Clustering versus non-clustering phase synchronizations Shuai Liu and Meng Zhan Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 013104 (2014); doi: 10.1063/1.4861685 View online: http://dx.doi.org/10.1063/1.4861685 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Hierarchical synchronization in complex networks with heterogeneous degrees Chaos 16, 015104 (2006); 10.1063/1.2150381 Synchronization in large directed networks of coupled phase oscillators Chaos 16, 015107 (2006); 10.1063/1.2148388 Phase Synchronization in Populations of Chaotic Electrochemical Oscillators AIP Conf. Proc. 676, 381 (2003); 10.1063/1.1612269 Stochastic Synchronization: Analogy with Systems Undergoing Phase Transition AIP Conf. Proc. 665, 94 (2003); 10.1063/1.1584879 Predicting Phase Synchronization from Nonsynchronized Chaotic Data AIP Conf. Proc. 622, 184 (2002); 10.1063/1.1487533
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CHAOS 24, 013104 (2014)
Clustering versus non-clustering phase synchronizations Shuai Liu1,2 and Meng Zhan1,a) 1
Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China 2 University of Chinese Academy of Sciences, Beijing 100049, China
(Received 28 May 2013; accepted 27 December 2013; published online 9 January 2014) Clustering phase synchronization (CPS) is a common scenario to the global phase synchronization of coupled dynamical systems. In this work, a novel scenario, the non-clustering phase synchronization (NPS), is reported. It is found that coupled systems do not transit to the global synchronization until a certain sufficiently large coupling is attained, and there is no clustering prior to the global synchronization. To reveal the relationship between CPS and NPS, we further analyze the noise effect on coupled phase oscillators and find that the coupled oscillator system can change from CPS to NPS with the increase of noise intensity or system disorder. These findings are expected to shed light on the mechanism of various intriguing self-organized behaviors in coupled C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4861685] systems. V
Path to phase synchronization (PS) in coupled nonlinear oscillators is one of the essential problems in nonlinear science and it is also closely connected with many applications in realistic systems. Different with the usual path to synchronization via various clustering processes, here we report a novel non-clustering path to phase synchronization, which is, however, expected to generally occur in coupled nonlinear oscillators with either strong systems nonlinearity or environmental disorder.
I. INTRODUCTION
Rich collective behaviors have been found in various nonlinear dynamical systems,1–4 including physical, chemical, and biological ones. Some examples include chemical pattern formation, laser systems, circadian rhythm, and neural networks, just to name a few. Their study has been a subject of great interest in the nonlinear dynamics field. The systematical investigation of coupled nonlinear oscillators is of crucial importance to their understanding and practical applications. PS2–13 is a basic phenomenon in coupled nonlinear systems and has been studied for many years. With the increase of coupling, the system coherence naturally increases. A universal path to the global phase synchronization shows various clustering processes. In this path, any two adjacent oscillators (or adjacent clusters of oscillators) with close frequencies can easily reach synchronization first and form a locally synchronous cluster. Along with the increase of coupling strength further, small clusters can naturally attract more elements and be merged into large clusters. Finally, all elements (and clusters) transit to a globally synchronous state after a critical coupling strength ec arrives. We call this process a clustering phase synchronization (CPS).9–13 a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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This clustering synchronization scenario can be regarded as a classical route to phase synchronization for coupled phase-coherent oscillators. For example, the coupled Kuramoto phase oscillators4,13,14 and coupled R€ossler systems12,15,16 with the nearest neighboring coupling are studied and the results are shown in Figs. 1(a) and 1(b), respectively. In these subfigures, clearly a bifurcation tree exhibits, showing complicated clusterings before the final largest cluster occurs. Without loss of generality, the number of coupled oscillators N ¼ 5 is chosen in this illustration, and it is found that the same feature appears qualitatively for other numbers. Recently, we have studied the collective dynamics of coupled excitable systems under external noisy driving and discovered a novel synchronization path, the non-clustering phase synchronization (NPS).17 In a contrast to the CPS, all elements in the coupled system transit to a synchronous state with an identical frequency simultaneously only after a certain coupling strength is achieved, and local synchronous clusters (or groups) cannot be found. In our study, the wellknown FitzHugh-Nagumo (FHN) excitable system18–21 has been considered. The bifurcation tree of the NPS is demonstrated in Fig. 1(c), which clearly shows a pattern different with that of the CPS in Figs. 1(a) and 1(b). But several questions still need to be answered, such as whether the NPS can also be generally observed in other systems. More importantly, what is the connection between these two different synchronization paths? To answer these interesting questions, we carry out further investigations on a few other models and explore noise effect. II. NPS IN OTHER SYSTEMS
We first consider a few different coupled nonlinear systems. To our great surprise, we find that the seemly unusual NPS route to global synchronization actually can be frequently observed. Therefore, based on this fact, it can be believed that both of them are intrinsic behaviors of coupled
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FIG. 1. Bifurcation trees of average frequencies in six different coupled systems: (a) Kuramoto phase oscillator, (b) R€ ossler, (c) FHN neuron, (d) Lorenz, (e) HR neuron, and (f) LV ecology systems. N ¼ 5. The insets in 1(a) and 1(b) show their detailed difference in transition to a cluster. The insets in 1(d)–1(f) show their detailed non-clustering structures. Clearly (a) and (b) belong to the CPS, whereas (c)–(f) belong to the NPS. For more details, see the text.
nonlinear systems. Figures 1(d)–1(f) depict the synchronization patterns of a few examples, such as the classical chaotic Lorenz oscillator,22 the Hindmarsh-Rose (HR) bursting neuron,23,24 and the Lotka-Volterra (LV) ecological system.25–28 Clearly, all of these systems give rise to NPS. It is interesting to note that different from the path of the CPS, the dynamics of the above systems becomes synchronous only after a certain large coupling strength arrives. There exists a common characteristic in the transition to the global phase synchronization: the value of frequency in the NPS has a long tail, although each of these models has a very distinct nature and its phase definition can be quite different. To show the lack of clusters before the global synchronization, we even magnify several parameter regions in the insets. For example, in Fig. 1(d) for the coupled chaotic Lorenz oscillators, the two seemly parallel lines do not cross at all. In Fig. 1(e) for the coupled HR bursting neurons, it shows the crossover of their average frequencies only at several single parameters, indicative of no clusters. It is notable that a cluster appears for identical frequencies at least for a continuous parameter interval. Again in Fig. 1(f) for the coupled LV ecological systems, the magnification of the small region clearly shows no cross, although it seems that a very small cluster appears there. To make these phenomena clearer, we plot the same bifurcation trees in Fig. 2, but with the data of the frequency difference fromPthe mean frequency: DXi ¼ Xi hXi i, where hXi i ¼ N1 Ni¼1 Xi . Obviously, the CPS with complicated clusterings in Figs. 2(a) and 2(b) show completely different patterns with the NPS in Figs. 2(c)–2(f), where they exhibit a common zero value only after the coupling is large
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FIG. 2. The same as Fig. 1 but for the frequency deviations DXi (DXi ¼ Xi hXi i) instead, showing a clear difference between the patterns of the CPS in (a) and (b) and the NPS in (c)–(f). The cross of two lines at a single coupling cannot be believed as a cluster, as shown in (e).
and do not show any further bifurcations prior to it. For the original phase bifurcation trees in Fig. 1, due to the small difference for each system within the small region, usually it is uneasy to judge without the aid of magnification whether some of systems exhibit identical (or very close) frequencies, i.e., constitute clusters or not. With this new plot, the small difference can be magnified in a comparably larger region, and thus we can clearly determine the occurrence of clusters or not. As a result, we demonstrate that the phenomenon of NPS occurs indeed and is quite distinct with the CPS without any ambiguity. Here, we also like to emphasize that although the NPS, which has been demonstrated for several examples in Figs. 1 and 2, is quite general, it does not mean the opposite part, the CPS, cannot be observed for these systems. We do find that any two oscillators in the coupled systems with slightly initial frequency mismatch can easily form a cluster, if the parameters are properly chosen. Hence, the phenomenon of the NPS is not exclusive. As another example, the case for N ¼ 20 is shown in Fig. 3, exhibiting the similar patterns as those in Fig. 1. Again, its frequency differences from the mean frequency are shown in Fig. 4. The difference between the two types of phase synchronization is clear. In particular, with the plots of frequency difference, different with the tree structure exhibiting clusters (branches) for exact identical frequency values for the CPS, the NPS shows completely different pattern: the onion structure, where each system independently and gradually damps to the common synchronous frequency with a much long tail. In this process, lack of clusters for identical frequencies over a finite parameter interval can be proved by magnifying the data in detail. The detailed descriptions of these oscillator models and their definitions of phase are given in the Appendix, where all the system parameters used in Fig. 1 are also provided.
