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Synchronization in nonlinear oscillators with conjugate coupling
Chaos, Solitons & Fractals 71 (2015) 1–6
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Synchronization in nonlinear oscillators with conjugate coupling Wenchen Han a, Mei Zhang b, Junzhong Yang a,⇑ a b
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 21 July 2014 Accepted 15 November 2014
a b s t r a c t In this work, we investigate the synchronization in oscillators with conjugate coupling in which oscillators interact via dissimilar variables. The synchronous dynamics and its stability are investigated theoretically and numerically. We find that the synchronous dynamics and its stability are dependent on both coupling scheme and the coupling constant. We also find that the synchronization may be independent of the number of oscillators. Numerical demonstrations with Lorenz oscillators are provided. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The study of synchronization in coupled periodic oscillators has been active since the early days of physics [1,2]. Chaos implies sensitive dependence on initial conditions, with nearby trajectories diverging exponentially, and the synchronization among chaotic oscillators has become a topic of great interest since 1990 [3,4]. The general theories on complete synchronization in which the distance between states of interacting identical chaotic units approaches zero for t ! 1 have been well framed [5–7]. In these theories, chaotic oscillators interact with each other through the same (nonconjugate) variables of different oscillators. However, coupling via dissimilar (conjugate) variables is also natural in real situations [8,9]. One example is the coupled-semiconductor-laser experiments by Kim and Roy [10], where the photon intensity fluctuation from one laser is used to modulate the injection current of the other, and vice versa. In the nonconjugate coupling case, the interaction vanishes with the buildup of complete synchronization and the synchronous state is a solution of isolated system. In contrast, the interaction ⇑ Corresponding author. E-mail addresses: [email protected] (M. Zhang), jzyang@bupt. edu.cn (J. Yang). http://dx.doi.org/10.1016/j.chaos.2014.11.013 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.
in the conjugate coupling case may stay nonzero even when oscillators are synchronized. The conjugate coupling has been used to realize the amplitude death [11,12] in coupled identical units, the phenomenon in which unstable equilibrium in isolated unit becomes stable with the assistance of coupling, in several recent works [13–15]. Interestingly, the realized amplitude death in those works is indeed one special type of synchronization. Then questions arise: Can synchronization in chaotic oscillators with conjugate coupling be realized? What is the synchronous state in chaotic oscillators with conjugate coupling and how about its stability? The main goal in this work is to theoretically investigate the synchronous dynamics and its stability in a ring of identical chaotic oscillators with conjugate coupling by following the methods in Refs. [5,7]. The statements are demonstrated through numerical simulations with the Lorenz oscillators. We also show that the statements are valid for regular random networks in which each oscillator has the same number of neighbors. 2. Analysis The model we consider takes the general form
x_ i ¼ fðxi Þ þ ðD2 xiþ1 D1 xi Þ þ ðD2 xi1 D1 xi Þ;
ð1Þ
2
W. Han et al. / Chaos, Solitons & Fractals 71 (2015) 1–6
Fig. 1. (a) The bifurcation diagram of one oscillator in a pair of coupled Lorenz oscillators. (b) The bifurcation diagram of the synchronous motion which follows Eq. (2) but with 2 replaced by . (c) The synchronization error D is plotted against the coupling constant, which shows that the synchronization ð2Þ ð2Þ error depends on in a non-monotonic way. (d) The first two largest Lyapunov exponents of the synchronous motion (K1 in red and K2 in green) and the ð1Þ largest Lyapunov exponent K1 of the transversal mode (in blue) are plotted against . r ¼ 10; r ¼ 28, and b ¼ 1. The matrices D1 and D2 are presented in the text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
constant matrices describing coupling schemes. When D1 ¼ D2 , the interaction terms become D1 ðxiþ1 þ xi1 2xi Þ and the ordinary non-conjugate coupled oscillators are recovered in which oscillators interact with each other through the same variables. Now we are interested in the synchronous state; the state resides on a synchronous manifold defined by M ¼ fðx1 ; . . . ; xN Þ : xi ¼ sðtÞg where sðtÞ obeys the equation of motion
s_ ¼ fðsÞ þ 2ðD2 D1 Þs:
ð2Þ
To be noted that the synchronous state is not the solution of the isolated oscillator any more and its dynamics depends on both the coupling constant and the matrices D1 and D2 . Introducing perturbation n ¼ n1 ; . . . ; nN to the synchronous state, we linearize equation (1) and have
