Accepted Manuscript Twisted states in nonlocally coupled phase oscillators with bimodal frequency distribution Yuan Xie, Shuangjian Guo, Lan Zhang, Qionglin Dai, Junzhong Yang PII: DOI: Reference: S1007-5704(18)30270-3 https://doi.org/10.1016/j.cnsns.2018.08.008 CNSNS 4622 To appear in: Communications in Nonlinear Science and Numerical Simulation Received date: Revised date: Accepted date: 15 April 2018 26 June 2018 20 August 2018 Please cite this article as: Yuan Xie, Shuangjian Guo, Lan Zhang, Qionglin Dai, Junzhong Yang, Twisted states in nonlocally coupled phase oscillators with bimodal frequency distribution, Communications in Nonlinear Science and Numerical Simulation (2018), doi: https://doi.org/10.1016/j.cnsns.2018.08.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Highlights • Twisted states in nonlocally coupled phase oscillators with bimodal frequency distribution are studied. CR IP T • Two different types of twisted states, twisted standing waves and stationary twisted states, appear successively with the increase of the coupling strength. AC CE PT ED M AN US • The twisted states and the stabilities are theoretically analyzed with the assistance of Ott-Antonsen ansatz. 1 ACCEPTED MANUSCRIPT Twisted states in nonlocally coupled phase oscillators with bimodal frequency distribution Yuan Xie, Shuangjian Guo, Lan Zhang, Qionglin Dai, Junzhong Yang∗ CR IP T School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China Abstract AN US Twisted states, referred to traveling waves in an array of nonlocally coupled phase oscillators, have drawn some attention in recent years. In this work, we study a one-dimensional ring of phase oscillators with nonlocal coupling and a bimodal natural frequency distribution. We show that twisted standing waves and stationary twisted states appear successively with the increase of the coupling strength. In the continuum limit, we derive a low-dimensional reduced equation using the Ott-Antonsen ansatz, which verifies the twisted states in the simulations of finite networks of oscillators. We also theoretically investigate the stationary twisted states and their stabilities by using the reduced equation. 1. Introduction PT AC 10 Network of coupled oscillators provides a platform for studying dynamical behaviors and spatiotemporal patterns in different areas of science [1, 2, 3]. One well-known system is the Kuramoto model [4, 5, 6] of phase oscillators. In an original Kuramoto model, N phase oscillators are globally coupled to each other by a sinusoidal function of phase difference with a same coupling strength. The natural frequencies of phase oscillators are randomly drawn from a given probability distribution. The Kuramoto model together with its variants have played a central role in the study of coupled oscillator systems in physics and biology since it was first proposed by Kuramoto in 1975 [7]. Over the last decade, a ring of identical phase oscillators coupled nonlocally has attracted much attention [4, 8, 9, 10, 11, 12]. Several interesting phenomena have been studied in this system. One of them is the chimera dynamics which shows the alternations between the regions of synchronous oscillators and asynchronous oscillators [4, 10, 11]. Another simpler dynamics is the twisted state, a traveling wave state where the phase difference between neighboring oscillators is fixed and a fixed profile of oscillators’ phases CE 5 ED M Keywords: Twisted standing waves, Nonlocal coupling, Twisted states, Ott-Antonsen ansatz 15 ∗ Corresponding author Email address: