ON THE GENERIC COMPLETE SYNCHRONIZATION OF THE DISCRETE KURAMOTO MODEL

. We study the emergent behavior of discrete-time approximation of the ﬁnite-dimensional Kuramoto model. Compared to Zhang and Zhu’s recent work in [38], we do not rely on the consistency of one-step foward Euler scheme but analyze the discrete model directly to obtain sharper and more explicit result. More precisely, we present the optimal condition for the convergence and order preserving for identical oscilators with generic initial data. Then, we give the exact convergence rate of the identical oscillators to their limit under the reasonable assuption on time step. Finally, we provide an alternative proof of the asymptotic phase-locking of nonidentical oscillators which can be applied whenever the given Lyapunov functional is continuous and all zeros are isolated.


1.
Introduction. Synchronization is one of the novel type of emergent behavior that has been frequently observed in nature. After Dutch physicist Christiaan Huygens' observation in 17th century, many types of synchronization of weakly coupled oscillators, such as firing neurons and flashing fireflies, were observed by several physicists and biologists [3,14,31]. Then, the systematic studies using explicit mathematical models were begun by Arthur Winfree [37] and Yoshiki Kuramoto [26]. The well-known phase synchronization model proposed by Kuramoto consists of the ODEs for finitely many weakly coupled oscillators: where the system parameter κ > 0 denotes the universal scale of coupling strengths between each pair of particles, and θ i , ν i ∈ R represent phase and natural frequency of i-th Kuramoto particle, respectively. The Kuramoto model and its variations have been extensively studied in the literature as a prototype model for synchronization, especially for the phase-transition behavior. To name a few, [1,23,33] provided comprehensive investigations on the Kuramoto model including numerical examples from the perspective of physicists. In [7,9,10,11,18,22], authors studied the Kuramoto model with extra structure such as inertia or frustration, and [20,21] studied an asymptotic complete phase/frequency synchronization for various initial data. Moreover, [5,12,34,35,36] estimated a critical coupling strength to guarantee the existence of phase locked states under several conditions, such as all-to-all and bipartite graph network topology, and [15] provided some sufficient conditions for the Kuramoto model with static network topology to exhibit asymptotic phase locking. For the kinetic description of the Kuramoto model, the well-posedness, spectral analysis, critical coupling strength and synchronization estimate of Kuramoto-Sakaguchi equation are proposed in [2,4,13,27,28,29,30,32], and [24,25] studied a nonlinear stability of Kuramoto-Sakaguchi equation with or without diffusion. However, the visualization of those results can only be done by discretization, which inevitably approximates the true model (1). For this reason, the applicability of the theory of (continuous) Kuramoto model (1) to the Euler type discrete time approximation scheme has also been studied. Here, the discrete-time version of (1) can be written as where h := ∆t > 0 is a given fixed time step. In [6], Choi and Ha first proved the complete consensus of the discrete-time Kuramoto model with identical oscillators (ν i ≡ ν) when initial phase configurations are confined in a half circle and h is sufficiently small. More precisely, (2) exhibits a complete consensus (complete phase synchronization) if 0 < κh < 1, max which is analogous to the result in [16]. Then, Ha et al. [17] verified the uniformin-time convergence of (2) to (1) during h → 0 and exponential synchronization of (discrete-time) identical oscillators, whose corresponding result was also provided in [8,16,19]. After [20] studied a synchronization and phase-locked state of the Kuramoto model for more generic setting, Zhang and Zhu established a corresponding stability theory in [38] for discrete gradient flow to show that (2) exhibits an asymptotic phase-locking for sufficiently small ν κ and h (see Definition 2.3 for the definition of phase-locked state). A notable remark is that [38] did not specify an explicit size of h to show the desired emergent behavior, while other previous results always provided the upper bound of h. This is because the main idea of [38] relys on the consistency of one-step foward Euler discretization to its continuous model, and the necessary h depends on the implicit converging speed of the discrete Kuramoto model to continuous Kuramoto model. Therefore, in this paper, we study a sufficient framework for the synchronization of the model (2) for generic initial data without using the classical consistency and convergence result of one-step foward Euler scheme. Then, the criteria for the synchronization of (2), especially the upper bounds for possible time step h are explicitly provided, and the result becomes more applicable in practice. Since system (2) depends only on κh, {ν i h} N i=1 , and initial configurations {θ i (0)} N i=1 , it is clear that all criteria must be able to be characterized by these independent variables, and all of our results are given by this way.
