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Chaos synchronization of resistively coupled Duffing systems: Numerical and experimental investigations E. Tafo Wembe, R. Yamapi * Department of Physics, Faculty of Douala, University of Douala, P.O. Box 24 157 Douala, Cameroon Received 12 October 2007; received in revised form 23 January 2008; accepted 23 January 2008

Abstract This paper studies chaos synchronization dynamics of two resistively coupled Duffing systems, through numerical and experimental investigations. Various bifurcation structures are derived and it is found that chaos appear suddenly, through period doubling cascades. The experimental study of these systems is carried out with appropriate software electronic circuit, proposed using the BSIMV3.3 parameters for the investigation of the dynamical behavior. The appropriate coupled coefficient for chaos synchronization is found using numerical and experimental simulations. The reliability of the analytical formulas is approved by the good agreement with the results obtained by both numerical and experiment simulations. Ó 2008 Elsevier B.V. All rights reserved. PACS: 05.45.Xt Keywords: Chaos synchronization; Duffing system

1. Introduction Synchronization phenomena have been actively investigated since the early days of physics. Initially, the attention was mainly devoted to the synchronization of periodic systems, while recently research for synchronization has moved to chaotic systems. One could wonder what happens when such systems interact. The main dynamical question is when such synchronous behavior is stable, especially in regard to coupling strengths in the system? And whether it is possible at all to find mechanisms of their interactions and explain how in many cases interconnected chaotic systems perform useful tasks like signal processing. Interest in this question has been high over several years [1–5] and especially since Pecora and Carroll [5] discovered,that chaotic systems can be synchronized, the intensive study has been carried in [5–8]. This paper deals with the synchronization dynamics of resistively coupled non-linear oscillators in the chaotic states using numerical and experimental investigations. For the experimental investigations, we will carry

*

Corresponding author. Tel.: +237 99 32 93 76; fax: +237 33 40 75 69. E-mail addresses: [email protected] (E.T. Wembe), [email protected] (R. Yamapi).

1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.01.019

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out the simulation of the systems through appropriate software electronic circuits [9], to confirm the results obtaining numerically. The comparison between numerical and experimental studies will be done. The structure of the paper is as follows: Section 2 deals with the dynamics of the Duffing electrical system. Bifurcation structures leading to chaotic behavior is found using numerical simulation of the Duffing equation. The results obtained here are confirmed experimentally by integrate the appropriate equivalent software electronic circuit. We analyze chaos synchronization of two coupled Duffing-type electrical systems and experiment through an analog simulation. The bifurcation diagrams as a function of the coupling parameter are derived to find the appropriate range of coupling parameter leading to chaos synchronization. These results are confirmed experimentally in the phase space. In Section 3, chaos synchronization of two coupled Duffing-type electrical systems are found both numerically and experimentally. Section 4 deals with the chaos control of the Duffing electrical system in the chaotic state. The chaotic behavior is converted into the desired periodic behavior, as the coupling coefficient is varied. The last Section is devoted to short conclusions.

2. Chaotic dynamics of the Duffing system 2.1. The electrical circuit and its differential equation The model shown in Fig. 1i is a non-linear electric circuit driven by a sinusoidal voltage source, described by the Duffing electrical oscillator, consisting of the linear resistor, in series with a sinusoidal source. These two components are connected in parallel with a capacitor and a non-linear inductor. The non-linear inductor is an inductor with a ferromagnetic core, which can be modelled, if an abstraction of the hysteresis phenomenon is made, by using i–u non-linear characteristic. The characteristic of the non-linear element is approximately described by the following relation: i ¼ a1 u þ a3 u 3 ;

ð1Þ

Fig. 1. (i) The electric circuit obeying to the Duffing equation and (ii) analog simulation circuit of the driven Duffing oscillator.

