Synchronization in chaotic oscillators by cyclic

Eur. Phys. J. Special Topics 222, 927–937 (2013) © EDP Sciences, Springer-Verlag 2013 DOI: 10.1140/epjst/e2013-01895-x

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Regular Article

Synchronization in chaotic oscillators by cyclic coupling O.I. Olusola1,2,a , A.N. Njah2 , and S.K. Dana1 1 2

CSIR-Indian Institute of Chemical Biology, Kolkata 700032, India Department of Physics, Federal University of Agriculture, Abeokuta, P.M.B 2240, Nigeria Received 22 March 2013 / Received in final form 3 May 2013 Published online 11 July 2013 Abstract. We introduce a type of cyclic coupling to investigate synchronization of chaotic oscillators. We derive analytical solutions of the critical coupling for stable synchronization under the cyclic coupling for the R¨ ossler system and the Lorenz oscillator as paradigmatic illustration. Based on the master stability function (MSF) approach, the analytical results on critical coupling are verified numerically. An enhancing effect in terms of lowering the critical coupling or enlarging the synchronization window in a critical coupling space is noticed. The cyclic coupling is also applied in other models, Hindmarsh-Rose model, Sprott system, Chen system and forced Duffing system to confirm the enhancing effect. The cyclic coupling allows tuning of two coupling constants in reverse directions when an optimal control of synchronization is feasible.

1 Introduction The concept of synchronization [1] is understood as a collective behavior when two or more oscillatory systems, periodic or chaotic, adjust each other in time, giving rise to a common rhythm. The main task in studies of synchronization in chaotic systems is to determine the critical coupling [2, 3] for which a synchronized state is stable [4]. The type of coupling and its topology [5, 6] plays key role in the stability of synchronization. The critical coupling differs for different type of coupling and its topology. The type of coupling most commonly used is unidirectional or bidirectional through one pair (scalar type) or pairs (vector type) of similar state variables of the interacting systems. In contrast, we assume a situation where one oscillator sends a signal to another oscillator via one pair of state variables and receives a feedback from the other through a different pair of state variables and thereby establish a mutual interaction which we define as a cyclic coupling. This is like two individuals, one pulling-by-hand and the other pushing-by-leg and thereby interacting in forward and backward push-pull motion in a cyclic order. Such a cyclic bidirectional interaction may arise in a natural situation such as neuronal interaction in the brain [7]. One neuron sends information to another neuron via one pair of dendrites (neurotransmitter based pre- and post-synaptic unidirectional communication) but may fail to a

e-mail: [email protected]

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The European Physical Journal Special Topics O1

O2

O1

O2

O1

O2

x1

x2

x1

x2

x1

x2

y1

y2

y1

y2

y1

y2

z1

z2

z1

z2

z1

z2

(a)

(b)

(c)

Fig. 1. Schematics of two oscillators (O1 and O2 ) interacting via cyclic coupling with different topologies, (a) x1 → x2 , y1 ← y2 , (b) x1 → x2 , z1 ← z2 and (c) y1 → y2 , z1 ← z2 variables.

synchronize with the other neuron until it receives a feedback signal at a suitable instant of time from the other neuron via another pair of dendrites (in absence of gap junction); it may work for a mutual benefit of the neurons to emerge into synchrony or desynchronization. We realize the cyclic coupling first in paradigm models, R¨ ossler system [8] and Lorenz system [9]. Using linear stability analysis, we analytically determine the critical coupling values for identical R¨ ossler systems and Lorenz systems under different topologies of cyclic coupling. Furthermore, we use the basic tool of master stability function (MSF) [10] to derive numerically the critical coupling for cyclic coupling and diffusive coupling as well for comparison. We also examine the critical coupling of synchronization in a number of other systems, Hindmarsh-Rose (HR) neuron model [11], Chen system [12], Sprott system [13] and forced Duffing oscillator [14] using the the cyclic coupling. Our extensive computational results lend credence to the fact that some of the topologies of the cyclic coupling enhance the critical coupling by lowering its value and, in some other cases, enlarge the critical coupling window of synchronization when compared to the conventional diffusive coupling. Additionally, it allows mutual tuning of two coupling constants in reverse directions to implement a control of synchronization or desynchronization. The organization of the paper is as follows: the cyclic coupling is described in the next section. The linear stability analysis for synchronization of R¨ ossler system and Lorenz system is presented in section 3. The MSF analysis is given in section 4 while section 5 focused on the numerical results. Results are concluded in section 6.