The main results of this paper are three-fold. First, we proceed a Lyapunov functional approach for discrete system (2) to characterize the asymptotic behavior for identical oscillators and find a critical size of h. (see Theorem 3.3). Then, we provide a sufficient criterion to preserve the order of identical oscillators and get an exact convergence rate to either complete phase synchronization state or bipolar state for generic initial data (see Lemma 3.4 and Theorem 3.7). Finally, we combine the Lyapunov functional approach with the finiteness of phase-locked states (see Theorem 4.4 and Theorem 4.6) to (2) to prove the emergence of phaselocked state for generic initial data. Although we follow the idea in [4,20] of the continuous Kuramoto model, the proof is completely parallel to their continuous version and never use the consistency with their discretization as in [38].
The rest of this paper is organized as follows. In Section 2, we briefly review some previous results on the continuous-time Kuramoto model (1), such as order parameters, asymptotic behavior and finiteness of phase-locked states. In Section 3, we provide a synchronization estimate and exponential convergence rate for the identical discrete Kuramoto model by using the monotoneness of order parameter and order preserving property of identical oscillators. In Section 4, we prove an asymptotic phase-locking for the nonidentical discrete Kuramoto model for generic initial data, when the coupling strength κ is sufficiently large compared to ν. Finally, Section 5 is devoted to a brief summary of our results.

2.
Preliminaries. In this section, we review some basic concepts and properties of (1)-(2) without proof. First of all, concerning the dynamics of deviationŝ loss of generality. In particular, it suffices to consider the case ν i ≡ 0 to see the behaviors of identical Kuramoto model. Note that this statement holds for both continuous and discrete Kuramoto model (1)-(2).

2.1.
Continuous-time Kuramoto model. In this part, we present some previous results on the continuous-time Kuramoto model (1). First, we introduce the definition of order parameters and their properties, which are frequently used in our whole discussion. Definition 2.1. Let Θ = (θ 1 , · · · , θ N ) ∈ R N be a given phase configuration. Then, the order parameter (r, φ) ∈ R 2 of the configuration Θ is defined as Now, for notational simplicity, we suppress Θ-dependence on r and φ, i.e., r := r(Θ), φ := φ(Θ).
Then, we introduce the following well-known properties of order parameter (r, φ) independent to the dynamics of Θ, and thus hold for both continuous and discrete Kuramoto model.
Lemma 2.2. The order parameters (r, φ) satisfy In particular, we have Note that the uniqueness of φ in (3) is only guaranteed up to modulo 2π. However, once the dynamics of Θ is given as (1), we can specify the angle φ ∈ R continuous in time and satisfying following differential equations: be a solution to the continuous time Kuramoto model (1). Then, (r, φ) satisfẏ In fact, this is a key estimate for the convergence of identical Kuramoto model (ν i ≡ 0). If ν i ≡ 0, (4) 1 simply gives a non-decreasingness of the order parameter r, and since it has an upper bound 1, each sin(θ j − φ) vanishes at infinity provided that r(Θ(0)) > 0. Then, we have the following: (1), where the natural frequencies ν i and initial configuration Then, for each i, j = 1, · · · , N , the limit exists and In [38], Zhang and Zhu provided an analogous result for Proposition 2 for the discrete Kuramoto model (2). The discrete gradient flow structure (see Proposition 6) and the convergence, consistency of one-step foward Euler scheme were used for the existence of Θ ∞ and the classification of asymptotic states. The weakness of this argument is that the consistency argument is only valid for uncertainly small h and therefore has a difficulty to apply the result to real numerical experiments. We here provide an alternative proof of the asymptotic synchronization by using (2) itself, so that the sufficient framework for the synchronization becomes more explicit (see Section 3).