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algorithms, such as the Euler scheme and the classical Heun algorithm (also known as the improved Euler method or two-stage Runge-Kutta method).29–32 III. CONNECTION BETWEEN CPS AND NPS
By carefully examining the bifurcation trees of the CPS in Figs. 1(a) and 1(b), one can still identify some differences. The bifurcation of any two Kuramoto phase oscillators is very clean, whereas the bifurcation of coupled R€ossler oscillators shows a slight tail. Therefore, a natural connection between the CPS and NPS might be that for the NPS, a sufficiently long tail is needed for any two neighboring elements to be synchronized, and the frequencies of all elements do not damp to an identical value until a sufficiently large coupling strength is applied. To test this hypothesis, let us study the noisy effect on collective behaviors of coupled nonlinear systems.33–35 First, we consider the simplest case with two coupled phase oscillators in the presence of noise h_ i ¼ xi þ e sinðhj hi Þ þ Dni ; FIG. 3. The same as Fig. 1 but for large coupled systems; N ¼ 20. Again (a) and (b) belong to the CPS, whereas (c)–(f) belong to the NPS, where the pattern for each system gradually damping to the common synchronous frequency with a much long tail is the same as Fig. 1 for N ¼ 5.
All the numerical computations for the deterministic differential equations in the present work adopt the standard fourth-order Runge-Kutta method for the time integration with a sufficiently small time stepping Dt ¼ 0.01. In contrast, for the stochastic differential p equations considered below, a ffiffiffiffiffi stochastic term scaled with Dt is added after each time step. Our results are verified with other stochastic
i; j ¼ 1; 2
(1)
where hi and xi indicate the phase and natural frequency of oscillator i, e represents the coupling strength, D is the noise strength, and ni’s (i ¼ 1, 2) are independent standard Gaussian white noises: hni ðtÞnj ðt0 Þi ¼ dij dðt t0 Þ. Without loss of generality, we may rescale xi with x1 ¼ 1.0 and x2 ¼ 1.0. Let us consider the simplest case in the absence of noise (D ¼ 0). With coupling, two oscillators reach the phase synchronization (X1 ¼ X2) after a critical coupling strength (denoted by e0) arrives. The value of e0 can be easily analyzed: e0 ¼ 1. We can derive the average frequency of the oscillator pffiffiffiffiffiffiffiffiffiffiffiffiffi X 1 ¼ 1 e2 (2) for e 1, and X1 ¼ 0
FIG. 4. The same as Fig. 3 but for the frequency deviations DXi (DXi ¼ Xi hXi i) instead.
(3)
for e > 1. Here, owing to the system symmetry (i.e., the only difference between two oscillators’ average frequency is their sign), we can simply consider the first oscillator only. If e is larger than the critical coupling strength (e > e0 ¼ 1), a pair of steady states exists: one is stable, Dh ¼ h1 h2 ¼ arcsinð1=eÞ, and the other is unstable, Dh ¼ h1 h2 ¼ p arcsinð1=eÞ. Clearly, a classical saddle-node bifurcation on the loop occurs at e0 ¼ 1. With the onset of noise (D 6¼ 0), the above deterministic picture changes. Generally, X1 still decreases with the increase of coupling strength. It, however, does not vanish even after the critical coupling strength (e0 ¼ 1 for D ¼ 0) is reached. When e > 1, due to the noise effect, the phase difference of oscillators will not disappear any more, and the noise will always kick oscillators out of the steady state. Thus, a non-zero value of average frequency should always be found, and the kick effect should be stronger under larger noise.
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To verify the above analysis, Eq. (1) was numerically integrated for different noise strengths. For obtaining reliable numerical results, a sufficiently long integration time 400 000 was used with a transient time 10 000 discarded. As several examples, the values of X1 for different D’s are shown in Fig. 5(a); from left to right, D ¼ 0, 0.1, 0.6, and 1, respectively. Clearly with the increase of D, the value of X1 first shows in the absence of a tail (for D ¼ 0), changes to a small tail (e.g., D ¼ 0.1), and further changes to a much larger tail (e.g., D ¼ 1.0). To quantitatively characterize these changes, we zoom-in the small value part of X1 in Fig. 5(b) with D ¼ 0.6. The critical values of x1 and ec as a function of D are further illustrated with solid lines in Figs. 5(c) and 5(d), respectively. It is clearly shown that both x1 and ec monotonically increase with the increase of D value. In Ref. 16, Lee et al. have studied the intermittent phase dynamics of two classical chaotic systems, R€ossler and Lorenz, and shown that they share some common features: the eyelet intermittency near the phase synchronization critical point and the type-I intermittency far from the critical point. (Note that in their original work, the non-clustering phase synchronization of the Lorenz system was not mentioned.) Therefore, although the coupled systems may show different patterns of bifurcation tree according to different systems, they may have some common behaviors. One may naturally ask what happens for the current coupled phase oscillators under noisy driving. To understand this situation, it is necessary to study the intermittent behaviors of the phase difference Dh ¼ h2 h1 for different e values. As an example, D ¼ 0.05 is chosen. The result is shown in Fig. 6, indicating the same dichotomy of intermittent behavior. Figures 6(a) and 6(b) show Dh as the evolution of time for several different e values. In Fig. 6(a), when e < et (et 1.003), Dh increases with an intermittent sequence slip of an approximately 2p periodic behavior. However, in Fig. 6(b), when et < e < es (es 1.05), the intermittent sequences begin to become much longer and do not show a clear periodic behavior. Here in numerics, the value of es for the second critical point is not distinguishable with that of ec
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FIG. 6. The analysis of intermittent behavior of two coupled phase oscillators under noise. (a) and (b) The time series of phase difference Dh ¼ h2 – h1 for different e’s, representing type-I intermittency and eyelet intermittency, respectively. (c) and (d) Distributions of 2p jump interval l for different e’s. (e) hli vs ðet eÞ1=2 for the behavior of type-I intermittency, and (f) lnhli vs ðec eÞ1=2 for that of eyelet intermittency.
attained in Fig. 5(d). Therefore, it is believed that they are the same: es ec. Furthermore, the distributions of Dh for these different e values within the two different parameter regimes are presented in Figs. 6(c) and 6(d), respectively. Comparing curves in the left and right columns, clearly one can find that a nearly symmetric distribution pattern has changed to a biased one. Finally, these two distinct behaviors can be well characterized by their different scale relationship hli / ðet eÞ0:5
(4)
lnhli / ðec eÞ0:5
(5)
for e < et, and
FIG. 5. Noise effect on two coupled phase oscillators. (a) The average frequency X vs coupling for different D’s. From left to right: D ¼ 0, 0.1, 0.6, and 1. (b) The magnification of (a) for D ¼ 0.6 as an example. (c) x1 vs D, and (d) ec (solid line) and et (open triangles) vs D.