jzyang@bupt.edu.cn (Junzhong Yang) Preprint submitted to Journal of LATEX Templates August 21, 2018 ACCEPTED MANUSCRIPT 35 40 CR IP T AN US 30 M 25 ED 20 travels at a constant speed. Generally, a uniformly twisted state is characterized by an integer winding number q in which the phase of oscillators varies linearly with spatial position in the form of θj = ωt + 2πqj/N + C, j = 1, . . . , N (ω is the natural frequency and C is any constant) [13]. Lately, partially coherent twisted states have been investigated in non-identical phase oscillators [14, 15, 16]. A partially coherent twisted state in heterogeneous networks is described by a spatially-dependent complex order parameter whose argument is a linear function of space position. The partially coherent twisted states and their stabilities have been theoretically investigated in the continuum limit by using the Ott-Antonsen (OA) ansatz [17, 18]. Up to now, twisted states have been investigated on phase oscillators whose natural frequency distribution g(ω) follows a unimodal one. For globally coupled phase oscillators with g(ω) symmetrical about its maximum at ω = 0, there exists a critical coupling strength Kc = 2π/g(0) above which the coupled oscillators transit from the incoherent states to partially synchronous states consisting of a unique synchronous cluster [19, 20, 21, 22]. On the other hand, if g(ω) follows a bimodal distribution, globally coupled oscillators allow for standing wave states consisting of two synchronous clusters with different average frequencies [23, 24, 25]. Increasing the coupling strength turns the incoherent state to standing wave states and further to partially synchronous states with a unique synchronous cluster. Now, it is natural to ask how bimodal frequency distribution influences the twisted states in nonlocally coupled phase oscillators. In this work, we investigate the effects of bimodal frequency distribution on twisted states in an array of nonlocally coupled phase oscillators. We find twisted standing wave states when incoherence states become unstable and stationary twisted states at strong coupling strength where the profile of spatially-dependent complex order is frozen in space. The paper is arranged as following. In Section 2, we describe the model. In Section 3, we present numerical results on different types of twisted states. In the continuum limit, we derive a low-dimensional reduced equation by using the OA ansatz and theoretically study these twisted states. The paper concludes with a summary in Section 4. We consider a one-dimensional ring of N nonlocally coupled phase oscillators in which the individual unit is coupled to M neighbors on each side with the coupling strength K, θ̇j = ωj + k=M X K sin(θj+k − θj ). 2M + 1 (1) k=−M AC CE 45 PT 2. Model The subscript j denotes the unit index and the periodic boundary condition is imposed. M denotes the coupling range. To be convenient, we introduce the coupling radius σ = M/N , which ranges from 1/N (nearest neighbour coupling) to 0.5 (global coupling). The natural frequency ωj is randomly chosen from the frequency distribution g(ω). We consider g(ω) to be a superposition of two Lorentzian distributions g(ω) = γ 1 1 [ + ], 2 2 2π (ω − ω0 ) + γ (ω + ω0 )2 + γ 2 3 (2) ACCEPTED MANUSCRIPT Wj = k=M X 1 eiθj+k , 2M + 1 k=−M we reformulate Eq. (1) as θ̇j = ωj + K [Wj e−iθj − W j eiθj ], 2i 3. Results AN US where the overbar indicates complex conjugate. 