2 Cyclic coupling scheme We describe the cyclic coupling in a schematic diagram in Fig. 1. Consider a pair of coupled dynamical systems, x˙ = f (x) + i GHi,j (x, y) y˙ = f (y) + j GHi,j (x, y)

(1)

where x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ); x ∈ Rn and y ∈ Rn are the dynamical variables, f : Rn → Rn is a continuously differentiable function and i,j , (i, j = 1, 2, . . . , n) are the coupling parameters. G stands for coupling matrix and H is the output function of each oscillator that is engaged in the coupling.

From Solitons and Chaos to Complex Systems – Perspectives and Trends

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As example, for two three dimensional systems, let us denote the state vectors by x = [x1 y1 z1 ]T and y = [x2 y2 z2 ]T . The cyclic coupling now requires that two pairs of variables are engaged in the coupling. There are six possible topologies of cyclic coupling for two 3D systems, out of which three are independent, other three topologies are symmetric for identical oscillators. Considering two pairs of variables, three independent options are: (i) x1 → x2 , y1 ← y2 when H1,2 = diag{1, 1, 0}, (ii) x1 → x2 , z1 ← z2 when H1,3 = diag{1, 0, 1} and (iii) y1 → y2 , z1 ← z2 when H2,3 = diag{0, 1, 1}. Figure 1 illustrates three possible topologies of the cyclic coupling for two oscillators. Arrows indicate the direction of coupling to establish the mutual interaction by the cyclic coupling. Consider two identical R¨ ossler oscillators to further illustrate the cyclic coupling, (i) x1 → x2 , y1 ← y2 (Eq. (2)), (ii) x1 → x2 , z1 ← z2 (Eq. (3)) and (iii) y1 → y2 , z1 ← z2 (Eq. (4)) as follows: x˙ 1 y˙ 1 z˙1 x˙ 2 y˙ 2 z˙2

= = = = = =

−y1 − z1 x1 + ay1 + 2 (y2 − y1 ) b + z1 (x1 − c) −y2 − z2 + 1 (x1 − x2 ) x2 + ay2 b + z2 (x2 − c),

x˙ 1 y˙ 1 z˙1 x˙ 2 y˙ 2 z˙2

= = = = = =

−y1 − z1 x1 + ay1 b + z1 (x1 − c) + 3 (z2 − z1 ) −y2 − z2 + 1 (x1 − x2 ) x2 + ay2 b + z2 (x2 − c),

x˙ 1 y˙ 1 z˙1 x˙ 2 y˙ 2 z˙2

= = = = = =

−y1 − z1 x1 + ay1 b + z1 (x1 − c) + 3 (z2 − z1 ) −y2 − z2 x2 + ay2 + 2 (y1 − y2 ) b + z2 (x2 − c),

(2)

(3)

(4)

where the system parameters are taken a = b = 0.2, c = 5.7.

3 Stability criteria and coupling threshold Let us examine the stability of a synchronized state under the cyclic coupling scheme in two identical systems (1). First, we recast Eq. (1) by separating the linear and nonlinear parts of the system and assume that δ1 and δ2 are the deviation from the synchronous state and their dynamics is governed by linearized variational equation of the form, ˙ δ˙1 = Aδ1 + g(x)δ 1 + i Hi,j (δ2 − δ1 ) (5) ˙δ2 = Aδ2 + g(y)δ ˙ 2 + j Hi,j (δ1 − δ2 ),

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The European Physical Journal Special Topics

where A and g are the linear matrix and the nonlinear part of the dynamical flow f (x) of the systems respectively and Hi,j (i, j = 1, 2, 3) is already defined above. In order to analyze the coupled system (5), we now made an approximation [15] that the time average of g(x) ˙ and g(y) ˙ is a constant ψ, i.e (g(x)+ ˙ g(y))/2 ˙ = ψ. The synchronization error is δ = δ1 − δ2 when the error dynamics is given by, δ˙ = (A + ψI − i Hi − j Hj )δ.