for et < e < ec, as shown in Figs. 6(e) and 6(f), respectively. In addition, the values of et for different D’s are superimposed in Fig. 5(d) by open triangles. Hence, it is clear that the addition of noise naturally fills the gap between CPS and NPS, two different paths to synchronization. To verify the generality of this finding, Figs. 7 and 8 illustrate two examples for coupled noisy phase oscillators with N ¼ 3 and N ¼ 5, respectively. Both of them show the transition from CPS to NPS with the increase of noise intensity. Here, the stochastic model with an input of noise on the coupled Kuramoto phase oscillators is considered. The insets in Figs. 7(a) and 7(b) indicate an obvious change when a small amount of noise is added. Finally, we further consider the coupled R€ossler oscillators under noise driving. The result is shown in Fig. 9 for different D values. Again, the crossover from CPS to NPS
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FIG. 7. Bifurcation trees of coupled phase oscillators for different D’s: (a) D ¼ 0, (b) D ¼ 0.1, (c) D ¼ 0.3, and (d) D ¼ 0.5. N ¼ 3.
occurs, showing a much longer tail of X with the increase of the noise intensity. Thus, the environmental noise can naturally induce the conversion of two seemingly different paths. IV. SUMMARY AND DISCUSSION
In summary, we have studied phase synchronization dynamics in coupled nonlinear elements and classified various phase synchronizations into two major types: CPS and NPS. We reveal the mechanism in which two phase synchronization pathways can be connected. All of these findings have been confirmed by our exhaustive numerical computations. Therefore, we expect that the present observations can shed light on our understanding of collective behaviors of nonlinear systems in physics, chemistry, and biology as well. Below it is worthwhile to further discuss these findings. (1) In contrast to our intuition, the NPS is also very common in dynamical systems. It occurs in coupled systems with intrinsic high nonlinearity. For example, clearly the Lorenz system is not phase-coherent and exhibits a strongly chaotic behavior. Additionally, for the excitable FHN and HR neurons, their spiking and bursting behaviors are also strongly chaotic, induced by either noise or
FIG. 8. Similar to Fig. 7 for N ¼ 5 instead.
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FIG. 9. Similar to Fig. 8 for coupled noisy R€ ossler oscillators considered instead. In all these panels, the frequency parameters: xi ¼ 1.005, 1.0115, 1.0, 0.99, 1.02 are fixed.
coupling. Therefore, it is not surprising to see all these coupled systems show a similar bifurcation route—NPS, which is quite different from that of the R€ossler oscillator, which shows the phase coherence and weak chaos much like a phase oscillator. In addition to strongly nonlinear models reported in the paper, many other dynamical systems with high nonlinearity are also expected to show NPS. As it is well-known that systems with ill-defined phase are less liable to phase synchronization, the different patterns of either CPS or NPS might come from intrinsic feature of systems: either well-defined phase or ill-defined phase. (2) On the other hand, the CPS and NPS may share the same scale relation in their intermittent behavior. Thus, they are closely connected. Furthermore, we find that the environmental noise may substitute the role of the system nonlinearity and induce the transition from one type of synchronization to another. (3) In Refs. 36 and 37, Pikovsky et al. have already reported cluster-free synchronization transitions in globally coupled phase oscillators, namely, they found that for small ensembles the transition of the type from “full synchrony” to “cluster state” to “periodic/quasiperiodic partially synchronous state” occurs, while for a large number of oscillators a direct transition from “full synchrony” to “periodic/quasiperiodic partially synchronous state” without showing clusters is typical. Here, we like to mention that there coupled identical oscillators are studied, and the concept of cluster states, which is defined as several clusters of identical phases of oscillators, is fundamentally different with ours, which is defined as that of identical frequencies of oscillators. Therefore, our findings of NPS for cluster-free synchronization transitions are different. (4) In some way, one may say the increase of coupling leads to certain synchronization in a generalized manner, but usually one cannot say this is generalized synchronization. The reason is simple: for the coupled systems with any non-zero coupling, the coupled elements are already intervolved; therefore, we cannot isolate any one element
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from the coupled systems and believe there is a pure functional relation between the states of any two elements. In the classical book of synchronization,3 it is typically believed that the generalized synchronization is only observed for unidirectional coupling, when the first (driving) system forces the second (driven) one, and there is no back-action. Recently, however, we do see some interesting extension of the concept of generalization synchronization to bidirectionally coupled oscillators, such as Refs. 38 and 39. In particular, in the latter paper, the explicit form of functional relation between states of the mutually coupled systems has to be substituted by an implicit functional relation, and thus the generalized synchronization can still be defined and characterized. Further detailed study within such a framework to explore the connection with the possibility of generalized synchronization in a more generalized manner is of interest. (5) Finally, although the simplest spatial ring connection with the nearest neighboring coupling is considered in the paper, these dynamical behaviors and classification can be applied to more complicated situations, such as complex networks with small-world or scale-free connections.40–46 A further study on how network structures may influence phase synchronization pathways is needed.
ACKNOWLEDGMENTS
We acknowledge the support of this work by the National Natural Science Foundation of China under Grant No. 11075202. In addition, we acknowledge with pleasure the critical reading and help of this work by Professor Guowei Wei, and helpful suggestions and comments of two anonymous referees. APPENDIX: MODEL DESCRIPTION 1. Coupled Kuramoto phase oscillators
The model of the classical Kuramoto phase oscillators is written as4,13,14 h_ i ðtÞ ¼ xi þ eðsinðhiþ1 hi Þ þ sinðhi1 hi ÞÞ; ¼ 1; …; N;
i (A1)
where hi denotes the phase angle of the ith oscillator, N is the number of oscillators, e indicates the coupling strength, and xi’s are natural frequency of oscillators, which are randomly chosen. Without losing generality, the periodic boundary condition (h0 ¼ hN, hNþ1 ¼ h1) is considered, and P N i¼1 xi ¼ 0 is taken. As the model of coupled Kuramoto phase oscillators focuses only on the phase (not amplitude) of the oscillators, it can be believed as the simplest model for coupled nonlinear oscillators. We may define the average frequency as ð 1 T_ h i ðtÞdt: (A2) Xi ¼ lim T!1 T 0
