Experimental realization of mixed-synchronization in counter-rotating coupled oscillators
EExperimental realization of mixed-synchronization incounter-rotating coupled oscillators
Amit Sharma and Manish Dev Shrimali
The LNM Institute of Information Technology,Jaipur 302 031, India
Abstract
Recently, a novel mixed–synchronization phenomenon is observed in counter–rotating non-linear coupled oscillators [1]. In mixed–synchronization state: some variables are synchronizedin–phase, while others are out–of–phase. We have experimentally verified the occurrence of mixed–synchronization states in coupled counter–rotating chaotic piecewise R ¨ ossler oscillator. Analyticaldiscussion on approximate stability analysis and numerical confirmation on the experimentallyobserved behavior is also given. a r X i v : . [ n li n . C D ] M a y . INTRODUCTION Huygens first describes the anti–phase synchronization in a pair of pendulum clocks [2].Later, the idea of synchronization of two identical chaotic system was introduced by Pecoraand Carroll [3]. Synchronization of chaotic systems has attracted much attention due to itspotential application in secure communication, chemical and biological system, informationscience, and so on [4]. Many different synchronization states have been studied in literature,namely complete or identical synchronization (CS) [3, 5, 6], in–phase (PS) [7, 8], anti–phase[9], lag synchronization (LS) [10], generalized synchronization (GS) [11, 12], intermittentlag synchronization(ILS) [13, 14], and anti-synchronization (AS) [15–17] in which one of thedynamical variable is synchronized then rest of variable follow the same. All these type ofsynchronization can be achieved with various type of interactions e.g. mismatch oscillators[18], conjugate [19, 20], delay [21], and nonlinear [22, 23], indirect [24, 25].If the directions of rotation of two oscillators are same, the system is co–rotating, whilesystem of oscillators rotating in opposite direction is called counter–rotating. Coupled co–rotating nonlinear oscillators have been extensively studied from both theoretical and exper-imental point of view [4]. Recently, a mixed–synchronization phenomenon was observed incoupled counter–rotating nonlinear oscillators [1], similar phenomena was engineered usinga general formulation of coupling function in co–rotating coupled oscillators [26, 27]. Inmixed–synchronization state, some variables are synchronized to in–phase state while othervariables are out–of–phase. The mixed–synchronization phenomenon is also studied in thecase of extended systems [1].In this Letter, we present the experimental observation of the mixed–synchronizationin two diffusive coupled counter–rotating chaotic piecewise R ¨ ossler oscillators. The ana-lytical discussion on approximate stability analysis and numerical simulations are in closeagreement with experimental results. The critical value of coupling strength, where counter–rotating coupled chaotic oscillators are synchronized, is larger in experiments as comparedto numerical simulations because of parameter mismatch in circuit implementation.The Letter is organized as follows: In the section II we numerically study the mixed–synchronization phenomenon in the coupled chaotic oscillators for piecewise R ¨ ossler system.The linear stability analysis and numerical results are presented. The experimental setupand the results of coupled counter–rotating chaotic oscillators are discussed in section III.2oncluding remarks and discussion about the mixed–synchronization in coupled counter–rotating chaotic oscillators are given in section IV. II. THE MODEL SYSTEM
