Chimera dynamics in nonlocally coupled moving phase oscillators
Wenhao Wang, Qionglin Dai, Hongyan Cheng, Haihong Li, Junzhong Yang
aa r X i v : . [ n li n . AO ] M a y Chimera dynamics in nonlocally coupled moving phase oscillators
Wenhao Wang, Qionglin Dai, Hongyan Cheng, Haihong Li, ∗ and Junzhong Yang † School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China (Dated: May 23, 2019)Chimera states, a symmetry-breaking spatiotemporal pattern in nonlocally coupled dynamicalunits, prevail in a variety of systems. However, the interaction structures among oscillators arestatic in most of studies on chimera state. In this work, we consider a population of agents. Eachagent carries a phase oscillator. We assume that agents perform Brownian motions on a ring andinteract with each other with a kernel function dependent on the distance between them. Whenagents are motionless, the model allows for several dynamical states including two different chimerastates (the type-I and the type-II chimeras). The movement of agents changes the relative positionsamong them and produces perpetual noise to impact on the model dynamics. We find that theresponse of the coupled phase oscillators to the movement of agents depends on both the phase lag α , determining the stabilities of chimera states, and the agent mobility D . For low mobility, thesynchronous state transits to the type-I chimera state for α close to π/ α . We investigate the statistical properties inthese different dynamical regimes and present the scaling laws between the transient time and themobility for low mobility and relations between the mean lifetimes of different dynamical states andthe mobility for intermediate mobility. I. INTRODUCTION
Chimera states refer to the type of spatiotemporal pat-tern consisting of coexisting coherent and incoherent re-gions. Since the first observation in nonlocally coupledidentical phase oscillators in 2002 [1], chimera states haveevolved in the last decade from surprising symmetry-breaking patterns to prevailing dynamical phenomenaranging from physics and chemistry to biology, from clas-sical to quantum systems [2–12]. Chimera states arenot only numerically found in mathematical models suchas periodic and chaotic maps [13], mechanical oscilla-tors [14], neuronal oscillators [15–17], but also exper-imentally realized in mechanical systems [7, 18], opti-cal systems [9] and chemical systems [8]. Recent workshave shown that chimera states may exist even in oscil-lators coupled globally [19–22] as well as locally [23, 24].Chimera states are not restricted to topologies such asrings, square lattices [25, 26], torus [27], spheres [28],and Erd¨os-R´enyi networks and scale-free networks [29].Chimera states are robust to random rewiring of edges[30] in a ring of symmetrically nonlocally coupled phaseoscillators. Though most of works dealt with coupledoscillators where units support self-oscillating solutions,chimera states have been studied in nonlocally coupledexcitable systems only allowing for an equilibrium. Se-menova et. al. realized chimera state in excitable unitsin the presence of noise through coherent resonance [31]and Dai et. al. found that chimera state may emergeout of excitable units through a coupling-induced collec-tive oscillation [32]. The connections between chimera ∗ Electronic address: [email protected] † Electronic address: [email protected] states and other dynamical phenomena have been inves-tigated. Motter et. al. proposed a connection betweenchimera states and cluster synchronization in networksof locally coupled chaotic oscillators [33]. Lai et. al.established a connection between chimera states and aquantum scattering phenomenon in 2-dimensional Diracmaterial systems where manifestations of classically inte-grable and chaotic dynamics coexist simultaneously [12].Dai et. al. reported a desynchronization transition fromsynchronous to asynchronous chimera states in a bicom-ponent phase oscillator system when the frequency mis-match among oscillators increases [34].Most of previous works on chimera states have as-sumed the interaction between oscillators to be static.However, it has been found in various fields, such