Any two oscillators (i and j) reach phase synchronization when their average frequencies become identical (i.e., Xi ¼ Xj). In Fig. 1(a), the natural frequency xi’s are randomly taken as 1.0, 0.7, 1.9, 0.1, 0.3; N ¼ 5. The integration time 20 000 with a transient time 5000 discarded is adopted here. For N ¼ 20 in Fig. 3(a), the natural frequency xi’s are randomly taken from a uniform distribution: xi 2 [1, 1] with the integration time 20 000 and transient time 10 000. For all other models, the average frequency is always calculated based on Eq. (A2), but with different definitions for the phase h(t) in the equation. € ssler oscillators 2. Coupled Ro
The classical coupled chaotic R€ossler system can be written as12,15,16 x_ i ¼ xi yi zi þ eðxiþ1 2xi þ xi1 Þ y_ i ¼ xi xi þ ayi z_ i ¼ b þ zi ðxi cÞ; i ¼ 1; …; N;
(A3)
where a ¼ 0.165, b ¼ 0.2, and c ¼ 10.0 are system parameters, and e denotes the coupling strength. Since single R€ossler oscillator rotates around the origin with a coherent phase, exhibiting the feature quite similar to the periodic phase oscillators, its phase can be easily defined47 hi ðtÞ ¼ arctanðyi ðtÞ=xi ðtÞÞ;
i ¼ 1; …; N:
(A4)
To avoid the limit of hi(t) between 0 and 2p, a 2p has to be added in phase if the attractor comes across the positive x-axis from the fourth quadrant to the first one, and oppositely 2p is subtracted. This technique has been widely applied in the study of chaotic phase synchronization.47,48 It is noticeable that all of these phase definitions given in the paper including the following ones come from the literature and are standard. Furthermore, its average frequency can be calculated based on the same definition in Eq. (A2). The result is shown in Fig. 1(b) with the parameters of xi randomly chosen: x1,…,5 ¼ 1.005, 1.0115, 1.0, 1.05, 1.02, and N ¼ 5 unchanged. The integration time is 100 000 and the transient time is 5000. For N ¼ 20 in Fig. 3(b), the natural frequency xi’s are randomly taken from xi 2 [0.95, 1.05], with the integration time 10 000 and transient time 5000. 3. Coupled FHN systems
Consider the following coupled system consisting of N FHN noisy excitable elements, described by18–20 x_ i ¼ xi x3i =3 yi þ eðxiþ1 2xi þ xi1 Þ; i ¼ 1; …; 5; y_ i ¼ xi þ ai þ Dni ðtÞ;
(A5)
where is small ( 1) allowing one to treat the xi and yi variables as the fast and slow variations, respectively, and the system parameter ai governs the stability of the isolated system: if jai j > 1, the system has only one stable fixed point, whereas for jai j < 1, it shows a limit cycle. In
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addition, ni(t) denotes the independent Gaussian white noise, with D representing the noise density. Again, the coupling strength is indicated by e. The essential dynamical behavior for excitable neuron is spike (action potential), which shows a rush peak for the membrane potential from negative to positive and lasts for only one or two milliseconds. For remaining most of time, the steady state for a constant negative membrane potential appears. These behaviors are quite similar to the “on” and “off” states in the digital circuit or transmission. Therefore, the phase dynamics of each excitable neuron can be catched by21 hi ðtÞ ¼ 2p
t sik þ 2pk; sik
sikþ1
(A6)
for sik t < sikþ1 , where sk is the time of kth firing of the ith neuron. Based on this definition, the phase and further the average frequency (firing rate) of excitable system can also be calculated. In the calculation of Fig. 1(c), ¼ 0.01, a1,…,5 ¼ 1.05, 1.01, 1.04, 1.08, 1.1, and D ¼ 0.2. The integration time is chosen as 200 000 and the transient time is 1000. For much larger N, e.g., N ¼ 20 in Fig. 3(c), ai’s are randomly chosen within ai 2 [1.01, 1.11], and to reduce fluctuation, the mean frequencies are averaged over 50 trails for different random initial conditions with the integration time 100 000 and transient time 10 000 for each trail. The detailed pattern of the average frequency in the paper exhibits a clear nonclustering phase synchronization transition with the long-tail effect.
Consider the HR neuron model23,24 e x_ i ¼ yi ax3i þ bx2i zi þ I þ ðxiþ1 2xi þ xi1 Þ 2 (A9) y_ i ¼ c dx2i yi z_ i ¼ r½Sðxi þ ji Þ zi ;
i ¼ 1; …; N
where the system parameters a ¼ 1.0, b ¼ 3.0, c ¼ 1.0, d ¼ 5.0, S ¼ 4.0, r ¼ 0.006, and I ¼ 3.0 are the same as in the literature,24 and ji is selected from a uniform distribution: ji 2 [1.56, 1.66]. For this classical HR model, which is capable of showing bursting behavior of neuron (firing stereotypical bursts of closely spaced spikes) and being chaotic, its phase can also be perfectly defined following the previous study in Ref. 24: _ sÞ=ðxðtÞ _ þ 0:1Þ; h ¼ arctan½xðt
(A10)
where s ¼ 0.5 is usually chosen. For N ¼ 5, the average frequencies are regarded as the mean value of 1000 trails with random initial conditions and the integration time 10 000 and transient time 5000. For N ¼ 20 in Fig. 3(e), ji’s are selected from a uniform distribution: ji 2 [1.5, 1.7], and the mean frequencies are averaged over 300 times with the integration time 10 000 and transient time 2000 for each time. From the numerical result in the paper, we can see that its phase transition belongs to NPS. 6. Coupled LV systems
4. Coupled Lorenz oscillators
For another classical chaotic oscillator, the model of coupled Lorenz systems is described by22 x_ i ¼ rðyi xi Þ þ eðxiþ1 2xi þ xi1 Þ; y_ i ¼ ri xi yi xi zi ; z_ i ¼ bzi þ xi yi ;
5. Coupled HR neurons
(A7)
i ¼ 1; …; N
where the “canonical” parameter values r ¼ 10.0 and b ¼ 8/3 are fixed. Although the Lorenz oscillator’s phase is not coherent showing two rotation centers, its phase can still be defined through the following simple transformation8,12 qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 hi ðtÞ¼arctan ½ xi þyi 2bðri 1Þ=½zi ðri 1Þ : (A8) Without losing generality, the system parameters r1,…,5 ¼ 36.5, 40.0, 41.0, 39.0, 35.0, and N ¼ 5 are chosen in Fig. 1(d). The average frequencies are regarded as the mean value of 1000 trails with different random initial conditions, while the integration time 5000 and transient time 2000 are adopted for each trail. In contrast, in Fig. 3(d) for N ¼ 20, ri’s are selected from a uniform distribution: ri 2 [36, 40], and the mean frequencies are averaged over 2000 tests with the integration time 10 000 and transient time 2000 for each time. A clear NPS transition shows that the average frequency of each oscillator transits to the synchronization only at the sufficiently large coupling strength.
In addition, we consider a well-known ecological LV system26–28 xi xi yi _x i ¼ axi 1 k þ eðxiþ1 2xi þ xi1 Þ K 1 þ jxi xi y i y_ i ¼ bi yi þ k ; i ¼ 1; …; 5 1 þ jxi (A11) where x and y stand for the prey and the predator species, separately. The parameters a and b denote the birth and death rates, respectively, K indicates the carrying capacity of the prey, k represents the rate of predation, and j is a constant of the functional response. Here, N ¼ 5, a ¼ 1, K ¼ 3, k ¼ 3, and j ¼ 1 are fixed, and bi’s are chosen from a uniform distribution: bi 2 [1 – 0.025, 1 þ 0.025]. The result is shown in Fig. 1(f), which again exhibits the pattern of the NPS. The integration time 100 000 and transient time 5000 are adopted here. For N ¼ 20 in Fig. 3(f), bi’s are selected from the same uniform distribution: bi 2 [1 – 0.025, 1 þ 0.025], while the mean frequencies are averaged over 100 times with the integration time 20 000 and transient time 5000, to reduce the fluctuation. All these results for various nonlinear systems including noisy FHN neuron, Lorenz oscillator, HR bursting neuron, and LV ecological system demonstrate that the path to phase synchronization in the absence of clusterings is very common, the same as the CPS, and could be extensively observed in many other coupled nonlinear systems as well.