Here, we illustrate the mixed–synchronization phenomena in two diffusive coupled piece-wise R ¨ ossler [28] systems given by following equations˙ x i = − γ i x i − α i y i − z i + (cid:15) ( x j − x i ) = f ( x i , y i , ω i ) + (cid:15) ( x j − x i )˙ y i = β i x i + a i y i = g ( x i , y i , ω i )˙ z i = h ( x i ) − z i (1)Here, in functions f and g , ω i ’s represents the internal frequency of two oscillators withopposite sign and it depends on the parameters α i and β i . The function h ( x ) = 0 if x ≤
3, or h ( x ) = µ ( x −
3) if x >
3. The rotation of Piecewise R ¨ ossler system (in x − y plane) can bechanged by changing the sign of α i and β i . The first system has counter clockwise rotationwhile second has clockwise rotation. The parameters values are: α = 0 . α = − . β = 1, β = − γ i = 0 . a i = 0 . µ i = 15. The coupling parameter is (cid:15) . Foridentical oscillators, a = a .The fixed points of the piecewise R ¨ ossler oscillators are ( x ∗ = µ i κ i , y ∗ = − µ i β i a i κ i , z ∗ = − µ i κ i ( − γ i + α i β i a i )), where κ i = − γ i + α i β i a i + β i depends on the sign of the α i and β i . Thechange in the dynamical behavior arises from the coupling between two identical piecewise R ¨ ossler oscillators. A. Numerical Results
We numerically study the mixed–synchronization of two coupled counter–rotating piece-wise R ¨ ossler oscillators. At the very small coupling strength the two oscillators are uncor-related. As the coupling strength increases, the phase synchronization set in when forthlargest Lyapunov exponent becomes negative and complete synchronization occurs whenthird largest Lyapunov exponent becomes negative as shown in Fig. 1(a).3o quantify synchronization, we use the following similarity function defined with respectto dynamical variables, x and y , of the chaotic oscillator [10] S ( x ) = (cid:115) < [ x ( t ) − x ( t )] > [ < x ( t ) >< x ( t ) > ] / (2) S ( y ) = (cid:115) < [ y ( t ) + y ( t )] > [ < y ( t ) >< y ( t ) > ] / (3)Synchronization (complete and anti) is characterized by S ( . ) = 0 for x and y variablesrespectively. The variables x and x are in–phase while y and y are out–of–phase. The twovariables x and y shows complete in–phase and out–of–phase synchronization respectivelyfor coupling strength (cid:15) > (cid:15) c , where (cid:15) c ∼ . z variable of the system also goesto complete synchronization state, where S ( z ) is defined similar to S ( x ). The in–phasesynchronization in x and x while out–of–phase in y and y is shown in Figure 1(c). Thecomplete synchronization in x and x with zero relative phase while out–of–phase state of y and y with phase difference of π is shown in Figure 1(d). B. Linear Stability Analysis
We analyze the stability of the mixed–synchronized state of two counter–rotating coupledchaotic systems given by Eq. (1) in x − y plane. The method of approximate linear stabilityanalysis is adopted for synchronization criteria [24]. If ξ and η represent the deviation ofcoordinates x and y respectively from the synchronization state, their dynamic is governedby the linearized equation as ˙ ξ i = f (cid:48) ( x i , y i , ω i ) + (cid:15) ( ξ j − ξ i ) , ˙ η i = g (cid:48) ( x i , y i , ω i ) (4)Where the f and g are functions in terms of coordinate and parameter. ω i , i=1,2 representthe frequency of the oscillators. The criteria for the stability is that synchronization statecorresponding to fixed point will be stable if all eigen values of the Eqs. (4) are negative.Dynamics of the deviation from the synchronization state is governed by the linearizedequation of Eqs (1). 