Synchronization of Diffusively Coupled Oscillators: Theory and Experiment

Transcription

1 American Journal of Electrical an Electronic Engineering 2015 Vol 3 No Available online at Science an Eucation Publishing DOI:12691/ajeee Synchronization of Diffusively Couple Oscillators: Theory an Experiment B Nana 1* P Woafo 2 1 Department of Physics Higher Teacher Training College University of Bamena PO Box 39 Bamena Cameroon 2 Laboratory of Moelling an Simulation in Engineering Biomimetics an Prototypes Faculty of Science University of Yaoune I PO Box 812 Yaoune Cameroon *Corresponing author: na1bo@yahoofr Receive February ; Revise March ; Accepte April Abstract In this paper complete synchronization of iffusively couple oscillators is consiere We present the results of both theoretical an experimental investigations of synchronization between two three an four almost ientical oscillators The metho of linear ifference signal has been applie The corresponing ifferential equations have been integrate analytically an the synchronization threshol has been foun Harware experiments have been performe an the measure synchronization error of less than 1% has been etermine Goo agreement is foun between theoretical an experimental results Keywors: Chaos Synchronization Oscillator Cite This Article: B Nana an P Woafo Synchronization of Diffusively Couple Oscillators: Theory an Experiment American Journal of Electrical an Electronic Engineering vol 3 no 2 (2015): 37-3 oi: 12691/ajeee Introuction In the last few years several researchers have focuse their attention on the problems relate to the synchronization of chaotic systems [1-5] Toay the potential of chaos theory is recognize in the worl-wie with research groups actively working on this topic [6-] One of the great achievements of the chaos theory is the application in secure communications The chaos communication funament is the synchronization of two chaotic systems uner suitable conitions if one of the systems is riven by the other Since Pecora an Carrol [11] have emonstrate that chaotic systems can be synchronize the research in synchronization of couple chaotic circuits is carrie out intensively an some interesting applications such as communications with chaos have come out of that research There are several methos for synchronizing chaotic oscillators escribe in literature [12-17] The simplest one employs the feeback in the form of linear ifference u t an the output between the output of the transmitter 1 of the receiver u2 ( t ) The ifference signal K( u2 u1) when applie with a certain weight K > Kmin to an appropriate input of the receiver synchronizes the latter to the transmitter [16] Although there are many papers escribing global synchronization of a network of couple oscillators less attention has been evote to experimental results for biirectional couple systems In this paper attention will be rawn to complete synchronization of two three an four couple chaotic oscillators The remainer of this paper is organize as follows In section 2 we present the use oscillator an stuy its ynamic behavior as a function of control parameter In section 3 synchronization of two three an four iffusively couple oscillators are analyze theoretically Section eals with the experimental setup an where experimental results are compare to the numerical ones Finally conclusions are rawn in section 5 2 Circuit Moel an Description 21 Circuit Moel The circuit iagram of the oscillator is shown in Figure 1 Figure 1 Electronic scheme of the oscillator