aspower transmission system [35], consensus problem [36],and person-to-person communication [37], that there aremany interesting scenarios while interaction patterns aretime-varying. The time-varying interaction patterns maybe realized either by rewiring connections in networks inthe course of time or by assuming oscillators to move inspace. In the past few years, the interest in the collec-tive dynamics in coupled oscillators has grown rapidlywhen oscillators may move in space. Different dynam-ical behaviors have been investigated, such as ampli-tude death and resurgence of oscillation [38], mobility-enhanced signal response [39], and synchronization [40–42]. It has been shown that synchronization time maydepend non-monotonically on the mobility of oscillatorsand the mechanism of driving synchronization are dif-ferent for different dynamical regimes [43–45]. Chimeradynamics in mobile oscillators has not been investigatedexcept for a recent work, in which an array of locallycoupled oscillators exchange positions and the persistentchimera states may be maintained under the interplaybetween the mobility and time delay [46].In this paper, we study how chimera dynamics reactsto the mobility in a ring of nonlocally coupled phase os-cillators and identify different regimes ruling the chimeradynamics. The rest of paper is arranged as follows. Insection 2, we present the model in which N agents, eachof which is associated with a phase oscillator, performBrownian motions on a ring. In section 3, we first presentthe dynamical states existing in the model when agentsare motionless. Then we study the response of these dy-namical states to the mobility and the statistical proper-ties of the responses. Finally, we conclude with a sum-mary in section 4. II. MODEL
We consider N agents performing independent Brow-nian motions on a ring with the length L . The positionof agent i , x i , evolves according to˙ x i ( t ) = η i ( t ) (1)with periodic boundary condition x i ( t ) + L = x i ( t ). Weassume that the noise η i ( t ) has a Gaussian probabilitydistribution with mean and correlation function h η i ( t ) i = 0 , h η i ( t ) η j ( t ′ ) i = Dδ ij δ ( t − t ′ ) . (2) D denotes the noise strength.Furthermore, each agent is associated with a phase os-cillator and, then, each agent is characterized by a phasevariable θ . Agents interact with each other with a nonlo-cal coupling strength depending on the distance betweenthem and the phase variable θ i of agent i evolves in termsof the following equation˙ θ i ( t ) = ω − N X j=1 G ( d ij ( t )) sin[ θ i ( t ) − θ j ( t ) + α ] . (3) α is the phase lag and ω is the natural frequency forall oscillators. Without losing generality, ω is set to bezero. d ij ( t ) = min ( | x i ( t ) − x j ( t ) | , L − | x i ( t ) − x j ( t ) | ) isthe distance between agents i and j . The kernel func-tion G ( d ij ( t )), the nonlocal coupling between oscillators i and j , is assumed to be even, nonnegative, decreas-ing with d ij ( t ). To be concrete, we consider widelyused kernel function G ( d ij ( t )) = [1 + A cos(2 πd ij ( t ) /L )]with 0 ≤ A ≤
1. Comparing with the exponential ker-nel G ( x ) ∝ exp ( κ | x | ) [1], the cosine kernel allows themodel to be solved analytically besides qualitatively sim-ilar results[2]. The parameter A controls the nonlocalcoupling, and set to be 1 in this paper. The other pa-rameters are chosen as N = 128 and L = 2 π . To ourknowledge, the model (3) with all motionless oscillatorsevenly distributed along the ring allows for two types ofchimera states besides the synchronous state. There isonly one coherent domain in the type-I chimera while (a1) x (a2) x (b1) t (b2) t R(t) (c1) t (c2) t X C (t) FIG. 1: The chimera states at α = 1 .
38 in the left columnand at α = 1 .
47 in the right column. (a) The snapshot of θ .(b) The evolution of R ( t ). (c) The evolution of the locationsof the coherent clusters X C ( t ). D = 0 and oscillators areassigned with positions x i = iL/N ( i = 1 , , · · · , N ). there are two coherent domains in the type-II chimerawith oscillators in different coherent domains being inantiphase. For A = 1, the type-I chimera is stable for α higher than around 1 .
