Criticality in transient behavior of coupled oscillator system toward chimera and synchronization

Chimera states in spatiotemporal dynamical systems have been investigated in physical, chemical, and biological systems, while how the system is steering toward different final destinies upon spatially localized perturbation is still unknown. Through a systematic numerical analysis of the evolution of the spatiotemporal patterns of multi-chimera states, we uncover a critical behavior of the system in transient time toward either chimera or synchronization as the final stable state. We measure the critical values and the transient time of chimeras with different numbers of clusters. Then, based on an adequate verification, we fit and analyze the distribution of the transient time, which obeys power-law variation process with the increase in perturbation strengths. Moreover, the comparison between different clusters exhibits an interesting phenomenon, thus we find that the critical value of odd and even clusters will alternatively converge into a certain value from two


I. INTRODUCTION
An chimera state is a spatiotemporal pattern, in which a system of identical oscillators is split into coexisting regions of coherent and incoherent oscillation.It was found by Kuramoto in 2002, 1 and named by Abrams and Strogatz. 2,3In real-word systems, this phenomenon might play a role in the hemispherical sleep of Marine mammals or birds, 4,5 in neural spikes, 6,7 in power grids, 8 in social systems, 9 or possibly in ventricular fibrillations 10 and epileptic seizures. 11Therefore, it is of high significance to study chimera states.At first, a great deal of research was focused on chimera states in regular networks of phase-coupled oscillators Chaos ARTICLE pubs.aip.org/aip/cha with one-dimensional chain or ring topology, 1,2,12 and networks with limit populations. 13,14Subsequent research found that this peculiar phenomenon can be found on two-dimensional planes, 3,15 three-dimensional lattices, 16 torus, 17,18 and systems with a spherical topology. 19With the in-depth research, many issues like effects of transient behaviors, [20][21][22] time delay, [23][24][25] phase lags, 26 criteria for state optimization, 27 coupling functions, [28][29][30][31][32] and the impacts of specific driven forces, 33 random perturbation and complex topology of coupling [34][35][36][37][38] have been solved, e.g., the emergence of chimera states in modular networks, 39 the association between synchronization dynamics and eigenvectors of coupling matrix, 40 and the optimal control strategies based on susceptible direction identification. 41hus, the survival conditions of chimera states have been continuously expanded.7][58][59] Besides widely numerical and theoretical studies, experimental evidence of chimera states has been presented in optical, 60,61 chemical, [62][63][64] mechanical, 65 electronic, 66,67 and electrochemical 68 setups.An interesting finding is that chimera states can be considered long living transients toward the in-phase synchronization.It was experimentally confirmed that the lifetime of a chimera grows exponentially with the system size. 21Since the size of the systems in nature is limited, various spatial characteristics of chimera states can be observed, such as Traveling chimera, 69,70 Breathing chimera, 71,72 Spiral wave chimera, 73,74 and Imperfect traveling chimera. 75Next to the classical chimera state, which exhibits one coherent phaselocked and one incoherent region, there exists a new class of dynamics that possesses multiple domains of incoherence.In this regard, two clusters can be observed in coupled oscillator systems, using symmetric Gaussian initial phase. 26Stable 4-cluster chimera states are obtained by adding an external force to the coupling system or introducing time-delay. 24,76Different phase lags α in the general non-locally coupled system, 30 for the special form of the nonlocally coupling function G(x − x ), 29,69 can also induced a variety of phase clustering.Moreover, these multi-chimera states can arise in a transition from classical chimera states, depending on the coupling strength and range. 28himera states in spatiotemporal dynamical systems have been investigated in physical, chemical, and biological systems, and have been shown to be robust against random perturbations.The robustness of chimera states have been discussed by studying the effects of random removal of links between oscillators. 35Iryna Omelchenko's research demonstrated that chimera states exhibit strong robustness in the presence of perturbations in coupling topologies.The properties of chimera states as the level of inhomogeneity increases were also analyzed. 77Malchow demonstrated that chimera patterns persist for inhomogeneous parameters. 78Recently, basin stability on chimera states and how to quantify their stability after even large perturbations have been investigated. 79Furthermore, it has been discovered that chimera states exhibit a self-adaptive behavior in the presence of spatially localized perturbations.The coherent centers of the chimera spontaneously drift toward an optimal location as far away from the perturbation as possible, aiming to minimize the impact of the perturbation. 80ctually, most of the previous researches predominantly focused on examining the properties, formation conditions, and patterns of chimera states.In this paper, we focus on the process of how does the system respond to perturbations before entering the new stable state.We will discuss for different final regimes of the system, how long does it take to enter a new stable state (chimera or synchronization).
The rest of the paper is organized as follows: In Sec.II, we introduce the model equations, the order parameter, and the perturbation method that we use.In Sec.III, after a brief review of results and new aspects on the chimera states, we do experiments on multi-cluster chimeras and analyze the critical behavior of a coupled oscillator system in transient time toward chimera and synchronization.In particular, we explore how does the system respond to perturbations before entering the new stable state on the basis of spatiotemporal patterns, transient time, and analyze the critical perturbation strength on the basis of the order parameter, plots of the transient time, and bifurcation diagrams.Then, we fit the changing trend to get further descriptions.In Sec.IV, we predict the unknown critical values of larger clusters and describe an interesting phenomenon based on the comparison plots and a convergence phenomenon.Finally, we summarize our study in Sec.V.