2 American Journal of Electrical an Electronic Engineering 38 The circuit is a thir orer nonlinear oscillator containing five operational amplifiers We assume that all the operational amplifiers operate in their linear omain In our moel the ioe acts like non linear component an we moel its voltage-current characteristic with an exponential function namely 0 exp v i I V 1 = (1) 0 where i is the current through the ioe v is the voltage across the ioe I 0 is the inverse saturation current an V 0 26 mv at the room temperature The ynamics of the oscillator is given by the following set of ifferential equations: v1 1 1 = v2 v3 (2) t RC RC v I0 v1 = v1 v2 v3 sinh (3) t RC 1 2 RC 2 2 RC 2 C2 5V0 v3 1 1 = v1+ v2 () t R5C3 R3C3 Introucing the following imensionless variables an parameters v1 v2 v3 1 x = y = z = t = ω0t a = 5V0 5V0 5V0 RC 1 2ω b = c = = (5) RC 2 2ω0 RC 2ω0 RC 3 3ω0 1 where ω0 = an 5VC 0 2 = 2 I0RC 6 1 RC 6 1 we come to the set of ifferential equations convenient for numerical integrations x = y z (6) y = ax by cz sinh x (7) z = x + y (8) ( 2n + 1! ) n! ( n+ 1! ) 2n+ 1 2n 0 = 0 x A x then 2n+ 1 2n ( x N 0 ) A x ( n ) + n ( n+ ) N sinh ( x0) 0 n= 0 2 1! n= 0! 1! () By substituting system (8) an equation (9) in the set of equations (5) (6) an (7) an equating the coefficients of j t e ω j ω t an e separately to zero we obtain the raian frequency of oscillatory orbit by While the amplitue of oscillation is obtaine by solving the following polynomial equation: n= 0 ( 1 ) N 2n A c b r = 0 with r = a+ (11) n! ( n+ 1! ) b+ In view to erive the analytical expression of amplitue A we fixe N = 3 an we obtain the following result: 2 1/3 A = 2 1 9r r+ 9r + 2 ( r r r ) (12) 1/3 1/ This system presents stationary perioic an chaotic attractors epening on the value of the parameters abc The bifurcation iagram as well as the Lyapunov exponent rawn in Figure 2 show that for certain sets of parameters the system exhibits chaotic oscillations We use c as control parameter an other use parameters are the following: a = 35 b = 05 = Dynamic behavior Assuming that in the oscillatory state variables x y an z can be replace by their corresponing virtual orbits x 0 y 0 an z 0 respectively we fin x 0 y 0 an z 0 in the following form: jωt jωt x0 P1cos ωt Q1sin ωt = x0 = Ae + Ae jωt jωt y0 = P2cos ωt Q2sin ωt y0 = Be + Be z0 P3cos ωt Q3sin ωt jωt jωt = z0 = Ce + Ce with (9) A = ( P1+ jq1) B = ( P2 + jq2) C = ( P3 + jq3) Using the above expressions an neglecting the higher harmonics we can show that Figure 2 a) One parameter bifurcation iagram in the ( c x) plane an b) maximal Lyapunov spectrum Lya The bifurcation iagram consists of quasiperioicity chaos winows perio aing sequences an the familiar perio oubling bifurcation sequence intermittency an so on As shown there is a goo agreement between the bifurcation iagram an its corresponing maximal Lyaponov exponent

3 39 American Journal of Electrical an Electronic Engineering 3 Synchronization of Couple Oscillators Recently Woafo an Kraenkel [18] consiere the problem of stability an uration of the synchronization process between classical Van er Pol oscillators an showe that the critical slowing-on behavior of the synchronization time an the bounaries of the synchronization omain can be estimate by analytical investigations The next subsections exten the calculations of Ref [18] to two three an four iffusely couple oscillators Only in our analytical treatment we assume that oscillators have ientical coefficients 31 Synchronization in a Case of two Oscillators Here we aim to etermine the threshol value for synchronization (the minimal value K such that practical synchronization occurs) of two oscillators The two oscillators are iffusely couple with a buffer an a variable resistor R c which give the coupling constant K= ( RCω ) 1 c 1 0 The ynamics of two iffusely couple oscillators can be escribe by the following set of equations: ( ) x j = yj zj + K xi j xj y j = ax j by j z j sh x j z j = x j + y j (13) where inex i an j represent the oscillator number ( i j {12} ) Assuming the three following vectors efine as u1( x1 y1 z 1) u2( x2 y2 z 2) an εε ( 1 ε2 ε2)=u1 u2 the stability of synchronization manifol is ecie by the asymptotic behavior of ε1 = x1 x2 ε2 = y1 y2 an ε3 = z1 z2 At a linear approximation ε1 ε2 an ε 3 obey to ε1 = ε2 ε3 2 Kε1 ε2 = a ch( x1) ε1 b ε2 cε3 ε3 = ε1+ ε2 (1) By replacing the chaotic variable x 1 by its virtual orbit x 0 the