Fig. 2. The dynamics of a pair of Lorenz oscillators with D1 and D2 presented in the text. (a) The synchronization error D is plotted against the coupling constant. (b) The first two largest Lyapunov exponents of the ð2Þ ð2Þ synchronous motion (K1 in red and K2 in green) and the largest ð1Þ Lyapunov exponent K1 of the transversal mode (in blue) are plotted against the coupling constant. r ¼ 10; r ¼ 28, and b ¼ 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where xi 2 Rn ði ¼ 1; 2; . . . ; NÞ, f : Rn ! Rn is nonlinear and capable of exhibiting rich dynamics such as chaos. The periodic boundary conditions are imposed on Eq. (1). The parameter is a scalar coupling constant. D1 and D2 are
d n ¼ I ðDfðsÞ 2D1 Þn þ C D2 n: dt
ð3Þ
DfðsÞ is the Jacobian matrix of f at s and I is the N N unit matrix. The coupling matrix C is an N N matrix with zero elements except that ci;iþ1 ¼ ci1;i ¼ 1, which describes the interaction among oscillators. The eigenvalues and eigenvectors of C satisfy C/i ¼ ki /i . By expanding n over the eigenvectors of C, we have n ¼ RNi¼1 gi /i where gi are time-dependent coefficients. Substituting the expansion into Eq. (3) and equating the coefficient for each /i , we have
g_ i ¼ ½DfðsÞ 2D1 þ ki D2 gi ; i ¼ 1; 2; . . . ; N;
ð4Þ
3
W. Han et al. / Chaos, Solitons & Fractals 71 (2015) 1–6
ð2Þ
ð2Þ
Fig. 3. The dynamics of a pair of Lorenz oscillators. The Lyapunov exponents of the synchronous motion (K1 in red and K2 in green) and the largest ð1Þ Lyapunov exponent K1 of the transversal mode (in blue) are plotted against the coupling constant. The matrices D1 and D2 in plots (a) and (b) are presented in the text. r ¼ 10; r ¼ 28, and b ¼ 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where ki ¼ 2 cos 2Npi is the eigenvalue of C. It can be demonstrated that the synchronous manifold coincides with the subspace spanned by the eigenvector of C with eigenvalue kN ¼ 2. The Lyapunov exponents given by the linear equation with kN ¼ 2 determine the dynamics of the synchronous state while the modes characterized by all other eigenvalues govern the dynamics transversal to the synchronous manifold. Assuming Lyapunov exponents ðiÞ ðiÞ KðiÞ 1 P K2 P P Kn for the mode with ki , we know that ðiÞ the stability of the synchronous state requires K1 < 0 for all i ranging from 1 to N 1. 3. Numerical simulations 3.1. The synchronization in a pair of oscillators To make above analysis clear, we take into consideration the Lorenz oscillator. We begin with a pair of oscillators coupled together which is a special case of a ring. For a pair of oscillators where N ¼ 2, there is only one coupling term in Eq. (1), therefore the motion equation of the synchronous state Eq. (2) and the linear stability equation Eq. (4) should be modified by replacing the factor 2 with . Correspondingly, the eigenvalues are changed to be ki ¼ 1 ði ¼ 1; 2Þ in which the mode with k2 ¼ 1 accounting for the synchronous motion. We consider two identical Lorenz oscillators coupled 0 1 0 0 0 @ and conjugately in which D1 ¼ 0 0 0 A 1 0 0 0 1 0 0 0 @ D2 ¼ 0 0 0 A. The motion of a single Lorenz oscillator 0 0 1 follows
x_ ¼ rðy xÞ; y_ ¼ rx y xz; z_ ¼ xy bz:
ð5Þ
Eq. (5) has a chaotic attractor for r ¼ 10; r ¼ 28, and b ¼ 1. Fig. 1(a) and (b) shows the bifurcation diagrams for coupled oscillators and for the synchronous dynamics against the coupling constant, respectively. With the variance of the coupling strength, a rich dynamics is found. Especially, the synchronous dynamics displays various periodic windows in which regular motions transit to chaotic ones. The resemblance between the diagrams for coupled oscillators and for the synchronous dynamics indicates that there is no other stable attractor other than the synchronous state even if the synchronous dynamics is unstable. Fig. 1(c) shows the synchronization error qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðx1 x2 Þ þ ðy1 y2 Þ þ ðz1 z2 Þ in which hi D¼ means the average over a long time interval after transient. D is a measure on desynchronization and the two oscillators get synchronized when D ¼ 0. The dependence of D on in Fig. 1(c) reveals an interesting feature that there exists one regime of in which the frequent alternation of synchronization and desynchronization is not quite common. Fig. 1(d) shows the first two Lyapunov exponents
Kð2Þ of the synchronous dynamics and the largest 1;2 ð1Þ
Lyapunov exponent K1 of the transversal mode against the coupling strength . The first two Lyapunov exponents of the synchronous dynamics show plenty of periodic windows in which the period-doubling bifurcation sequence ð1Þ
can be found. K1 stays at zero for > 0:41 and fluctuates around zero in the range of 2 ð0:26; 0:41Þ. As analyzed above, the largest Lyapunov exponent of the transversal model being negative indicates the synchronization of oscillators. The stability regimes of the synchronous ð1Þ
dynamics indicated by negative K1 are in agreement with those shown by D in Fig. 1(c). To be addressed, the synchronization between two oscillators does not rely on whether the synchronous dynamics is periodic or chaotic as shown in Fig. 1.