(6)

The error dynamics is stable if M = A + ψI − i Hi − j Hj is less than zero (i, j=1,2). The synchronous state δ1 = δ2 = constant is then stable. Notice that I is an identity matrix and the constant ψ can assume any arbitrary value. Now, we derive the criteria for stability of synchronization in two R¨ ossler systems and two Lorenz systems under the influence of cyclic coupling. First, let us explore the stability of the synchronized dynamics in two identical R¨ ossler oscillators coupled via a pair of variables x1 → x2 and y1 ← y2 in a cyclic order (see Eq. (2)). The matrix M for this configuration is: ⎞ −1 −1 ψ − 1 M = ⎝ 1 ψ + a − 2 0 ⎠ . 0 0 ψ−c ⎛

(7)

The eigenvalues of the M matrix are λ1 = ψ − c λ2,3

−(1 + 2 − 2ψ − a) ± = 2

√

P

,

(8)

where P = (1 + 2 − 2ψ − a)2 − 4[(ψ − 1 )(ψ + a − 2 ) + 1]. From Eq. (8), the stability criteria for complete synchronization (CS) are then derived from the negative real parts of the eigenvalues of the matrix M , i.e. (i) ψ < c, (ii) if (1 + 2 − 2ψ − a)2 < 4[(ψ − 1 )(ψ + a − 2 ) + 1], λ2,3 are complex and the stability condition is 1 + 2 > 2ψ + a, (iii) if (1 + 2 − 2ψ − a)2 > 4[(ψ − 1 )(ψ + a − 2 ) + 1], λ2,3 are real and the stability condition takes the form 1 + 2 > 2ψ + a and (ψ − 1 )(ψ + a − 2 ) + 1 > 0. Thus, the critical coupling for synchronization in two identical R¨ ossler oscillators using a particular topology of the cyclic coupling (defined by Eq. 2) is derived from the following equations, ψ 2ψ + a (ψ − 1 )(ψ + a − 2 ) + 1 > 0.

(9)

With the system parameters as defined earlier and by a choice [15] of ψ = 0.019, the transition to CS occurs at critical coupling constants (1 and 2 ) satisfying the conditions, 1 + 2 > 0.238 and (0.019 − 1 )(0.219 − 2 ) + 1 > 0. The range of critical values vary with the system parameter a of the R¨ ossler oscillator. Next, we analyze the stability of CS in two identical Lorenz systems by using cyclic coupling via the pair of variables x1 → x2 and y1 ← y2 . The governing equations for


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the coupled system, x˙ 1 y˙ 1 z˙1 x˙ 2 y˙ 2 z˙2

= = = = = =

σ(y1 − x1 ) x1 (ρ − z1 ) − y1 + 2 (y2 − y1 ) x1 y1 − βz1 σ(y2 − x2 ) + 1 (x1 − x2 ) x2 (ρ − z2 ) − y2 x2 y2 − βz2 .

(10)

Here, the M matrix is, ⎞ σ 0 ψ − σ − 1 ρ ψ − 2 − 1 0) ⎠ . M =⎝ 0 0 ψ−β ⎛

(11)

The eigenvalues of M matrix are λ1 = ψ − β √ (2ψ − σ − 1 − 2 − 1) ± Q λ2,3 = , 2

(12)

where Q = (2ψ − σ − 1 − 2 − 1)2 − 4[(ψ − σ − 1 )(ψ − 2 − 1) − ρσ]. From Eq. (12), the following stability conditions are obtained, (i) ψ < β, (ii) if (2ψ − σ − 1 − 2 − 1)2 < 4[(ψ − σ − 1 )(ψ − 2 − 1) − ρσ], λ2,3 are complex and the stability condition is 2ψ − σ − 1 − 2 − 1 < 0 (iii) if (2ψ − σ − 1 − 2 − 1)2 > 4[(ψ − σ − 1 )(ψ − 2 − 1) − ρσ], λ2,3 are real and the stability condition becomes (ψ − σ − 1 )(ψ − 2 − 1) − ρσ > 0. Again, ψ is an arbitrary constant and the transition to CS occurs at critical coupling obtained from, ψ 0.