ynamics of the synchronization errors is then escribe by the linear system ε1 2K 1 1 ε1 ε = α b c ε 2 2 ε ε 3 with N 2n A α = a ch( x0 ) = a 2 n=0 ( n! ) (15) Using the Lyapunov criteria the synchronization is stable if the real part of all eigenvalues is negative Assuming that λ is the eigenvalue of system (15) then it obeys the following algebraic thir orer equation: ( b 2K ) ( 2Kb c 1) ( c + b + 2 Kc + α ) = λ + + λ + + α + λ+ (16) The etermination of signs of the real parts of the root λ may be carrie out by making use of the Routh- Hurwitz criterion In applying this criterion we fin that the real parts of the roots are negative if b+ 2 K > 0 2Kb + c α + 1 > 0 c + b + 2 Kc + α > 0 (17) 2 2 K b + ( 2b 2α + 2 ) K + ( bc bα α c) > 0 Then the synchronization is sai to be stable if K > K 2 min where the new parameter K 2 min is efine as follow: 2 min = c+ b+α K 2c (18) As shown in Figure 3 the result obtaine from equation (18) is verifie by a irect numerical simulation of system (13) Figure 3 Synchronization bounaries in the case of two couple oscillators Numerically we use the fourth orer Runge Kutta algorithm an we fin the first value of K for which quantity x1 x2 < 005 for t 00 In Figure 3 the numerical result (points an ashe-line) fluctuates aroun the analytic curve (soli line) 32 Synchronization in a Case of three Oscillators In this case the ynamics of three iffusely couple oscillators can be escribe by the following set of equations: ( 1 + 1) x j = yj zj + K xj 2 xj + xj y j = ax j by j z j sh x j z j = x j + y j (19) Inex j (with 1 j 3 ) represents the oscillator number an the two perioic bounary conitions x0 = x 3 an x = x 1 are use Assuming the five following

4 American Journal of Electrical an Electronic Engineering 0 vectors efine as u1( x1 y1 z 1) u2( x2 y2 z 2) u3( x3 y3 z 3) ηη ( 1 η2 η2)= u1 2u2 + u3 an εε ( 1 ε2 ε2)=u1 u3 the stability of synchronization manifol is ecie by the asymptotic behavior of η an ε At a linear approximation the components of η an ε obey to η1 = η2 η3 3 Kη1 η2 = a ch( x1) η1 b η2 cη3 η3 = η1+ η2 ε1 = ε2 ε3 3 Kε1 ε2 = a ch( x1) ε1 b ε2 cε3 ε3 = ε1+ ε2 (20) (21) The form of systems (20) an (21) brings the following comment: the three oscillators fall together in the synchronization Proceeing in the same manner as the above subsection we obtain the synchronization bounary as K > K where 3min 3min = c+ b+α K 3c (22) Figure shows comparison between analytical result (full line) an numerical result (points an ashe-line) Numerically we fin the first value of K for which 05 x x + x x < 005 for t 00 quantity an x5 = x 1 are use Assuming the three following vectors efine as η = ( u + u ) ( u + u ) ε = u u an ξ = u u (2) The stability of synchronization manifol is ecie by the asymptotic behavior of η ε an ξ At a linear approximation the components of η ε an ξ obey to η = M η ε = M ε ξ = M ξ where K 1 1 2K 1 1 (25) M1 = α b c an M2 = α b c c+ b+α Choosing K min = if c Kmin < K <2 Kmin = K 2min the ring falls in the cluster synchronization ( u1 u3 u2 u while u1 u2 ) If K >2K min the complete synchronization ( u1 u2 u3 u ) occurs in the ring To verify our assumption we plot as shown in Figure 5 our analytical result ( 2K min ) as function of c an our numerical result obtaine while recoring the first value of K for which 1 ( ) < x x + x x + x x (ashe lines) Figure Synchronization bounaries in the case of three couple oscillators 33 Synchronization in a Case of four Oscillators Four systems are iffusely couple in a ring structure with a coupling constant K The ynamics of four iffusely couple oscillators can be escribe by the following set of equations: ( 1 + 1) x j = yj zj + K xj 2 xj + xj y j = ax j by j z j sh x j z j = x j + y j (23) Inex j (with 1 j ) represents the oscillator number an the two perioic bounary conitions x0 = x Figure 5 Synchronization bounaries in the case of four couple oscillators Although the cluster omain obtaine analytically is verifie numerically the oscillators lost their chaotic state Experimental an Numerical Results 1 Experimental Setup An experimental setup consisting of a network of four oscillators is shown in Figure 6 The networks of two an three oscillators can be erive from Figure 6 by choosing the suitable connections In the set of equations (13) (19) an (23) the variables x j y j an z j are the voltages across the capacitors C 1 j C 2 j an C 3 j respectively The coupling strength between oscillators is controlle by four variable resistors R cj The circuits are built using (TL082) Operational Amplifiers (BBY0) Dioes