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Fig. 4. The dynamics of a ring of Lorenz oscillators with r ¼ 10; r ¼ 28, and b ¼ 1. The dependence of the largest Lyapunov exponent K1 on the coupling constant and the parameter k for different coupling schemes (see the text). The black curves in the plots denote the set of k and at which K1 ¼ 0. The inset in (a) shows the synchronization error against the coupling constant for N ¼ 7. The orange horizontal lines in (a) denote all eigenvalues except for k7 ¼ 2 for N ¼ 7. The orange lines and the white line in (b) denote the eigenvalues except for the one characterizing the synchronous mode for N ¼ 4 and N ¼ 3, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The second example is the pair of Lorenz oscillators 0 1 0 1 0 1 0 1 0 0 @ A @ with D1 ¼ 0 0 0 and D2 ¼ 0 0 0 A. The transi0 0 0 0 0 0 tion to the synchronization at ¼ 2:5 can be found either ð1Þ
from D in Fig. 2(a) or from the largest K1 of the transversal mode in Fig. 2(b). Fig. 2(b) shows that the stable synchronous dynamics could be chaotic or time-independent. It can be found that there are two stable time-independent synchronous solutions which satisfy x ¼ y – 0 and are symmetrical under transformation of ðx; y; zÞ ! ðx; y; zÞ. The properties of the stable time-independent synchronous solutions indicate that they are actually the pair of unstable fixed points in the isolated Lorenz oscillator. In terminology, amplitude death refers to the phenomenon in which individual oscillators cease to oscillate when coupled and settle down to their unstable equilibria. Previous studies have shown that systems with different chaotic oscillators non-conjugately coupled can give rise to time-independent solution either. However, these realized solutions do not set systems onto unstable equilibria of their own and such a phenomenon is always named as phase death [16–18] or quenched death. It is an interesting finding that using conjugate coupling may generate amplitude death in coupled chaotic oscillators in its original sense. The state of amplitude death in Fig. 2 loses its stability with the decrease of the coupling constant through a subcritical Hopf bifurcation, which gives rise to a synchronous chaotic dynamics.
There is only one nonzero element in the matrices D1 and D2 in the above two examples. However, the coupling scheme in Eq. (1) may be more complicated than them. For 0 1 0 1 0 1 0 1 0 0 example, D1 ¼ @ 0 0 0 A and D2 ¼ @ 0 0 0 A in 1 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 @ A @ Fig. 3(a), and D1 ¼ 1 0 0 and D2 ¼ 0 1 0 A in 1 0 0 0 1 0 Fig. 3(b). In these two examples, there are two and three non-zero elements in the matrices D1 and D2 , respectively. In each case, we find that coupling strength may tune the system from desynchronization to synchronization. The synchronization in these two situations can be realized no matter whether the synchronous dynamics is chaotic or periodic. Different from Figs. 1, 3 shows only one parameter regime for synchronization. It is worth mentioning that oscillators with conjugate coupling may provide plenty of coupling schemes. For example, in coupled Lorenz oscillators, there are eighteen coupling schemes even if there is only one non-zero elements in D1 and D2 . Large number of coupling schemes may give rise to rich synchronous dynamics and de-synchronous dynamics, which may render the oscillators with conjugate coupling a platform for investigating exotic nonlinear dynamics and pattern formation. 3.2. A ring of oscillators with N > 2 Now we consider a ring of Lorenz oscillators in which N > 2. In a ring structure, the eigenvalues of the coupling
W. Han et al. / Chaos, Solitons & Fractals 71 (2015) 1–6
5
Fig. 5. The dynamics of Lorenz oscillators with conjugate coupling on different random networks. r ¼ 10, r ¼ 28, and b ¼ 1. The top panel shows the bifurcation diagram of oscillators and the bottom shows the synchronization error. The inset in the plots (A)–(C) in the bottom panel shows the underlying network where each oscillator has three neighbors. N ¼ 8; D1 and D2 are the same as those in Fig. 1.
