FORMATION, STABILITY AND BASIN OF PHASE-LOCKING FOR KURAMOTO OSCILLATORS BIDIRECTIONALLY COUPLED IN A RING

. We consider the dynamics of bidirectionally coupled identical Kuramoto oscillators in a ring, where each oscillator is inﬂuenced sinusoidally by two neighboring oscillator. Our purpose is to understand its dynamics in the following aspects: 1. identify all the phase-locked states (or equilibria) with stability or instability; 2. estimate the basins for stable phase-locked states; 3. identify the convergence rate towards phase-locked states. The crucial tool in this work is the celebrated theory of (cid:32)Lojasiewicz inequality.


1.
Introduction. Collective behaviors in coupled nonlinear oscillators have attracted numerous attentions owing to its significance in both dynamical theory and various applications. Among coupled nonlinear systems, we concern the sinusoidally coupled oscillators which was pioneered by Kuramoto [12], and recently it has been a hot topic in many scientific disciplines such as neuroscience, nonlinear dynamics, statistical physics, engineering and network theory [2,18]. In this work, we study dynamical behavior of a finite group of Kuramoto oscillators bidirectionally coupled in a ring by performing nonlinear stability analysis. The classic Kuramoto model was set as a population of sinusoidally coupled oscillators with all-to-all coupling. A lot of studies have been done for this model, see [4,5,7,8,9] for example, and a nice property that is useful in analysis is the meanfiled property. For identical Kuramoto oscillators with all-to-all coupling, it is well known that the phase synchronization (in short, sync) is the only stable phase-locked state, which denotes the collapse of all phases into a single phase, see [17]. Hence, almost all initial configurations of phases converge to the phase sync asymptotically. It is reasonable to guess that different asymptotic patterns for Kuramoto oscillators can emerge depending on different network topologies. For example, the literature [17] studies stability properties of the Kuramoto model with identical oscillators by linear stability analysis and the authors presented a six-node example to point out that a stable non-sync equilibrium arises for oscillators bidirectionally coupled in a ring. Recently, Wiley, Strogatz, and Girvan [20] addressed the problem of "the size of the sync basin" for the identical Kuramoto oscillators with k-neighbor couplinġ by setting θ N +j := θ j . One can assume ω = 0 by applying substitution θ i → θ i +ωt.
In [20] they found that when k/N is greater than a critical value, then the phase sync is the only stable phase-locked state; as k/N passes below this critical value, other stable phase-locked states are born which takes the form of twisted waves (or splay-state). They also numerically investigate how the size of basin of phase sync or splay-state depends on the winding number of the state. In [11,19], the stability of phase-locking was considered for identical Kuramoto oscillators unidirectionally coupled in a ring, which takes the form of a system on an asymmetric network: In particular, in [19], Rogge and Aeyels used an extended Gershgorin disc theorem to derive the linear stability/instability of phase locking when the phase difference of neighboring oscillators is in either (− π 2 , π 2 ) or ( π 2 , 3π 2 ); however, when phase differences equal to π 2 or 3π 2 , the system is not hyperbolic and zero eigenvalues arise so that the linearization approach does not work well. In [11], Ha and Kang performed nonlinear stability analysis and presented nontrivial proper subsets of synchronization and splay-state basins with positive Lebesgue measure in N -phase space. In this paper, we will concern the identical Kuramoto oscillators bidirectionally coupled in a ring, which is given by the following systeṁ The main tool is the theory of Lojasiewicz inequality of analytic potential, by which we can determine the stability/instability of all possible phase locked states of (1.1), including those with phase differences equal to π 2 or 3π 2 . The main contributions of this paper are as follows. First, we will identify the formation and stability/instability of all phase-locked states for system (1.1) (see Theorems 3.1 and 3.2). This also enables us to determine the exact number of (asymptotically) stable phase-locked states (see Remark 3.4). Second, for the stable phase-locked states, we present their basins with positive Lebesgue measure in Nphase space (see Theorem 4.3). Third, we calculate the Lojasiewicz exponent for the stable equilibriums of (1.1) and clarify that the convergence towards the stable phase-locked states is exponentially fast (see Theorems 3.6 and 4.3).
Organization of paper.-In Section 2, we give some preliminaries. In Section 3, we identify the formation and stability/instability of all phase-locked states. The Lojasiewicz exponent of stable phase-locked states is also presented. In Section 4, we prove the convergence of (1.1) and give an estimate on the basin of stable phase-locked state. Section 5 is devoted to be a brief summary of this paper.
2. Preliminaries. We consider N (N ≥ 3) coupled oscillators. The dynamics of the i-th phase θ i is governed by the following system: where we set θ N +1 := θ 1 . It is easy to see that N i=1θ i = 0, so the phase-locked solutions of the system (2.1) are its equilibrium points. Let then system (2.1) can be written as a gradient systeṁ Next, we present some results for gradient systems with analytic potential. Consider the following systemẋ (t) = −∇f (x(t)).