31 [2] while the type-II chimeraexists for α higher than around 1 . III. RESULTS AND DISCUSSION
We consider three types of initial conditions preparedat the corresponding α with the positions of motionlessagents x i = iL/N ( i = 1 , , · · · , N ), the synchronousstate with all oscillators in phase [ θ i ( t ) = θ ( t )], thetype-I chimera with one coherent domain in which coher-ent oscillators are almost in phase(shown in left columnof Fig. 1), and the type-II chimera with two coherentdomains (shown in right column of Fig. 1). The syn-chronous state is always locally stable when α < π/ α close to π/ α = 1 . α = 1 .
47. The differenceof the two chimera states in the number of coherentdomains is obvious in the snapshots of the spatiotem-poral plots of oscillators’ phases. To further distin-guish the two chimera states, we consider two quanti-ties. The first is the global order parameter defined as Z ( t ) = R ( t ) e i Θ( t ) = P Ni =1 e iθ i ( t ) . R ( t ), the amplitudeof global order parameter, fluctuates around 0 .
65 for thetype-I chimera. In contrast, R ( t ) for the type-II chimerais much lower since oscillators in different coherent do-mains are in antiphase, as shown in Fig.1(b2), R ( t ) isalways less than 0 .
35 for the type-II chimera. The sec-ond quantity is the location of the coherent domains, X c ,represented by the center of the coherent domain. Tomeasure it, we consider position-dependent order param-eter Z ( x, t ) = R ( x, t ) e i Θ( x,t ) = P Nj=1 G ( d j ) e iθ j ( t ) with d j = min ( | x − x j | , L − | x − x j | ). R ( x, t ) roughly reachesits local maximum at X c . The locations of the coher-ent domains, X c , are defined as the positions at whichthe amplitude R ( x ) reaches its local maximum. Sincethere are two coherent domains in the type-II chimera,as shown in Fig. 1(c2), the coherent domains for thesetwo chimera states move along the ring in an irregularway, which is due to the finite size effects [47]. It hasbeen concluded theoretically for the type-I chimera thatthe size of the coherent domain decreases with α increase[2], which suggests that the global order parameter R de-creases with α . It can be numerically confirmed that thetype-II chimera against α behaves in a similar way.Then we consider how the model Eqs. (1-3) reacts tothe movement of agents. With nonzero D , agents walkrandomly along the ring. Between successive times t and t ′ , the displacement of an agent , | x ( t ) − x ( t ′ ) | = p D ( t − t ′ ) is given as square root for short time in-terval t − t ′ , which suggests that the mobility of agentsmay be measured by the noise strength D and, large D means high mobility. The relative positions betweenagents change as time goes on, which takes effects on thedynamics of agents in two aspects. Firstly, the mobilityof agents may act as perpetual perturbation to chimerastates when coherent oscillators may move out of coher-ent domains and incoherent oscillators may move intocoherent domains. The higher the mobility, the strongerthe perturbation. It is well known that, for the modelhere, chimera states in a finite number of motionlessagents are always transient to the synchronous state [47]but with transient time increasing with the number of os-cillators in an exponential way. Therefore, the mobilityof agents may drive a chimera state to the synchronousstate. Secondly, the summation P Nj=1 G ( d ij ) is no longera constant and, it becomes both time-dependent andagent-dependent. Consequently, the time-varying sum-mation for each agents actually act as a perpetual per-turbation to the synchronous state, may prevent chimerastates from being attracted by the synchronous state.For the convenience of illustration in the following, wepresent an example of dynamical process with movingagents. Fig. 2(a) shows that R spends most time on thesections with R close to 1, R close to 0, and R close to0 .
65 and these sections are connected by frequent jumpsamong them. The insets in Fig. 2(a) show the snapshotsof oscillators in these three time sections with different R . Clearly, the sections with R close to 1 is in the syn-chronous state while the time sections with R close to0 .
65 and 0 are featured by the type-I and the type-IIchimeras, respectively. We also plotted the averaged R in the moving window with the length of 50 time unitsin order to reduce the fluctuation in R in value and thequalitative features are the same. Figure 2(b) plots X c (b) X c t (a) R (t) FIG. 2: (a) The evolutions of R ( t ) (black) and the windowaveraged R ( t ) (red). The insets show the snapshots of θ fordifferent dynamical states, the synchronous state (blue) where R close to 1, the type-I chimera (green) with R close to 0 . R close to 0. The ar-rows denote the time at which snapshots are taken. (b) Theevolution of the locations of the coherent domains. A = 1, α = 1 .