Remark 1.
There are two main obstacles to adapt the above argument to (2).
1. Clearly, the first one is that we have to find an alternative way to get the non-decreasingness of r instead of Proposition 2.2. The other difficulty is that we cannot deduce the convergence of θ j − φ from sin(θ j − φ) → 0 immediately because of the absence of the continuity of θ j and φ. 2. Another unique feature of continuous-in-time model is collision avoidance of identical oscillators, i.e., order preserving. This can be shown by the uniqueness and analyticity of solution {θ j } N j=1 for (1), which cannot be guaranteed for the discrete Kuramoto model. Still, we can show the order preserving property for the identical discrete Kuramoto model by using different argument (see Section 3). Now, we review two previous results on the phase-locked states of nonidentical oscillators, namely, emergence of phase-locked state and finiteness of phase-locked states. Below, we introduce a definition of phase-locked states and other relative notions for continuous and discrete Kuramoto model. 1. We say that Θ ∞ = (θ ∞ 1 , · · · , θ ∞ N ) is a phase-locked state of continuous (or discrete) Kuramoto model if and only if it is a stationary solution of (1) (or (2)), i.e., 2. We say that the phase configurations Θ = (θ 1 , · · · , θ N ) andΘ = (θ 1 , · · · ,θ N ) are equivalent if and only if for each 1 ≤ i, j ≤ N there exists an integer m ij satisfying (θ i −θ i ) − (θ j −θ j ) = 2m ij π. 3. The phase-locked state Θ is called standardizable if and only if it has a positive order parameter r(Θ) > 0.
Then, the following proposition tells us that the asymptotic phase-locking emerges in a strong coupling regime for generic initial configuration. Proposition 3. [20] Suppose that the initial configuration Θ in and natural frequen- Then there exists a large coupling stregth κ ∞ > 0 such that if κ ≥ κ ∞ the solution of (1) with the initial data Θ in exhibits asymptotic phase-locking.
Although the Kuramoto model (1) can exhibit asymptotic phase-locking for generic initial configuration, the number of possible asymptotic state is finite for given initial data, which comes from the proposition below. Proposition 4. [21] There are at most 2 N non-equivalent phase-locked states for the nonidentical Kuramoto model (1).

Remark 2.
From the above definition, the phase-locked state of the discrete Kuramoto model (2) is equivalent to that of continuous-in-time Kuramoto model (1). Therefore, the Proposition 4 also tells us that there are at most 2 N non-equivalent phase-locked states for the nonidentical discrete Kuramoto model (2).
2.2. Discrete Kuramoto model. In this part, we summarize some previous synchronization results on the discrete Kuramoto model (2) provided in [6,38]. All results in this subsection are at least slightly improved in Section 3 and Section 4.
Then, we have an asymptotic complete phase synchronization (see Definition 3.1 for the definition of complete phase synchronization): In Section 3, we extend this result to 0 < κh ≤ 1, though the proof is completely different with [6]. In [6], Choi and Ha proved Proposition 5 by adapting the idea in [16] for the discrete Kuramoto model (2). However, we prove the extended result by showing the convergence of the identical discrete Kuramoto model for 0 < κh ≤ 2 and the order preserving property for 0 < κh ≤ 1 (see Lemma 3.6).
Next proposition below and its corollary are the main result of [38], which follows the idea of gradient flow argument in [20] to the discrete Kuramoto model (2).
where h is the mesh size. Then, for sufficiently small h, there exists a point Remark 3. For the identical discrete Kuramoto model, the sufficient h to achieve the convergence given by Proposition 6 is To see this, we here present the brief introduction to the proof of Proposition 6. For the Kuramoto model, the potential function f is given as and there exists ξ(n) satisfying where each element H ij of the Hessian H is bounded by Therefore, we have However, N −independent result of Proposition 5 indicates that the sufficient condition (5) might not be optimal, and in fact, this contidion is improved in Theorem 3.3.
Finally, as a Corollary of Proposition 6, one has the emergence of phase-locked state for generic initial data.
