Design of Coupling for Mixed Synchronization in Chaotic Oscillators Syamal K. Dana1, E.Padmanaban1, Ranjib Banerjee1, Prodyot K.Roy2, Ioan Grosu3 1
Central Instrumentation, Indian Institute of Chemical Biology (Council of Scientific and Industrial Research), Kolkata 700032, India 2 Departement of Physics, Presidency College, Kolkata 700073, India 3 Faculty of Bioengineering, University of Medicine and Pharmacy,“Gr.T.Popa”, Iasi, Romania E-mail:
[email protected]. Abstract— We report a design of coupling in chaotic oscillators for realizing a desired response: complete synchronization, antisynchronization and amplitude death simultaneously in different state variables of a system and thereby targeting a control of synchronization. This is robust to parameter mismatch and the route of transition to synchrony obeys a scaling law. Experimental evidence of the coupling is presented using an electronic circuit.
I.
INTRODUCTION
Chaos synchronization [1] ushers in potential applications in living [2] and engineering systems [3]. Recently, the concept of feedback control is used [4] in synchronization for engineering various dynamical structures in a population of oscillators. Importance of engineering synchronization is explained in a recent article: Rhythm engineering [5]. Formulating a design scheme of coupling is also important and another step towards targeting a desired response like complete synchronization (CS) [6], antisynchronization (AS) [7] and amplitude death [89] in chaotic oscillators. Chaotic oscillators are in opposite phase in AS unlike in CS when they are in-phase but in common they have identical amplitudes. In amplitude death, the response oscillator is driven to a steady state. We report a design approach of unidirectional coupling to target desired responses like CS, AS and amplitude death in chaotic oscillators. We used the open-plus-closed-loop (OPCL) coupling [10-12] based on Hurwitz matrix stability to realize and to control such synchronization processes. We have generalized the OPCL coupling here to engineer the broad variety of synchronization. With this we are able to induce mixed synchronization between a driver and a response: a pair of similar state variables of the driver and the response develops AS while the other pair is at CS state and another response variable at resting state or amplitude death. Other combinations of mixed correlated dynamics like simultaneous amplification with AS, attenuation with CS and amplitude death can also be attained. A method [13] was reported very recently which attempted hybrid synchronization based on Lyapunov function stability where complete synchronization and antiphase were induced in a response oscillator. However, this method is limited to theoretical and numerical studies. Instead, we have physically realized our design of coupling and
it is found robust to parameter mismatch. Moreover, the coupling is more flexible since amplification (attenuation) and amplitude death is allowed as desired response. A transition route to synchrony with parameter mismatch is also found that obeys a unique scaling law. II.
GENERALIZED OPCL COUPLING
The OPCL coupling was used to realize CS [10-12] and AS [11-12] in chaotic oscillators and complex networks [14, 15]. We generalize the coupling using mismatched chaotic oscillators: a driver is defined by y = f ( y ) + Δf ( y ), y ∈ R n , where Δf(y) represents the mismatch terms. It drives another oscillator to target a goal ~ (t ) = α y (t ) , where α (i=1, 2, 3,…, n) is a dynamics, g i i i constant Δ
and
T g~ (t ) =[g 1 (t ) g 2 (t ) g 3 (t ) ... g n (t )]
(1)
= [α 1 y1 α 2 y 2 α 3 y 3 .... α n y n ]
T
T denotes transpose of a matrix. The response system after coupling becomes
x = f ( x) + D ( x, g~ ).
