Instability of mixing in the Kuramoto model: From bifurcations to patterns
IInstability of mixing in the Kuramoto model: From bifurcations topatterns
Hayato Chiba, ∗ Georgi S. Medvedev, † and Matthew S. Mizuhara ‡ September 2, 2020
Abstract
We study patterns observed right after the loss of stability of mixing in the Kuramoto model ofcoupled phase oscillators with random intrinsic frequencies on large graphs, which can also be random.We show that the emergent patterns are formed via two independent mechanisms determined by the shapeof the frequency distribution and the limiting structure of the underlying graph sequence. Specifically, weidentify two nested eigenvalue problems whose eigenvectors (unstable modes) determine the structureof the nascent patterns. The analysis is illustrated with the results of the numerical experiments with theKuramoto model with unimodal and bimodal frequency distributions on certain graphs.
Models of interacting dynamical systems come up in different areas of science and technology. Modern ap-plications ranging from neuroscience to power grids emphasize models with spatially structured interactionsdefined by graphs. Identifying dynamical mechanisms underlying pattern formation in such networks is aninteresting problem with many important applications. In this paper, we study patterns emerging near theloss of stability of mixing in the Kuramoto model (KM) with random intrinsic frequencies on large graphs.We show that by varying the frequency distribution and the graph structure, one can generate a rich varietyof spatiotemporal patterns and identify a precise mathematical mechanism underlying pattern formation inthis model.The KM is one of the most widely used models in the theory of synchronization. In this paper, we studythe KM on graphs, which is formulated as follows. Let (Γ n ) be a sequence of graphs and consider ˙ θ i = ω i + K ( α n n ) − n (cid:88) j =1 a nij sin( θ j − θ i ) , i ∈ [ n ] := { , , . . . , n } , (1.1) ∗ Advanced Institute for Materials Research, Tohoku University, Sendai, 980-8557, Japan, [email protected] † Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, [email protected] ‡ Department of Mathematics and Statistics, The College of New Jersey, [email protected] a r X i v : . [ n li n . C D ] A ug here θ i : R + → T := R / π Z stands for the phase of oscillator i ; ω i , i ∈ [ n ] , are independent randomintrinsic frequencies drawn from the distribution with density g ( ω ) , and K is the strength of coupling. ( a nij ) is an n × n symmetric (weighted) adjacency matrix of graph Γ n . The scaling sequence α n is either identicallyequal to if (Γ n ) is a sequence of dense graphs, or α n (cid:38) subject to condition nα n → ∞ if (Γ n ) is asequence of sparse graphs with edge density O ( nα n ) . For more details on the KM on sparse graphs we referan interested reader to [15].Suppose (Γ n ) is a convergent sequence of graphs whose limiting behavior is described by a symmetricgraphon W ∈ L ([0 , ) (cf. [15]). Then under fairly general assumptions, the dynamics of (1.1) for large n can be approximated by the Vlasov equation ∂ t f ( t, θ, ω, x ) + ∂ θ { V ( t, θ, ω, x ) f ( t, θ, ω, x ) } = 0 , (1.2)where f ( t, θ, ω, x ) is a probability density function in ( θ, ω ) ∈ T × R of an oscillator at point x ∈ I := [0 , at time t ∈ R + . The velocity field is defined as follows V ( t, θ, ω, x ) = ω + K i (cid:16) κ ( t, x ) e − iθ − κ ( t, x ) e − iθ (cid:17) , (1.3)where κ ( t, x ) = (cid:90) T e iθ (cid:90) R ( W f ( t, θ, ω, · )) ( x ) dωdθ, (1.4)is called a (local) order parameter. The following self-adjoint compact operator on L ([0 , will play animportant role ( W φ ) ( x ) := (cid:90) I W ( x, y ) φ ( y ) dy, I := [0 , . (1.5)Rigorous justification of the Vlasov