II. THE MODEL
In this article, we consider a one-dimensional network of nonlocally coupled, identical phase oscillators with periodic boundary condition (a ring configuration).The system is mathematically described as where φ(x i ) is the phase of the ith oscillator at the spatial location x i and N denotes the size of the oscillator system.The angular velocity ω and phase lag α of the oscillators are constants in space.Without loss of generality, we set ω = 0 and α = π/2 − 0.18, which are slightly less than π/2.To induce multi-chimera states, we consider the widely used kernel function, with an extra parameter "n" for n-cluster chimera: , which is a non-negative even function that characterizes the nonlocal coupling among the oscillators.In this function, A is set to A = 0.995, which was chosen based on early references, to ensure the occurrence of chimera states and prevent the kernel function from becoming 0 for any (x i − x j ) value.As the value of A decreases, the existence domain of the 2-cluster chimera becomes smaller, and when A < 0.8, this domain no longer exists. 2,12,26The initial condition is generated using the function f(x) = 6r • exp(−0.76x 2 ), where r is a random variable uniformly distributed in [−1/2, 1/2].The complex order parameter Z(x i ) defined for oscillator i can be used to characterize the dynamics of the system, 1

Chaos ARTICLE pubs.aip.org/aip/cha
The global order parameter R, which varies with time and space, is a macroscopic quantity that can be interpreted as the collective rhythm produced by the whole population.The value of R represents the level of synchronization of the system, with 0 R(t) 1.A higher value of R indicates that the oscillators in the system are more synchronized, and R = 1 corresponds to a global synchronization state (phase locking).While the lower values indicate more disorder, and when R = 0, the oscillators run incoherently.Looking closer at the phase dynamics, the observed behavior of 0 < R < 1 is the consequence of complex spatiotemporal patterns, in which inphase synchronous domains coexist with asynchronous ones.In this case, the coherent centers are identified by calculating the local order parameters R(x i ).The oscillator with local maximum value of R(x i ) is considered the coherent centers, and in a n-cluster chimera, we would find n coherent centers.
Since the oscillator network has periodic boundary conditions, choosing different coherent centers will yield similar experimental results.Therefore, in Secs.III A-III C, we randomly select one of the coherent centers (m c ) as a representative for subsequent perturbation experiments and analysis.To assess how perturbations affect the chimera states, we apply a phase perturbation to four tar- where m represents the index of the oscillators, ranging from 1 to N. The nature of the perturbation is to force upon the oscillator a constant phase difference φ with respect to the local mean phase φlocal of its neighbors, with an equal number of neighbors on the left and right sides.Namely, φ (x m , t) = φlocal (x m ) + φ for those perturbation points.The evolution and relaxation process of chimera states under phase perturbations can be analyzed by observing spatiotemporal patterns, phase plots, and snapshots, and by measuring the phase velocities, the relaxation, and the transient time, etc.