matrix C take the form of 2 cos 2iNp and distribute between 2 and 2. The larger N is, the denser distribution of the eigenvalues of C is. The stability of the synchronous dynamics can be treated in a two-step procedure. In the first step, the largest transversal Lyapunov exponent for each k is calculated using Eqs. (4) and (2). Then the region in the plane of k, in which the largest transversal Lyapunov exponent is positive, is obtained. In the second step, the eigenvalues of the matrix C are calculated. If there is any eigenvalue except for the one accounting for synchronous dynamics falling onto the region, the synchronous dynamics will be unstable. Otherwise, the synchronous dynamics is stable. Following the procedure, the stability of the synchronous dynamics for a ring of oscillators with the size N can be determined. A specific example is given in Fig. 4(a) where the dependence of K1 on k and is presented with 0 1 0 1 0 0 0 0 0 0 D1 ¼ @ 0 0 0 A and D2 ¼ @ 0 0 0 A. The black curve 1 0 0 0 0 1 in the plot denotes the set of k and at which K1 ¼ 0. Then we consider N ¼ 7. The seven eigenvalues are represented by the orange lines in Fig. 4(a). To be noted, the eigenvalue p 2 cos 2iNp is the same as 2 cos 2ðNiÞ and there are only four N orange lines besides kN ¼ 2 in the plot. According to the condition that stable synchronous dynamics requires K1 < 0 for all transversal modes (or in another word, at any , the seven eigenvalues except for k ¼ 2 should fall into the region with K1 < 0), the figure shows that there exists a large range of for stable synchronous dynamics. As a comparison, we plot the synchronization error against for N ¼ 7 in the inset, which shows disconnected regimes of for stable synchronous dynamics and is in agreement with the analysis based on the dependence of K1 on k
and . In the case of oscillators with nonconjugate coupling, there is a size instability for synchronous dynamics: At given oscillator parameters and coupling constant, the synchronous dynamics can only be realized when the number of oscillators is below a threshold [5]. However, an extraordinary feature in Fig. 4(a) is that the stability of the synchronous dynamics may be independent of the number of oscillators in certain ranges of . For example, > 0:25 in which K1 < 0 provided that k – 2. Consequently, the size instability for the synchronization among 0 1 oscillators is absent. 0 1 0 Then, we consider the case with D1 ¼ @ 0 0 0 A and 0 1 0 0 0 1 0 0 D2 ¼ @ 0 0 0 A, which are the same as those in Fig. 2. 0 0 0 The results in Fig. 4(b) shows that the stability of the synchronous dynamics becomes independent of the number of oscillators when > 12. On the other hand, at < 12, the synchronous dynamics can only be realized for small number of oscillators, for example N < 4. For N ¼ 3, Fig. 4(b) also shows that stability of the synchronous motion is non-monotonically dependent on since the eigenvalues k1 ¼ 2 cos 23p and k2 ¼ 2 cos 43p cross the black curves denoting K1 ¼ 0 several times. Furthermore, we 0 1 0 1 0 @ and consider the case with D1 ¼ 0 0 0 A 0 0 0 0 1 0 0 1 D2 ¼ @ 0 0 0 A. The synchronous dynamics realized in 0 0 0 this case is a quenched state which is stationary but not the solution of isolated oscillator. Fig. 4(c) and (d) shows the results for increasing and decreasing , respectively.
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W. Han et al. / Chaos, Solitons & Fractals 71 (2015) 1–6
The two figures show that oscillators conjugate coupled in this way exhibit strong hysteresis and there exists a range of in which the synchronous dynamics coexists with desynchronous dynamics. Additionally,the synchronous dynamics in these two figures is independent of the number of oscillators once it is realized.
the size instability exists. We also show that the theoretical analysis in this work is applicable to regular random networks. However, the synchronization among oscillators sitting on an arbitrary complex network with conjugate coupling pose a question which worths further investigation.
3.3. Regular random networks
References
The theoretical frame proposed here is not limited to ring structures and it may be applied to regular random networks in which each oscillator has the same number of neighbors. The synchronous dynamics in a regular random network with degree d follows
[1] Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D. Complex networks: structure and dynamics. Phys Rep 2006;424(175): 175–308. [2] Arenas A, Daz-Guilera A, Kurths J, Moreno Y, Zhou C. Synchronization in complex networks. Phys Rep 2008;469(93):93–153. [3] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64(821):888–96. [4] Pikovsky AS. On the interaction of strange attractors. Z Phys B Condens Matter 1984;55(149):149–54. [5] Yang J, Hu G, Xiao J. Chaos synchronization in coupled chaotic oscillators with multiple positive Lyapunov exponents. Phys Rev Lett 1998;80(3):496–9. [6] Hu G, Yang J, Liu W. Instability and controllability of linearly coupled oscillators: eigenvalue analysis. Phys Rev E 1998;58(4): 4440–53. [7] Pecora LM, Carroll TL. Master stability functions for synchronized coupled systems. Phys Rev Lett 1998;80(10):2019–112. [8] Goleniewski G. Modelling cultivar mixtures using SEIR compartmental models. Biom J 1996;38(3):281–97. [9] Ueta T, Kawakami H. Chaos in cross-coupled BVP oscillators. In: Proceedings – IEEE international symposium on circuits and systems, vol. 3, no. III-68; 2003. pp. III-68–71. [10] Kim MY, Roy R, Aron JL, Carr TW, Schwartz IB. Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment. Phys Rev Lett 2005;94(088101):1–4. [11] Bar-Eli K, bar-eli K. Physica D 1985;14(2):242–52. [12] Reddy DVR, Sen A, Johnston GL. Time delay induced death in coupled limit cycle oscillators. Phys Rev Lett 1998;80(23):5109–12. [13] Karnatak R, Ramaswamy R, Prasad A. Amplitude death in the absence of time delays in identical coupled oscillators. Phys Rev E 2007;76(035201(R)):1–4. [14] Singla T, Pawar N, Parmananda P. Exploring the dynamics of conjugate coupled Chua circuits: simulations and experiments. Phys Rev E 2011;83(026210):1–8. [15] Sharma A, Shrimali MD, Prasad A, Ramaswamy R, Feudel U. Phaseflip transition in relay-coupled nonlinear oscillators. Phys Rev E 2011;84(016226):1–5. [16] Bar-Eli K, Reuveni S. Stable stationary states of coupled chemical oscillators. Experimental evidence. J Phys Chem 1983;89(8): 1329–30. [17] Zeyer KP, Mangold M, Gilles ED. Experimentally coupled thermokinetic oscillators? Phase death and rhythmogenesis. J Phys Chem 2001;105(30):7216–24. [18] Zhu Y, Qian X, Yang J. A study of phase death states in a coupled system with stable equilibria. Europhys Lett 2008; 82(40001):1–5.