55 CR IP T 50 where γ is the half width of each Lorentzian and ±ω0 are the center frequencies of each Lorentzian. The distribution g(ω) is a bimodal one if ωγ0 > √13 and becomes a unimodal one otherwise [23]. In this work, we focus on the case with bimodal distribution g(ω) and set γ = 0.5 and ω0 = 1.5. By defining spatially-dependent complex order parameter Wj = Rj eiΘj as (3) (4) In this section, we start with the numerical simulations of Eq. (1) by using fourthorder Runge-Kutta algorithm with a time step δt = 0.01. Throughout this work, we let N = 2000. Traditionally, the coherence in coupled phase oscillators is characterized by the global order parameter defined as iΘ M Re N 1 X iθj = e . N j=1 Here R (0 ≤ R ≤ 1) measures the coherence of the oscillator population and Θ is the average phase. However, as shown by Eq. (4), each oscillator is driven by a unique spatially-dependent order parameter. P To reflect this fact, we consider another global quantity h|W |i, defined as h|W |i = j |Wj |, to characterize the coherence in the model. We investigate the transition scenario in Eq. (1) from the incoherence to synchronous behaviors with the coupling strength K. For each coupling strength, we consider dozens of realizations with random initial conditions. For each realization, we calculate R and h|W |i. Fig. 1(a) presents R and h|W |i against the coupling strength K at the coupling radius σ = 0.035. The incoherence becomes unstable for the coupling strength above a threshold around K = 2. For strong coupling strength such as K > 5, R and h|W |i, acquired from different realizations, against K fall onto several curves, which suggests the coexistence of several distinct states at a given coupling strength. Actually, these distinct states represent partially synchronized twisted states with different winding numbers. To verify it, we calculate R and h|W |i against K for prepared initial conditions θj = 2πqj/N with q = 0, ±1, ±2, ±3, · · · . The corresponding results presented in Fig. 1(a2) reproduce the characteristics in Fig. 1(a1). Figs. 1(a1) and (a2) suggest that twisted states with the winding number |q| ≥ 4 do not exist when K < 10 and the critical coupling strengths for twisted states with different winding numbers increase with |q|. In these figures, we find that h|W |i is always larger than R except for the twisted state with q = 0. The reason is that the contributions to R by two antiphased oscillators in a twisted state with q 6= 0 are always offset by each other. As shown below, the partially coherent states in 4 AC 70 CE 65 PT ED 60 75 80 (5) ACCEPTED MANUSCRIPT 1 1 (a1) 0 2 4 6 8 10 0 0 2 4 6 1 1 (b2) (a2) W R, W R, q= 2 q= 1 0 0 q= 3 2 4 6 8 10 0 0 2 4 6 K q= 2 8 10 AN US K 10 q=0 q=0 q= 1 8 CR IP T 0 R, R, W W (b1) M Figure 1: The global order parameter R (in black) and the average of spatially-dependent order parameter h|W |i (in red) are plotted against the coupling strength K for the coupling radius σ = 0.035 in (a1,a2) and for σ = 0.1 in (b1,b2). The top row shows the results for dozens of random initial conditions at each K while the bottom row shows the results for prepared initial conditions θj = 2πqj/N with q = 0, ±1, ±2, ±3, · · · . The green arrows mark the different twisted states. Blue dashed arrows show critical coupling strength for the partially synchronized twisted states with different winding number q. The critical coupling strength K = 4.6, 5.4, 5.8, 7.2 for q = 0, ±1, ±2, ±3 in (a2) and K = 4, 4.5, 9.1 for q = 0, ±1, ±2 in (b2), respectively. Other parameters: ω0 = 1.5 and γ = 0.5. ED AC CE 90 PT 85 the intermediate coupling strength, 2 < K < 5, are also twisted