III. RESULTS
In this paper, numerical simulations are conducted in a onedimensional network of N = 240 non-locally coupled identical phase oscillators with periodic boundary condition.Through setting n = 1, 2, 3, 4, 5 in the coupling function G(x i − x j ), one can generate chimera states consisting of n clusters.We apply phase perturbations to four oscillators located at the center of the system, where the initial condition is set to be a coherent center of the chimera.Figure 1 shows the spatiotemporal patterns of 1, 2, 3, 4, 5-cluster chimera, in response to perturbations at four target oscillators with increasing strength φ.

A. Two different destinies of the system under perturbation
According to the spatiotemporal pattern of oscillators, it can be found that after a period of transient time, the system evolves from the initial condition to a certain stable state.The final stable state would either be chimera or synchronization, depending on the strength of the perturbation φ (see Fig. 1).
For the case with small perturbation strength φ (as is shown in the left-hand side panels in Fig. 1), the system will enter the chimera state after a self-adaption transient process, in which the global order parameter of the n-cluster chimera and the boundary between incoherent and coherent regions gradually evolved to its stability.
However, for the case with large enough perturbation strength φ (as is shown in the right-hand side panels in Fig. 1), the chimera state will collapse and the in-phase synchronized oscillatory state takes over.We conjecture that this phenomenon occurs for the reason that the basin of chimera is typically relatively small, with respect to that of the synchronization.Strong perturbations on oscillators destabilize these multi-cluster chimera states, and thus promote global synchronization.

B. The critical value of the perturbation strength
As mentioned above, stronger perturbations lead to a phase transition from chimera to synchronization.But how to describe the influence of perturbations φ is worth to be further studied.
First, the transient time (T) is defined as the duration from the moment of perturbation to the occurrence of a stable chimera or synchronization states.By observing the spatiotemporal pattern and the variation of R, we have found that when the chimera state is reached, R will fluctuate regularly around a certain value as a breathing chimera with a stable rhythm.Therefore, the transient time T of a chimeras is from the initial moment to the moment that R starts to behave as a breathing chimera.However, if the global order parameter R cannot exhibit regular fluctuations and instead gradually converges to R = 1, it indicates that the system enters a fully synchronized state instead of a chimera.Hence, from the initial moment to R approaches 1 is the transient time before reaching the fully synchronized state.
We sweep the perturbation strength from 0 to π with a step size of 0.01π and apply the perturbation to the target oscillator in the chimera state.Then, we observe the spatiotemporal patterns of the system's evolution at each perturbation strength.From the spatiotemporal patterns, we can clearly see the state in which the system maintains after a certain period of evolution.As the perturbation strength increases, the time for the system to recover the chimera state gradually lengthens and extends to infinity, until it exceeds a certain perturbation threshold φ c .Beyond this threshold, the system cannot recover the chimera state and transitions to a fully synchronized state.This pertubation threshold can be considered the critical value φ c between these two possible final states.When the perturbation strength is less than φ c , the system can resist the perturbation through self-adaptations; however, when the perturbation strength is greater than φ c , chimera collapses and is replaced by synchronization, instead of making adaptive adjustments.
For each cluster's chimera, we calculated the critical values of perturbation strength under ten different initials, denoted as ( φ cn 1 , . . ., φ cn 10 , where n represents the cluster number, and 1-10 correspond to the ten initial groups).The critical value of that cluster's chimera is the average of these ten initial groups: φ cn = avg( φ cn 1 , . . ., φ cn 10 ).Since the results were obtained from multiple experiments with ten sets of initial values, and the transient time curves exhibit consistent trends for different initial values, averaging all the trials provides a more accurate estimation of the critical value for that specific cluster's chimera.In addition, we have also conducted a comparative analysis of critical values, procedures of phase transitions, and the transient time among different cluster structures.These difference well describe antiinterference capabilities and robustness of different chimera structures.We conjecture that the larger the perturbation critical value φ c is, the stronger anti-interference ability of the structure would have.