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A. T. Winfree, The Geometry of Biological Time (Springer-Verlag, New York, 1980). 2 S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Perseus Books Publishing, MA, 1994). 3 A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001). 4 Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (SpringerVerlag, New York, 1984). 5 S. H. Strogatz, Physica D 143, 1 (2000). 6 M. Zhan, Z. G. Zheng, G. Hu, and X. H. Peng, Phys. Rev. E 62 , 3552 (2000). 7 S. G. Guan, C. H. Lai, and G. W. Wei, Phys. Rev. E 72, 016205 (2005). 8 A. S. Pikovsky, M. G. Rosenblum, G. Osipov, and J. Kurths, Physica D 104, 219 (1997). 9 Z. H. Liu, Y. C. Lai, and F. C. Hoppensteadt, Phys. Rev. E 63, 055201(R) (2001). 10 Y. Wu, J. H. Xiao, G. Hu, and M. Zhan, EPL 97, 40005 (2012). 11 W. Q. Liu, Y. Wu, J. H. Xiao, and M. Zhan, EPL 101, 38002 (2013). 12 Z. G. Zheng, G. Hu, C. S. Zhou, and B. B. Hu, Acta Phys. Sin. 49, 2320 (2000) (in Chinese); see http://wulixb.iphy.ac.cn/CN/Y2000/V49/I12/2320. 13 Z. G. Zheng, G. Hu, and B. B. Hu, Phys. Rev. Lett. 81, 5318 (1998). 14 Y. Kuramoto, Lect. Notes Phys. 39, 420 (1975). 15 O. E. R€ossler and Z. Naturforsch, Phys. Lett. A 57, 5 (1976). 16 K. J. Lee, Y. Kawk, and T. K. Lim, Phys. Rev. Lett. 81, 321 (1998). 17 S. Liu, Z. W. He, and M. Zhan, Front. Phys. 9, 120–127 (2014). 18 R. FitzHugh, Biophys. J. 1, 445 (1961). 19 W. Y. Chiang, P. Y. Lai, and C. K. Chan, Phys. Rev. Lett. 106, 254102 (2011). 20 A. S. Pikovsky and J. Kurths, Phys. Rev. Lett. 78, 775 (1997). 21 B. Hu and C. S. Zhou, Phys. Rev. E 61, R1001 (2000). 22 E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963). 23 J. L. Hindmarsh and R. M. Rose, Proc. R. Soc. London, Ser. B 221, 87 (1984).
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CHAOS 24, 013104 (2014)
Clustering versus non-clustering phase synchronizations Shuai Liu1,2 and Meng Zhan1,a) 1
Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China 2 University of Chinese Academy of Sciences, Beijing 100049, China
(Received 28 May 2013; accepted 27 December 2013; published online 9 January 2014) Clustering phase synchronization (CPS) is a common scenario to the global phase synchronization of coupled dynamical systems. In this work, a novel scenario, the non-clustering phase synchronization (NPS), is reported. It is found that coupled systems do not transit to the global synchronization until a certain sufficiently large coupling is attained, and there is no clustering prior to the global synchronization. To reveal the relationship between CPS and NPS, we further analyze the noise effect on coupled phase oscillators and find that the coupled oscillator system can change from CPS to NPS with the increase of noise intensity or system disorder. These findings are expected to shed light on the mechanism of various intriguing self-organized behaviors in coupled C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4861685] systems. V
Path to phase synchronization (PS) in coupled nonlinear oscillators is one of the essential problems in nonlinear science and it is also closely connected with many applications in realistic systems. Different with the usual path to synchronization via various clustering processes, here we report a novel non-clustering path to phase synchronization, which is, however, expected to generally occur in coupled nonlinear oscillators with either strong systems nonlinearity or environmental disorder.
I. INTRODUCTION
Rich collective behaviors have been found in various nonlinear dynamical systems,1–4 including physical, chemical, and biological ones. Some examples include chemical pattern formation, laser systems, circadian rhythm, and neural networks, just to name a few. Their study has been a subject of great interest in the nonlinear dynamics field. The systematical investigation of coupled nonlinear oscillators is of crucial importance to their understanding and practical applications. PS2–13 is a basic phenomenon in coupled nonlinear systems and has been studied for many years. With the increase of coupling, the system coherence naturally increases. A universal path to the global phase synchronization shows various clustering processes. In this path, any two adjacent oscillators (or adjacent clusters of oscillators) with close frequencies can easily reach synchronization first and form a locally synchronous cluster. Along with the increase of coupling strength further, small clusters can naturally attract more elements and be merged into large clusters. Finally, all elements (and clusters) transit to a globally synchronous state after a critical coupling strength ec arrives. We call this process a clustering phase synchronization (CPS).9–13 a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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This clustering synchronization scenario can be regarded as a classical route to phase synchronization for coupled phase-coherent oscillators. For example, the coupled Kuramoto phase oscillators4,13,14 and coupled R€ossler systems12,15,16 with the nearest neighboring coupling are studied and the results are shown in Figs. 1(a) and 1(b), respectively. In these subfigures, clearly a bifurcation tree exhibits, showing complicated clusterings before the final largest cluster occurs. Without loss of generality, the number of coupled oscillators N ¼ 5 is chosen in this illustration, and it is found that the same feature appears qualitatively for other numbers. Recently, we have studied the collective dynamics of coupled excitable systems under external noisy driving and discovered a novel synchronization path, the non-clustering phase synchronization (NPS).17 In a contrast to the CPS, all elements in the coupled system transit to a synchronous state with an identical frequency simultaneously only after a certain coupling strength is achieved, and local synchronous clusters (or groups) cannot be found. In our study, the wellknown FitzHugh-Nagumo (FHN) excitable system18–21 has been considered. The bifurcation tree of the NPS is demonstrated in Fig. 1(c), which clearly shows a pattern different with that of the CPS in Figs. 1(a) and 1(b). But several questions still need to be answered, such as whether the NPS can also be generally observed in other systems. More importantly, what is the connection between these two different synchronization paths? To answer these interesting questions, we carry out further investigations on a few other models and explore noise effect. II. NPS IN OTHER SYSTEMS
We first consider a few different coupled nonlinear systems. To our great surprise, we find that the seemly unusual NPS route to global synchronization actually can be frequently observed. Therefore, based on this fact, it can be believed that both of them are intrinsic behaviors of coupled
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C 2014 AIP Publishing LLC V
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FIG. 1. Bifurcation trees of average frequencies in six different coupled systems: (a) Kuramoto phase oscillator, (b) R€ ossler, (c) FHN neuron, (d) Lorenz, (e) HR neuron, and (f) LV ecology systems. N ¼ 5. The insets in 1(a) and 1(b) show their detailed difference in transition to a cluster. The insets in 1(d)–1(f) show their detailed non-clustering structures. Clearly (a) and (b) belong to the CPS, whereas (c)–(f) belong to the NPS. For more details, see the text.
nonlinear systems. Figures 1(d)–1(f) depict the synchronization patterns of a few examples, such as the classical chaotic Lorenz oscillator,22 the Hindmarsh-Rose (HR) bursting neuron,23,24 and the Lotka-Volterra (LV) ecological system.25–28 Clearly, all of these systems give rise to NPS. It is interesting to note that different from the path of the CPS, the dynamics of the above systems becomes synchronous only after a certain large coupling strength arrives. There exists a common characteristic in the transition to the global phase synchronization: the value of frequency in the NPS has a long tail, although each of these models has a very distinct nature and its phase definition can be quite different. To show the lack of clusters before the global synchronization, we even magnify several parameter regions in the insets. For example, in Fig. 1(d) for the coupled chaotic Lorenz oscillators, the two seemly parallel lines do not cross at all. In Fig. 1(e) for the coupled HR bursting neurons, it shows the crossover of their average frequencies only at several single parameters, indicative of no clusters. It is notable that a cluster appears for identical frequencies at least for a continuous parameter interval. Again in Fig. 1(f) for the coupled LV ecological systems, the magnification of the small region clearly shows no cross, although it seems that a very small cluster appears there. To make these phenomena clearer, we plot the same bifurcation trees in Fig. 2, but with the data of the frequency difference fromPthe mean frequency: DXi ¼ Xi hXi i, where hXi i ¼ N1 Ni¼1 Xi . Obviously, the CPS with complicated clusterings in Figs. 2(a) and 2(b) show completely different patterns with the NPS in Figs. 2(c)–2(f), where they exhibit a common zero value only after the coupling is large
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FIG. 2. The same as Fig. 1 but for the frequency deviations DXi (DXi ¼ Xi hXi i) instead, showing a clear difference between the patterns of the CPS in (a) and (b) and the NPS in (c)–(f). The cross of two lines at a single coupling cannot be believed as a cluster, as shown in (e).