38, and D = 4 × − . with time. In the time sections of the type-II chimera,there are two possible X c at the same time. In contrast,the synchronous state and the type-I chimera state arehard to be distinguished by X c though X c varies slowlywith time in the sections.The response of the model Eqs. (1-3) to the mobilityof agents depends on α . We consider three cases: 1) α = 1 .
38 close to the stability boundary of the type-I chimera where the type-II chimera cannot be main-tained; 2) α = 1 .
47 where both the type-I chimera andthe type-II chimera can be maintained; 3) α = 1 .
54 closeto α = π/ α with the positions of motionless agents x i = iL/N , and the type-II chimera in Fig. 1(a2). Foreach parameter combination, α and D , and each typeof initial conditions, we run tens of realizations and thetypical results are presented in Fig. 3. The left column inFig. 3 shows the results for weak mobility D = 10 − . For α = 1 .
38, both the type-I and the type-II chimeras evolvetowards the synchronous state though the transient timefor the type-II chimera is much longer than that for thetype-I chimera. However for α = 1 .
47, only the type-Ichimera will transit to the synchronous state while thetype-II chimera is maintained for the simulation time upto 6 × time units. Interestingly, weak mobility maytransfer the synchronous state to the type-I chimera at α = 1 .
54 while the type-II chimera is robust to the per-turbation induced by the mobility of agents. For inter-mediate mobility D = 4 × − , the coupled oscillators
100 1000 10000 1000000.00.51.0 1000 10000 100000 1000 10000 1000000.00.51.00.00.51.0 t t t R (i)(h)(g) (f)(e)(d) (c)(b) D=4*10 -3 =1.47 =1.54 D=4*10 -2 R =1.38 D=10 -5 R (a) FIG. 3: The evolutions of R starting with the synchronous state (black), the type-I chimera (red), and the type-II chimera(blue) for different α and D . D = 10 − in the left column, D = 4 × − in the middle column, and D = 4 × − in the rightcolumn. α = 1 .
38 in the top row, α = 1 .
47 in the middle row, and α = 1 .
54 in the bottom row. response to the mobility of agents at α = 1 .
38 by jumpingamong the synchronous states, the type-I and the type-II chimeras in an irregular way regardless of the initialstates. In contrast, the coupled oscillators jump only be-tween the type-I and the type-II chimeras at α = 1 . α = 1 .
54. The right column in Fig. 3 shows the situa-tion with high mobility D = 4 × − . Independent of α and initial conditions, R fluctuates around the valueclose to the type-I chimera. Increasing α reduces thefluctuation. Actually, there is no clear signature for thetype-I chimera to be identified, which may be caused byfast movement of X c if the dynamical state does belongto the type-I chimera.The response of dynamical states to the mobility ofagents may be related to the basin stability [48]. Thebasin stability is a method to account for the stabilityof a dynamical state by using its attraction basin butnot the linear stability analysis. Generally, the basinstability of a dynamical state is strong if the fraction ofinitial conditions leading to the state is high. There aretwo competing processes when a dynamical state is sub-jected to perpetual perturbation, the perturbation drivesthe system away from the state and even to exit its at- -7 -6 -5 -4 -5 -4 -3 -0.4 (a) T D -0.4 (b) D (c) T -0.56 (d) D FIG. 4: The transient time T is plotted against the mobility D at α = 1 .
38 in (a) and at α = 1 .
47 in (b), respectively, withthe type-I chimera as the initial state. The blue line denotesthe fitting curve T ∼ D − . . (c) T against α with the type-Ichimera as the initial state, D = 10 − . (d) T against D withthe type-II chimera as the initial state, α = 1 .
