A Study on the Synchronization Aspect of Star Connected Identical Chua Circuits
aa r X i v : . [ n li n . C D ] J un A Study on the Synchronization Aspect of StarConnected Identical Chua’s Circuits
Sishu Shankar Muni ∗ , Subhransu Padhee † and Kishor Chandra Pati ∗∗ Department of MathematicsNational Institute of Technology, Rourkela, Odisha, IndiaEmail: [email protected], [email protected] † Department of Electronics and Communication EngineeringNational Institute of Technology, Rourkela, Odisha, IndiaEmail: [email protected]
Abstract —This paper provides a study on the synchronizationaspect of star connected N identical chua’s circuits. Differentcoupling such as conjugate coupling, diffusive coupling andmean-field coupling have been investigated in star topology.Mathematical interpretation of different coupling aspects havebeen explained. Simulation results of different coupling mecha-nism have been studied. Keywords — Chua’s Circuit ; Coupling ; Synchronization
I. I
NTRODUCTION
Dutch scientist Christian Huygens described the first doc-umented work about synchronization using two pendulumshanging from a beam and the system provides anti-phase syn-chronization. Synchronization can be roughly said as the rhyth-mic adjustment of oscillating objects (objects which possesnonlinear dynamics). A significant research has been goingon to formulate the mathematics behind the synchronization ofmultiple identical as well as non-identical nonlinear oscillators.Synchronization of nonlinear oscillator finds wide spread usein different engineering applications where researchers use theconcept of chaotic synchronization in communication [1]–[3]and wireless sensor and actuator network (WSAN).There are variety of nonlinear oscillators such as nonlinearpendulum, Van der Pol oscillator, R¨ossler oscillator, Lorenzoscillator, Fitzhugh Nagumo oscillator and Duffing oscillator.On the other hand there are electronic circuits which giveschaotic output. Chua’s circuit is one of the well-known nonlin-ear oscillator which provides chaotic output. As the prototypeof electronic nonlinear oscillator (such as chua’s circuit) canbe developed in laboratory and its nonlinear behavior canbe studied, it has been widely accepted by the academiccommunity. The chaotic oscillators are sensitive to initialconditions. The behavior of the system is chaotic and difficultto predict.One of the first investigation of synchronization of twoidentical nonlinear oscillators having chaotic behavior in dis-sipative system can be found in [4]. The numerical and experi-mental investigation of synchronization of chua’s circuit can befound in [5], [6]. Synchronization of Van der Pol oscillator andFitzhugh Nagumo oscillator and ring coupled four oscillatorshave been studied in [7], [8]. Adaptive observer design for adaptive synchronization of chua’s circuit [9], synchroniza-tion of chua’s circuit using adaptive control [10], adaptivebackstepping control [11] and H ∞ adaptive synchronization[12] have been reported in the literature. Different couplingsuch as diffusive coupling, conjugate coupling and mean-fieldcoupling in star network topology with N identical R¨ossleroscillator and Lorenz oscillator have been studied in [13]. Theauthors have shown the chimera states in end nodes of the starnetwork.Many papers have investigated the synchronization aspectof chua’s circuits which are in master-slave configuration. InWSAN applications, different network topologies are used.One of the most basic network topology is star networktopology. This paper investigates the mathematical aspect ofsynchronization of N identical chua’s circuits connected instar network configuration (bidirectional coupling). Differentbidirectional coupling aspects such as diffusive coupling,conjugate coupling and mean field coupling are investigated.Simulation results have been provided to validate the mathe-matical derivation of synchronization.This paper is organized as follows. Section II providessystem modeling and dynamics of chua’s circuit. Section IIIprovides star network topology and different coupling aspects.Section IV provides simulation results and Section V providesthe concluding remarks.II. S YSTEM M ODELING AND D YNAMICS OF C HUA ’ S C IRCUIT