states though they are hardly identified in Figs. 1(a1) and (a2). The transition scenario described above holds at larger coupling radius, for example σ = 0.1 in Figs. 1(b1) and (b2). Then, we examine the twisted states for σ = 0.035 in details by using prepared initial conditions θj = 2πqj/N (q = 1, 2). The top two panels in Fig. 2 show the results for K = 3, an intermediate coupling strength. The snapshots of θj in Figs. 2(a1,b1) show that oscillators distribute their phases in the range of 2π non-uniformly and oscillators tend to crowd themselves around the mean phase Θj of the spatially-dependent complex order parameter Wj (the red curves). According to the definition of the partially synchronized twisted states [15, 16], the states in Figs. 2(a1,b1) are the twisted ones with q = 1 and q = 2. The partial coherence of the states is demonstrated by plotting the effective frequency ωe , defined as ωe = h dθ dt it with h·it the average over a long time interval after transient, against the natural frequency ω. As illustrated in Figs. 2(a2,b2), the graph of ωe (ω) for each twisted state exhibits two plateaus with opposite ωe . That is, there exist two synchronous clusters with different average frequencies in the states, a typical feature of standing wave states. The periodic variation of |Wj | in Figs. 2(a3,b3) and that Θj oscillates periodically without drifting along space in Figs. 2(a4,b4) further demonstrate the existence of twisted standing wave states in the model (1) at an intermediate coupling strength. The bottom two panels in Fig. 2 show the results at K = 10, a strong coupling strength. The snapshots in Figs. 2(c1,d1) demonstrate the twisted states with q = 1 and q = 2 where most oscillators condensate their phases to the mean phase Θj of spatially5 95 100 0 (a1) 0.5 (a3) 650 t t -5 -5 5 2000 (d3) 2000 0 6000 ED j 2000 -20 t 20 0 PT 0 2000 -20 0 K=3,q=2 2000 (c4) t 0 j , 2000 6000 e j 20 600 0 6000 K=10,q=1 2000 (d4) t -20 (d2) 20 M (d1) 2000 t , 2000 0 600 0 6000 j -20 0 2 2000 (b4) 650 (c3) (c2) e j 20 0 650 t e , 2000 (c1) 2 2000 (b3) j 0 0 600 600 0 AN US 5 (b2) 5 j 2 -5 -5 K=3,q=1 t (b1) j 650 , 2000 0 (t) (a4) j j 0 2 0 1 |W |(t) e j 2 (a2) 5 CR IP T ACCEPTED MANUSCRIPT K=10,q=2 2000 j 2000 0 j 2000 AC CE Figure 2: The partially synchronized twisted states in the phase oscillators model Eq. (1) with bimodal natural frequency distribution Eq. (2). (a,b) The twisted standing wave states with q = 1 and q = 2 at K = 3. (c,d) The stationary twisted state with q = 1 and q = 2 at K = 10. The first column shows the snapshots of oscillators’ phases and red solid line shows the argument Θj of the spatially-dependent order parameter Wj . The second column shows the graphs of the effective frequencies of oscillators ωe (ω). The third and the fourth columns show the evolutions of the spatially-dependent order parameter |Wj | and its argument Θj , respectively. Other parameters: σ = 0.035, ω0 = 1.5, and γ = 0.5. 6 ACCEPTED MANUSCRIPT 110 ∂ ∂f + (f v) = 0, ∂t ∂θ v=ω+ 115 AN US where CR IP T 105 dependent order parameter. Only one plateau with the frequency at around zero, the mean natural frequency, in the graphs of ωe (ω) in (c2,d2) indicates that there is only one synchronous cluster in the oscillators. The evolutions of |Wj | in (c3,d3) and Θj in (c4,d4) suggest that the twisted states at K = 10 are roughly stationary. The twisted states in Eq. (1) can be studied with the assistance of the Ott-Antonsen ansatz (OA ansatz) [17, 18]. Supposing the length of the ring to be 2π and the position of oscillator j to be xj = 2πj/N , we consider the continuum limit (N → ∞) with the coupling radius σ unchanged. We consider the probability density function f (x, ω, θ, t) defined as the fraction of oscillators with phases between θ and θ + dθ and natural frequencies between ω and ω + dω at time t and position x. The density f (x, ω, θ, t) satisfies the continuity equation K [W e−iθ − W eiθ ]. 