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where a1 and a3 are constants depend on the type of the inductor [8], and u is the flux over inductor. Using the electrical laws, it is found that the non-linear differential equation of the circuit is given by V R þ V L ¼ E00 cos x0 s;

ð2Þ

iR ¼ i þ iC ;

where the voltage of the resistance and the inductor VR and VL are given by the following two relations: V R ¼ RiR ; du : VL ¼ ds

ð3Þ ð4Þ

iR and iC are, respectively, the currents of the resistor and condenser. iC ¼ C

dV L : ds

ð5Þ

Substituting relations (1), (3), (4), and (5) in Eq. (2), we obtain the following differential equation: d2 u 1 du a1 a3 3 E00 þ u þ u cos w0 s: þ ¼ ds2 RC ds C C RC

(i)

ð6Þ

2 1

x

0 -1 -2 -3 -4 0

5

10

15

20

25

30

35

40

25

30

35

40

E0

(ii)

0.15 0.1 0.05

Lya

0 -0.05 -0.1 -0.15 -0.2 0

5

10

15

20

E0

Fig. 2. Bifurcation diagram (i) and variation of the Lyapunov exponent (ii) versus the amplitude E0 = B with the parameters: b = 1, e = 0.180 and x = 0.8.

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1 Set t ¼ we s; u ¼ u0 x; e ¼ RCw ; w2e ¼ aC1 ; b ¼ e following differential Duffing equation:

d2 x dx þ e þ x þ bx3 ¼ B cos xt: dt2 dt

a3 u20 Cw2e

E0

; B ¼ Ra10u0 , one finds that the electrical circuit is described by the ð7Þ

As we mentioned in Section 1, our aim in this section is to find some bifurcation structures and transition to chaos in the Duffing electrical system, following numerical and experimental investigations. In the following subsection, we present a design of the analog simulator before process to the chaotic dynamics of the Duffing equation.

Fig. 3. Chaotic phase portrait of the Duffing oscillator obtained through analog simulation with the parameters, b = 1, e = 0.180 and x = 0.8: (a) Double-band chaos for B = 28.5; (b) double-band chaos for B = 26.7 and (c) chaos phase for B = 23.5.

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2.2. Design of the analog simulator The electronic analog simulator can be easily constructed as in Ref. [7–11]. The equivalent schematic simulation circuit of non-linear differential equation (see Eq. (7)) of the circuit is shown in Fig. 1ii, which is a scheme of the complete software electronic simulator. The electronic multipliers Mi(i = 1, 2, 3) are the software analog devices (Laplace multiplication operator or AD633JN) (see Ref. [9]). They operate over a dynamic range of ±11 V, while the spice pins used are X1, X2, Y1, Y2, VS+, W, Z, VS. The integrators are operational amplifiers (lA741, LF353, or LM358) with feedback capacitors and summations, and are obtained using operational amplifiers with feedback resistors and multiple input resistors. Using an appropriate time scaling, the simulator outputs can be viewed directly on a software oscilloscope (see Ref. [9] called Proteus). 2.3. Bifurcation structures and onset of chaos Our aim in this subsection is to find various bifurcation structures which appear in the single Duffing-type electrical system before the onset of chaos. For this purpose, the periodic stroboscopic bifurcation diagram of the coordinate x is used to map the transitions (the stroboscopic time period is T = 2p/w). We have found that chaos appears in the Duffing equation for the physical parameters b = 1; w = 0.8 and e = 0.8. Fig. 2 shows a representative bifurcation diagram and the variation of the corresponding Lyapunov exponent versus the amplitude of the external excitation B. We have done the numerical calculations very accurately in double

(i)

(ii)

Fig. 4. (i) Two Duffing circuits bi-directionally coupled via a linear resistor and (ii) analog simulation circuit of two Duffing-type electrical circuits bi-directionally coupled via a linear resistor.