equation (1.2) in the context of the KM with all–to–all coupling wasgiven in [14]. It is based on the classical theory for kinetic equations (cf. [16, 1, 9]). For the KM on densegraphs, the Vlasov equation was further justified in [12, 4]. For the KM on sparse graphs with unboundeddegree, the results in [12, 4] continue to hold when combined with [15, Theorem 4.1].Equation (1.2) has a steady state solution f mix = g ( ω )2 π . (1.6)It describes the regime when all phases are uniformly distributed over T , which corresponds to mixing.Numerical experiments show that mixing is stable for small | K | . In his classical work on synchronization,Kuramoto identified the critical value K c = 2 ( πg (0)) − marking the loss of stability of mixing [13]. Thisformula assumes all–to–all coupling (i.e., W ≡ ) and continuous even unimodal density g ( ω ) . Kuramoto’sfindings started a new area of research, which culminated in a rigorous analysis of the loss of stability ofmixing in the KM with all–to–all coupling in [2]. For the KM on graphs, bifurcations of the mixing statewere analyzed in [4, 5]. Interestingly, the analysis of the spatially extended model along with the pitchforkbifurcation at positive value of K leading to synchronization reveals the possibility of a bifurcation for K < . For instance, it was shown that for a network with nonlocal nearest-neighbor coupling there is abifurcation to so–called twisted states at a certain K − c < [5, 7]. Thus, network structure plays a role in theloss of stability of mixing and affects the emerging patterns.2rom the beginning, the studies of the KM have been mainly focused on the transition to synchronization,i.e., on the pitchfork bifurcation of mixing. This turned out to be a challenging problem. The main technicalobstacle to the bifurcation analysis was the presence of continuous spectrum on the imaginary axis. It wasovercome in [2] with the help of the generalized spectral theory. The method of [2] was further applied tothe analysis of Andronov–Hopf bifurcation in [3] and was extended to the KM on graphs in [4, 5]. Thetechnical difficulty of the bifurcation analysis for a long time obstructed its pattern formation aspect, whichis perhaps even more interesting from the nonlinear science point of view. The examples in [7] already givea glimpse into pattern formation capacity of spatially extended KM. In this note, we develop this themefurther. We show that the combination of frequency distribution and graph structure provides a flexiblemechanism for controlling spatiotemporal patterns arising at the loss of stability of mixing in the KM ongraphs. Surprisingly, the contributions of the frequency distribution and the graph structure are independentfrom each other, which results in a variety of possible patterns obtained by combining features controlledby these two (vector–valued) parameters (Figures 2-5). Furthermore, we show that asymmetric frequencydistribution results in asymmetric chimera like patterns (Figure 7). To keep the presentation simple, werestrict to linear stability analysis, which is sufficient to relate bifurcations to patterns. Readers interestedin the analysis beyond linear stability are referred to [5] for the treatment of the pitchfork bifurcation in theKM on graphs. The Andronov–Hopf bifurcation is analyzed similarly by extending the results in [3] to thespatially extended model following the lines of [5].The outline of the paper is as follows. In the next section, we perform a linear stability analysis ofmixing. This is done for completeness, as the stability of mixing in the KM on graphs was already analyzedin [4, 5]. To complement the presentation in [4], this time we adapt the approach based on the theoryfor Volterra equations from [8] to the KM on graphs. It