(13)

The definition of the M -matrix can be used, as a general stability criterion, to derive the critical coupling for other cyclic topologies in R¨ ossler system and Lorenz system and, in fact, it can also be used for other continuous dynamical systems but details are not presented. However, numerical values of the critical coupling for all different topologies in many systems are tabulated (Table 1) below using the concept of MSF.

4 Master stability function The dynamics of a network of coupled oscillators is expressed by, x˙ i = F(xi ) −

N j=1

Gi,j H(xj ),

(14)

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Table 1. MSF based critical couplings for diffusive and cyclic coupling in oscillators. Oscillators

x1 ↔ x2

y1 ↔ y2

R¨ ossler

c1 = 0.139, c1 = 0.166 c2 = 3.566

Lorenz

c1 = 1.478 c1 = 0.662

HR Chen

z1 ↔ z2 NS

x1 → x2 , y1 → y2 , x1 → x2 y2 ← y1 z2 ← z1 z2 ← z1 c1 = 0.116 c1 = 0.175, c1 = 0.253, c2 = 7.283 c2 = 7.413

c1 = 2.524 1 = 1.318, c2 = 28.256, c3 = 58.219

c1 = 1.418 c1 = 0.532

c1 = 0.537 c1 = 0.0137 c1 = 0.0358 c1 = 0.0107 c1 = 0.0142 c1 = 0.0276 NS

c1 = 0.563

Sprott

c1 = 0.0965 c1 = 0.0945

Duffing

c1 = 0.145 c1 = 0.260

c1 = 2.324

c1 = 0.637 c1 = 0.775 c1 = 6.712

NS

c1 = 0.088 c1 = 0.176 c1 = 0.173 c1 = 0.183

where x, F(x), , G and H are already defined above. The coupling matrix, G, N satisfies the condition j=1 Gij = 0 (i.e. row sum zero) implying that G has at least one null eigenvalue for any i, where N is the number of oscillators in the network. As a consequence, the synchronization manifold x1 = x2 = · · · = xN = s is invariant when all the real parts of the eigenvalues (αi , i = 1, . . . N ) associated with the transversal modes are negative. The network is connected so there is only one zero eigenvalue such that the eigenvalues can be sorted as 0 = α1 ≤ α2 ≤ · · · ≤ αN . Based on the concept of MSF, the tendency to synchronization of the network is a function of the eigenvalues αk of the coupling matrix G, k = 0, 1, 2, . . . N − 1. Hence, for the system described by Eq. (14), the variational equations governing the time evolution of the infinitesimal transverse vectors φi (t) = xi (t) − s(t) from the synchronous state s(t) are, φ˙ i = DF(s)φi −

N

Gi,j H(s)φi ,

(15)

j=1

where DF(s) and DH(s) are the n × n Jacobian matrices of the corresponding vector functions evaluated at s(t). After block diagonalization of Eq. (15), there appear N −1 separate blocks, (16) φ˙ i = [DF(s) − αk DH(s)]φi . By assuming ηi = αk (k = 2, . . . N ) as a set of normalized coupling parameters, one is left with the generic variational equation of all decoupled blocks as given by, φ˙ i = [DF(s) − ηDH(s)]φi .

(17)

Now, we can define the MSF [10] as the largest transversal Lyapunov exponent, λmax , for a normalized coupling parameter η, and it may be calculated from the Eq. (17). If λmax is negative, a small disturbance from the synchronization manifold will decay exponentially so that the synchronization solution is asymptotically stable. In a network of coupled oscillators defined by the Eq. (14), a necessary condition for synchronization is that all the coupling parameters are located in the interval of η values where λmax is negative. Note that, in case of our proposed cyclic coupling, not all the diagonal elements of the output function H(s) are engaged in the coupling as described above. For instance, in a coupled oscillators (say, R¨ ossler system or Lorenz system), the interaction involves a pair of variables, say, xi → xj , yi ← yj ; or yi → yj , zi ← zj , or xi → xj , zi ← zj . Thus, the matrix DH has two non-zero components only for each choice.