(2.4)
A crucial tool in this study is the nice theory for gradient inequality which was first developed by Lojasiewicz [16]. In his earlier work he proved the following result.
Proposition 2.1. Let f : R N → R be a real analytic function. 1. For any x * ∈ R N , there exist a neighborhood N (x * ) of x * and some constants c > 0 and r ∈ (0, 1 2 (2.5) 2. Let x(·) be a solution of (2.4). If {x(t)} ∞ t=0 is bounded, then there exists an equilibrium x ∞ such that x(t) → x ∞ .
The inequality (2.5) is referred as the celebrated Lojasiewicz's inequality and the constant r ∈ 0, 1 2 is called the Lojasiewicz exponent of f at x 0 . We note that for a non-equilibrium, i.e., ∇f (x) = 0, the inequality (2.5) certainly holds. However, when an equilibrium is concerned, the above inequality reveals a fundamental relation between the potential and its gradient near the equilibrium. Owing to this inequality, he could prove the second assertion in Proposition 2.1. Thus, the Lojasiewicz inequality provides a powerful tool to analyze the convergence of a trajectory towards a single equilibrium. To catch some idea from Lojasiewicz inequality to convergence, we refer to [3] for the so-called finite-length argument.
Furthermore, the value of Lojasiewicz exponent gives some information on the convergence rate; more precisely, the convergence is at least algebraically slow if r ∈ 0, 1 2 , and exponentially fast if r = 1 2 (see [3]). Using similar argument as in [3] with Lojasiewicz inequality, the following result can be obtained.
Proposition 2.2. [3,14] Let f : R N → R be a real analytic function and x(·) be a bounded trajectory of (2.4) which converges to an equilibrium x ∞ . Let r * ∈ (0, 1 2 ] be the Lojasiewicz exponent of f at x ∞ . Then we have: 1. If r * = 1 2 , then there exist constants C, T, λ > 0 such that 2. If r * < 1 2 , then there exist constants C, T > 0 such that Based on Lojasiewicz inequality, Absil and Kurdyka [1] gave a sufficient and necessary condition for the stability of equilibrium of gradient system. We restate it here and this will be the main tool to identify the stability/instability for each equilibrium of (2.1).
Let f be real analytic in a neighbourhood of z * ∈ R n . Then, z * is a stable equilibrium of (2.4) if and only if z * is a local minimum of f . Furthermore, it is asymptotically stable if and only if it is a strict local minimum.
Before we close this section, we present an inequality which will be useful in this paper. Let us consider a symmetric and connected network G = (V, E) where V = {1, 2, . . . , N } and E ⊂ V × V are vertex and edge sets, respectively. For a given network or graph G, we assume that Kuramoto oscillators are located at the nodes of the network, and they interact symmetrically through the interacting channels registered by the interaction topology E. We say G = (V, E) is connected if for any two nodes i and j, there exists a path The distance between i and j is the number of arcs in a shortest path connecting i and j. We now state a lemma from [10] as follows.
Lemma 2.4. [10] Suppose that the graph G = (V, E) is connected and the set {γ i : i = 1, 2, . . . , N } has zero mean: Then, we have where diam(G) denotes the diameter of a graph which is the shortest distance between any pair of nodes in G.
Theorem 3.2. The equilibrium in (3.6) is the only stable equilibrium of (3.2). Moreover, The equilibrium in (3.6) is asymptotically stable.

XIAOXUE ZHAO, ZHUCHUN LI AND XIAOPING XUE
We denote the index set For any θ ∈ N (θ * ), we have We now consider several cases depending on α and the configurations of G 1 and G 2 .
The sign of f (θ ) − f (θ * ) depends on the sign of . This means that θ * is not a minimum of f in N (θ * ). Therefore, it is unstable. (6) If α = 3π 2 , we note that π − α = − π 2 ∼ 3π 2 , so θ * is corresponding to φ * = 3π 2 , 3π 2 , . . . , 3π 2 . In this case, we consider θ ∈ N (θ * ) corresponding to Similar to Case (5), we find the equilibrium is unstable. We now summarize the above cases (1)- (6) to conclude that the only stable equilibrium is the state φ = (α, α, . . . , α) with α ∈ − π 2 , π 2 . Furthermore, we can easily see from Case (1) that θ * is a strict local minimum. Therefore, by Lemma 2.3 we conclude that it is asymptotically stable. Remark 3.3. A classic method for stability/instability analysis is based on the linearization and eigenvalues. In [19], Rogge and Aeyels applied Gershgorin disc theorem to perform the linear stability analysis for phase locking with phase differences in (− π 2 , π 2 ) or ( π 2 , 3π 2 ), when the oscillators are unidirectionally coupled in a ring. However, this approach based on linearization does not work when the phase difference is π 2 or 3π 2 which produces zero eigenvalue (system (3.2) is not hyperbolic). In contrast, our analysis in Theorem 3.2 does not rely on any information on the algebraic spectrum of linearization, and we can determine the stability/instability for all possible phase locked states. Actually, the number of the stable non-phase-sync equilibriums of (2.1) is exactly the number of integer k's with which is given by 2 N −1