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precision for different initial conditions and for different variations of parameters, and verified our results. In order to obtain reliable numerical results, the step size has been chosen to be equal to 104, and the first 107 steps are discarded to avoid the transient regime. These curves are obtained by numerically solving Eq. (2) and the corresponding variational equations, the Lyapunov exponent being defined by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! dx2 þ dv2x Lya ¼ Lim In ; ð8Þ t!1 t where dx and dvx are, respectively, the variations of x and vx. That is a measure of the rate of divergence between initially closed trajectories in the two dimensional phase spaces. As it appears, different types of bifurcations take place before the onset of chaos. In Fig. 2, we find that as B increases from zero, the amplitude of the periodic oscillations increases until B = 19.7 where the periodic behavior bifurcates into a 2T-periodic states. Then after the values of B = 21.7 and B = 22.05, a double periodic transition appears and the system passes into a chaotic state at B = 22.12. As B increases further, the chaotic orbit bifurcates to another chaotic

(i)

(ii)

Fig. 5. (i) Two Duffing circuits unidirectionally coupled via a linear resistor and (ii) analog simulation circuit of two Duffing-type electrical circuits unidirectionally coupled via a linear resistor.

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Fig. 6. Bifurcation diagrams y1  x1 versus K in the case of bi-directional coupling, with the parameters of Fig. 3.

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state through a band of 3T-periodic states. In summary three bands of chaotic states can be observed and the three values of B chosen in the chaotic band used in the paper are: B = 23.5, B = 26.7 and B = 28.5. We also find the types of behavior which appears in the Duffing system for the fixed value of B, by using the analog simulation to find the correlated states x and vx. For this aim, let us find the equivalence between the components of both circuits. In terms of circuit component, the parameters of Eq. (2) are defined as follows: B¼

V ; 10 R1 R5 C 1 C 2 8



1 ; 10 R4 C 1 4



1 10 R2 R5 C 1 C 2 E2 8

In order to control each parameter of the equation of motion of Fig. 1ii, by varying only one resistor, we set the following values of the circuit components: C1 = C2 = 10 nF, as b = 1 then R1 = R3 = R5 = 10 kX, R2 = 100 X, E = 10, e = 0.180 and R4 = 55555.6 X. In the experiment investigations, we fixed the frequency w0 of the external sinusoidal signal at 2p ¼ 1; 273; 239 Hz, so that the corresponding redefined frequency 0 x = R1C1x = 0.8. The circuit shown in Fig. 1ii will give the waveforms x and vx and they can be observed through an oscilloscope to show the phase portrait in the (vx  x) plane. The experimental results are given in Fig. 3, and we find that for the analog simulation, chaos appears in the three bands of chaotic behavior as it is found from numerical simulation.

3. Chaos synchronization of two coupled Duffing-type electrical systems 3.1. The coupled Duffing electrical systems The system of two identical Duffing-type electrical circuits bi-directionally coupled via a linear resistor is shown in Fig. 4i. The differential equation of the system is

Fig. 7. Correlated states between the variables x1 and y1 for the case of bi-directional coupling: (a) B = 23.5, K = 0.48, the system is out of synchronization; (b) B = 23.5, K = 1.0, the system is in chaotic synchronization; (c) B = 26.7, K = 0.60, the system is also out of synchronization and (d) B = 26.7, K = 1.60, the system is in chaotic synchronization. The other parameters are defined in Fig. 3.

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Fig. 8. Bifurcation diagrams y1  x1 versus K in the case of unidirectional coupling. The order parameters are defined in Fig. 3.

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dx1 dt dx2 dt dy 1 dt dy 2 dt

¼ x2 ; ¼ eðK þ 1Þx2  x1  x31 þ eKy 2 þ B cos xt;