s_ ¼ fðsÞ þ dðD2 D1 Þs:
ð6Þ
The stability of the synchronous dynamics on a regular random network can be determined in the method presented Fig. 4. To be noted, the range of the eigenvalues k of the coupling matrix depends upon the degree d on a regular random network. We consider the system consisting of eight Lorenz oscillators with the same D1 and D2 as those in Fig. 1. We presented the bifurcation diagrams and the synchronization errors for the oscillators on three regular random networks in Fig. 5. As shown by the top panel in Fig. 5, the synchronous dynamics in these three cases is the same regardless of the different network topologies, which is in agreement with the above analysis. The discrepancies on the range of for stable synchronous dynamics in these three cases result from the different eigenvalue spectrums of the coupling matrices. 4. Conclusion In discussion, we have considered the synchronization among oscillators with conjugate coupling in which oscillators interact through the coupling of dissimilar variables. We proposed a general theoretical frame work for the synchronous dynamics and its stability. We found that the synchronous dynamics and its stability are dependent on both coupling scheme and the coupling constant. We found that the stability of synchronous dynamics may be independent of the number of oscillators, which is different from non-conjugate coupled oscillators where
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Synchronization in nonlinear oscillators with conjugate coupling Wenchen Han a, Mei Zhang b, Junzhong Yang a,⇑ a b
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 21 July 2014 Accepted 15 November 2014
a b s t r a c t In this work, we investigate the synchronization in oscillators with conjugate coupling in which oscillators interact via dissimilar variables. The synchronous dynamics and its stability are investigated theoretically and numerically. We find that the synchronous dynamics and its stability are dependent on both coupling scheme and the coupling constant. We also find that the synchronization may be independent of the number of oscillators. Numerical demonstrations with Lorenz oscillators are provided. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The study of synchronization in coupled periodic oscillators has been active since the early days of physics [1,2]. Chaos implies sensitive dependence on initial conditions, with nearby trajectories diverging exponentially, and the synchronization among chaotic oscillators has become a topic of great interest since 1990 [3,4]. The general theories on complete synchronization in which the distance between states of interacting identical chaotic units approaches zero for t ! 1 have been well framed [5–7]. In these theories, chaotic oscillators interact with each other through the same (nonconjugate) variables of different oscillators. However, coupling via dissimilar (conjugate) variables is also natural in real situations [8,9]. One example is the coupled-semiconductor-laser experiments by Kim and Roy [10], where the photon intensity fluctuation from one laser is used to modulate the injection current of the other, and vice versa. In the nonconjugate coupling case, the interaction vanishes with the buildup of complete synchronization and the synchronous state is a solution of isolated system. In contrast, the interaction ⇑ Corresponding author. E-mail addresses: [email protected] (M. Zhang), jzyang@bupt. edu.cn (J. Yang). http://dx.doi.org/10.1016/j.chaos.2014.11.013 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.
in the conjugate coupling case may stay nonzero even when oscillators are synchronized. The conjugate coupling has been used to realize the amplitude death [11,12] in coupled identical units, the phenomenon in which unstable equilibrium in isolated unit becomes stable with the assistance of coupling, in several recent works [13–15]. Interestingly, the realized amplitude death in those works is indeed one special type of synchronization. Then questions arise: Can synchronization in chaotic oscillators with conjugate coupling be realized? What is the synchronous state in chaotic oscillators with conjugate coupling and how about its stability? The main goal in this work is to theoretically investigate the synchronous dynamics and its stability in a ring of identical chaotic oscillators with conjugate coupling by following the methods in Refs. [5,7]. The statements are demonstrated through numerical simulations with the Lorenz oscillators. We also show that the statements are valid for regular random networks in which each oscillator has the same number of neighbors. 2. Analysis The model we consider takes the general form
x_ i ¼ fðxi Þ þ ðD2 xiþ1 D1 xi Þ þ ðD2 xi1 D1 xi Þ;
ð1Þ
2
W. Han et al. / Chaos, Solitons & Fractals 71 (2015) 1–6
Fig. 1. (a) The bifurcation diagram of one oscillator in a pair of coupled Lorenz oscillators. (b) The bifurcation diagram of the synchronous motion which follows Eq. (2) but with 2 replaced by . (c) The synchronization error D is plotted against the coupling constant, which shows that the synchronization ð2Þ ð2Þ error depends on in a non-monotonic way. (d) The first two largest Lyapunov exponents of the synchronous motion (K1 in red and K2 in green) and the ð1Þ largest Lyapunov exponent K1 of the transversal mode (in blue) are plotted against . r ¼ 10; r ¼ 28, and b ¼ 1. The matrices D1 and D2 are presented in the text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