Then, there exists a large coupling strength κ ∞ > 0 and a small mesh size h 0 > 0 such that, if κ > κ ∞ and 0 < h < h 0 , the emergence of phase-locked state will asymptotically occur, i.e., we can find a phase locked state Θ ∞ such that the solution to system (2) with initial data {Θ in } satisfies 3. Emergent dynamics of identical oscillators. In this section, we study the emergent dynamics of discrete Kuramoto model (2) with identical oscillators ν i ≡ ν: For this, we consider the alternative sequence: , so that the dynamics of Θ(n) n≥0 is given by : Therefore, as we pointed out in Section 2, we study the model (7) in the whole Section 3 instead of (6). Note that the total sum of phases N i=1θ i is preserved in (7).

Asymptotic behavior.
In this subsection, we study a sufficient framework to guarantee the monotonicity of order parameter r for identical oscillators. Then, we show that for generic initial data, the system exhibits either bipolar configuration or complete phase synchronization asymptotically as n → ∞.
First, we begin this subsection with the definition of bipolar configuration and complete phase synchronization: [20] Let Θ = (θ 1 , · · · , θ N ) be a given phase configuration in R N .
1. The phase configuration Θ = (θ 1 , · · · , θ N ) is a bipolar configuration if and only if the following two conditions hold: According to [38], the solution {Θ(n)} n≥0 of (7) will exhibit either the asymptotic complete synchronization or the convergence to some bipolar configuration for sufficiently small h. On top of this, we here provide the critical time step h to achieve the monotonicity of order parameter r, which is given as follows: Lemma 3.2. Let {Θ(n)} n≥0 be a vector-valued sequence satisfying (7). Then, if the system parameters κ and h satisfy then the order parameter r n := r(Θ(n)) increases monotonically as n increases.
Proof. It suffices to prove that the functional A n below is always nonnegative: To see this, we first write the functional A n explicitly by using Lemma 2.2: Now, we introduce the function f a motivated from the summand in the representation of A n above. More precisely, we define the function f a as below: Then, one can easily see that f a (x) is a 2π-periodic even function with respect to x and a sin x, x − a sin x ∈ [0, π], 0 ≤ x ≤ π, 0 ≤ a ≤ 1. Therefore, we obtain the non-negativity of f a and conclude Remark 4. Let us evaluate the optimality of the criteria (8) by checking a specific case. To do this, assume that N, ν, Θ(0) and κh are given as Then, we have θ 1 (n) = −θ 2 (n), φ(Θ(n)) = 0, n ≥ 0, and when ε is a small positive number satisfying Therefore, condition (8) is optimal to guarantee the non-decreasing property of order parameter r without any information on intial configurations. Now, we are ready to show our first main result.
Theorem 3.3. Let {Θ(n)} n≥0 be a vector-valued sequence satisfying the recurrence relation (7), where the system parameter κh satisfies the condition 0 < κh ≤ 2. If the initial order parameter r 0 := r(Θ(0)) is strictly positive, then for each i, j = 1, · · · , N , exists and satisfies Since {r n } n≥0 is monotonically increasing with upper bound 1, the sequence {r n } n≥0 converges to some positive real number r ∞ (≤ 1) and therefore We now claim: Indeed, since f a is 2π-periodic even function, it suffices to show for x ∈ [0, π]. Then the above inequality holds for where we used the following basic relations: Similarly, for x ∈ [ π 2 , π], we take y = π −x and complete the proof of aforementioned claim f a (x) = sin(a sin y) sin(y + a sin y) ≥ 2 π (a sin y) · sin( π 2 + 1) where we used the following relations in the first inequality: Thus, for each 1 ≤ j ≤ N , we have and here the former case also implies the latter case. To see this, recall that Lemma 2.2 gives a following representation of order paramter r n : Then, the convergence of order parameter r n to 1 is equivalent to cos(θ j (n) − φ(n)) → 1 for each j, which implies the latter case in (9). Therefore, in any case, we have [κhr n sin(φ(n)−θ j (n))] = 0. Now, although φ(n) always defined uniquely up to modulo 2π, we may employ the inequality to find a unique φ(n + 1) from φ(n) satisfying Therefore, for this inductively determined sequence {φ(n)} n≥0 , we have lim sup From this observation, the convergence of sin(θ j (n) − φ(n)) to 0 now implies the convergence of θ j (n) − φ(n), and there exist N integers k 1 , · · · , k N determined by the relation lim n→∞ (θ j (n) − φ(n)) = k j π.