affords a quick derivation of the equation for thecritical values of K and provides a nice geometric picture of the loss of stability of mixing in the KM. Afterlocating the instabilities, we compute the unstable modes, which determine the bifurcating patterns. Asshown in [4] the loss of stability in the KM on graphs is captured by two nested eigenvalue problems. Thefirst one is obtained via the Fourier transform of the linearized system and is the same as in the stabilityanalysis of the KM with all–to–all coupling [2, 8]. We refer to this problem as the principal problem. Thesecond one is that for W (cf. (1.5)) and is called a secondary eigenvalue problem. It turns out that eachproblem is responsible for specific features of the bifurcating solutions: the principal modes determine thelocal structure of the emerging patterns, whereas the secondary modes capture their spatial organization.In particular, the principal eigenvalue problem determines whether mixing loses stability via a pitchforkor an Andronov–Hopf bifurcation. The former results in stationary patterns, whereas the latter producespatterns traveling with a nonzero speed. However, it is the secondary eigenvalue problem that decides theactual pattern. Depending on the form of the eigenfunctions corresponding to the critical eigenvalue ofthe secondary problem, one can observe spatially uniform clusters or patterns with more complex spatialorganization like twisted states. Importantly, the two spectral problems are independent in the sense thatone is determined by the shape of the frequency distribution while the other - by the graph structure. Afterderiving the necessary analytical tools in Section 2, we turn to the detailed discussion of the bifurcatingpatterns in the KM with unimodal and bimodal g in Section 3. To this end, we compare solutions bifurcatingfrom the mixing state in the KM on all–to–all and nonlocal nearest–neighbor graphs. These examples clearlydemonstrate the contributions of the principal and secondary unstable modes to the structure of the emergingpatterns. 3 Linear stability
In this section, we rewrite (1.2) in Fourier variables and linearize the resultant system about the mixingsteady state. Then we reduce the problem of stability to the Volterra equation for vector–valued functionsand derive the equation for the eigenvalues of the linearized operator. Here, we extend the method in [8]to the KM on graphs. Then we compute the corresponding eigenvalues following [5]. This information issufficient to explain the patterns emerging at the loss of stability of mixing, which is the main focus of thispaper. For more details on stability analysis of mixing in the KM on graphs, we refer an interested reader to[4, 5].We start by applying the Fourier transform in ( θ, ω ) u ( t, l, ξ, x ) = (cid:90) T (cid:90) R e ilθ e iξω f ( t, θ, ω, x ) dωdθ, ( l, ξ, x ) ∈ Z × R × I (2.1)to (1.2). Note that by the definition of f,g ( ω ) = (cid:90) T f ( t, θ, ω, x ) dθ ∀ ( x, t ) ∈ I × R + . (2.2)Thus, by Fubini’s theorem, u ( t, , ξ, x ) = (cid:90) R e iξω g ( ω ) dω =: ( F g )( ξ ) , (2.3)where F g stands for the Fourier transform in ω throughout this paper.Following [8], we assume F g ∈ C a ( R ) = { φ ∈ C ( R ) : (cid:107) φ (cid:107) a = sup t ∈ R e at | φ ( t ) | < ∞} for some a > . (2.4)Further, since f is real, u ( t, l, ξ, x ) = u ( t, − l, − ξ, x ) it is sufficient to consider ∂ t u ( t, , ξ, x ) = ∂ ξ u ( t, , ξ, x ) + K (cid:16) κ ( t, x )( F g )( ξ ) − κ ( t, x ) u ( t, , ξ, x ) (cid:17) , (2.5) ∂ t u ( t, l, ξ, x ) = l∂ ξ u ( t, l, ξ, x ) + Kl (cid:16) κ ( t, x ) u ( t, l − , ξ, x ) − κ ( t, x ) u ( t, l + 1 , ξ, x ) (cid:17) , (2.6)for ≥ . Note that κ defined in (1.4) can be rewritten as κ ( t, x ) = (cid:90) I W ( x, y ) u ( t, , , y ) dy. (2.7)The equilibrium f mix corresponds to u mix = ( F g, , , . . . ) in the Fourier space for l ∈ { , , , . . . } .The linearization of about u mix is thus given by ∂ t u ( t, , ξ, x ) = ∂ ξ u ( t, , ξ, x ) + K κ ( t, x )( F g )( ξ ) , (2.8) ∂ t u ( t, l, ξ, x ) = l∂ ξ u ( t, l, ξ, x ) , l ≥ . (2.9) Note that ∂ t u ( t, , ξ, x ) = 0 , ( ξ, x ) ∈ R × I by (2.3). ∂ t φ = T φ. (2.10) T is viewed as a linear operator densely defined on L ([0 , C a ( R )) . Integrating (2.8) along characteristics and recalling (2.7), we have u ( t, , ξ, x ) = u (1 , ξ + t, x ) + K (cid:90) t ( W u ( s, , , · )) ( F g )( ξ + ( t − s )) ds, (2.11)By plugging ξ = 0 in (2.11), we obtain the following Volterra equation u ( t, x ) = u ( t, x ) + K (cid:90) t ( W u ( s, · )) ( F g )( t − s ) ds, (2.12)where by abuse of notation u ( t, x ) := u ( t, , , x ) . We recast (2.12) in a more general form ν ( t ) + (cid:90) t A ( t − s ) ν ( s ) ds = φ ( t ) , (2.13)where A : R + → L ( H ) and φ , ν : R → H . By L ( H ) we denote the space of linear bounded operators on H . For the problem at hand, H = L ([0 , , A ( t ) = − K F g )( t ) W and φ ( t ) = u ( t, · ) , ν ( t ) = u ( t, · ) . (2.14)From now on, we will use the bold font to denote vector–valued functions along with operators. Theorem 2.1. [11, Theorems 1 & 2] Let A : R + → L ( H ) be strongly measurable and (cid:107) A ( · ) (cid:107) ∈ L loc ( R + ; R ) , where (cid:107) · (cid:107) stands for the operator norm. Then there exists a strongly measurable resolvent R : R + → L ( H ) R ( t ) = A ( t ) − (cid:90) t R ( t − s ) A ( s ) ds = A ( t ) − (cid:90) t A ( t − s ) R ( s ) ds. (2.15) For any φ ∈ L loc ( R + , H ) the unique solution of (2.13) can be expressed as ν ( t ) = φ ( t ) − (cid:90) t R ( t − s ) φ ( s ) ds. (2.16) Moreover, R ∈ L ( R + ; L ( H )) if and only if I + ( L A )( z ) is invertible as a bounded operator on H for all z ∈ C with (cid:60) z ≥ . Here, L A ( z ) = (cid:90) t e − zt A ( t ) dt. (2.17) Remark . For (2.12) following the lines of the analysis in the finite–dimensional case [10, Theorem 3.1],the resolvent can be obtained constructively as a Neumann series
R h = − ∞ (cid:88) j =1 k ∗ j ∗ ( W j h ) , k ( t ) = K F g )( t ) ,
5r in expanded form ( R h )( t, x ) = − ∞ (cid:88) j =1 (cid:90) t ( k ∗ j ( t − s ) (cid:0) W j h ( s, · ) (cid:1) ( x ) ds. Here, k ∗ a stands for the convolution of two functions ( k ∗ a )( t ) = (cid:82) t k ( t − s ) a ( s ) ds . Similarly, k ∗ j ∗ a = k ∗ ( k ∗ · · · ∗ ( k ∗ (cid:124) (cid:123)(cid:122) (cid:125) j times a ) . . . ) . The data in (2.12) clearly satisfy the assumptions of Theorem 2.1. Our next goal is to understand invert-ability of M ( z ) = I + ( L A )( z ) . By (2.14), M ( z ) is invertible unless z is a root of G ( z ) = 2 Kµ , G ( z ) := L ( F g ) ( z ) , µ ∈ Spec( W ) / { } . (2.18)Below, we will need the following observation G ( z ) = (cid:90) ∞ ( F g )( t ) e − tz dt = (cid:90) ∞ (cid:90) ∞−∞ e i ( ξ + t ) η e − tz g ( η ) dηdt = (cid:90) ∞−∞ g ( η ) z − iη dη, (2.19)where we used Fubini’s theorem.Equation (2.18) will be used to compute the eigenvalues of T . The corresponding eigenfunctions can befound by extending the corresponding results of the theory for Volterra equations on a finite–dimensionalspace (cf. [10, Theorem 2.1]) to the problem at hand. For simplicity of presentation, we will compute theeigenfunctions directly using the results in [4].Let z = λ be a root of (2.18) corresponding to µ ∈ Spec( W ) . By Lemma 3.2 in [4] (see also [8,Lemma 27]), λ is an eigenvalue of T . The corresponding eigenfunction is w λ = F v λ , (2.20)where v λ ( ω, x ) = K g ( ω ) λ − iω (cid:90) I × R W ( x, y ) v ( λ, y ) dydλ. (2.21)By integrating both sides of (2.21) with respect to ω, we have V = K G ( λ ) W V, where we used (2.19). V ( x ) = (cid:82) R v ( λ, x ) dλ is an eigenfunction of W corresponding to µ = K G ( λ ) . We conclude that w λ ( η, x ) = (cid:90) R e iηω g ( ω ) λ − iω dω V ( x ) (2.22)6s an eigenfunction of T corresponding to eigenvalue λ . Here, we dropped the factor Kµ/ since eigen-functions are defined up to a multiplicative constant.Thus, v λ ( ω, x ) = (cid:0) F − w λ ( · , x ) (cid:1) ( ω ) = Υ λ ( ω ) V ( x ) , Υ λ ( ω ) := g ( ω ) λ − iω . (2.23) Remark . The separable structure of v λ has important implications for pattern formation. Υ and V aredetermined by the intrinsic frequency distribution g and the graph limit W respectively. Equation (2.23)shows that the frequency distribution and the graph structure shape the unstable modes independently fromeach other.Below we will need to know the structure of Υ = lim λ → Υ λ = lim λ → g ( ω ) λ − iω . For ( F g ) ∈ C a ( R ) , Υ can be viewed as a tempered distribution. Indeed, for any φ from the Schwartz space S ( R ) , by Sokhotski–Plemelj formula (cf. [17]), we have (cid:104) Υ , φ (cid:105) = lim λ → (cid:90) ∞−∞ g ( ω ) φ ( ω ) λ − iω dω = πg (0) − i P . V . (cid:90) ∞−∞ g ( ω ) ω dω. Thus, as an element of S (cid:48) ( R ) , Υ can be written as Υ = πg (0) δ − i P g, , (2.24)where δ stands for the delta function and (cid:104) P g,α , φ (cid:105) = P . V . (cid:90) ∞−∞ g ( ω + α ) φ ( ω + α ) ω dω. Similarly, we compute Υ ± iy ∗ := lim λ (0+0) ± iy ∗ g ( ω ) λ − iω = πg ( ± y ∗ ) δ ± y ∗ − i P g, ± y ∗ , (2.25)where δ β = δ ( · + β ) . Next we turn to the bifurcations in the KM (1.1) and the corresponding patterns. To illustrate the typicalscenarios realized in this model we will consider unimodal ( U ) and bimodal ( B ) g ∈ C a ( R ) combined withall–to-all ( aa ) and nonlocal nearest-neighbor connectivity ( nn ). We will code the corresponding models by( Xy ) where X ∈ { U , B } and y ∈ { aa , nn } . 7 Re zIm K u z b Re zIm K b z Figure 1: The critical curves generated by a unimodal ( a ) and bimodal ( b ) intrinsic frequency distribution.As a first step, we locate the bifurcations in (1.1) by solving (2.18). To this end, recall (2.19) and notethat G is analytic in Π := { z ∈ C : (cid:60) z > } and is continuous in ¯Π . By Sokhotski–Plemelj formula, lim y →±∞ lim x → G ( x + iy ) = lim y →±∞ (cid:18) πg ( y ) − i P . V . (cid:90) ∞−∞ g ( s ) dsy − s (cid:19) = 0 . (3.1)Thus, G maps the imaginary axis to a bounded closed curve. As in [8], we use the Argument Principle[17] to conclude that (2.18) has a root in Π if and only if Kµ ) − ∈ G (Π) . (3.2)In Figure 1, we plot G ( ∂ Π) for symmetric unimodal and bimodal g . The critical curve G ( ∂ Π) alwaysintersects the real axis at the origin (cf. (3.1)). In addition, it has another point of intersection with the realaxis at K u = G (0) > , (3.3)if g is unimodal. In the bimodal case, there is a point of double intersection K b = G ( ± iy ∗ ) > . (3.4)Having understood the qualitative features of the critical curves for the unimodal and bimodal densities,we now turn to bifurcations. (Uaa) We start with the all–to–all coupling first. In this case, W ≡ , the largest eigenvalue of W is µ = 1 , and the corresponding eigenfunction is constant [4]. Since the critical curve G ( ∂ Π) is bounded, K − / ∈ Π for K (cid:29) . Thus, there are no roots of (2.18) for large K . As we decrease K , K − hits G ( ∂ Π) when K − = G (0) = πg (0) . Note that the corresponding root of (2.18) is z = 0 . Thus, at K c = 2 / ( πg (0)) the system undergoesa pitchfork bifurcation. The emerging pattern is determined by the unstable mode v , which has asingularity at ω = 0 . This implies that the emerging pattern contains a stationary cluster. Further, For φ vanishing in a neighborhood of the origin (cid:104) Υ , φ (cid:105) is a regular functional.