constant matrices describing coupling schemes. When D1 ¼ D2 , the interaction terms become D1 ðxiþ1 þ xi1 2xi Þ and the ordinary non-conjugate coupled oscillators are recovered in which oscillators interact with each other through the same variables. Now we are interested in the synchronous state; the state resides on a synchronous manifold defined by M ¼ fðx1 ; . . . ; xN Þ : xi ¼ sðtÞg where sðtÞ obeys the equation of motion
s_ ¼ fðsÞ þ 2ðD2 D1 Þs:
ð2Þ
To be noted that the synchronous state is not the solution of the isolated oscillator any more and its dynamics depends on both the coupling constant and the matrices D1 and D2 . Introducing perturbation n ¼ n1 ; . . . ; nN to the synchronous state, we linearize equation (1) and have
Fig. 2. The dynamics of a pair of Lorenz oscillators with D1 and D2 presented in the text. (a) The synchronization error D is plotted against the coupling constant. (b) The first two largest Lyapunov exponents of the ð2Þ ð2Þ synchronous motion (K1 in red and K2 in green) and the largest ð1Þ Lyapunov exponent K1 of the transversal mode (in blue) are plotted against the coupling constant. r ¼ 10; r ¼ 28, and b ¼ 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where xi 2 Rn ði ¼ 1; 2; . . . ; NÞ, f : Rn ! Rn is nonlinear and capable of exhibiting rich dynamics such as chaos. The periodic boundary conditions are imposed on Eq. (1). The parameter is a scalar coupling constant. D1 and D2 are
d n ¼ I ðDfðsÞ 2D1 Þn þ C D2 n: dt
ð3Þ
DfðsÞ is the Jacobian matrix of f at s and I is the N N unit matrix. The coupling matrix C is an N N matrix with zero elements except that ci;iþ1 ¼ ci1;i ¼ 1, which describes the interaction among oscillators. The eigenvalues and eigenvectors of C satisfy C/i ¼ ki /i . By expanding n over the eigenvectors of C, we have n ¼ RNi¼1 gi /i where gi are time-dependent coefficients. Substituting the expansion into Eq. (3) and equating the coefficient for each /i , we have
g_ i ¼ ½DfðsÞ 2D1 þ ki D2 gi ; i ¼ 1; 2; . . . ; N;
ð4Þ
3
W. Han et al. / Chaos, Solitons & Fractals 71 (2015) 1–6
ð2Þ
ð2Þ
Fig. 3. The dynamics of a pair of Lorenz oscillators. The Lyapunov exponents of the synchronous motion (K1 in red and K2 in green) and the largest ð1Þ Lyapunov exponent K1 of the transversal mode (in blue) are plotted against the coupling constant. The matrices D1 and D2 in plots (a) and (b) are presented in the text. r ¼ 10; r ¼ 28, and b ¼ 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where ki ¼ 2 cos 2Npi is the eigenvalue of C. It can be demonstrated that the synchronous manifold coincides with the subspace spanned by the eigenvector of C with eigenvalue kN ¼ 2. The Lyapunov exponents given by the linear equation with kN ¼ 2 determine the dynamics of the synchronous state while the modes characterized by all other eigenvalues govern the dynamics transversal to the synchronous manifold. Assuming Lyapunov exponents ðiÞ ðiÞ KðiÞ 1 P K2 P P Kn for the mode with ki , we know that ðiÞ the stability of the synchronous state requires K1 < 0 for all i ranging from 1 to N 1. 3. Numerical simulations 3.1. The synchronization in a pair of oscillators To make above analysis clear, we take into consideration the Lorenz oscillator. We begin with a pair of oscillators coupled together which is a special case of a ring. For a pair of oscillators where N ¼ 2, there is only one coupling term in Eq. (1), therefore the motion equation of the synchronous state Eq. (2) and the linear stability equation Eq. (4) should be modified by replacing the factor 2 with . Correspondingly, the eigenvalues are changed to be ki ¼ 1 ði ¼ 1; 2Þ in which the mode with k2 ¼ 1 accounting for the synchronous motion. We consider two identical Lorenz oscillators coupled 0 1 0 0 0 @ and conjugately in which D1 ¼ 0 0 0 A 1 0 0 0 1 0 0 0 @ D2 ¼ 0 0 0 A. The motion of a single Lorenz oscillator 0 0 1 follows
x_ ¼ rðy xÞ; y_ ¼ rx y xz; z_ ¼ xy bz:
ð5Þ
Eq. (5) has a chaotic attractor for r ¼ 10; r ¼ 28, and b ¼ 1. Fig. 1(a) and (b) shows the bifurcation diagrams for coupled oscillators and for the synchronous dynamics against the coupling constant, respectively. With the variance of the coupling strength, a rich dynamics is found. Especially, the synchronous dynamics displays various periodic windows in which regular motions transit to chaotic ones. The resemblance between the diagrams for coupled oscillators and for the synchronous dynamics indicates that there is no other stable attractor other than the synchronous state even if the synchronous dynamics is unstable. Fig. 1(c) shows the synchronization error qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðx1 x2 Þ þ ðy1 y2 Þ þ ðz1 z2 Þ in which hi D¼ means the average over a long time interval after transient. D is a measure on desynchronization and the two oscillators get synchronized when D ¼ 0. The dependence of D on in Fig. 1(c) reveals an interesting feature that there exists one regime of in which the frequent alternation of synchronization and desynchronization is not quite common. Fig. 1(d) shows the first two Lyapunov exponents
Kð2Þ of the synchronous dynamics and the largest 1;2 ð1Þ
Lyapunov exponent K1 of the transversal mode against the coupling strength . The first two Lyapunov exponents of the synchronous dynamics show plenty of periodic windows in which the period-doubling bifurcation sequence ð1Þ
can be found. K1 stays at zero for > 0:41 and fluctuates around zero in the range of 2 ð0:26; 0:41Þ. As analyzed above, the largest Lyapunov exponent of the transversal model being negative indicates the synchronization of oscillators. The stability regimes of the synchronous ð1Þ
dynamics indicated by negative K1 are in agreement with those shown by D in Fig. 1(c). To be addressed, the synchronization between two oscillators does not rely on whether the synchronous dynamics is periodic or chaotic as shown in Fig. 1.
