¼ 3:6 the system is still a period-1 system, while inFig. 24(b)the system is already a chaotic system when Fr > 2:6:
4. Conclusions
In this paper, we study the dynamic system of the suspended track with moving load system. The synchronization of the master and slave system is studied. It is easy to increase the coupling strength A of the uni-directional and mutual coupling term to synchronize the coupled systems. These phenomena (synchronization or non-synchronization) can be proved by the Lyapunov exponent. One of the Lyapunov exponents transverses the zero value from positive to negative at synchronization. But in some conditions of the non-linear mutual coupling term, some error is found. The critical values of synchronization match with the Lyapunov exponent criterion. The phenomena of synchronization of autonomous and non-autonomous system are studied. The synchronization is impossible. Even if coupling strength is very large, error system behavior exists. However, the phenomena of phase locking and phase synchronization can be presented. In the same condition the slave system is phase locking to the frequency of exterior excitation, and master and slave system has phase synchronization. The phenomena of transient times are studied, when master and slave system are synchronizing. If we change the initial conditions of master and slave, the critical value of synchronization remains unchanged. Finally, the application of synchronization in the secure communication is given. We encrypted the private information-bearing signal and the chaotic signal of the master system from the encryption function. We decrypted the encrypted signal from the chaotic signal of slave system and encrypted function. It is found that if systems are not synchronized, the encrypted signal cannot be recovered.
The various anticontrol methods for the system have been studied. In order to increase the chaotic region, the constant torque, periodic torque and periodic impulse input are added to the system. On the other hand, the delay feedback control can also be used. Finally, by the adaptive control, period-1 or period-2, to chaos is successfully presented.
Acknowledgements
This research was supported by the National Science Council, Republic of China, under Grant Number NSC 91-2212-E-009-025.
References
[1] Z.-M. Ge, C.-C. Fang, Dynamic analysis and control of chaos for a suspended track with moving load, Transactions of Canadian Society for Mechanical Engineering 25 (1) (2001) 79–105.
[2] Y. Liu, P. Davis, Dual synchronization of chaos, Physical Review E 61 (3) (2000) 2176–2179. [3] Eth. Mosekilde, Complexity, Chaos and Biological Evolution, Nato Series, Plenum, New York, 1991.
[4] T.E. Vadivasova, A.G. Balanov, O.V. Sosnovtseva, D.E. Postnov, E. Mosekilde, Synchronization in driven chaotic systems: diagnostics and bifurcations, Physics Letters A 253 (1999) 66–77.
[5] L. Junge, U. Parlitz, Synchronization using dynamic coupling, Physical Review E 64 (1997) 055204-1–055204-4. [6] P. Parmananda, Generalized synchronization of spatiotemporal chemical chaos, Physical Review E 56 (2) (1997)
5402–5405.
[7] M. Zhan, Z.-G. Zheng, G. Hu, X.-H. Peng, Nonlocal chaotic phase synchronization, Physical Review E 62 (3) (2000) 3552–3557.
[8] J.-W. Shuai, D.M. Durand, Phase synchronization in two coupled chaotic neurons, Physics Letters A. 264 (1999) 289–297.
[9] Z.-P. Jiang, A Note on chaotic secure communication system, IEEE Transactions on Circuits and Systems 49 (1) (2002) 92–96.
[10] J.J. Terry, G.D. Vanwiggeren, Chaotic communication using generalized synchronization, Chaos Solitons and Fractals 11 (2001) 457–461.
[11] S. Sinha, R. Ramaswamy, J.S. Rao, Adaptive control in nonlinear dynamics, Physica D 43 (1991) 118–128. [12] R. He, P.G. Vaidya, Time delayed chaotic systems and their synchronization, Physical Review E 59 (4) (1998)
4048–4051.
[13] T. Kapitaniak, Controlling Chaos, Academic Press, London, 1996.
[14] S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison, Reading, MA, 1994.
[15] J.-Q. Fang, Y. Hong, G. Chen, Switching manifold approach to chaos synchronization, Physical Review E 59 (3) (1999) 2523–2526.
[16] R.S. Paul, S. Rajasekar, K. Murali, Coexisting chaotic attractors, their basin of attractions and synchronization of chaos in two coupled duffing oscillators, Physics Letters A 264 (1999) 283–288.
[17] G. Santobont, S.R. Bishop, A. Varone, Transient time in unidirectional synchronization, International Journal of Bifurcation and Chaos 9 (12) (1999) 2345–2352.