εc1

εc2

εc1

0

λmax

λmax

0

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(a) -1.2 0

5

(d) 10

-9

εc1

5

10 εc2

εc1

λmax

0

λmax

0

0

(b) -1.4 0 0.16

5

10

-1

(e) 0

5 εc2

εc1

λmax

λmax

0

10

(c) 0

0

5 ε

10

-1

(f) 0

5 ε

10

Fig. 2. MSF (λmax ) vs. coupling parameter, (1 = 2 = ), for coupled chaotic chaotic R¨ ossler systems interacting via diffusive coupling in (a) x1 ↔ x2 , (b) y1 ↔ y2 , (c) z1 ↔ z2 , and cyclic coupling in (d) x1 → x2 , y1 ← y2 , (e) y1 → y2 , z1 ← z2 and (f) x1 → x2 , z1 ← z2 .

5 Numerical analysis Using the above theory of MSF [10], we numerically determine the critical coupling c above which a stable synchronous behavior can be observed for both the coupled R¨ ossler system and Lorenz system. In Fig. 2, we display the results of MSF (λmax ) for the conventional diffusive coupling and the cyclic coupling in the left and right columns respectively for R¨ ossler systems (Eqs. (2)–(4)) for a symmetric coupling (1 = 2 = ). It is seen that the λmax crosses over to a negative value at lower critical coupling (c1 ≈ 0.116) in the first case in Fig. 2(d) when compared to Fig. 2(a) (c1 ≈ 0.139) and maintains synchrony for large cyclic coupling in contrast to the diffusive coupling that allows a smaller window of critical coupling for synchronization (also see Table 1). Looking at the Fig. 2(e), λmax has two thresholds, c1 and c2 , however the window is quite large although it is a smaller window compared to what is seen Fig. 2(b). c2 stands for a transition from synchrony to desynchronization (λmax crosses to a positive value) which may be useful for inducing desynchronization as and when require. Furthermore, in Fig. 2(c), MSF result shows that the diffusive coupling does not produce synchronization when two R¨ ossler systems interact via z1 ↔ z2 . However, when the cyclic coupling is activated via (z1 ← z2 , y1 → y2 ), or (z1 ← z2 , x1 → x2 ) variables (see Fig. 2(e) & (f)),

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0.15

0.1

0.1

0.2 0

ε

ε

2

2

0.15

−0.2

NS 0 0

−0.4 ε

0.1

0 0

0.15

ε

1

0.1

0.15

1

Fig. 3. R¨ ossler systems via cyclic coupling on x1 → x2 , y1 ← y2 variables. (a) Transition from nonsynchronous (NS) to synchronous (blue) regime based on the criteria in Eq. (9), (b) MSF (λmax ) plot in the 1 − 2 plane; the CS region (blue) is separated from NS regime (red) by a critical coupling boundary when λmax changes from positive to negative. εc1

-3

(d)

(a) 0

50

100

-12

εc1

0

0

50

100

50

100

50 ε

100

εc1

λmax

λmax

0

-9

(e)

(b) 0

50

100

-18

1.5

0 εc1

0

-1.5

εc3 εc2

εc1

(f)

(c) 0

0

λmax

λmax

εc1

λmax

λmax

0

0

50 ε

100

-7

0

Fig. 4. MSF (λmax ) vs. coupling parameter (1 = 2 = ) for coupled Lorenz oscillators interacting via diffusive coupling on (a) x1 ↔ x2 , (b) y1 ↔ y2 , (c) z1 ↔ z2 , and cyclic coupling, (d) x1 → x2 , y1 ← y2 , (e) y1 → y2 , z1 ← z2 , and (f) x1 → x2 , z1 ← z2 .