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Fig. 4. The dynamics of a ring of Lorenz oscillators with r ¼ 10; r ¼ 28, and b ¼ 1. The dependence of the largest Lyapunov exponent K1 on the coupling constant and the parameter k for different coupling schemes (see the text). The black curves in the plots denote the set of k and at which K1 ¼ 0. The inset in (a) shows the synchronization error against the coupling constant for N ¼ 7. The orange horizontal lines in (a) denote all eigenvalues except for k7 ¼ 2 for N ¼ 7. The orange lines and the white line in (b) denote the eigenvalues except for the one characterizing the synchronous mode for N ¼ 4 and N ¼ 3, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The second example is the pair of Lorenz oscillators 0 1 0 1 0 1 0 1 0 0 @ A @ with D1 ¼ 0 0 0 and D2 ¼ 0 0 0 A. The transi0 0 0 0 0 0 tion to the synchronization at ¼ 2:5 can be found either ð1Þ
from D in Fig. 2(a) or from the largest K1 of the transversal mode in Fig. 2(b). Fig. 2(b) shows that the stable synchronous dynamics could be chaotic or time-independent. It can be found that there are two stable time-independent synchronous solutions which satisfy x ¼ y – 0 and are symmetrical under transformation of ðx; y; zÞ ! ðx; y; zÞ. The properties of the stable time-independent synchronous solutions indicate that they are actually the pair of unstable fixed points in the isolated Lorenz oscillator. In terminology, amplitude death refers to the phenomenon in which individual oscillators cease to oscillate when coupled and settle down to their unstable equilibria. Previous studies have shown that systems with different chaotic oscillators non-conjugately coupled can give rise to time-independent solution either. However, these realized solutions do not set systems onto unstable equilibria of their own and such a phenomenon is always named as phase death [16–18] or quenched death. It is an interesting finding that using conjugate coupling may generate amplitude death in coupled chaotic oscillators in its original sense. The state of amplitude death in Fig. 2 loses its stability with the decrease of the coupling constant through a subcritical Hopf bifurcation, which gives rise to a synchronous chaotic dynamics.
There is only one nonzero element in the matrices D1 and D2 in the above two examples. However, the coupling scheme in Eq. (1) may be more complicated than them. For 0 1 0 1 0 1 0 1 0 0 example, D1 ¼ @ 0 0 0 A and D2 ¼ @ 0 0 0 A in 1 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 @ A @ Fig. 3(a), and D1 ¼ 1 0 0 and D2 ¼ 0 1 0 A in 1 0 0 0 1 0 Fig. 3(b). In these two examples, there are two and three non-zero elements in the matrices D1 and D2 , respectively. In each case, we find that coupling strength may tune the system from desynchronization to synchronization. The synchronization in these two situations can be realized no matter whether the synchronous dynamics is chaotic or periodic. Different from Figs. 1, 3 shows only one parameter regime for synchronization. It is worth mentioning that oscillators with conjugate coupling may provide plenty of coupling schemes. For example, in coupled Lorenz oscillators, there are eighteen coupling schemes even if there is only one non-zero elements in D1 and D2 . Large number of coupling schemes may give rise to rich synchronous dynamics and de-synchronous dynamics, which may render the oscillators with conjugate coupling a platform for investigating exotic nonlinear dynamics and pattern formation. 3.2. A ring of oscillators with N > 2 Now we consider a ring of Lorenz oscillators in which N > 2. In a ring structure, the eigenvalues of the coupling
W. Han et al. / Chaos, Solitons & Fractals 71 (2015) 1–6
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Fig. 5. The dynamics of Lorenz oscillators with conjugate coupling on different random networks. r ¼ 10, r ¼ 28, and b ¼ 1. The top panel shows the bifurcation diagram of oscillators and the bottom shows the synchronization error. The inset in the plots (A)–(C) in the bottom panel shows the underlying network where each oscillator has three neighbors. N ¼ 8; D1 and D2 are the same as those in Fig. 1.