4 ξ = ( − γξ − α η ) f (cid:48) ( x , y ) + (cid:15) ( ξ − ξ ) , ˙ η = ( β ξ + aη ) g (cid:48) ( x , y ) , ˙ ξ = ( − γξ − α η ) f (cid:48) ( x , y ) + (cid:15) ( ξ − ξ ) , ˙ η = ( β ξ + aη ) g (cid:48) ( x , y ) (5)Where γ, α, β , and a are parameters. For the Perfect synchronization in counter rotatingcoupled system , i.e. x = x (complete) and y = − y (Anti-synchronization), we can define µ = ξ − ξ ,µ = η + η (6)Then Eqs (5) can be written as˙ µ = − γf (cid:48) ( x , y ) µ − ( α η − α η ) f (cid:48) ( x , y ) − (cid:15)µ , ˙ µ = ( β ξ + β ξ ) g (cid:48) ( x , y ) + ag (cid:48) ( x , y ) µ (7)If we assume that the time average values of Jacobian matrix elements f (cid:48) ( x i , y j ) and g (cid:48) ( x i , y i ), where i=1,2 are approximately the same and can be replaced by an effectiveconstant value λ and λ .In the case of counter rotating R ¨ ossler systems, frequency of the coupled systems are ofopposite sign: α = − α and β = − β . Then˙ µ = − ( γλ + 2 (cid:15) ) µ − α λ µ , ˙ µ = β λ µ + aλ µ (8)Eliminating µ from above equations, we get¨ µ = ( aλ − ( γλ + 2 (cid:15) )) ˙ µ − ( α β λ λ − aλ ( γλ + 2 (cid:15) )) µ (9)Solution of the equation µ = Ae mt , we get5 = ( aλ − ( γλ + 2 (cid:15) )) ± (cid:112) ( aλ − ( γλ + 2 (cid:15) ))) − α β λ λ − aλ ( γλ + 2 (cid:15) ))2 (10)The synchronization state define by µ = ξ − ξ = 0 and µ = η + η = 0, is stable ifRe[ m ] is negative for both the solutions. • If ( aλ − γλ − (cid:15) )) < α β λ λ − aλ ( γλ + 2 (cid:15) )), m is complex and the stabilitycondition becomes ( γλ + 2 (cid:15) ) > aλ . • If ( aλ − γλ − (cid:15) )) > α β λ λ − aλ ( γλ + 2 (cid:15) )), m real and the stability conditionbecomes ( γλ + 2 (cid:15) ) > aλ .The transition to stable synchronization is given by the threshold values of the parameterssatisfying the condition (cid:15) c = 12 ( aλ − γλ ) (11)Figure 2 shows the transition from the unsynchronized to mixed–synchronization statein the (cid:15) − a space. A linear relations is clearly seen and the solid line is drawn with theeffective λ = − .
45 and λ = 0 .
55, thus validating the transition criterion of Eq. (11)obtained from the stability theory. The condition given by Eq. (11) is necessary but notsufficient for synchronization.
III. EXPERIMENTAL SETUP AND RESULTS
Experiments are conducted using a pair of electronic oscillators whose dynamics mimicthat of the chaotic R ¨ ossler oscillator [28]. One of the oscillators rotate clockwise whileanother anti–clockwise. The two oscillators are approximately identical since in reality itis not possible to ensure that parameters are exactly equal. Further, unlike the piecewise R ¨ ossler system (Eq. (1)) discussed above, the coupling is asymmetric and frequencies of theoscillators are not equal in experiment. Hence, we observe lag and phase synchronization incoupled piecewise R ¨ ossler oscillators as coupling is increased.The Piecewise R ¨ ossler oscillator circuit shows the dynamics of rotation in counter clock-wise. We have to connect two inverting amplifier U and U (as shown in Fig. 3) for changing6he direction of the rotation in circuit. Both piecewise R ¨ ossler oscillators are consisting ofthe passive components like resistance R − , capacitors C − , diodes D − and operationalamplifier uA741 U − . We use simple linear scheme for the coupling between x variablesof the two piecewise R ¨ ossler oscillators. The OPAMP U , , , in circuit are used for linearcoupling scheme. RF and RF are the variable resistors characterizing the coupling param-eter. The electronic components in circuits are carefully chosen and values are mentionedin the diagram (Fig. 3). The typical oscillating frequencies of the circuits are in the audiofrequency range. Both oscillators are operated by a low-ripple and low noise power supply.The output voltages form both oscillators are monitored using digital oscilloscope 100MHz2 channel (Agilent DSO1012A) with maximum sampling rate of 2 GSa/s.Transitions from asynchronous chaos to lag synchronization and then to in–phase syn-chronization is observed at the critical values of variable resistance, R c and R c , respectively.The lag synchronization occurs in the interval [ R c , R c ], where R c = 0.5kΩ and R c = 9kΩ.It has been observed in experiments that the variables of one of the oscillator tends to fellowthe variables of the another oscillator in some range of the coupling strength [29]. Here, itis due to the parameter mismatch in the coupled oscillators. The Similarity function of x and y variables (Eq. (2) and (3)) of the coupled piecewise R ¨ ossler oscillator with variableresistance R is shown in Fig. 4. At RF = RF = 1.4kΩ the output voltage of x and x shows in–phase dynamics while y and y are out–of–phase. Phase relationship of x and y variables with lag synchronization are shown in Fig. 5(c-d). Further increase of the couplingstrength shows the transition from lag to phase synchronization. Phase relationship at RF = RF = 11.2kΩ is shown in Fig. 5(e-f). IV. CONCLUSION
We presented the experimental evidence of mixed–synchronization in the piecewise R ¨ ossler oscillators circuit via diffusive type of coupling under the parameter mismatch. Theexperimental results are in close agreement with the numerical results. The critical value ofcoupling strength for onset of mixed–synchronization is calculated using approximate linearstability analysis. We have also studied the sprott circuit [30] and obtained similar results formixed-synchronization. The natural emergence of novel mixed–synchronization phenomenonin chaotic as well as limit cycle counter–rotating coupled oscillators has possible applications7n secure communication and chaos based computing. Acknowledgments
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06 and ε = 0 . x, y , and z are marked by black, red, and green color respectively. Figure 2: (Color online) Transition from unsynchronized to mixed–synchronized region isshown in the parameter plane ( a, (cid:15) ) for coupled piecewise R ¨ ossler oscillators. Figure 3: (Color online) Schematic diagram of two bidirectional coupled (counter clockwiseand clockwise) piecewise chaotic R ¨ ossler oscillator. Variable resistors are used to changethe coupling. The OPAMP are type of uA741. All resistors are metal film type withtolerance 1% and capacitors are polyester type with tolerance 5%. The circuit is run by ± V source. Figure 4: (Color online) Similarity function S for x (circle) and y (triangle) variables ofthe experimental system of two coupled counter rotating piecewise R ¨ ossler oscillators withvariable resistance RF = RF = R . Figure 5: (Color online) Dynamic of the piecewise R ¨ ossler oscillator in (a) counter clock-wise rotation (b) clockwise rotation. The phase relationship of x and y variables respectivelyat RF = RF = 1 . k Ω for lag–synchronization in (c) and (d). mixed–synchronization at RF = RF = 11 . k Ω in (e) and (f). 10 ε -0.1-0.0500.05 λ ε S -6 -4 -2 0 2 4 x , y , z -4-20246 x , y , z -6 -4 -2 0 2 4 x , y , z -4-20246 x , y , z λ λ λ λ (a)(b)(c) (d) FIG. 1: (Color online) (a) The largest four Lyapunov exponents of identical coupled counter–rotating piecewise R ¨ ossler oscillators. (b) the similarity functions for x, y and z variables of thecoupled oscillators. (c) and (d) the phase relationship between the variables x, y and z at ε = 0 . ε = 0 . x, y , and z are markedby black, red, and green color respectively. IG. 2: (Color online) Transition from unsynchronized to mixed–synchronized region is shown inthe parameter plane ( a, (cid:15) ) for coupled piecewise R ¨ ossler oscillators. IG. 3: (Color online) Schematic diagram of two bidirectional coupled (counter clockwise andclockwise) piecewise chaotic R ¨ ossler oscillator. Variable resistors are used to change the coupling.The OPAMP are type of uA741. All resistors are metal film type with tolerance 1% and capacitorsare polyester type with tolerance 5%. The circuit is run by ± V source. Variable resistance R (k ) S Ω Lag Syanchronization Phase Synachronzation