Chua’s circuit (Figure 1) is one of the simple yet well-known chaotic oscillator circuit which can be easily built usingdifferent laboratory components. [14]–[16]. Chua’s circuitcomprises of an inductor, two capacitors, a resistor and achua’s diode. Chua’s diode is a negative conductance piece-wise linear element. The behavior of chua’s diode can beeasily implemented using operational amplifier but the use ofoperational amplifier makes the frequency a constraint.The state equation of chua’s circuit can be represented as dv dt = C ( G ( v − v ) − g ( v )) dv dt = C ( G ( v − v ) + i L ) di L dt = L ( − v − R o i L ) (1) $ (cid:13) C2 C1R Nr i L i r Fig. 1. Circuit diagram of chua’s circuit where, v is the voltage across capacitor C , v is the voltageacross capacitor C and i L is the current across inductor L , G is the conductance of R (cid:0) G ≈ R (cid:1) , g ( . ) is the non-linearvoltage-current ( v − i ) characteristics of chua’s diode N R . g ( . ) is formulated as piecewise-linear function.The nonlinear characteristics of the chua’s diode can berepresented as g ( v R ) = G b v R + ( G b − G a ) E v R ≤ − E G a v R | v R | ≤ − E G b v R + ( G a − G b ) E v R ≥ E (2)where, G a , G b and E are known real constant which satisfythe following conditions G b < G a < and E > L C2 C1R i L R4R5R6 R3 R2R1+- +-
Fig. 2. Circuit diagram of chua’s circuit used in developing experimentalprototype
Circuit diagram of chua’s circuit used in developing exper-imental prototype is shown in Figure 2. The chua’s diode canbe emulated using operational amplifier.Chua’s circuit can be represented using dimensionless equa-tions dxdτ = α ( y − x − f ( x )) dydτ = x − y + z dzdτ = − βy (3)where, x , y and z represents the state variable of the system, α and β are the system parameters and f ( x ) is the nonlinearfunction. x = v C E , y = v C E , z = i L ( E G ) , τ = tGC , a = RG a , b = RG b , α = C C , β = C R L Some of the widely used nonlinear functions are representedas f ( x ) = bx + 0 . a − b ) ( | x + c | − | x − c | ) f ( x ) = h x − h x f ( x ) = − a tanh ( bx ) f ( x ) = d x + d x | x | (4)III. S TAR N ETWORK T OPOLOGY AND C OUPLING
Star network of N nodes comprises of a central node andother end nodes. The central node and the end nodes of thenetwork are connected using bidirectional coupling. In starnetwork, there is a central hub node (site index as i = 1 ) and N − peripheral end nodes connected to this central hub node.This can also be interpreted as a set of uncoupled identicaloscillators powered through a common drive. Our motivation isto study the dynamical patterns arising in these N − identicalend nodes. End NodeSource Node (Chua Circuit)(Chua Circuit)
Fig. 3. Star network configuration of N identical chua circuits A. Diffusive Coupling
The dynamical equations of the diffusive coupling throughsimilar variables can be represented as ˙ x i = f x ( x i , y i , z i ) + N P j =1 K ij ( x j − x i )˙ y i = f y ( x i , y i , z i )˙ z i = f z ( x i , y i , z i ) (5)Where K = ( k ij ) is the coupling matrix of order N × N k . . . k k ... k where k is the coupling strength. B. Conjugate Coupling
The dynamical equations of conjugate coupling where cou-pling involves dissimilar variable can be represented as ˙ x i = f x ( x i , y i , z i ) + N P j =1 K ij ( y j − x i )˙ y i = f y ( x i , y i , z i )˙ z i = f z ( x i , y i , z i ) (6)SCEECS 2018 . Mean-Field Coupling The dynamical equations of central node in mean-fieldcoupling can be represented as ˙ x = f x ( x , y , z ) + k ( x m − x )˙ y = f y ( x , y , z )˙ z = f z ( x , y , z ) (7)where, x m = N − P j =2 ,..,N x j is the mean field of end-nodes.The dynamical equations of the remaining end nodes inmean-field coupling can be represented as x i = f x ( x i , y i , z i ) + k ( x − x i ) y i = f y ( x i , y i , z i ) z i = f z ( x i , y i , z i ) (8)IV. S IMULATION R ESULTS
This section provides simulation results for chua’s circuitand different coupling aspects of star network connectedchua’s circuit.