matrix C take the form of 2 cos 2iNp and distribute between 2 and 2. The larger N is, the denser distribution of the eigenvalues of C is. The stability of the synchronous dynamics can be treated in a two-step procedure. In the first step, the largest transversal Lyapunov exponent for each k is calculated using Eqs. (4) and (2). Then the region in the plane of k, in which the largest transversal Lyapunov exponent is positive, is obtained. In the second step, the eigenvalues of the matrix C are calculated. If there is any eigenvalue except for the one accounting for synchronous dynamics falling onto the region, the synchronous dynamics will be unstable. Otherwise, the synchronous dynamics is stable. Following the procedure, the stability of the synchronous dynamics for a ring of oscillators with the size N can be determined. A specific example is given in Fig. 4(a) where the dependence of K1 on k and is presented with 0 1 0 1 0 0 0 0 0 0 D1 ¼ @ 0 0 0 A and D2 ¼ @ 0 0 0 A. The black curve 1 0 0 0 0 1 in the plot denotes the set of k and at which K1 ¼ 0. Then we consider N ¼ 7. The seven eigenvalues are represented by the orange lines in Fig. 4(a). To be noted, the eigenvalue p 2 cos 2iNp is the same as 2 cos 2ðNiÞ and there are only four N orange lines besides kN ¼ 2 in the plot. According to the condition that stable synchronous dynamics requires K1 < 0 for all transversal modes (or in another word, at any , the seven eigenvalues except for k ¼ 2 should fall into the region with K1 < 0), the figure shows that there exists a large range of for stable synchronous dynamics. As a comparison, we plot the synchronization error against for N ¼ 7 in the inset, which shows disconnected regimes of for stable synchronous dynamics and is in agreement with the analysis based on the dependence of K1 on k
and . In the case of oscillators with nonconjugate coupling, there is a size instability for synchronous dynamics: At given oscillator parameters and coupling constant, the synchronous dynamics can only be realized when the number of oscillators is below a threshold [5]. However, an extraordinary feature in Fig. 4(a) is that the stability of the synchronous dynamics may be independent of the number of oscillators in certain ranges of . For example, > 0:25 in which K1 < 0 provided that k – 2. Consequently, the size instability for the synchronization among 0 1 oscillators is absent. 0 1 0 Then, we consider the case with D1 ¼ @ 0 0 0 A and 0 1 0 0 0 1 0 0 D2 ¼ @ 0 0 0 A, which are the same as those in Fig. 2. 0 0 0 The results in Fig. 4(b) shows that the stability of the synchronous dynamics becomes independent of the number of oscillators when > 12. On the other hand, at < 12, the synchronous dynamics can only be realized for small number of oscillators, for example N < 4. For N ¼ 3, Fig. 4(b) also shows that stability of the synchronous motion is non-monotonically dependent on since the eigenvalues k1 ¼ 2 cos 23p and k2 ¼ 2 cos 43p cross the black curves denoting K1 ¼ 0 several times. Furthermore, we 0 1 0 1 0 @ and consider the case with D1 ¼ 0 0 0 A 0 0 0 0 1 0 0 1 D2 ¼ @ 0 0 0 A. The synchronous dynamics realized in 0 0 0 this case is a quenched state which is stationary but not the solution of isolated oscillator. Fig. 4(c) and (d) shows the results for increasing and decreasing , respectively.
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The two figures show that oscillators conjugate coupled in this way exhibit strong hysteresis and there exists a range of in which the synchronous dynamics coexists with desynchronous dynamics. Additionally,the synchronous dynamics in these two figures is independent of the number of oscillators once it is realized.
the size instability exists. We also show that the theoretical analysis in this work is applicable to regular random networks. However, the synchronization among oscillators sitting on an arbitrary complex network with conjugate coupling pose a question which worths further investigation.
3.3. Regular random networks
References
The theoretical frame proposed here is not limited to ring structures and it may be applied to regular random networks in which each oscillator has the same number of neighbors. The synchronous dynamics in a regular random network with degree d follows
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s_ ¼ fðsÞ þ dðD2 D1 Þs:
ð6Þ
The stability of the synchronous dynamics on a regular random network can be determined in the method presented Fig. 4. To be noted, the range of the eigenvalues k of the coupling matrix depends upon the degree d on a regular random network. We consider the system consisting of eight Lorenz oscillators with the same D1 and D2 as those in Fig. 1. We presented the bifurcation diagrams and the synchronization errors for the oscillators on three regular random networks in Fig. 5. As shown by the top panel in Fig. 5, the synchronous dynamics in these three cases is the same regardless of the different network topologies, which is in agreement with the above analysis. The discrepancies on the range of for stable synchronous dynamics in these three cases result from the different eigenvalue spectrums of the coupling matrices. 4. Conclusion In discussion, we have considered the synchronization among oscillators with conjugate coupling in which oscillators interact through the coupling of dissimilar variables. We proposed a general theoretical frame work for the synchronous dynamics and its stability. We found that the synchronous dynamics and its stability are dependent on both coupling scheme and the coupling constant. We found that the stability of synchronous dynamics may be independent of the number of oscillators, which is different from non-conjugate coupled oscillators where