(2)
where
⎛ ∂f ( g~ ) ⎞ D( x, g~ ) = g~ − f ( g~ ) + ⎜⎜ H − ~ ⎟⎟( x − αy ) ∂( g ) ⎠ ⎝ and ( x − αy )
(3) (4) T
= [( x1 − α 1 y1 ) ( x 2 − α 2 y 2 ) ..... ( x n − α n y n )] ~ ∂f ( g ) is the Jacobian of the model system and H is an ∂g~ arbitrary constant Hurwitz matrix (nxn), whose eigenvalues all have negative real parts. The error signal of the coupled ~ when f (x) can be written, system is defined by e = x − g using the Taylor series expansion, as
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∂f ( g~) f ( x) = f ( g~ ) + ~ ( x − g~ ) + ... ∂g
x1 = −ax2 − a(α 2 − α 1 ) y 2 − ( Δa )(1 / ε )α 2 y 2 , x 2 = x1 + x3 + (α 2 − α 1 ) y1 + (α 2 − α 3 ) y 3 ,
(5)
Keeping the first order terms in (5) and substituting in (2), the error dynamics is obtained as e = He . This ensures e → 0 as t → ∞ and that the synchronization is asymptotically stable. The Hurwitz matrix H is derived from the Jacobian of the oscillator model. The elements of the Hurwitz matrix H ij are
(8)
2 2 2 x 3 = x1 + x 2 − x3 + (α 3 − α 2 ) y 2 + (α 3 − α 1 ) y1
+ ( p − 2α 2 y 2 )( x 2 − α 2 y 2 )
⎛ ∂f ( g~ ) ⎞ ⎛ ∂f ( g~ ) ⎞ H ij = ⎜⎜ ~ ⎟⎟ when ⎜⎜ ~ ⎟⎟ is a ⎝ ∂g ⎠ ij ⎝ ∂g ⎠ ij
chosen such that
⎛ ∂f ( g~ ) ⎞
constant. If ⎜⎜ ~ ⎟⎟ involves a state variable, we define ⎝ ∂g ⎠ ij
H ij = p ij where pij is a constant. The design of coupling is based on systematic selections of the pij parameters. They are so appropriately chosen that the Routh-Hurwitz criterion [1012] be fulfilled to ensure asymptotic stability of synchronization even in presence of parameter mismatch. The multiplying constant αi can be used to control mixed correlated dynamics: for example, CS (α1=1), AS (α2=-1) and amplitude death (α3=0) simultaneously in different state variables of 3D response systems. Other combinations of correlated dynamics like attenuation and CS ( 0 < α 1 < 1 ), AS and amplification ( α 2 > 1 ) and amplitude death (α3=0) are also realizable. MIXED SYNCHRONIZATION
III.
We first present a numerical example using a 3D system to demonstrate mixed synchronization and then show how instabilities due to mismatch is taken care of by the coupling. We choose a simple Sprott system as given by 2 x1 = − ax 2 , x 2 = x1 + x3 , x 3 = x1 + x 2 − x3 .
(6)
It has a single quadratic nonlinearity when the coupling becomes simpler. For systems with higher order nonlinearity, the coupling is more complex, however, it is not difficult to derive [11, 12] the coupling for such systems. We choose another Sprott system with a mismatch (Δa) as the driver oscillator
y1 = −ay 2 − Δay 2 , y 2 = y1 + y 3 ,
(7)
2 y 3 = y1 + y 2 − y 3 .
The
Jacobian
of
the
model
(6)
is
given
by
T
J= [0 - a 0; 1 0 - 1; 1 2 x2 - 1] when the Hurwitz matrix is derived as H= [0 - a 0; 1 0 - 1; 1 p - 1] T ; p is a
~ ) in parameter as described above. The OPCL coupling D ( x, g (3) is then derived to obtain the response Sprott system after coupling,
Figure 1. Mixed synchronization in Sprott system. [a=0.225, Δa=0.025]. Time series x1 (red) and y1 at upper row left (blue) in AS, x3 and y3 at middle row left (blue) in CS. Plots of x1 vs. y1 at upper row right) in AS, x3 vs. y3 at middle row right in CS. Lower row: time series y2, x2 of chaotic oscillatory driver (blue) and response at amplitude death (red horizontal zero line) respectively.
ε is a constant introduced deliberately and taken as unity for current simulation. Figure 1 confirms mixed synchronization and that the generalized coupling allows choices of response dynamics as controlled by αi : AS in one pair of state variables (α1=-1), CS in the other (α3=1) and amplitude death (α2=0) in another. Similarly, partial attenuation/amplification can also be realized by controlling the magnitude of αi. Such a coupling is interesting for practical applications like in processing industry where concentration of any of the constituents of reaction systems can be enhanced or reduced and even made zero to obtain a desired product output. If we assume α1=α2=α3=α (say), we can achieve the conventional correlated dynamics like CS, AS or amplitude death, one in all state variables. The response Sprott system then becomes more simple,
x1 = −ax 2 − (Δa )(1 / ε )αy 2 , x 2 = x1 + x3 , 2 2 x 3 = x1 + x 2 − x3 + α (1 − α ) y 2
+ ( p − 2αy 2 )( x 2 − αy 2 ).