5 1 American Journal of Electrical an Electronic Engineering Capacitances an Resistances The nominal values of the components can be foun in Table 1 Due to the tolerances of the components oscillators are slightly ifferent Therefore synchronization in the sense that u () t u () =0 t is not possible an practical i j synchronization is efine as ui() t uj() t δ with δ 1 Figure 7 Phase portrait of one oscillator before the coupling a) Experimentally obtaine b) Numerically plot Maintaining the same parameters use in Figure 7 we show in Figure 8 the phase portraits ( x 1 x 2 ) to illustrate the absence of synchronization in the system before the coupling Figure 8 Phase portrait in the plane ( x 1 x 2 ) before the coupling a) Experimental result b) Numerical result After setting the coupling we ecrease the values of the resistances R cj to fin experimentally the synchronization omain Table 1 Values of capacitors an resistances use in the overall electronic circuit Resistances ( Ω) First Secon Thir Fourth an Oscillator Oscillator Oscillator Oscillator Capacitances nf =1 =2 =3 j = ( ( j ) ( j ) ( j ) Figure 6 Electronic schematic of overall system 2 Experimental an numerical results Before the coupling Figure 7(a) shows a phase portrait of the first oscillator which correspons to a chaotic phase portrait (Figure 7(b)) obtaine numerically with the following normalize parameters: a 1 = 3522 b 1 = 0513 c 1 = 2511 an 1 = 126 C 1 j C 2 j C 3 j R 1 j R 2 j R 3 j R j R 5 j R 6 j R 7 j R8j R 9j 21 Two Oscillators In this case we fin that for Rcj 00Ω ( j = 12) the system falls in the synchronization Figure 9(a) an 9(b) illustrate the complete synchronization state of the

6 American Journal of Electrical an Electronic Engineering 2 ring for Rc 1 = 3150Ω Rc 2 = 3187 Ω an K 1 = 0813 K 2 = 0815 Another use parameters are the following: a = 3522 b = 0513 c = 2511 = a 2 = = = 2508 b c an 2 = 1266 Figure 9 Phase portraits in the plane ( x 1 x 2 ) illustrating synchronization in the ring of two couple oscillators a) Experimental result b) Numerically plot 22 Three oscillators In the case of three couple oscillators when the coupling resistances satisfy Rcj 298Ω ( j = 123) the ring is in the complete synchronization To prove it we plot in Figure some phase portraits: Figure (a) (resp Figure (c)) represents our experimental result in the plane ( x 1 x 2 ) (resp ( x 1 x 3 )) while Figure (b) (resp Figure ()) is its numerical analogous The coupling resistances are Rc1 = 2372 Ω Rc 2 = 205 Ω Rc3 = 2378 Ω an K 1 = 053 K 2 = 052 K 3 = 055 Other parameters are keep constant: a1 = 3522 b1 = 0513 c 1 = = 126 a 2 = 3518 b 2 = 0517 c 2 = = 1266 a 3 = 3525 b 3 = 0515 c 3 = 2513 an 3 = Four Oscillators Although the analytical stuy foresaw a cluster synchronization the numerical an the experimental stuies showe only a complete synchronization state This is obtain experimentally when Rcj 35Ω ( j = 123) Proceeing as the above subsection we plot in Figure 11 ifferent phase portraits: Figure 11(a) Figure 11(c) an Figure 11(e) represent our experimental results rawn in the planes ( x 1 x 2 ) ( x 1 x 3 ) an ( x 1 x ) respectively while Figure 11 (b) Figure 11 () an Figure 11 (f) are their numerical corresponing These following coupling parameters are use: Rc 1 = 3272Ω Rc1 = 3255 Ω Rc1 = 328 Ω Rc1 = 328 Ω an K 1 = 803 K 2 = 8025 K 3 = 8027 K = 8031 a 1 = 3522 b 1 = 0513 c 1 = = 126 a 2 = 3518 b 2 = 0517 c 2 = = 1266 a3 = 3525 b3 = 0515 c 3 = = 1261 a = 3515 = 0520 = 2511 = 1265 Figure Phase portraits illustrating the complete synchronization in the ring of three couple oscillators a) Experimental result in the plane ( x 1 x 2 ) b) Numerically plot in the plane ( x 1 x 2 ) c) Experimental result in the plane ( x 1 x 3 ) ) Numerically plot in the plane ( x 1 x 3 ) Figure 11 Phase portraits illustrating the complete synchronization in the ring of four couple oscillators a) Experimental result in the plane ( x 1 x 2 ) b) Numerically obtaine in the plane ( x 1 x 2 ) c) Experimental result in the plane ( x 1 x 3 ) ) Numerically obtaine in the plane ( x 1 x 3 ) e) Experimental result in the plane ( x 1 x ) f) Numerically obtaine in the plane ( x 1 x )

7 3 American Journal of Electrical an Electronic Engineering 5 Conclusion In this paper theoretical an experimental complete synchronization of iffusively two three an four couple oscillators are presente With the experimental setup it is impossible to achieve a zero synchronization error ue to the tolerances of the electrical components We obtain that three an four iffusely couple oscillators synchronize or esynchronize together provie initial values are chosen in the vicinity of the synchronization manifol Despite the fact that the single harmonic response give in equation (9) may be questionable the analytical treatment gives a goo inication on the bounary of K for synchronization to be achieve The presente experimental results are qualitative comparable with numerical simulations Acknowlegement This work is supporte by the Acaemy of Sciences for the Developing Worl (TWAS) uner Research grant N RG/PHYS/AF/AC References [1] Ling L an Le M Parameter ientification an synchronization of spatiotemporal chaos in uncertain complex network Nonlinear Dynamics 66 () Dec 2011 [2] Carroll TL an Pecora LM Synchronizing chaotic circuits IEEE Trans Circ Syst 38 (2) Apr 1991 [3] Yu PE an Kuznetsov AP Synchronization of couple van er pole an Kislov-Dmitriev self-oscillators Technical Physics 56 () 35-2 Apr 2011 [] Peng JH Ding EJ Ding M an Yang W Synchronizing hyperchaos with a scalar transmitte signal Phys Rev Lett 76 (6) [5] Nijmeijer H Control of chaos an synchronization Systems an Control Letters 31 (5) Oct 1997 [6] Kenney MP Rovatti R an Setti G Chaotic Electronics in Telecommunications CRC Press USA 2000 [7] Gamal MM Shaban AA an AL-Kashif MA Dynamical properties an chaos synchronization of a new chaotic complex nonlinear system Nonlinear Dynamics Oct 2008 [8] Tanmoy B Debabrata B an Sarkar BC Complete an generalize synchronization of chaos an hyperchaos in a couple first-orer time-elaye system Nonlinear Dynamics 71 (1) Jan 2013 [9] Buric N an Vasovic N Global stability of synchronization between elay-ifferential systems with generalize iffusive Chaos Solitons an Fractals 31 (2) [] Fotsin H an Bowong S Aaptive control an synchronization of chaotic systems consisting of Van er Pol oscillators couple to linear oscillators Chaos Solitons an Fractals 27 3) Apr 2006 [11] Carroll TL an Pecora LM Nonlinear Dynamics in Circuits Worl Scientific Publishing Singapore 1995 [12] Juan C Jun-an L an Xiaoqun W Biirectionally couple synchronization of the generalize Lorenz systems Journal of Systems Science an Complexity 2 (3) 33-8 Jun 2011 [13] Pecora LM an Carroll TL Driving systems with chaotic signals Physl Rev A () Aug 1991 [1] Cuomo KM an Oppenheim AV Circuit implementation of synchronize chaoswith application to communications Phys Rev Lett 71 (1) Jul 1993 [15] Nana B an Woafo P Synchronization in a ring of four mutually couple Van er pol oscillator: Theory an experiment Phys Rev E [16] Yu YH Kwak K Lim TK Synchronization via small continuous feeback Phys Lett A 191 (3) [17] Li-feng Z Xin-lei A an Jian-gang Z A new chaos synchronization scheme an its application to secure communications Nonlinear Dynamics 73 (2) Jan 2013 [18] Woafo P an Kraenkel RK Synchronization stability an uration time Phys Rev E