47 and the datais fitted to be T ∼ D − . (the blue line). (b) DD (a)
FIG. 5: The mean lifetimes of the synchronous state (black),the type-I chimera (red), and the type-II chimera (green) areplotted against D . α = 1 .
38 in (a) and α = 1 .
47 in (b). traction basin while the nonlinear stability draws thesystem within the attraction basin back towards it. Inthis view, how a system owning multistable states reactsto perpetual noise is determined by the basin stabilitiesof these dynamical states and the strength of perpetualnoise. The stronger the basin stability of a state, themore possibly the state to be an absorbing state. Thestronger the perpetual noise, the more possibly the sys-tem to be taken away from a dynamical state. For themodel Eqs. (1-3), the basin stabilities of the type-I andthe type-II chimeras increase with α while the one of thesynchronous state decreases with α . Considering that theglobal parameter R for the type-I chimera is in betweenthe synchronous state and the type-II chimera, it seemsthat it is easier for the type-II chimera to transit to thetype-I chimera than to the synchronous state, which isalso observed in extensive simulations. We first considerthe case with weak mobility. For α = 1 .
38, the basin sta-bility of the synchronous state is much stronger than thatof the type-I chimera and, the type-II chimera is not sus-tainable, which leads to the synchronous state to be theonly absorbing one. For α = 1 .
47, the type-II chimerais stable and the results in Fig. 3(d) suggest that thetype-I and the type-II chimeras have comparable basinstabilities. In comparison with the synchronous state, the type-I and the type-II chimeras have weaker basinstabilities, which results in that initial type-I chimera isabsorbed by the synchronous state. However, the type-II chimera is sustained since the type-I chimera plays arole of buffer to keep the type-II chimera away from thesynchronous state. For the case with intermediate mobil-ity, the noise is strong enough to take the system out ofthe attraction basin of any dynamical state. As a result,we find the frequent jumps of the system among differentstates for α = 1 .
38. It is also interesting to note that, thesystem is just only between the two chimeras for α = 1 . α = 1 .
54, whichsuggest that the driving away from dynamical states byperpetual perturbation and drawing back to the state byits nonlinear stability contributes to the response of thesystem in a collective way but not in a simple way. Whennoise is sufficiently strong, the perpetual noise dominatesand it drives the system away from the three states. Un-der such strong noise, the effect of nonlinear stability canhardly be seen.To get a clearer view on the response of the system tothe mobility of agents, we investigate the transient time T of a given initial state before it is captured by the syn-chronous state for weak mobility (small D ). Bearing theBrownian motions of agents in mind, we know that tran-sient time T may not be the same in different simulationruns even for the same initial conditions. For this sake,we record the transient time for those realizations leadingthe system to the absorbing synchronous state. The tran-sient time T for each D is averaged over 20 realizations.Figures 4(a) and (b) show T against D for α = 1 .
38 and α = 1 .
47 with the type-I chimera as initial conditions, re-spectively. Interestingly, T seems to scale with D roughlyas T ∼ D − γ . Fitting the curve with a linear function inthe double-log plot, we find that the exponent γ ≃ . α . To be mentioned, the scalingrelation between T and D is fitting well just for weak mo-bility. For D > × − at α = 1 .
38 and for
D > × − at α = 1 .
47, the synchronous state is competing with thejumping between different dynamical states, which re-sults in the violation of the scaling relation. Actually,the transient time T here amounts to the first passagetime of the coupled oscillators to the boundary of theattraction basin of the type-I chimera. For a Brownianparticle in a homogeneous space, the first passage time ofthe particle to leave a given domain scales with D with anexponent − γ = 0 . T against D .The transient time T against D tends to deviate from thescaling law at large D . The comparison between Fig. 4(a)and (b) also shows that the transient time T at large α is much larger than that at small α , behind which is thatthe basin stability of the synchronous state (the chimerastates) decreases (increase) with α . Then, we investigatethe transient time T against α at D = 10 − . As shownin Fig. 4(c), T increases with α . When α goes down be-low 1 . T decreases to 0 since chimera states becomeunstable at sufficient low α . On the other hand, T showsa sharp rise and large fluctuation at around α = 1 . T to the synchronous state also exhibits a scaling lawwith D . The exponent γ ≃ .