4
. Therefore, the total number of stable phase-locked states is given by 2 N −1 4 + 1.
Remark 3.7. Theorem 3.6 gives the first result for Lojasiewicz exponent of Kuramoto model when the phases of equilibrium are distributed in an arc which is larger than a quarter of circle. Actually, in [14,15] it was proved that the exponent is 1 2 when the phases are distributed inside a quarter of circle. On the other hand, an example in [14] also shows that the exponent can be less that 1 2 when they are distributed on the boundary of the quarter of circle, which implies the algebraic convergence shown in [11]. In this paper, we show that for system (2.1) the exponent is still 1 2 . This will enable us to conclude the exponential rate by Proposition 2.2 once we can show the convergence towards such an equilibrium.

4.
Convergence and a basin for stable phase locking. In this section, we will concern the global dynamics of system (2.1). We will show that any trajectory of (2.1) must converge and we present a non-trivial subset of the basin for the stable phase-locked state in (3.6).

Convergence.
A general form of the following theorem was presented in [13] for gradient system with analytic and periodic potential. For readers' convenience, we state it for system (2.1) and give a proof here.
Theorem 4.1. For any initial data, the solution of system (2.1) converges to some equilibrium.

4.2.
Basin. For a given configuration φ = (φ 1 , φ 2 , . . . , φ N ),we introduce the indices then we have φ M = max 1≤i≤N φ i and φ m := min 1≤i≤N φ i . Let φ(t) be the solution of (3.2). For time-varying configuration φ(t), the indices M and m depend on t and the extremal phase differences φ M − φ m is Lipschitz continuous and piecewise differentiable. We will show that the set: Lemma 4.2. Let φ be the smooth solution to system (3.2) with initial condition φ 0 ∈ B con . Then φ M is noninceasing and φ m is nondecreasing for all t ∈ [0, +∞).
Proof. • Step 1. Suppose for some T ∈ (0, ∞] we have Then for t ∈ [0, T ), we have Let φ be the smooth solution to system (3.2) with initial condition φ 0 ∈ B con which satisfies Then we have Furthermore, the convergence is exponentially fast. Proof.
Step 1. We refine the estimate in Lemma 4.2 and claim that: (i) φ m is strictly increasing; (ii) φ M is strictly decreasing. We will prove (i) and ignore the proof for (ii) since they are similar. Note thaṫ If φ m is not strictly increasing, then we can find some open interval I such that Then it follows from the graph of sinusoidal function on − π 2 , π 2 that we should have The second case certainly contradicts the definition of φ m . So we should have This implies that for t ∈ I 0 =φ m =φ m+1 = (sin φ m+2 −sin φ m+1 )−(sin φ m+1 −sin φ m ) = sin φ m+2 −sin φ m+1 .
Using similar argument we can obtain φ l = φ m , ∀ t ∈ I, ∀ 1 ≤ l, m ≤ N, which implies that φ is an equilibrium. However, this contradicts the initial condition which is not an equilibrium. Thus, φ m is strictly increasing.

XIAOXUE ZHAO, ZHUCHUN LI AND XIAOPING XUE
On the other hand, we have a conservation law that Thus, we have φ i (t) = 2kπ, i.e., α = 2kπ N .
That is, φ(t) converges to the splay-state 2kπ N 1 N as t → ∞.
Finally, we use Theorem 3.6 and Proposition 2.2 to conclude the exponential rate (see Remark 3.7). That is, α = 0. In other words, the trajectory exponentially converges to a phase sync state.

5.
Conclusion. In this paper, we considered the dynamical properties of identical Kuramoto oscillators bidirectionally coupled in a ring. By rigorous analysis we can describe the formation of all phase-locked states and prove that the phase sync state and splay states with phase difference less than π 2 are the only stable ones. Furthermore, we gave estimates for the basins for these stable phase-locked states and we showed that the convergence rate is exponential fast by investigating the Lojasiewicz exponent of its potential.