(9)
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C1
Simulating the driver (7) and response (9), we can easily find evidences [11, 12] either of the correlated dynamics CS, AS or amplitude death by controlling α as demonstrated in the experiment in section IV.
R5 1k
10n U3 3
+
2
3
R1 6
U1 uA741OUT
0
1k
2
R6
R2
A. Transition Route to Synchronization
2
< ( x1 (t ) − αy1 (t − τ )) >
10k
3
R4
(10)
-
C3
10k
10k
2
U12 1 2 3 4 6
+ U2 uA741OUT
0
6
-
U4
X1 X2 Y1 W Y2 Z
3
R8 7
+ uA741
0
1k
2
-
C4
R13
OS-2
10n 3
U5
2
1k
+ uA741 OUT
0
C5
U6
0 2
10k
2
R14
R15
10k
10k
U13
1 2 3 4 6
+ uA741 OUT
+ 6
-
C6 10n
10n 3
U7
0 uA741 OUT
1k
R10
R11
3
R9
6
-
50k
6
OUT
AD633/AD
0
A
R12
(< x1 (t ) 2 >< y1 (t ) 2 >)1 / 2
6
OUT
10n 10n
10k
δε =
10k
R3
As seen from (9), the OPCL coupling is nonlinear type. In addition, a linear coupling term appears due to the mismatch parameter. This additional linear coupling help nullify the destabilizing effect on synchronization due to the mismatch. In fact, for each of mismatch parameter, an additional linear coupling term appears and they take care of the instabilities [11, 12]. To explain their role on synchrony, we tune the parameter ε in (9) from both sides of a critical value (ε =εc=1): higher and lower values as positive and negative variation in mismatch. A similarity measure is then used [1] to estimate the corresponding error between the driver and the response dynamics for varying ε,
uA741
R7
C2
50k
+
0
6
-
X1 X2 Y1 W Y2 Z
R16 7
U8 +
0 1k
AD633/AD
0
B
3
2
OUT
6
-
uA741
C
V+
OPCL Coupling
R17 40k
(a)
R18 20k
(b)
R19 40k
Figure 2. Coupled Sprott system: (a) transition to synchrony with tuning parameter ε in (a), slope γ ~1.57 in (b). Open circles are for numerical data points.
7
+ uA741 OUT
0
R20
2
1k
6
-
R24
R22
R23
10k
10k
10k
1 2 3 4 6
Δ
A global minimum of δε = δmin= 0 stands for synchronization either CS or AS depending on the sign of α-value (delay τ=0 for CS or AS). For the Sprott system, ε is varied from 0.5 to 1.5 to obtain δmin=0 at ε=εc for a mismatch, Δa=0.025 and the corresponding ln(δ) vs. (ε-εc) plot is shown in Fig.2(a). A sharp dipping into a minimum is observed at a critical value, εc=1, when stable synchronization is attained. Effectively, ε acts as a strength of the linear coupling. Any compromise with the strength of this linear coupling will induce degradation in synchrony. Interesting to note that the transition to synchrony obeys a scaling law δ =(ε-εc)γ where γ ∼1.57 and this is also valid [11, 12] for other systems like Lorenz system and Rössler oscillator.
U11
AD633/AD
0
V-
10k 3
U14 1 2 X1 3 X2 4 Y1 W 6 Y2 Z
A
R21
R25 R26
1k
0
U15 X1 X2 Y1 W Y2 Z
7
R30 1k
C
AD633/AD
U10
5k
3
0 2
+ uA741
OUT
6
R27 U9
3 R28
B
uA741 OUT
0 2
10k
1k
+
R29 6 10k
-
Figure 3. Coupled Sprott circuit: driver OS-1 coupled to response OS-2 via OPCL coupling. Component values (1% tolerance) are noted
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IV.