Finally, once we obtain the convergence of {φ(n)} n≥0 ⊂ R from the conservation of j θ j (n), i.e., we conclude that the limit θ ∞ i := lim n→∞ θ i (n) exists for all i and Remark 5. If the order parameter r is zero at time k, then Lemma 2.2 implies that for 1 ≤ i ≤ N, we have Therefore, the phase configuration remains constant from the time k and never changes. From this observation, one can conclude that the identical discrete Kuramoto model (7) always converges if 0 < κh ≤ 2.

3.2.
Order preserving and exponential convergence. In this subsection, we show the order preserving property of the identical discrete Kuramoto model (7) and characterize the bipolar states that can be reached from given initial data. Then, we provide the exact convergence rate of the oscillators to the bipolar state.  (7) with 0 < κh ≤ 1, r 0 = r(Θ in ) < 1. Then, the order of the oscillators {θ i } N i=1 is invariant, i.e., θ in i < θ in j ⇒ θ i (n) < θ j (n) ∀n ≥ 0, 1 ≤ i, j ≤ N, and the complete phase synchronization cannot be achieved in any finite time step.
Proof. We split the proof into two steps.
• Step 1: We first show that r n is strictly less than 1 for every n < ∞.
Suppose in the contrary that there exists an integer n 0 ≥ 0 satisfying r n0+1 = 1 and r k < 1 for every k ≤ n 0 .
From the above conditions, there exist N integers (k 1 , · · · , k N ) satisfying Now, consider the following alternative intial configuration Θ(0) = θ j (0) and denote Θ(n) n≥0 as a solution to (7) with initial data Θ(0). Then, one can easily see from the induction argument that θ j (n) = θ j (n) − 2k j π holds for any n ≥ 0 and 1 ≤ j ≤ N . Therefore, we obtain where the existence ofξ 1j between φ −θ 1 and φ −θ j is guaranteed from the mean value theorem. Since r k is assumed to be smaller than 1 for k ≤ n 0 and 0 < κh ≤ 1, each multiplicand 1−κhr k cos ξ 1j (k) in the above equation is strictly positive, which implies θ j (0) = θ 1 (0) for all 1 ≤ j ≤ N and contradicts to the initial condition r 0 < 1.
Similar to the previous step, the difference of phases θ i and θ j at time (n + 1) has a multiplicative representation where the existence of ξ ij between φ − θ i and φ − θ j is again guaranteed from the mean value theorem. Then, since 0 < κh ≤ 1, r k < 1 and | cos ξ ij | ≤ 1, the sign of θ j − θ i is invariant in time n.
Remark 6. Below, we comment on two facts of the order preserving property.
1. For any k ∈ Z N , if {Θ(n)} n≥0 is a solution to (7), 2πk+Θ(n) is also a solution of (7). Therefore, the order preserving property also implies that 2. The order preserving property has been partially shown in [38], when θ j 's are initially highly aggregated. Now, we classify the asymptotic states of the identical discrete Kuramoto model. We here used a similar argument with [20] for continuous Kuramoto model, and the admissible set of h is again extended from [38].
Next, we will recover another well-known result on the continuous time Kuramoto model under discrete time setting: identical Kuramoto oscillators confined in half circle exhibits complete synchronization. As we noted in Section 2.2, this result was already shown for 0 < κh < 1 by Choi and Ha in [6].
Proof. From Theorem 3.3 and order preserving property, it suffices to show that {(θ N − θ 1 )(n)} n≥0 is a nonincreasing sequence of nonnegative real numbers. If {(θ N − θ 1 )(n)} is nonincreasing for all n ≤ k, then we have Now, recall that the diameter θ N − θ 1 at time k + 1 has a following formula: Therefore, we conclude our desired result by applying (13) to (14).