and do not show any further bifurcations prior to it. For the original phase bifurcation trees in Fig. 1, due to the small difference for each system within the small region, usually it is uneasy to judge without the aid of magnification whether some of systems exhibit identical (or very close) frequencies, i.e., constitute clusters or not. With this new plot, the small difference can be magnified in a comparably larger region, and thus we can clearly determine the occurrence of clusters or not. As a result, we demonstrate that the phenomenon of NPS occurs indeed and is quite distinct with the CPS without any ambiguity. Here, we also like to emphasize that although the NPS, which has been demonstrated for several examples in Figs. 1 and 2, is quite general, it does not mean the opposite part, the CPS, cannot be observed for these systems. We do find that any two oscillators in the coupled systems with slightly initial frequency mismatch can easily form a cluster, if the parameters are properly chosen. Hence, the phenomenon of the NPS is not exclusive. As another example, the case for N ¼ 20 is shown in Fig. 3, exhibiting the similar patterns as those in Fig. 1. Again, its frequency differences from the mean frequency are shown in Fig. 4. The difference between the two types of phase synchronization is clear. In particular, with the plots of frequency difference, different with the tree structure exhibiting clusters (branches) for exact identical frequency values for the CPS, the NPS shows completely different pattern: the onion structure, where each system independently and gradually damps to the common synchronous frequency with a much long tail. In this process, lack of clusters for identical frequencies over a finite parameter interval can be proved by magnifying the data in detail. The detailed descriptions of these oscillator models and their definitions of phase are given in the Appendix, where all the system parameters used in Fig. 1 are also provided.
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algorithms, such as the Euler scheme and the classical Heun algorithm (also known as the improved Euler method or two-stage Runge-Kutta method).29–32 III. CONNECTION BETWEEN CPS AND NPS
By carefully examining the bifurcation trees of the CPS in Figs. 1(a) and 1(b), one can still identify some differences. The bifurcation of any two Kuramoto phase oscillators is very clean, whereas the bifurcation of coupled R€ossler oscillators shows a slight tail. Therefore, a natural connection between the CPS and NPS might be that for the NPS, a sufficiently long tail is needed for any two neighboring elements to be synchronized, and the frequencies of all elements do not damp to an identical value until a sufficiently large coupling strength is applied. To test this hypothesis, let us study the noisy effect on collective behaviors of coupled nonlinear systems.33–35 First, we consider the simplest case with two coupled phase oscillators in the presence of noise h_ i ¼ xi þ e sinðhj hi Þ þ Dni ; FIG. 3. The same as Fig. 1 but for large coupled systems; N ¼ 20. Again (a) and (b) belong to the CPS, whereas (c)–(f) belong to the NPS, where the pattern for each system gradually damping to the common synchronous frequency with a much long tail is the same as Fig. 1 for N ¼ 5.
All the numerical computations for the deterministic differential equations in the present work adopt the standard fourth-order Runge-Kutta method for the time integration with a sufficiently small time stepping Dt ¼ 0.01. In contrast, for the stochastic differential p equations considered below, a ffiffiffiffiffi stochastic term scaled with Dt is added after each time step. Our results are verified with other stochastic
i; j ¼ 1; 2
(1)
where hi and xi indicate the phase and natural frequency of oscillator i, e represents the coupling strength, D is the noise strength, and ni’s (i ¼ 1, 2) are independent standard Gaussian white noises: hni ðtÞnj ðt0 Þi ¼ dij dðt t0 Þ. Without loss of generality, we may rescale xi with x1 ¼ 1.0 and x2 ¼ 1.0. Let us consider the simplest case in the absence of noise (D ¼ 0). With coupling, two oscillators reach the phase synchronization (X1 ¼ X2) after a critical coupling strength (denoted by e0) arrives. The value of e0 can be easily analyzed: e0 ¼ 1. We can derive the average frequency of the oscillator pffiffiffiffiffiffiffiffiffiffiffiffiffi X 1 ¼ 1 e2 (2) for e 1, and X1 ¼ 0
FIG. 4. The same as Fig. 3 but for the frequency deviations DXi (DXi ¼ Xi hXi i) instead.
(3)
for e > 1. Here, owing to the system symmetry (i.e., the only difference between two oscillators’ average frequency is their sign), we can simply consider the first oscillator only. If e is larger than the critical coupling strength (e > e0 ¼ 1), a pair of steady states exists: one is stable, Dh ¼ h1 h2 ¼ arcsinð1=eÞ, and the other is unstable, Dh ¼ h1 h2 ¼ p arcsinð1=eÞ. Clearly, a classical saddle-node bifurcation on the loop occurs at e0 ¼ 1. With the onset of noise (D 6¼ 0), the above deterministic picture changes. Generally, X1 still decreases with the increase of coupling strength. It, however, does not vanish even after the critical coupling strength (e0 ¼ 1 for D ¼ 0) is reached. When e > 1, due to the noise effect, the phase difference of oscillators will not disappear any more, and the noise will always kick oscillators out of the steady state. Thus, a non-zero value of average frequency should always be found, and the kick effect should be stronger under larger noise.
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To verify the above analysis, Eq. (1) was numerically integrated for different noise strengths. For obtaining reliable numerical results, a sufficiently long integration time 400 000 was used with a transient time 10 000 discarded. As several examples, the values of X1 for different D’s are shown in Fig. 5(a); from left to right, D ¼ 0, 0.1, 0.6, and 1, respectively. Clearly with the increase of D, the value of X1 first shows in the absence of a tail (for D ¼ 0), changes to a small tail (e.g., D ¼ 0.1), and further changes to a much larger tail (e.g., D ¼ 1.0). To quantitatively characterize these changes, we zoom-in the small value part of X1 in Fig. 5(b) with D ¼ 0.6. The critical values of x1 and ec as a function of D are further illustrated with solid lines in Figs. 5(c) and 5(d), respectively. It is clearly shown that both x1 and ec monotonically increase with the increase of D value. In Ref. 16, Lee et al. have studied the intermittent phase dynamics of two classical chaotic systems, R€ossler and Lorenz, and shown that they share some common features: the eyelet intermittency near the phase synchronization critical point and the type-I intermittency far from the critical point. (Note that in their original work, the non-clustering phase synchronization of the Lorenz system was not mentioned.) Therefore, although the coupled systems may show different patterns of bifurcation tree according to different systems, they may have some common behaviors. One may naturally ask what happens for the current coupled phase oscillators under noisy driving. To understand this situation, it is necessary to study the intermittent behaviors of the phase difference Dh ¼ h2 h1 for different e values. As an example, D ¼ 0.05 is chosen. The result is shown in Fig. 6, indicating the same dichotomy of intermittent behavior. Figures 6(a) and 6(b) show Dh as the evolution of time for several different e values. In Fig. 6(a), when e < et (et 1.003), Dh increases with an intermittent sequence slip of an approximately 2p periodic behavior. However, in Fig. 6(b), when et < e < es (es 1.05), the intermittent sequences begin to become much longer and do not show a clear periodic behavior. Here in numerics, the value of es for the second critical point is not distinguishable with that of ec
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FIG. 6. The analysis of intermittent behavior of two coupled phase oscillators under noise. (a) and (b) The time series of phase difference Dh ¼ h2 – h1 for different e’s, representing type-I intermittency and eyelet intermittency, respectively. (c) and (d) Distributions of 2p jump interval l for different e’s. (e) hli vs ðet eÞ1=2 for the behavior of type-I intermittency, and (f) lnhli vs ðec eÞ1=2 for that of eyelet intermittency.