EXEPRIMENTAL OPCL COUPLING
We describe a physical realization of the OPCL coupling in Sprott system (9). Figure 3 shows the coupled circuit. The Opamps U1-U4 (U5-U8) with analog multiplier U12 (U13) and resistances R1-R8 (R9-R16) and capacitances C1-C3 (C4-C6) represent the driver OS-1 (response OS-2). The OPCL coupling in eq.(9) is designed using Op-amps U9-U11 and multipliers U14-U15 and resistances R17-R30. Voltage measurements of the driver and the response variables (analogs of y1,2 and x1,2) are made at the output of Op-amps (U1, U2) and (U5, U6) respectively using a 4-channel digital oscilloscope (Yokogawa, DL9140, 1GHZ, 5GS/s). The components of the driver and the response circuits are chosen for chaotic regime before and after coupling. The coupling circuit components are selected for α1=α2=α3=α , which is controlled by varying the resistance R18. R18 is turned from one to the other end for changing α-values from –2 to 2 to realize AS to CS, and AD at its middle. Oscilloscope pictures in Fig.4 show CS, AS and AD for different α values. Amplification with AS (CS) is also realized for R18=20kΩ (R18=0) while attenuation is achieved for intermediate values of R18 but pictures are not shown.
Figure 4. Oscilloscope pictures. Upper row: 2D driver attractor; plot of output (U1 vs.U2) at left, response attractor (output, U5 vs. U6) at middle for CS (α=1, R18=5kΩ). Output of U1 vs.U5 confirms CS at right. Lower row: left panel confirms AD (R18=10kΩ) by the steady state response at origin. Response attractor (outputs of U5 vs. U6) in AS (α=-1, R18=15kΩ) at middle and confirmed by the outputs of U1 vs.U5).
V.
ACKNOWLEDGEMENTS I.G., P.K.R. and S.K.D. acknowledge support by the Ministry of Education and Research, Romania and Ministry of Science & Technology, DST, India. R.B. is supported by the CSIR, India as Research Fellow. REFERENCES [1] A.Pikovsky, M.Roseblum, J.Kurths, Synchronization: A Concept in Nonlinear Sciences, CUP, New York, 2001. [2] S.K.Dana, P.K.Roy and J.Kurths (Eds.), Complex dynamics in physiological systems: From heart to brain, Springer, 2009. [3] A.Argyris, D. Syvridis, L. Larger, V.Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C.R. Mirasso, L. Pesquera, K. A.Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links” Nature 438, 343-346 (2005). [4] István Z. Kiss, Craig G. Rusin, Hiroshi Kori, John L. Hudson, “Engineering dynamical structures: sequential patterns and desynchronization”, Science 316 (5833), 1886 (2007). [5] W.L.Kath, J.M.Ottino, “Rhythm engineering, Science”, 316, 1857 (2007). [6] L.Pecora, T.Carroll, “Synchronizing chaotic circuits”, Phy.Rev.lett. 64, 821 (1990). [7] L-Y.Cao, Y-C.Lai, “Antiphase synchronization of chaotic systems”, Phys.Rev.E 58, 382 (1998). [8] D.V.R.Reddy, A.Sen and G.L.Johnston, “Time delay induced death in coupled limit cycle oscillators”, Phy.Rev.Lett. 80, 5109 (1998). [9] A.Prasad, J.Kurths, S.K.Dana, R.Ramaswamy, “Phase-flip bifurcation in delay coupled system”, Phy.Rev.E 74, 03502R (2006). [10] I.Grosu, “Robust synchronization”, Phy.Rev.E 56, 3709 (1997). [11] I.Grosu, E.Padmanaban, P.K.Roy and S.K.Dana, “Designing coupling for synchronization and amplification of chaos” , Phy.Rev.Lett. 100, 0234102 (2008). [12] I.Grosu, R.Banerjee, P.K.Roy, S.K.Dana, “Design of coupling for synchronization in chaotic oscillators”, Phy.Rev.E (accepted). [13] C.Li, Q.Chen, T.Huang, “Coexistence of antiphase and complete synchronization in coupled Chen system via single variable”, Chaos Solitons & Fractals, 38, pp. 461-464 (2008). [14] C.Li, W.Sun, J.Kurths, “Synchronization in two complex networks”, Phy.Rev.E 76, 046204 (2007). [15] R.Banerjee, I.Grosu, S.K.Dana, “Antisynchronization in two complex dynamical networks”, Lectures notes inComputer Science, Springer-Verlag (in Press).
SUMMARY
A generalized unidirectional OPCL coupling is proposed to introduce mixed synchronization: one can target a mixed response of AS, CS and amplitude death simultaneously in different state variables of a response oscillator. The OPCL coupling takes care of instabilities due to parameter mismatch. The route to synchrony with parameter mismatch obeys a scaling law. The OPCL coupling is physically realized. Circuit implementation of mixed synchronization is checked in PSPICE circuit simulator, real experiment is under progress.
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