2i (6) (7) The spatially-dependent complex order parameter W (x, t) can be reformulated as Z 2π Z ∞ Z 2π W (x, t) = G(x − y) f eiθ dθdωdx (8) −∞ 0 1 4πσ , M with G(x) = 0, 0 |x| < 2πσ otherwise (9) ED Using OA ansatz, we write the probability density function f (x, ω, θ, t) as (10) PT ∞ X g(ω) [a(ω, x, t)]n einθ + c.c.] f (θ, ω, x, t) = [1 + 2π n=1 where c.c. is the complex conjugate of the previous term. By substituting Eqs. (7,10) into Eq. (6), we have For the bimodal frequency distribution Eq. (2), we have Z ∞ Z 2π 1 f (θ, ω, x, t)eiθ dθdω = [a(ω0 − iγ, x, t) + a(−ω0 − iγ, x, t)]. 2 −∞ 0 AC 120 CE da(x, t) K = iωa(ω, x, t) + [W (x, t) − a2 (ω, x, t)W (x, t)]. dt 2 Defining u1 (x, t) = a(ω0 − iγ, x, t) = b1 eiϕ1 and u2 (x, t) = a(−ω0 − iγ, x, t) = b2 eiϕ2 , we have ∂u1 (x, t) = (−γ + iω0 )u1 (x, t) + ∂t ∂u2 (x, t) = (−γ − iω0 )u2 (x, t) + ∂t K [W (x, t) − W (x, t)u21 (x, t)] 2 K [W (x, t) − W (x, t)u22 (x, t)] 2 7 (11) 0.0 (a1) 0 (b2) 650 600 2 650 2 (c2) 0 2 (d2) PT t 700 600 0 x CE 0 600 600 0 2 (d3) 700 t 1,2 700 2 (c3) t (d1) 600 0 t M t 600 2 (b3) 700 ED 700 600 0 arg[W(x,t) ] t 0 700 1,2 0 600 2 (a3) AN US t 0 600 (c1) 2 600 650 1,2 2 650 (b1) 0 650 t 0 600 2 1.0 W(x,t) t 650 1,2 2 0.50 (a2) CR IP T ACCEPTED MANUSCRIPT 2 600 0 x 2 AC Figure 3: The partially synchronized twisted states in the reduced OA model Eq. (11). (a,b) The twisted standing wave states with q = 1 and q = 2 at K = 3. (c,d) The stationary twisted states with q = 1 and q = 2 at K = 10. The first column shows the time evolutions of ϕ1 (in black) and ϕ2 (in red) for one of OA oscillators. The second and the third columns show the evolutions of the spatially-dependent order parameter |W (x, t)| and its argument arg[W (x, t)], respectively. Other parameters: σ = 0.035, ω0 = 1.5, and γ = 0.5. 8 ACCEPTED MANUSCRIPT 135 140 ED M 145 CR IP T 130 AN US 125 R 2π with W (x, t) = 12 0 G(x − y)[u1 (y, t) + u2 (y, t)]dy. The dynamical behaviors in Fig. 2 can be reproduced by simulating Eq. (11) (the OA oscillators model). We take the coupling radius σ = 0.035 as an example. To build twisted states with the winding number q, we prepare initial conditions taking the form qx iqx )e . The top two panels in Fig. 3 show the twisted standing wave states u1,2 (x, 0) ∼ ( 2π with q = 1 and q = 2 at the intermediate coupling strength K = 3, respectively. The time sequences of one of OA oscillators in Figs. 3(a1) and (b1) show that ϕ1 (x, t) and ϕ2 (x, t) oscillate at the same frequency but in opposite directions. Based on the simulation results, we may express u1,2 as u1 (x, t) = b(t)eiΩt+qx and u2 (x, t) = b(t)e−iΩt+qx . That is, u1 and u2 support their own traveling waves and the two waves propagate in opposite directions. Then we have W (x, t) = b(t) sin(2πσ) cos(Ωt)eiqx /2πσ, which is in agreement with the homogeneous oscillating |W (x, t)| in Figs. 3(a2) and (b2) and spatially-dependent and time-independent argument of W (x, t) in Figs. 3(a3) and (b3). All of these demonstrate the presence of the twisted standing wave states at K = 3. The winding numbers q of these two twisted states can be read from the variation of the argument of Z(x, t) along