C. Criticality observed from transient time
The transient time (T) that denotes the duration for the system to achieve a stable state of either chimera or synchronization is plotted in Fig. 2 as a function of perturbation strength φ.The variation trend of T on both sides of the critical value φ c behaves generally the same for the chimera with different numbers of clusters.Figure 2 shows the results from the systems with ten different initial states, and the qualitative tendency for the transient time T is basically the same.To eliminating the influence of initials and make the changing characteristic clearer, we subtract the critical values φ cn from the X axis of the transient time curve and perform a translation.In this way, curves were centralized, and X axis describes the distance between the perturbation strength and the critical value (X = φ − φ c ) as shown in Fig. 3, and X = 0 corresponds to the critical point.Specifically speaking, when the perturbation strength approaches the critical value on the left, the transient time toward chimera increases rapidly according to a power law.However, when the perturbation strength is larger than the critical value and keeps increasing, the transient time of synchronization decays rapidly in the form of another power-law distribution.Moreover, the transient time approaches infinity around the critical point.In order to find the power-law distribution that the transient time toward two states obeys, we average the original data and present it in a log-log plot.We use least squares fitting to estimate the parameters in the power-law relationship on the logarithmic scale, i.e., log( T) = −αlog(X) + log(C) (see Fig. 4) or T ∼ C( φ − φ c ) −α , and the fitting error R 2 can be adopted to judge the goodness of the fitting (shown in Tables I and II).One can see that the magnitude of their exponents is in the same order of the critical value: α 1 < α 3 < α 5 < α 4 < α 2 .Moreover, the order of T's variation range is also the same: T1 < T3 < T5 < T4 < T2 .It means that the larger the critical value, the larger the variation range of the transient time, and the higher the stability and robustness of the system.Meanwhile, the transient time before entering the synchronization can also be well fitted by a power-law distribution, but more interestingly, for each cluster case, when the perturbation strength increases, the transient time decreases in a power law with similar power exponents, which are all approximately equal to 0.27.However, values of the fitted intercepts have the same order as the critical values, which is

IV. THEORETICAL ANALYSIS OF THE CRITICAL VALUE A. Comparison and assumption
After centered the perturbation strength and averaged the transient time of ten initial groups, we can compare the critical value of n-cluster chimera (n = 1, 2, 3, 4, 5) in the same coordinate system as shown in Fig. 5.
According to the above results and figures, we found an interesting phenomenon, that is, as the number of clusters increases, the difference between critical values becomes smaller and smaller, which reminds us of the Feigenbaum constant in the period-doubling bifurcation, which expresses the limit of the ratio of distances between consecutive bifurcation diagram. 81Therefore, we suppose that as the cluster number approaches infinity, the critical values of odd and even clusters will converge into the same value from two sides in two proportional sequences.A crude estimation of the common ratio of critical values of odd clusters is q 1 , which can be calculated by Then, based on the summation formula of the proportional series: S 1 = a 1 (1 − q n 1 )/(1 − q 1 ), as n → +∞, we can predict the critical value of the n-cluster chimera by = φ c1 + a 1 /(1 − q 1 ).After that, the common ratio of even clusters q 2 can be obtained from ).First, using critical values from our experiments, we have a 0.022π , and q 1 = a 2 a 1 = 0.4234 consequently.Thus, the predicted critical value of the n-cluster chimera with n → ∞ is S 2 = φ c2 − = 0.068π and q 2 = 1 − 0.022π /S 2 = 0.6763 are obtained.Moreover, based on the relationship , we can derive a relation equation between q 1 and q 2 , that is, 0.222/(1 − q 1 ) + 0.022/(1 − q 2 ) = 0.453.As the assumption stated above, this convergence process is demonstrated in Fig. 6.