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Figure 2: The pitchfork bifurcation leading to the formation of spatially homogeneous synchronous solutionin the KM with all–to–all coupling and unimodal g . κ ∞ stands for the time asymptotic value of the L –normof the local order parameter (1.4). The data presented in this plot was obtained after a long transient time.Further, the L –norm of the local order parameter was averaged over a certain time interval to smoothen thedata. The same method was used in all subsequent computations of the order parameter.since V ≡ , the bifurcating solution is uniform in space. We conclude that the instability leads to theformation of a stationary coherent cluster (see Figure 2). This is a classical scenario of the onset ofsynchronization. (Baa) Next we discuss the case of the bimodal density and all–to–all coupling. In this case K − hits G ( ∂ Π) at the point of double intersection of the critical curve with the real axis: K − = G (0) = πg ( ± y ∗ ) . The roots of (2.18) z = ± iy ∗ . The system undergoes Andronov–Hopf bifurcation at K c = 1 / ( πg ( y ∗ )) The eigenfunctions v ± iy ∗ have singularities at ω = ± y ∗ respectively (cf. (2.25)), while V is stillconstant. Thus, the emerging pattern consists of two spatially homogeneous clusters rotating withconstant speed in opposite directions (see Figure 3). (Unn) It remains to consider the nonlocal nearest–neighbor coupling (see [4, § W in this case). The new feature here is that along with the largest positive eigenvalue µ + = 1 (corresponding to V + ≡ ), there can be one or more negative eigenvalues (see [4] for a detailed dis-cussion). Suppose µ − < is the smallest negative eigenvalue of W . The corresponding eigenspaceis spanned by V − , = e ± πiqx for some q ∈ N . Thus, there are two bifurcation points K + c = 2 / ( πµ + g ( y ∗ )) > and K − c = 2 / ( πµ − g (0)) < . The KM with all–to–all coupling and bimodal frequency distribution was discussed in [8], but the Andronov–Hopf bifurcationwas not identified there. component of the unstable mode v is the same for the bifurcations at K − c and K + c . This impliesthat the bifurcating patterns gravitate towards stationary clusters. However, the spatial organizationis different. The pattern emerging at K + c is uniform in space, whereas those emerging at K − c areorganized as q –twisted states (see Figure 4).A new feature of this example is that in addition to positive eigenvalues of W there are negativeeigenvalues. Denoting the largest postive and smallest negative eigenvalues of W by µ + and µ − < respectively (see [4] for explicit formulae of the eigenvalues of W ). For µ + the correspondingeigenfunction is constant as in the all–to–all coupling case. For µ − , the eigenfunctions are linearcombinations of so–called q –twisted states: e ± πiqx for the appropriare q ∈ N . Thus, in the unimodalcase the mixing state bifurcates into a spatially homogeneous solutions at K + = 2 / ( πµ + g (0)) > and into a twisted state at K − = 2 / ( πµ − g (0)) < (see Figure 4). In the bimodal case, the mixingstate bifurcates into a two-cluster at K + = 2 / ( πµ + g ( y ∗ )) > and into a pair of twisted states at K − = 2 / ( πµ − g ( y ∗ )) < (see Figure 4). (Bnn) The only difference of this case with the one that we just discussed is that the principal unstablemodes v ± iy ∗ are localized around ω = ± y ∗ . Thus, the stationary patterns in (Unn) turn into rotatingones: rotating clusters at K + c and twisted states traveling in opposite directions at K − c (see Figure 5).This concludes the description of the bifurcation scenarios in the KM with symmetric unimodal andbimodal frequency distribution on complete and nonlocal nearest–neighbor graphs. Breaking the symmetryof the distribution (see Figure 6) results in new patterns including chimera like patterns shown in Figure 7.They can be understood using the techniques of this paper. The situation is even more interesting for thesecond–order KM, which will be covered in the future work [6]. The complete and nonlocal nearest–neighbor graphs were used in this paper as representative examples. The analysis of this paper applieswithout any changes to the KM on a variety of convergent graph sequences including Erd˝os–R´enyi, small-world, and power–law graphs (cf. [7, 15]). Acknowledgements.
This work was supported in part by NSF grants DMS 1715161 and 2009233 (toGSM). Numerical simulations were completed using the high performance computing cluster (ELSA) at theSchool of Science, The College of New Jersey. Funding of ELSA is provided in part by National ScienceFoundation OAC-1828163. MSM was additionally endorsed by a Support of Scholarly Activities Grant atThe College of New Jersey.
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Figure 3: Andronov–Hopf bifurcation leading to the formation of two–cluster in the KM on complete graphsand bimodal g . TWISTEDSTATES MIXING SYNCHRONOUS
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Figure 5: Bifurcations in the KM with nonlocal nearest–neighbor coupling and bimodal frequency distribu-tion. The Andronov–Hopf bifurcation at K + c > leads to formation of 2–cluster, whereas the bifurcation at K − c < results in a pair of 2–twisted states traveling in opposite directions. K b Re zIm z
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