space at an arbitrary time, for example q = 1 in Fig. 3(a3) and q = 2 in Fig. 3(b3). The bottom two panels in Fig. 3 show the twist states with q = 1 and q = 2 at the strong coupling strength K = 10. Clearly, either the time sequences of one of OA oscillators in Figs. 3(c1) and (d1) or the spatiotemporal evolutions of Z(x, t) in Figs. 3(c2,c3) and (d2,d3) demonstrate that the twisted states are stationary. Using the OA oscillators model Eq. (11), the twisted state at strong coupling strength can be solved rigorously and their stabilities can be analyzed. Partially synchronized twisted states with winding number q can be written as u1,2 (x, t) = b1,2 ei(qx+ξ1,2 t+ν1,2 ) where b1,2 > 0, ξ1,2 and ν1,2 are real numbers. Substituting the solution into Eq. (11), we have K iξ1,2 b1,2 = (−γ ± iω0 )b1,2 + [Ze−i(qx+ξ1,2 t+ν1,2 ) 2 2 i(qx+ξ1,2 t+ν1,2 ) −Zb1,2 e ], (12) where ξ1 = ξ2 is required. By letting ν1 − ν2 = φ, we have = (−γ + iω0 )b1 + b K G(q) [b1 + b2 e−iφ − (b1 + b2 eiφ )b21 ], 4 = (−γ − iω0 )b2 + b K G(q) [b1 eiφ + b2 − (b1 e−iφ + b2 )b22 ] 4 CE PT iξb1 (13) R 2π 2πσq b with G(q) = 0 G(x) cos(qx)dx = sin2πσq . Equating real and imaginary parts of these equations, we have two sets of solutions. One is a symmetrical solution with b1 = b2 and φ < π/2, determined by AC 150 iξb2 ξ = 0, 0 = b ω0 γ K G(q) + − , sin φ 1 + cos φ 2 0 = ( b ω0 2 γ Kγ G(q) ) +( )2 − , 2 2 1+b 1−b 2(1 − b2 ) 9 (14) 1 1 <|W|> <|W|> ACCEPTED MANUSCRIPT q=2 q=1 q=3 q=1 0.6 0.8 10 15 20 9 K CR IP T q=2 12 15 K AN US Figure 4: h|W |i of different stationary twisted states are plotted against the coupling strength at the coupling radius σ = 0.035 in (a) and σ = 0.1 in (b). Numerical results are in black for N = 2000 and blue for N = 10000, and theoretical results are in red. ω0 = 1.5, γ = 0.5. and the other is an asymmetrical one with b1 6= b2 and π/2 < φ < 3π/2, determined by 0 = 0 = b1 b2 + cosφ, b sin φ K G(q) (b22 − b21 ). = 8 cos φ ED PT AC 165 (15) The stationary twisted states at strong coupling strength in Figs. 2 and 3 are symmetrical ones where φ is close to zero and their spatially-dependent order parameters are frozen in space. For theseRstationary twisted states with different winding numbers 2π q, h|W |i (defined as h|W |i = 0 |W (x, t)|dx) against the coupling strength K are presented in Figs. 4(a) and (b) for the coupling radius σ = 0.035 and σ = 0.1, respectively. As a comparison, we also plot h|W |i acquired by simulating Eq. (1). We find that the deviation of h|W |i between theoretical and numerical results is weak and increases with K decrease. Moreover, we have carried out the numerical simulations for two different N . As shown by Fig. 4, the results for N = 10000 are closer to the theoretical results when compared with those for N = 2000. Thus, it can be inferred that the deviation between theoretical and numerical results will go away as N goes to infinity. Furthermore, we study the linear stability of stationary twisted states. We introduce perturbation v(x, t) to a stationary twisted state with the winding number q such that u1,2 = [b + v1,2 (x, t)]ei(qx+ν1,2 ) . Inserting u1,2 into Eq. (11), we have the evolution of v1,2 CE 160 2ω0 + M = ξ 155 b sin φ 4γ K G(q) [1 + 3 cos2 φ − ], b 4 cos φ K G(q) 4γ 1 + cos2 φ − − b21 − b22 , b K G(q) 0 10 ACCEPTED MANUSCRIPT 1 q=0 CR IP T b q=1 q=2 q=3 0.5 9 6 3 12 15 AN US K Figure 5: The amplitude |b| as a function of the coupling strength K for stationary twisted state solutions of Eq. (14) with q = 0, 1, 2 and 3. Their stabilities are determined by Eq. (18). Solid: stable; dashed: unstable. Pink dashed arrows show the critical coupling strength. Here, K = 4, 4.6, 8.9 for q = 0, 1, 2, respectively, which agree well with the results shown in Fig. 1(b2). Other parameters: ω0 = 1.5, γ = 0.5, σ = 0.1. v̇1 (x, t) K −η1 v1 (x, t) + 4 Z 2π + e−iφ ) G(x − y)v1 (y, t)ei(y−x)q dy − = ED [(1 M as PT b2 (1 v̇2 (x, t) CE [(1 + eiφ ) 0 2π Z 0 G(x − y)v 1 (y, t)e−i(y−x)q dy], K −η2 v2 (x, t) + 4 Z 2π + eiφ ) G(x − y)v2 (y, t)ei(y−x)q dy − = 0 b2 (1 + e−iφ ) Z 0 2π G(x − y)v 2 (y, t)e−i(y−x)q dy] ∓iφ b + b2 + b2 e±iφ ). with η1,2 = K 4 G(q)(1 + e Then, we look for solutions of Eq. (16) with the form v1,2 (x, t) = v1,2 eiqx eλt . Substituting this into Eq. (16) and after some algebras, we have the spectrum following q 1 λ± (q) = (trB1,2 ± (trB1,2 )2 − 4detB1,2 ), (17) 2 AC 170 (16) 11 ACCEPTED MANUSCRIPT (1 − b2 )M1 Imη1 + K (1 − b2 )H1 −Reη1 + K 8 8 B1 = , 2 2 −Imη1 − K −Reη1 + K 8 (1 + b )H1 8 (1 + b )M1 2 2 −Reη2 + K Imη2 + K 8 (1 − b )M2 8 (1 − b )H2 B2 = K K −Imη2 − 8 (1 + b2 )H2 −Reη2 + 8 (1 + b2 )M2 and M1 = (1 + cos φ)h+ − ih− sin φ, H1 = h+ sin φ + ih− (1 + cos φ), M2 = (1 + cos φ)h+ + ih− sin φ, H2 = −h+ sin φ + ih− (1 + cos φ), b + q0 ) + G(q b − q0 ), h+ (q, q0 ) = G(q AN US b + q0 ) − G(q b − q0 ). h− (q, q0 ) = G(q CR IP T where The boundary of the stable stationary twisted states can be determined by 00 Re[λ+ (0)] = 0 00 M 180 with the second derivative of λ+ (q) with respect to its argument at q = 0. As shown in Fig. 5, the critical coupling strength for the onset of stable stationary twisted states increases with q and no stable twisted state with q > 2 exists in the parameter regimes investigated here. Comparing with Fig. 1 where the critical values of K for different stationary twisted states are marked by blue dashed line, theoretical results agree well with the simulations. On the other hand, similar stability analysis can be made on the asymmetrical twisted states Eq. (15) and we find that these states are always unstable. 4. Conclusion PT AC 190 In summary, we studied a ring of nonlocally coupled phase oscillators with bimodal natural frequency distribution and zero mean natural frequency. We found the existence of partially coherent twisted states. Depending on the coupling strength, there are two types of twisted states, the twisted standing wave states for intermediate coupling strength which consist of two synchronous clusters oscillating in opposite directions and the twisted states for strong coupling strength which consist of only one synchronous clusters oscillating at the mean natural frequency. It is worthy to mention that, when the frequency distribution of the oscillators is unimodal [16], partially synchronized twisted states can be found whereas the twisted standing wave states do not exist. Moreover, these twisted states are investigated with the assistance of OA ansatz and the stability of the stationary twisted states are theoretically analyzed. CE 185 ED 175 (18) Acknowledge 195 This work was supported by the National Natural Science Foundation of China under Grants No. 11575036 and No. 11505016. 12 ACCEPTED MANUSCRIPT References References AC CE PT ED M AN US CR IP T [1] Barreto E, Hunt B, Ott E, So P. Synchronization in networks of networks: The onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators. Phys Rev E 2008;77:036107. [2] Ren L, Ermentrout B. Phase locking in chains of multiple-coupled oscillators. Physica D 2000;143:5673. [3] Pikovsky A, Rosenblum M, Kurths J. 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