A. Dynamics of chua’s circuit
The parameters for chua’s circuit (Figure 1) are selectedas, C = 10 nF, C = 100 nF, L = 18.75 mH, R = 1 k Ω .Using the above mentioned parameters the chua’s circuit issimulated using MATLAB and the system exhibits a double-scroll chaotic attractor (Figure 4). The double scroll chaoticattractor can be seen for different nonlinear functions. Figure5 presents the double scroll behavior of chua’s circuit for non-linear function f . Similarly, Figure 6 and Figure 7 presents thedouble scroll behavior of chua’s circuit for non-linear function f and f respectively. −6 −4 −2 0 2 4 6−1.5−1−0.500.511.5 V c1 V c Fig. 4. Double scroll chaotic attractor of a chua’s circuit
B. Diffusive Coupling in Star Network
The synchronization depends on three parameters such as(a) number of nodes N , (b) coupling strength k and (c) initialconditions. For diffusive coupling, the following parametersare considered. Coupling strength k = 27 . , time step size dt = 0 . , number of nodes N = 100 , the phase spacedynamics of some of the end node oscillators are shown inFigure 8. From Figure 9, it can be seen that the 2nd and 4thend nodes as well as 3rd and 4th end nodes are in complete t x t y z t (a) −4 −3 −2 −1 0 1 2 3 4−0.8−0.6−0.4−0.200.20.40.60.8 x y (b)Fig. 5. (a) Chaotic dynamics of chua’s circuit with non-linear function f (b) Double-scroll attractor with non-linear function f t x t y z t (a) −6 −4 −2 0 2 4 6−1.5−1−0.500.51 x y (b)Fig. 6. (a) Chaotic dynamics of chua’s circuit with non-linear function f (b) Double-scroll attractor with non-linear function f SCEECS 2018
20 40 60 80 100−505 t x t y z t (a) −4 −3 −2 −1 0 1 2 3 4−0.5−0.4−0.3−0.2−0.100.10.20.30.40.5 x y (b)Fig. 7. (a) Chaotic dynamics of chua’s circuit with non-linear function f (b) Double-scroll attractor with non-linear function f −3 −2 −1 0 1 2 3−0.500.5−4−2024 X(2) −−−−> plot of dynamics of 1st endnode:Y(2) −−−−> Z ( ) −−−−> −3 −2 −1 0 1 2 3−0.500.5−4−2024 X(3) −−−−> plot of dynamics of 2nd endnode:Y(3) −−−−> Z ( ) −−−−> −3 −2 −1 0 1 2 3−0.500.5−4−2024 X(4) −−−−> plot of dynamics of 3rd endnode:Y(4) −−−−> Z ( ) −−−−> −3 −2 −1 0 1 2 3−0.500.51−10−505 X(5) −−−−> plot of dynamics of 4th endnode:Y(5) −−−−> Z ( ) −−−−> Fig. 8. Phase space dynamics of some end nodes synchronization as evident from the sharp straight line plotbetween the x state variable of the end nodes. Also it isseen that the 1st end node and 3rd end node are in partialsynchrony where as the 2nd and 3rd end nodes are not insynchrony. The straight line plots between x i vs x j whichrepresents the end nodes in star network (Figure 9) confirmsthe complete synchronization behavior between different endnodes in diffusive coupling.Figure 10 shows the difference plot between the x statevariables of different end nodes which tend to zero as timeprogresses implying synchronization. C. Conjugate Coupling
For simulation of conjugate coupling the number of endnodes considered are N = 100 , coupling strength k = 1 . −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−3−2−10123 Straightline plot showing Synchronization b/w 2nd and 3rd endnodeX(3) −−−−> X ( ) −−−−> −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−3−2−10123 Straightline plot showing Synchronization b/w 1st and 3rd endnodeX(4) −−−−> X ( ) −−−−> −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−3−2−10123 Straightline plot showing Synchronization b/w 3rd and 4th endnodeX(4) −−−−> X ( ) −−−−> −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−3−2−10123 Straightline plot showing Synchronization b/w 2nd and 4th endnodeX(3) −−−−> X ( ) −−−−> Fig. 9. Synchronization plots between x i vs x j X ( ) − X ( ) −−−−> X ( ) − X ( ) −−−−> X ( ) − X ( ) −−−−> X ( ) − X ( ) −−−−> Fig. 10. Difference plots of the end nodes in diffusive coupling with initial conditions x ∈ [0 . , . , y ∈ [ − . , − . , z ∈ [ − . , − . In conjugate coupling, it can be observed that for a lowvalue of coupling strength, the end nodes get synchronized.The phase plot dynamics of say 4th end node is shownin Figure 11. Synchronization of nodes in star network inconjugate coupling is evident from Figure 12. −2 −1 0 1x 10 −2−1012x 10 −3−2−1012 x 10
Fig. 11. Phase space of the 4th end node
Figure 13 presents the phase space of end node in conju-gately coupled star network of chua’s circuit. Figure 14 showsthe difference plots of the end nodes in conjugate coupling.SCEECS 2018 −4−3−2−101234 x 10