attained in Fig. 5(d). Therefore, it is believed that they are the same: es ec. Furthermore, the distributions of Dh for these different e values within the two different parameter regimes are presented in Figs. 6(c) and 6(d), respectively. Comparing curves in the left and right columns, clearly one can find that a nearly symmetric distribution pattern has changed to a biased one. Finally, these two distinct behaviors can be well characterized by their different scale relationship hli / ðet eÞ0:5
(4)
lnhli / ðec eÞ0:5
(5)
for e < et, and
FIG. 5. Noise effect on two coupled phase oscillators. (a) The average frequency X vs coupling for different D’s. From left to right: D ¼ 0, 0.1, 0.6, and 1. (b) The magnification of (a) for D ¼ 0.6 as an example. (c) x1 vs D, and (d) ec (solid line) and et (open triangles) vs D.
for et < e < ec, as shown in Figs. 6(e) and 6(f), respectively. In addition, the values of et for different D’s are superimposed in Fig. 5(d) by open triangles. Hence, it is clear that the addition of noise naturally fills the gap between CPS and NPS, two different paths to synchronization. To verify the generality of this finding, Figs. 7 and 8 illustrate two examples for coupled noisy phase oscillators with N ¼ 3 and N ¼ 5, respectively. Both of them show the transition from CPS to NPS with the increase of noise intensity. Here, the stochastic model with an input of noise on the coupled Kuramoto phase oscillators is considered. The insets in Figs. 7(a) and 7(b) indicate an obvious change when a small amount of noise is added. Finally, we further consider the coupled R€ossler oscillators under noise driving. The result is shown in Fig. 9 for different D values. Again, the crossover from CPS to NPS
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FIG. 7. Bifurcation trees of coupled phase oscillators for different D’s: (a) D ¼ 0, (b) D ¼ 0.1, (c) D ¼ 0.3, and (d) D ¼ 0.5. N ¼ 3.
occurs, showing a much longer tail of X with the increase of the noise intensity. Thus, the environmental noise can naturally induce the conversion of two seemingly different paths. IV. SUMMARY AND DISCUSSION
In summary, we have studied phase synchronization dynamics in coupled nonlinear elements and classified various phase synchronizations into two major types: CPS and NPS. We reveal the mechanism in which two phase synchronization pathways can be connected. All of these findings have been confirmed by our exhaustive numerical computations. Therefore, we expect that the present observations can shed light on our understanding of collective behaviors of nonlinear systems in physics, chemistry, and biology as well. Below it is worthwhile to further discuss these findings. (1) In contrast to our intuition, the NPS is also very common in dynamical systems. It occurs in coupled systems with intrinsic high nonlinearity. For example, clearly the Lorenz system is not phase-coherent and exhibits a strongly chaotic behavior. Additionally, for the excitable FHN and HR neurons, their spiking and bursting behaviors are also strongly chaotic, induced by either noise or
FIG. 8. Similar to Fig. 7 for N ¼ 5 instead.
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FIG. 9. Similar to Fig. 8 for coupled noisy R€ ossler oscillators considered instead. In all these panels, the frequency parameters: xi ¼ 1.005, 1.0115, 1.0, 0.99, 1.02 are fixed.
coupling. Therefore, it is not surprising to see all these coupled systems show a similar bifurcation route—NPS, which is quite different from that of the R€ossler oscillator, which shows the phase coherence and weak chaos much like a phase oscillator. In addition to strongly nonlinear models reported in the paper, many other dynamical systems with high nonlinearity are also expected to show NPS. As it is well-known that systems with ill-defined phase are less liable to phase synchronization, the different patterns of either CPS or NPS might come from intrinsic feature of systems: either well-defined phase or ill-defined phase. (2) On the other hand, the CPS and NPS may share the same scale relation in their intermittent behavior. Thus, they are closely connected. Furthermore, we find that the environmental noise may substitute the role of the system nonlinearity and induce the transition from one type of synchronization to another. (3) In Refs. 36 and 37, Pikovsky et al. have already reported cluster-free synchronization transitions in globally coupled phase oscillators, namely, they found that for small ensembles the transition of the type from “full synchrony” to “cluster state” to “periodic/quasiperiodic partially synchronous state” occurs, while for a large number of oscillators a direct transition from “full synchrony” to “periodic/quasiperiodic partially synchronous state” without showing clusters is typical. Here, we like to mention that there coupled identical oscillators are studied, and the concept of cluster states, which is defined as several clusters of identical phases of oscillators, is fundamentally different with ours, which is defined as that of identical frequencies of oscillators. Therefore, our findings of NPS for cluster-free synchronization transitions are different. (4) In some way, one may say the increase of coupling leads to certain synchronization in a generalized manner, but usually one cannot say this is generalized synchronization. The reason is simple: for the coupled systems with any non-zero coupling, the coupled elements are already intervolved; therefore, we cannot isolate any one element
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from the coupled systems and believe there is a pure functional relation between the states of any two elements. In the classical book of synchronization,3 it is typically believed that the generalized synchronization is only observed for unidirectional coupling, when the first (driving) system forces the second (driven) one, and there is no back-action. Recently, however, we do see some interesting extension of the concept of generalization synchronization to bidirectionally coupled oscillators, such as Refs. 38 and 39. In particular, in the latter paper, the explicit form of functional relation between states of the mutually coupled systems has to be substituted by an implicit functional relation, and thus the generalized synchronization can still be defined and characterized. Further detailed study within such a framework to explore the connection with the possibility of generalized synchronization in a more generalized manner is of interest. (5) Finally, although the simplest spatial ring connection with the nearest neighboring coupling is considered in the paper, these dynamical behaviors and classification can be applied to more complicated situations, such as complex networks with small-world or scale-free connections.40–46 A further study on how network structures may influence phase synchronization pathways is needed.