56 is different from thatwith the type-I chimera as initial conditions, which fur-ther supports the collective contributions to the transienttime from the mobility of agents and the nonlinear stabil-ity of the current dynamical state. Comparing with thecase with the type-I chimera as initial conditions, thecase with the type-II chimera as initial conditions showslonger transient time and stronger fluctuation, both ofthem result from the facts that the route from the type-II chimera to the synchronous state has to involve thetype-I chimera.For intermediate mobility, the coupled phase oscil-lators jump among different dynamical states and thelengths of the time sections holding a certain dynami-cal state may fluctuate greatly. We are interested in thejumping dynamics which may be characterized by themean lifetimes τ of different dynamical states. To calcu-late τ , we set two R , R = 0 . R = 0 .
4. The dy-namical state is thought to be in the synchronous statewhen
R > R , in the type-II chimera when R < R ,and the type-I chimera when R < R < R . To re-duce the fake jumps among these states, we monitorthe window averaged R . The coupled phase oscillatorsjump among the synchronous state and the two chimerastates at α = 1 .
38, and Fig. 5(a) presents τ against D forthese states. With D increasing, we find that the meanlifetimes of both the synchronous state and the type-IIchimera decrease. On the contrary, the mean lifetime ofthe type-I chimera increases with the mobility D and sat-urates at large D such as D = 10 − . The observationsthat τ tends to zero for both the synchronous state andthe type-II chimera and that τ stays at around hundredsof time units for the type-I chimera at large D indicatethat the jumps among the three states become frequentwith D increase. Then we consider the jump dynamicsat α = 1 .
47 where the jumps occur mainly between thetwo chimera states. We treat the time sections holdingthe type-I chimera as those with
R > R and holding thetype-II chimera otherwise. The mean lifetimes against D are presented in Fig. 5(b). Increasing the mobility alwaysleads the rise (fall) of the lifetime of the type-I (the type-II) chimera. The abnormal behavior of the lifetime of thetype-I chimera that τ is much high for D < − hintsthat the existence the synchronous state in some timesections cannot be ignored. Additionally, Fig. 5 shows that the time sections supporting either the synchronousstate or the type-II chimera do not exist for high mobility. IV. CONCLUSION
In this work, we considered N agents performing Brow-nian motions on a ring. Each agent carries a phase os-cillator and interacts with others with a kernel functiondepending on the distance between them. When agentsare motionless, the model reduces to a standard one inthe field of chimera state and allows for three dynamicalstates, the synchronous state, the chimera state with onecoherent domain (the type-I), and the chimera state withtwo coherent domains (the type-II). The motion of agentsimpacts on the model dynamics by the perpetual distur-bances induced by the change of relative positions amongagents. We found that the response of the coupled phaseoscillators to the movement of agents depends on boththe phase lag α and the agent mobility D . For low mobil-ity, the type-I chimera state transits to the synchronousstate for α more less than π/ α . We also investigated the sta-tistical properties in these different dynamical regimes.We found the scaling laws between the transient timeand the mobility for low mobility and presented the rela-tions between the mean lifetimes of different dynamicalstates and the mobility for intermediate mobility. In thiswork, we did not discuss the situation with sufficientlyhigh mobility. Uriu et. al. [49] have shown that mo-bile coupled oscillators behave as a mean field system athigh mobility of oscillators and Petrungaro et. al. [50]have showed that chimera states disappear for very highmobility and the global synchronization dominates thesystem. For the model with Eqs. (1-3) in this work, thesynchronous state with all oscillators being in phase is nolonger the solution to the model [see the inset in fig2(a)].Though the chimera dynamics is lost at sufficiently highmobility, the seemingly disorder states with R = 1 ap-pear. Acknowledgments
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