Straightline plot showing Synchronization b/w 3rd and 2nd endnodeX(4) −−−−> X ( ) −−−−> Fig. 12. Synchronization in case of conjugate coupling −1 −0.5 0 0.5 1x 10 −101x 10 −3−2−10123 x 10 X(4) −−−−> plot of dynamics of 3rd endnode:Y(4) −−−−> Z ( ) −−−−> Fig. 13. Phase space of end node in conjugately coupled star network ofchua’s circuit −6−4−20246x 10 difference b/w 2nd and 3rd endnode t −−−> X ( ) − X ( ) −−−−> −6−4−20246x 10 difference b/w 3rd and 4th endnode t−−−−> X ( ) − X ( ) −−−−> −1−0.500.511.5x 10 difference b/w 2nd and 4th endnode t −−−−> X ( ) − X ( ) −−−−> −1.5−1−0.500.51x 10 difference b/w 2nd and 6th endnode t −−−−> X ( ) − X ( ) −−−−> Fig. 14. Difference plots of the end nodes in conjugate coupling
It is observed that a spiral phase space is obtained in thecase of conjugately coupled chua’s circuit in the 2nd end node(Figure 13) indicating that the system dynamics get spiraldown to steady state as time progresses. Also for the samecoupling strength k , in random initial conditions over thesame range in 2nd end node we get the double scroll attractor(Figure 15). −3 −2 −1 0 1 2 3 −0.5 0 0.5−4−2024 Y(3) −−−−> X(3) −−−−> plot of dynamics of 2nd endnode: Z ( ) −−−−> Fig. 15. Double scroll in 2nd end node in conjugate coupling of chua’s circuit
D. Mean-Field Coupling
Figure 16 shows the difference plots of the end nodesin mean-field coupling. The plot converges to zero, whichindicates synchronization. −1−0.500.51x 10 difference b/w 2nd and 3rd endnode t −−−> X ( ) − X ( ) −−−−> −6−4−2024x 10 difference b/w 3rd and 4th endnode t−−−−> X ( ) − X ( ) −−−−> −6−4−2024x 10 difference b/w 2nd and 4th endnode t −−−−> X ( ) − X ( ) −−−−> −6−4−2024x 10 difference b/w 2nd and 6th endnode t −−−−> X ( ) − X ( ) −−−−> Fig. 16. Difference plots of the end nodes in mean-field coupling
Figure 17 presents the phase space of end nodes in mean-field coupling (which is spiral phase space) in star connectedchua’s circuit. It is observed after simulations that over awide range of coupling strength values the mean field coupledsystem synchronizes.Figure 18 presents the straight line synchronization plot ofend nodes in mean-field coupling in star connected chua’scircuit. V. C
ONCLUSION
This paper provides a mathematical interpretation of syn-chronization of N identical chua’s circuit connected in startopology with bidirectional coupling. Different bidirectionalcoupling such as diffusive coupling, conjugate coupling andmean-field coupling have been used. Synchronization of chua’scircuit in these coupling have been mathematically validatedand simulation results have been provided to authenticatethe mathematical formulation. From the simulation results,it is observed that the synchronization occurs over a widerange of values of the coupling strength k in case of mean-field coupling. In case of conjugate coupling, the nodes getSCEECS 2018 −2−1012x 10 −1−0.500.511.52 x 10 Fig. 17. Phase space of end nodes in mean-field coupling in star connectedchua’s circuit −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5−2−1.5−1−0.500.511.522.5 Straightline plot showing Synchronization b/w 4th and 3rd endnodeX(5) −−−−> X ( ) −−−−> Fig. 18. Synchronization of end nodes in mean-field coupling in starconnected chua’s circuit synchronized for low values of coupling strength up to acertain critical coupling strength and gets destabilized forhigher values of coupling strength. In diffusive coupling,synchronization takes place in larger values of k than othercoupling forms. It is observed that some end nodes getsynchronized and remaining remain out of synchronizationwhich provides a hint of prevalence of chimera states. In futurework, stability of synchronization and stability of chimerastates can be studied in details.A CKNOWLEDGEMENT
The first author would like to express his deepest apprecia-tion to his guide Prof. Kishor Chandra Pati, HOD of Mathe-matics, NIT, Rourkela who has shown the attitude and the sub-stance of a genius. He also expresses his deep gratitude to Prof.Amit Apte, ICTS, Bangalore without whose supervision andconstant support this work would not have been possible. Heis very much thankful to Subbhransu Padhee, Ph.D. scholar,NIT, Rourkela and Suman Acharyya, Postdoctral fellow, ICTSfor their constant support and encouragement. R
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