the oscillators emerge into CS with a large window of critical coupling. The cyclic coupling is therefore able to achieve CS via some pairs of variables where diffusive coupling does not work in favor of synchrony. The onset of CS is thus enhanced, in general, by this scheme. Additional evidence of synchronization via cyclic coupling is explored by plotting the MSF (λmax ) in the parameter plane (1 -2 ) for asymmetric coupling (1 = 2 ). For


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example, by applying cyclic coupling via x1 → x2 , y1 ← y2 , we display in Fig. 3(b) the MSF (λmax ) plot in the (1 -2 ) plane and showed regions of asynchronous (red) and synchronous behaviour (blue). We also verify the synchronization regime in the (1 2 ) plane of the critical coupling using the analytical criterion in Eq. (9) and it is seen, in Fig. 3(a), that (taking ψ = 0.019) the analytic critical coupling (c =0.119) is in good agreement with the numerical result in Fig. 3(b). A small fluctuation across the interface of synchronization (blue) and non-synchronization (red) is due to numerical errors in MSF estimation. However, we observe a negative slope of the boundary line delineating the synchronous regime from the asynchronous regime in both the cases. This negative slope reveals an interesting feature of the cyclic coupling. It is clear that the critical value of either the coupling strengths (1 or 2 ) has a maximum value when the other is zero (i.e. the systems are in unidirectional coupling mode). The cyclic coupling then opens up an option for tuning control of synchronization: if the critical value of 2 increases, that of 1 decreases and vice versa along the boundary of the synchronization and the nonsynchronous (NS) regimes. A practical advantage is that an optimal control of synchronization is possible by tuning two coupling constants in reverse directions. In natural systems too, such as in neuronal interaction, systems may have adaptability to build new cyclic links and thereby enabling to adjust the strength of coupling either way as and when necessary for inducing faster synchronization or desynchronization. Next we examine the Lorenz system [9], x˙ = σ(y − x) y˙ = x(ρ − z) − y z˙ = xy − βz,

(18)

where σ = 10.0, ρ = 28, and β = 8/3. For three topologies of diffusive coupling, x1 ↔ x2 , y1 ↔ y2 and z1 ↔ z2 , we compute the MSF as shown in Figs. 4(a)-(c) respectively for symmetric coupling. Next, we compute the MSF for three cyclic topologies, (i) x1 → x2 , y1 ← y2 , (ii) y1 → y2 , z1 ← z2 , and (iii) x1 → x2 , z1 ← z2 and display the results in Figs. 4(d)-(f) respectively. We observe that MSF (λmax ) crosses to a negative value at a lower coupling in case of cyclic coupling via (i) x1 → x2 , y1 ← y2 and, (ii) x1 → x2 , z1 ← z2 (see Table 1). For instance, comparing Figs. 4(a) and 4(d), the oscillators evolve into CS state at c1 = 1.478 and c1 = 1.318 respectively for diffusive coupling via x1 ↔ x2 and cyclic coupling via x1 → x2 , y1 ← y2 . In contrast, diffusive coupling via y1 ↔ y2 achieves synchronization at lower threshold when compared to cyclic coupling (cf. Figs. 4(b) and 4(e)). In Fig. 4(c), MSF diagram reveals three critical couplings when two identical Lorenz systems coupled diffusively via z1 ↔ z2 variables. In the interval of coupling strengths ∈ [28.256, 58.219], the oscillator exhibits mixed synchronization (coexisting CS and anti-synchronization). On the other hand, for cyclic coupling as engaged via x1 → x2 , z1 ← z2 variables, the oscillators evolve into CS at a much lower critical coupling (c1 ≈ 0.532) and it continues for larger coupling. For the chaotic Lorenz system, (taking ψ = 2.5), we discussed above the symmetric cases (1 = 2 = ) only. Now, for the asymmetric case, we plot in Figs. 5(a) & 5(b) the analytical values of critical coupling using Eq. 13 and the MSF (λmax ) of two Lorenz systems using one of the topologies (x1 → x2 , y1 ← y2 ) of cyclic coupling. The regions of nonsynchronous (NS) and synchronous behaviors are clearly shown in the 1 − 2 plane. Notice that the qualitative feature of the critical coupling derived from Eq. 13 is similar to the numerical results of coupled Lorenz system. A negative slope of the boundary of critical coupling for synchronization is also evident from the analytical results and the MSF plot.