B. Deviation analysis
From the assumption above, we can predict the critical value of the n-cluster chimera, using φ c1 , φ c2 , φ c3 , φ c4 , φ c5 that we obtained in experiments.However, there exists a standard deviation for each calculated critical values σ c1 = 0.0327, σ c2 = 0.0126, σ c3 = 0.0135, σ c4 = 0.0117, σ c5 = 0.0574.Therefore, the predicted has a standard deviation σ c either, which can be calculated by using the error transfer formula of mathematical statistics.The procedures are as follows: We already know that = φ c1 + a 1 1 1−q 1 .Thus, based on the error transfer formula, we have The standard deviation of n-cluster's critical value with n → ∞, i.e., the standard deviation of is σ c = 0.9858.

C. Extension
To confirm the universality of the critical behavior of the transient time and the transion of the final stable state from chimera to sycronization, we conducted more simulation for the system of larger sizes.It was observed that with an increasing φ, the transition behavior from a 1-cluster chimera or a 2-cluster chimera to synchronization was also quantitatively identical in larger-sized systems.Figure 7 illustrates the results of a system with N = 480.Furthermore, on both sides of the critical point φ c , which corresponds to the transition from the initial state to either the chimera state or the synchronized state, the transient behavior exhibits scaling characteristics.When the network size becomes larger, we find that an equivalent perturbation cannot be achieved by proportionally increasing the number of perturbed oscillators.The study of the number of perturbed oscillators is a topic that deserves further investigation in the future.In this paper, we only discuss the impact of increasing perturbation strength on transient behavior.
We have also attempted a scenario closer to the real Kuramoto model, in which the self-frequencies ω i of each oscillator were not constant but randomly selected from a Gaussian distribution with different standard deviations.The formation conditions for the chimera state are quite stringent, as the chimera state can only be formed when the ω i values are completely identical (homogeneous).
As the standard deviation σ of the Gaussian distribution N(0, σ 2 ) used for sampling the ω i values increases, it indicates a higher degree of heterogeneity, and the disorderliness of the system also gradually increases.From the spatiotemporal pattern in Fig. 8, we may find that when the standard deviation of the Gaussian distribution (representing the heterogeneity of ω i values) is non-zero, the system cannot form a stable chimera state.Moreover, as σ becomes larger, the system enters a disordered state.Therefore, due to the strict conditions for the existence of chimera, the transient time behavior of chimera and the transition from chimera to synchronization observed in simulations of identical oscillator systems would be difficult to observe in real Kuramoto systems, where the oscillators have non-identical ω i values.

V. CONCLUSION AND DISCUSSION
In summary, we have numerically solved the coupled phase oscillator system of multi-chimera states (1, 2, 3, 4, 5-cluster chimera) upon phase perturbations, in order to investigate the critical behavior of the coupled oscillator system in the transient time toward chimera and synchronization.
First, we have introduced the definition of the transient time, experimental procedures, and measurement methods that we use.Next, based on our results, we have discussed how does the system respond to perturbations before entering the stable state and found that there is a critical value between chimera and synchronization that the system will get into.Once the perturbation strength is greater than the critical value, chimera will be taken over by synchronization.After fitting the changing trend of the transient time based on a log-log verification and the linear regression model, we have found that with the increase in the disturbance strength, the transient time obeys an increasing power-law distribution on the left-hand side of the critical value, and a decreasing power-law distribution on the right-hand side.In addition, we have shown that the parameters in the fitted model are associated with the critical value of this chimera.
Particularly, based on the comparison plots and the interesting phenomenon that the critical value of odd and even clusters will converge into a certain value from two sides in two proportional sequences, we have calculated the common ratio, made a prediction for the critical value of n-cluster chimera, and performed a deviation analysis finally.The mechanism for the transitions can be summarized as follows: chimera, asynchronization, and synchronization states have their own attraction basins, respectively, in the state space.For the classical synchronization transition through increasing coupling strength among oscillators, 82 the increase in coupling strength actually leads to the expansion of the attraction basin of the synchronization state, which, in turn, produces transition from asynchronization to synchronization.In our study, the transition from chimera to synchronization is induced by applying phase driving upon a small set of oscillators and, in fact, the phase driving does not change the attraction basins of chimera and synchronization, but directly pulls the state of the system from the attraction basin of chimera to that of synchronization.Briefly speaking, the mechanism of transitions can be classified into two types, one is due to the change of the attraction basins, and the other is due to the change of state crossing the boundary of two different basins.The transitions in our work belong to the latter case.