In our second main result, we provide the exact convergence rate to two asymptotic states described in Lemma 3.5.
Moreover, the condition I 1 = ∅ implies that the order parameter r n converges to 1, i.e., lim n→∞ r n = 1, and ξ 1N → 0 can be obtained from the definition of I 0 . Therefore, we have • Case 2 (I 1 = {N}): In this case, there exists a common limit θ ∞ only for θ 1 , · · · , θ N −1 : and this common limit θ ∞ can be determined by the relation If θ N (n) − π is smaller than θ 1 (n) or larger than θ N −1 (n) for some n, then the solution {Θ(n)} n≥0 exhibits an asymptotic complete phase synchronization according to Lemma 3.6. Since this contradicts to the condition I 1 = {N }, we have Now, we combine (17)- (19) and order preserving property to obtain Thus, we have and the order parameter r n , angle ξ 1N −1 converge to where we used Lemma 2.2 and (17) to find the limit of r n . Therefore, we have Remark 7. In [38], the above convergence rates are implicitly given as the upper bounds of convergence rates when h is sufficiently small.

4.
Phase-locking for nonidentical oscillators. In this section, we provide a Lyapunov functional for the Kuramoto model with nonidentical oscillators and show the asymptotic phase-locking for generic initial data for sufficiently large coupling strength. Proof. For simplicity, we now set Below, we compute the increase of the functional F (Θ) at time step n.
First, similar to Lemma 3.2, we write the difference F (Θ(n+1))−F (Θ(n)) explicitly: Then, we split J 2 into three terms and claim: To prove this, we again simplify J 1 + J 21 + J 23 by using a j := ν j − κr n sinθ j (n).
Then, J 1 and J 21 + J 23 can be written as: ha 2 j + κhr n a j sinθ j (n) , where we used r n = 1 N N i=1 cosθ i (n) in the last equality.
On the other hand, from the Taylor's theorem, there exists c j ∈ [0, 1] satisfying cos θ j (n) + a j h = cosθ j (n) − a j h sinθ j (n) − (a j h) 2 2 cos θ j (n) + c j a j h .
Once we have a Lyapunov functional F (Θ), the unifom boundedness of F (Θ(n)) in time n will give a convergence of F (Θ). The following lemma says that if coupling strength κ is sufficiently large and the majority of phases θ i are initially confined in some small interval, then all phases θ i are uniformly bounded in time n. In fact, an analogous result of Lemma 4.2 is invented by [20], and this is a first step to achieve the uniform boundedness of Θ.
Proof. The basic idea of this lemma is similar to [20]. We assume the finite first hitting time of the diameter of majority set to the prescribed diameter . Once we deduce a contradiction, the majority set is then contained in small moving arc whose length is . Then, if the distance from these majorities to one of the other phases is assumed to be unbounded, it will be captured eventually to this length arc. The continuity of {θ i (t)} t≥0 in the argument in [20] is replaced by the boundness of θ i (n + 1) − θ i (n), and the majority diameter bound has to be larger than this to establish the above capturing argument. Although some of the ideas are similar to [38], we here provide a full proof of our lemma for the completeness of the paper.
First, note that m cannot be zero (i.e., m * ≥ 0) due to the condition (22). If m < ∞, we can find a index pair (j m , k m ) satisfying from the definition of m. Then, we consider a following inequality and condition 0 < κh ≤ 1 to derive a lower bound of θ jm − θ km at time n = m * : However, we can use (2) and (24) to obtain Here, we have the lower bound of cosines in (25) from the following observations: Thus, it follows from (25), (26) and the condition on κ in (22) that which gives a contradiction. Therefore, we have m = ∞, i.e., diamΣ 0 (n) ≤ , n ≥ 0.
Now, for the times steps z and z * , we introduce two indices k z * and k z in {1, · · · , n 0 } satisfying and we claim: θ n0+1 (z) < 2π + θ kz (z) + . To see this, we again use (2) and condition (D(ν)+2κ)h < in (22). More precisely, we can prove the above claim as below: Therefore, θ n0+1 (z) lies on a length-arc containing Σ 0 (z), and by using same argument with Case A, we have This gives a contradiction to (27), and hence the diameter D(Θ(n)) is uniformly bounded.