ACKNOWLEDGMENTS
We acknowledge the support of this work by the National Natural Science Foundation of China under Grant No. 11075202. In addition, we acknowledge with pleasure the critical reading and help of this work by Professor Guowei Wei, and helpful suggestions and comments of two anonymous referees. APPENDIX: MODEL DESCRIPTION 1. Coupled Kuramoto phase oscillators
The model of the classical Kuramoto phase oscillators is written as4,13,14 h_ i ðtÞ ¼ xi þ eðsinðhiþ1 hi Þ þ sinðhi1 hi ÞÞ; ¼ 1; …; N;
i (A1)
where hi denotes the phase angle of the ith oscillator, N is the number of oscillators, e indicates the coupling strength, and xi’s are natural frequency of oscillators, which are randomly chosen. Without losing generality, the periodic boundary condition (h0 ¼ hN, hNþ1 ¼ h1) is considered, and P N i¼1 xi ¼ 0 is taken. As the model of coupled Kuramoto phase oscillators focuses only on the phase (not amplitude) of the oscillators, it can be believed as the simplest model for coupled nonlinear oscillators. We may define the average frequency as ð 1 T_ h i ðtÞdt: (A2) Xi ¼ lim T!1 T 0
Any two oscillators (i and j) reach phase synchronization when their average frequencies become identical (i.e., Xi ¼ Xj). In Fig. 1(a), the natural frequency xi’s are randomly taken as 1.0, 0.7, 1.9, 0.1, 0.3; N ¼ 5. The integration time 20 000 with a transient time 5000 discarded is adopted here. For N ¼ 20 in Fig. 3(a), the natural frequency xi’s are randomly taken from a uniform distribution: xi 2 [1, 1] with the integration time 20 000 and transient time 10 000. For all other models, the average frequency is always calculated based on Eq. (A2), but with different definitions for the phase h(t) in the equation. € ssler oscillators 2. Coupled Ro
The classical coupled chaotic R€ossler system can be written as12,15,16 x_ i ¼ xi yi zi þ eðxiþ1 2xi þ xi1 Þ y_ i ¼ xi xi þ ayi z_ i ¼ b þ zi ðxi cÞ; i ¼ 1; …; N;
(A3)
where a ¼ 0.165, b ¼ 0.2, and c ¼ 10.0 are system parameters, and e denotes the coupling strength. Since single R€ossler oscillator rotates around the origin with a coherent phase, exhibiting the feature quite similar to the periodic phase oscillators, its phase can be easily defined47 hi ðtÞ ¼ arctanðyi ðtÞ=xi ðtÞÞ;
i ¼ 1; …; N:
(A4)
To avoid the limit of hi(t) between 0 and 2p, a 2p has to be added in phase if the attractor comes across the positive x-axis from the fourth quadrant to the first one, and oppositely 2p is subtracted. This technique has been widely applied in the study of chaotic phase synchronization.47,48 It is noticeable that all of these phase definitions given in the paper including the following ones come from the literature and are standard. Furthermore, its average frequency can be calculated based on the same definition in Eq. (A2). The result is shown in Fig. 1(b) with the parameters of xi randomly chosen: x1,…,5 ¼ 1.005, 1.0115, 1.0, 1.05, 1.02, and N ¼ 5 unchanged. The integration time is 100 000 and the transient time is 5000. For N ¼ 20 in Fig. 3(b), the natural frequency xi’s are randomly taken from xi 2 [0.95, 1.05], with the integration time 10 000 and transient time 5000. 3. Coupled FHN systems
Consider the following coupled system consisting of N FHN noisy excitable elements, described by18–20 x_ i ¼ xi x3i =3 yi þ eðxiþ1 2xi þ xi1 Þ; i ¼ 1; …; 5; y_ i ¼ xi þ ai þ Dni ðtÞ;
(A5)
where is small ( 1) allowing one to treat the xi and yi variables as the fast and slow variations, respectively, and the system parameter ai governs the stability of the isolated system: if jai j > 1, the system has only one stable fixed point, whereas for jai j < 1, it shows a limit cycle. In
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addition, ni(t) denotes the independent Gaussian white noise, with D representing the noise density. Again, the coupling strength is indicated by e. The essential dynamical behavior for excitable neuron is spike (action potential), which shows a rush peak for the membrane potential from negative to positive and lasts for only one or two milliseconds. For remaining most of time, the steady state for a constant negative membrane potential appears. These behaviors are quite similar to the “on” and “off” states in the digital circuit or transmission. Therefore, the phase dynamics of each excitable neuron can be catched by21 hi ðtÞ ¼ 2p
t sik þ 2pk; sik
sikþ1
(A6)
for sik t < sikþ1 , where sk is the time of kth firing of the ith neuron. Based on this definition, the phase and further the average frequency (firing rate) of excitable system can also be calculated. In the calculation of Fig. 1(c), ¼ 0.01, a1,…,5 ¼ 1.05, 1.01, 1.04, 1.08, 1.1, and D ¼ 0.2. The integration time is chosen as 200 000 and the transient time is 1000. For much larger N, e.g., N ¼ 20 in Fig. 3(c), ai’s are randomly chosen within ai 2 [1.01, 1.11], and to reduce fluctuation, the mean frequencies are averaged over 50 trails for different random initial conditions with the integration time 100 000 and transient time 10 000 for each trail. The detailed pattern of the average frequency in the paper exhibits a clear nonclustering phase synchronization transition with the long-tail effect.
Consider the HR neuron model23,24 e x_ i ¼ yi ax3i þ bx2i zi þ I þ ðxiþ1 2xi þ xi1 Þ 2 (A9) y_ i ¼ c dx2i yi z_ i ¼ r½Sðxi þ ji Þ zi ;
i ¼ 1; …; N
where the system parameters a ¼ 1.0, b ¼ 3.0, c ¼ 1.0, d ¼ 5.0, S ¼ 4.0, r ¼ 0.006, and I ¼ 3.0 are the same as in the literature,24 and ji is selected from a uniform distribution: ji 2 [1.56, 1.66]. For this classical HR model, which is capable of showing bursting behavior of neuron (firing stereotypical bursts of closely spaced spikes) and being chaotic, its phase can also be perfectly defined following the previous study in Ref. 24: _ sÞ=ðxðtÞ _ þ 0:1Þ; h ¼ arctan½xðt
(A10)
where s ¼ 0.5 is usually chosen. For N ¼ 5, the average frequencies are regarded as the mean value of 1000 trails with random initial conditions and the integration time 10 000 and transient time 5000. For N ¼ 20 in Fig. 3(e), ji’s are selected from a uniform distribution: ji 2 [1.5, 1.7], and the mean frequencies are averaged over 300 times with the integration time 10 000 and transient time 2000 for each time. From the numerical result in the paper, we can see that its phase transition belongs to NPS. 6. Coupled LV systems
4. Coupled Lorenz oscillators
For another classical chaotic oscillator, the model of coupled Lorenz systems is described by22 x_ i ¼ rðyi xi Þ þ eðxiþ1 2xi þ xi1 Þ; y_ i ¼ ri xi yi xi zi ; z_ i ¼ bzi þ xi yi ;
5. Coupled HR neurons
(A7)
i ¼ 1; …; N
where the “canonical” parameter values r ¼ 10.0 and b ¼ 8/3 are fixed. Although the Lorenz oscillator’s phase is not coherent showing two rotation centers, its phase can still be defined through the following simple transformation8,12 qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 hi ðtÞ¼arctan ½ xi þyi 2bðri 1Þ=½zi ðri 1Þ : (A8) Without losing generality, the system parameters r1,…,5 ¼ 36.5, 40.0, 41.0, 39.0, 35.0, and N ¼ 5 are chosen in Fig. 1(d). The average frequencies are regarded as the mean value of 1000 trails with different random initial conditions, while the integration time 5000 and transient time 2000 are adopted for each trail. In contrast, in Fig. 3(d) for N ¼ 20, ri’s are selected from a uniform distribution: ri 2 [36, 40], and the mean frequencies are averaged over 2000 tests with the integration time 10 000 and transient time 2000 for each time. A clear NPS transition shows that the average frequency of each oscillator transits to the synchronization only at the sufficiently large coupling strength.
In addition, we consider a well-known ecological LV system26–28 xi xi yi _x i ¼ axi 1 k þ eðxiþ1 2xi þ xi1 Þ K 1 þ jxi xi y i y_ i ¼ bi yi þ k ; i ¼ 1; …; 5 1 þ jxi (A11) where x and y stand for the prey and the predator species, separately. The parameters a and b denote the birth and death rates, respectively, K indicates the carrying capacity of the prey, k represents the rate of predation, and j is a constant of the functional response. Here, N ¼ 5, a ¼ 1, K ¼ 3, k ¼ 3, and j ¼ 1 are fixed, and bi’s are chosen from a uniform distribution: bi 2 [1 – 0.025, 1 þ 0.025]. The result is shown in Fig. 1(f), which again exhibits the pattern of the NPS. The integration time 100 000 and transient time 5000 are adopted here. For N ¼ 20 in Fig. 3(f), bi’s are selected from the same uniform distribution: bi 2 [1 – 0.025, 1 þ 0.025], while the mean frequencies are averaged over 100 times with the integration time 20 000 and transient time 5000, to reduce the fluctuation. All these results for various nonlinear systems including noisy FHN neuron, Lorenz oscillator, HR bursting neuron, and LV ecological system demonstrate that the path to phase synchronization in the absence of clusterings is very common, the same as the CPS, and could be extensively observed in many other coupled nonlinear systems as well.
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