ð9Þ

¼ y2; ¼ eðK þ 1Þy 2  y 1  y 31 þ eKx2 þ B cos xt;

where K ¼ RR1c is the resistively coupling parameter. As we are also interested in the analog simulator, Fig. 4ii shows Analog circuit of two Duffing-type electrical circuits bi-directionally coupled via a linear resistor. The system of two identical Duffing circuits unidirectional or one-way coupled via a linear resistor is shown in Fig. 5i. The buffer in the branch coupling the two Duffing-type electrical circuits isolates the dynamics of the left circuit from the influence of the dynamics of the right circuit. In this case, the system of unidirectional by coupled oscillators is described by dx1 ¼ x2 ; dt dx2 ¼ ex2  x1  x31 þ B cos xt; dt dy 1 ¼ y2; dt and dy 2 ¼ eðK þ 1Þy 2  y 1  y 31 þ eKx2 þ B cos xt: dt

ð10Þ

Fig. 5ii shows analog simulation circuit of two Duffing-type electrical circuits unidirectionally coupled via a linear resistor. 3.2. Chaos synchronization Considering the case that the two coupled circuits are identical and are driven by signals of the same amplitude, we have simulated chaotic synchronization as the coupling coefficient K is changed. The bifurcation diagram y1  x1 versus K is provided to find the dynamical states of the coupled Duffing-type electrical systems. When the difference y1  x1 becomes equal to zero, this means that the two circuits are in a chaotic synchronization.

Fig. 9. Correlated states between the variables x1 and y1 for the case of unidirectional coupling: (a) B = 26.7, K = 0.95, the system is in the chaotic state, i.e. out of synchronization and (b) B = 26.7, K = 1.20, the system is in chaotic synchronization with the parameters defined before in Fig. 3.

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Fig. 10. Bifurcation diagrams y1  x1 versus the coupling coefficient K in the one-way coupling case showing how the chaos behavior is converted to the period-1 state. The parameters used are shown in Fig. 3.

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In the case of bi-directional coupling case, Fig. 6 shows the bifurcation diagrams y1  x1 versus the coupling coefficient K for B = 23.5 and B = 26.7. As we can observe, with B = 23.5, a chaotic synchronization appears in the range defined as K > 0.909, while for B = 26.7, chaotic synchronization is observed for K > 0.886. In both cases, a phase-locked state of period 2T is created in some range of K, before the onset of chaotic synchronization. To confirm these numerical results, let us experimentally find the correlated states between the variables y1 and x1. Fig. 7 shows the phase portraits y1 versus x1, for some ranges of K and B. The periodic states (without chaos synchronization) are shown in Fig. 7a and c, while the chaotic synchronization state for (B, K) = (23.5; 1.0) and (B, K) = (26.7; 1.6) are shown in Fig. 7b and d. In the case of unidirectional coupling, Fig. 8 also shows the bifurcation diagrams y1  x1 versus K, and the following results are obtained. The coupled chaotic system is on the chaotic synchronization state for K > 0.93 (for B = 23.5) and K > 1.14 (for B = 26.7). The correlated states between y1 and x1 are provided in Fig. 9 for B = 26.7 following experimental investigations and one finds that the coupled system is in the chaotic state for K = 0.95, while the chaotic synchronization state appears for K = 1.2.

4. Suppression of chaos in the Duffing-type electrical system In recent year, particular attention has been devoted on the suppression of chaos in the non-linear system in the chaotic regime, for example, Patidar et al. [12] have used mutual coupling systems to suppress chaos in the non-linear system. They have shown that in the case of two mutually coupled non-linear systems of the same kind, one periodic and one chaotic, chaotic behavior is converted into the desired periodic behavior, as the coupling coefficient is varied. We have applied this method of suppression of chaos in both coupled schemes, unidirectional and bi-directional. We assume that the two coupled oscillators have the same circuit parameters except the amplitude of the external excitation. The first oscillator has the amplitude E0, which can take

Fig. 11. Effects of the control on the chaotic states in the case of one-way coupling with E0 = 28.5, B = 5. The system is converted to the periodic state as the coupling parameter increasing. The parameters used are shown in Fig. 3.

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Fig. 12. Bifurcation diagrams y1  x1 versus the coupling coefficient K in the two-way coupling case showing how the chaos behavior is converted to the period-1 state. The parameters used are shown in Fig. 3.