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1.5

ε2

1

NS

0.5 0

0

1

ε1

2

2.5

Fig. 5. (a) Transition from nonsynchronous (NS) regime to synchronous (blue) regime based on the criteria in Eq. (13), (b) MSF (λmax ) plot in the 1 − 2 plane for coupled chaotic Lorenz systems interacting via cyclic coupling on x1 → x2 , y1 ← y2 variables. The MSF (λmax ) has two regions NS (red) and CS (blue) regimes, the boundary line separating the two regions has a negative slope.

Further, we investigate other chaotic oscillators, namely, HR model [11], Chen system [12], Sprott system [13] and forced Duffing oscillator [14] and confirms the enhancement of synchrony in the coupling parameter space as numerically estimated using the MSF, results are detailed in Table 1.

6 Conclusion We studied synchronization in two identical chaotic oscillators under cyclic coupling. We showed an enhancement in synchrony when the CS is achieved with a reasonably larger window of critical coupling or at a lower critical value compared to the conventional diffusive coupling. We confirm using both analytical and MSF plots that the cyclic coupling can ensure CS in coupled oscillators where some topology of diffusive coupling does not work. Particularly, for the asymmetric cyclic coupling when the coupling constants in reverse directions are different, one can tune two coupling constants to induce a control of synchronization and thereby realize optimal values of critical coupling constant to enhance synchrony/desychrony. This is a major advantage of the cyclic coupling and can be used for practical purposes. Such an adaptable coupling scheme is expected to exist in natural systems. We presented a feasible coupling scheme which is working in favor of synchrony. This flexibility in topology and critical coupling may be used for enhancing synchrony of complex networks in the future. O.I.O. acknowledges the FICCI and the DST India, for the award of CV Raman International Fellowship. He is grateful to the Federal University of Agriculture Abeokuta (FUNAAB), Nigeria for granting the study leave and personally to D. Ghosh, R. Banerjee and C.R. Hens for their constant help and discussions. S.K.D. acknowledges support by the Abdus Salam ICTP, Trieste, Italy for Visiting Scholar Award to Nigeria and the FUNAAB, Nigeria for local hospitality for two visits in 2011 and 2012.

References 1. A. Pikovsky, M. Rosemblum, J. Kurths, Synchronization: A universal concept in nonlinear science (Cambridge University Press, NY, 2001) 2. L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64, 821 (1990)

From Solitons and Chaos to Complex Systems – Perspectives and Trends 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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D.G. Aronson, G.B. Ermentrout, N. Koppel, Physica D 41, 403 (1990) H. Fujisaka, T. Yamada, Prog. Theor. Phys. 69, 32 (1983) C. Letellier, L.A. Aguirre, Phys. Rev. E 82, 016204 (2010) L. Huang, Q. Chen, Y.-C. Lai, L.M. Pecora, Phys. Rev. E. 80, 036204 (2009) E. Kandel, J. Schwartz, T. Jessell, Principles of neural science (McGraw-Hill, USA, 2000) O.E. R¨ ossler, Phys. Lett. A 57, 397 (1976) E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963) L.M. Pecora, T.L. Carrol, Phys. Rev. Lett. 80, 2109 (1998) J.L. Hindmarsh, R.M. Rose, Proc. R. Soc. London, Ser. B 221, 87 (1984) G. Chen, T. Ueta, Int. J. Bifurcation Chaos Appli. Sci. Eng. 9, 1465 (1999) J.C. Sprott, Phys. Rev. E 50, 647R (1994) A. Stefa´ nski, P. Perlikowski, T. kapitaniak, Phys. Rev. E 75, 016210 (2007) V. Resmi, G. Ambika, R.E. Amritkar, Phys. Rev. E 81, 046216 (2010)