(ii) (Uniform boundness of Θ(n)) Note that the condition which is uniformly bounded in n.
Proof. Define P as a intersection of G −1 (0) with (−2π, 4π) m , i.e., Since P is finite, we may impose the order for the elements of P and denote as Then, there exists a positive integer M such thatB 1/M (x 1 ), · · · ,B 1/M (x s ) are disjoint closed sets. Note that the disjoint condition implies that R > 2 M . Now, the set G([0, 2π] m − s i=1 B 1/M (x i )) is a compact set which does not contain 0, and then the distance r M from 0 is strictly positive. Therefore, we obtain Moreover, concerning the periodicity of G, we have Now, we are ready to prove the phase-locking of two oscillator system. In fact, for two oscillator system, there is no need to specify the condition for initial data Θ(0). Theorem 4.4. Suppose that the natural frequencies and coupling strength satisfy the following conditions: Then, each sequence {θ 1 (n)} n≥0 and {θ 2 (n)} n≥0 of the system (2) converges: i.e., the solution {Θ(n)} n≥0 exhibits an asymptotic phase locking.
It suffices to show that (32) implies the desired result. In fact, the asymptotic phase-locking can be obtained from the convergence θ 1 − θ 2 , as we already know that the total sum θ 1 + θ 2 is conserved for all n ≥ 0. We can obtain the convergence of θ 1 − θ 2 by using Lemma 4.3 to (32).
Therefore, we obtain max 1≤i≤N |θ N I i (n + 1) − θ I i (n + 1)| ≤ (1 + 2κh) max where we used j ν j = 0 in the last inequality. Since the initial phases Θ I (0) and Θ N I (0) are same, we have Remark 9. If we set h := t n and n → ∞, we have which coincides with the estimate in [20].
Now, consider the solution set Then, there are finitely many non-equivalent phase-locked states for given κ, ν from Proposition 4, and the intersection of each equivalence class and S has N elements modulo 2π. To see this, consider two equivalent configurations Θ 1 , Θ 2 in S . Then, for every 1 ≤ i, j ≤ N , we have θ 1 i − θ 2 i ≡ θ 1 j − θ 2 j mod 2π, and therefore Now, in order to apply Lemma 4.3, we consider the following function: Then, G is a continuous 2π-periodic function, and for every X ∈ G −1 (0), x k is an element of S . Since X 1 ≡ X 2 mod 2π implies X 1 ≡ X 2 mod 2π, G −1 (0) has finitely many elements modulo 2π. Finally, by applying Lemma 4.3 to (35) and function G, we obtain the convergence of the configuration (θ 1 (n), · · · , θ N −1 (n)) to an element of G −1 (0), which implies the asymptotic phase-locking. 5. Conclusion. In this paper, we have presented a direct approach to see the emergence of synchronization of the discrete-time Kuramoto model. For the identical oscillators, we first show that the condition 0 < κh ≤ 2 is enough to obtain the asymptotic equilibrium for every initial configuration by using the nondecreasing property of order parameter r. This is the sharpest condition to guarantee the nondecreasingness of r. Then, to get the exact convergence rate to the equilibrium, we show the order preserving property and classification of all asymptotic states that can be reached from generic initial data under the condition 0 < κh ≤ 1. In particular, we slightly extended the condition to achieve the complete phase synchronization in [6] to 0 < κh ≤ 1 when the initial configuration is confined in half circle. Combining all results under the condition 0 < κh ≤ 1, we presented the exact convergence rate lim n→∞ n Θ(n) − Θ ∞ in terms of N and κh. Finally, for nonidentical oscillators, we provided a criteria to obtain the asymptotic phase-locking in terms of κh, D(ν)h, Θ(0). We here used a convergence of Lyapunov functional and the finiteness of non-equivalent phase-locked states instead of discrete gradient flow argument in [38].