FIG. 2 .
FIG. 2. The transient time under perturbations.The solid lines describe transient times toward chimera, while the dotted lines describe those toward synchronization.The parameter n represents the cluster number of chimera states: (a) transient time of the 1-cluster chimera in response to perturbations, (b) transient time of the 2-cluster chimera, (c) transient time of the 3-cluster chimera, (d) transient time of the 4-cluster chimera, and (e) transient time of the 5-cluster chimera.Each figure has ten pairs of lines, representing the averaged measurement from different initial groups.

FIG. 3 .
FIG. 3. Averaged transient time, corresponding to the distance between perturbation strength and the critical value (X = φ − φ c ), which is zero-centered.The solid lines describe the transient time toward chimera, while the dotted lines describe the transient time toward synchronization.The parameter n in figures represents the cluster number of the chimera system: (a) transient time of the 1-cluster chimera in response to perturbations, (b) transient time of the 2-cluster chimera, (c) transient time of the 3-cluster chimera, (d) Transient time of the 4-cluster chimera, and (e) transient time of the 5-cluster chimera.Each figure has ten pairs of lines, representing the averaged measurement from different initial groups.

FIG. 4 .
FIG. 4. Transient time in logarithmic coordinates.(a) The transient time toward chimera in logarithmic coordinates, where points are measured values and the solid lines are fitted models.(b)The transient time toward synchronization in logarithmic coordinates, where points are measured values and the solid lines are fitted models.The red dotted line is the predicted distribution in n-cluster chimera (n → ∞).

FIG. 5 .
FIG. 5.Comparison of the critical values and the changing trend of the transient time toward chimera and synchronization.In this way, the order of critical values is represented clearly: φ c1 < φ c3 < φ c5 < φ c4 < φ c2 .Moreover, the red dotted line is a prediction of n-cluster chimera's critical value with n → ∞, denoted by , which is obtained based on the convergence phenomenon.

FIG. 6 .
FIG.6.The convergence process.(a) The X axis is the number of clusters, while the Y axis corresponds to the critical value.As the cluster number approaches infinity, the critical value of odd and even clusters will converge into a certain value from two sides in two proportional sequences.(b) is a schematic illustration of the convergence process.φ ci is the critical value of the i-cluster chimera, a k is the difference between the critical value of the (2k + 1) and (2k − 1)-cluster, and b j is the difference between the critical value of the 2j and (2j + 2)-cluster(i, k, j = 1, 2, 3, • • •).q 1 and q 2 are common ratios of proportional sequences of odd and even clusters, respectively.The convergence direction is represented by red arrows.

FIG. 7 .
FIG. 7. Transient time for the system with N = 480.(a) Transition from chimera to synchronization as φ increases through the critical value marked by the dashed line, for the system with the number of clusters to be 1 (blue curves) and 2 (green curves), respectively.(b) The transient time toward chimera in logarithmic coordinates, where points are measured values and the solid lines are the linear fittings.The fitted slopes are −0.2685(blue line) and −0.2134 (green line).(c) The transient time toward synchronization in logarithmic coordinates, where points are measured values and the solid lines are the linear fittings.The fitted slopes are −0.2570(blue line) and −0.1990 (green line).

FIG. 8 .
FIG. 8. Spatiotemporal patterns of systems with non-identical ω i and different sizes.(a) N = 240, (b) N = 480.The self-frequencies ω i of the oscillators are randomly selected from Gaussian distributions N(0, σ 2 ) with different standard deviation σ .The x axis represents the oscillator index, and the y axis represents the evolution time steps.