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different values being in the chaotic state, while the second oscillator has the amplitude B = 5.0 being in a period-1 state. In the unidirectional case, Fig. 10 shows numerical results of the bifurcation diagrams x1  y1 versus the coupling parameter K. It appears that the chaotic behavior is converted to the period-1 state follow the reverse period-doubling sequence scenario. Fig. 11 confirm the scenario through the experimental results of

Fig. 13. Effects of the control on the chaotic states in the case of two-way coupling with B = 5, K = 1 and (i) E0 = 23.5, (ii) E0 = 26.7 and (iii) E0 = 28.5. The system is converted to the periodic state as the coupling parameter increasing. The parameters used are shown in Fig. 3.

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the phase portrait vx versus x in the three chaotic bands. Figs. 12 and 13 show respectively the numerical results of the bifurcation diagrams and the experimental results of the phase portraits in the bi-directional coupling. 5. Conclusions In this paper, we have simulated the synchronization dynamics of resistively coupled non-linear Duffingtype electrical oscillators. The numerical and analog simulation have been used to scan the parameter range in order to find those corresponding to sensitive and complex behavior of the coupled system, such as chaos synchronization dynamics. The results from software electronic circuit were compared with the numerical results and we found good agreement. Because of the recent advances in the theory of synchronization phenomena, it would be interesting to extend the study of induced-noise chaos synchronization of two or more coupled self-sustained systems. Acknowledgements Part of this work has done during the visit of E. Tafo Wembe to the Chinese Academy of Sciences (CAS), the Institute of Modern Physics (IMP). He would like to thanks the Third World Academy of sciences TWAS. He would also thank Liang Qiang, Xie Ming for their hospitality. The authors thank G. Filatrella for enriching discussions. References [1] Winful HG, Rahman L. Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers. Phys Rev Lett 1990;65:1575–8. [2] Ashwin P, Buescu J, Stewart I. Bubbling of attractors and synchronisation of chaotic oscillators. Phys Lett A 1994;193:126–39. [3] Heagy JF, Carroll TL, Pecora LM. Experimental and numerical evidence for riddled basins in coupled chaotic systems. Phys Rev Lett 1994;73:3528–31. [4] Heagy JF, Carroll TL, Pecora LM. Desynchronization by periodic orbits. Phys Rev E 1995;52:R1253–6. [5] Carroll TL, Pecora LM. Synchronization in chaotic system. Phys Rev Lett 1990;64:821–4. [6] Lakshmanan M, Murali K. Drive-response scenario of chaos synchronization in identical nonlinear system. Phys Rev E 1994;49:4882–5. [7] Lakshmanan M, Murali K. Chaos in nonlinear oscillators: Controlling and synchronization, World Scientific of Non-linear Science, Series A, Vol. 13, 1996. [8] Kyprianidis IM, Volos Ch, Stouboulos IN, Hadjidemetriou J. Dynamics of two resistively coupled Duffing type electrical oscillators. Int J Bifurcat Chaos 2006;16(6):1765–75. [9] ProteusPRO6.7SP3.exe. Labcenter; 2005. [10] Fotsin HB. Phenomenes coherents et incoherents dans le microphone a condensateur et dans les circuits electroniques a diode varicap. The`se de 3eme cycle de Physique, Universite´ de Yaounde´; 1999. [11] Chedjou JC, Kyamakya K, Moussa I, Kuchenbecker H-P, Mathis W. Behavior of a self-sustained electromechanical transducer. Trans ASME 2006;128:282–93. [12] Patidar V, Pareek NK, Sud KK. Suppression of chaos using mutual coupling. Phys Lett A 2002;304:121–9.

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Chaos synchronization of resistively coupled Duffing ...

doi:10.1016/j.cnsns.2008.01.019. * Corresponding author. Tel.: +237 99 32 93 76; fax: +237 33 40 75 69. E-mail addresses: [email protected] (E.T. Wembe), ...

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