REMARKS ON THE CRITICAL COUPLING STRENGTH FOR THE CUCKER-SMALE MODEL WITH UNIT SPEED

. We present a non-trivial lower bound for the critical coupling strength to the Cucker-Smale model with unit speed constraint and short-range communication weight from the viewpoint of a mono-cluster(global) ﬂocking. For a long-range communication weight, the critical coupling strength is zero in the sense that the mono-cluster ﬂocking emerges from any initial conﬁgurations for any positive coupling strengths, whereas for a short-range communication weight, a mono-cluster ﬂocking can emerge from an initial conﬁguration only for a suﬃciently large coupling strength. Our main interest lies on the condi- tion of non-ﬂocking. We provide a positive lower bound for the critical coupling strength. We also present numerical simulations for the upper and lower bounds for the critical coupling strength depending on initial conﬁgurations and compare them with analytical results.


1.
Introduction. Collective self-driven synchronized motions such as the aggregation of bacteria, flocking of birds and swarming of fish are often observed in biological complex system [19,20,21,37,44,48,49,50,51]. They have been extensively studied in an engineering domain, because of their potential applications to unmanned aerial vehicles and client network equipments, etc. [37,45,44]. In a half century ago, Winfree and Kuramoto [34,51] proposed several agent-based models which are studied extensively both numerically and analytically. Recently, several Vicsek type particle models with unit speed were proposed in literatures [11,18,25] for the study of velocity alignment. Motivated by a quantum synchronization model [38,39], one of the authors [14] discussed the mono-cluster flocking for the Cucker-Smale model with unit speed constraint. As far as the authors know, constant speed constraint matters a lot in flocking modeling due to the historical development of flocking in physics literature. The modeling of flocking phenomena was first introduced by Vicsek's group [50] in the physics community and the unit speed constraint was employed in relation with the phase models for synchronization, e.g., the Kuramoto model [35,36]. This Vicsek's work advanced further research [23,32,33,37,40,43,44,46] on the collective dynamics of interacting multi-agent systems in engineering and physics.
In this paper, we provide the existence of the critical coupling strength κ c for short-range communication weights. More precisely, we provide a positive lower bound κ 0 for κ c depending on the initial configurations. For a given initial configuration, our strategy for estimating a critical coupling strength can be summarized. We first classify an ensemble of particles into several sub-ensembles according to the initial velocities, i.e., two particles are in the same sub-ensemble if and only if they have the same initial velocities, and we then prove that if κ < κ 0 , then any two particles in different groups will not flock forever. An interesting question is whether the particles in the same sub-ensemble will flock or not. This motives us to study the phenomena of multi-cluster flocking. We allow different particles to have different initial velocities and we assume that the differences among particles in same groups is small comparing to the differences between particles in different groups, then multi-cluster flocking will appear if each group departs from the others (see Theorem 4.2).
The rest of the paper is organized as follows. In Section 2, we review some basic properties of the Cucker-Smale model with a short-range communication weight ψ which will be useful in the following sections. In Section 3, we show that monocluster flocking will not emerge in a small coupling strength regime. In Section 4, we prove the emergence of multi-cluster flocking for suitable small coupling strength. In Section 5, we provide several numerical simulations and compare them to our analytic results in previous sections. Finally, Section 6 is devoted to the summary of our main results.
Notation. Throughout this paper, we use the superscript to denote the component of vector and subscript to denote the ordering of particles. For vectors and inner product are defined as follows:

2.
Preliminaries. In this section, we first recall the concepts of multi-cluster flocking, critical coupling strength, basic a priori estimates for (1.1), and then summarize previous results on the flocking estimate.
2.1. The Cucker-Smale model with unit speed. In this subsection, we briefly discuss the basic a priori estimates for (1.1). We set , v i (t) be a global solution to system (1.1) with initial data with unit speed constraint: Then, we have Proof. (i) We multiply equation (1.1) by 2v i (t) and sum the results together to have Hence we conclude that v i (t) = v 0 i , for all t ≥ 0 and i = 1, · · · , N . (ii) For the proof of non-increasing property of A(v), we refer to Lemma 2.2 in [14].
Next, we briefly discuss the relationship between our C-S model with unit speed constraint and the flocking model introduced in [ We take an inner product with (− sin θ i , cos θ i ), then we obtain Thus, our proposed model (1.1) becomes the model in [27]:

2.2.
Review on the previous results. In this subsection, we recall definitions of mono-cluster flocking and multi-cluster flocking, and summarize the result in [14] for the mono-cluster flocking.
be an ensemble of the Cucker-Smale flocking group.
We next present a concept of the critical coupling strength for the emergence of mono-cluster flocking as follows. Definition 2.3. For a given initial configuration (x 0 , v 0 ), a nonnegative constant κ c = κ c (x 0 , v 0 ) is a critical coupling strength for mono-cluster flocking if and only if the following two criterions hold.
Before the end of this subsection, we recall the flocking estimates on the monocluster formation for (1.1). We set Theorem 2.4. [14] Suppose that the coupling strength and initial configuration Then, for any solution x(t), v(t) to system (1.1), there exists a positive constant Remark 2.5. Note that Theorem 2.4 yields a sufficient condition for a mono-cluster flocking. For a small coupling strength κ 1, bi-cluster and multi-cluster flockings can emerge from some initial configurations. It has been shown that local flocking, in particular bi-cluster flocking, can emerge from some well-prepared configurations close to bi-cluster configurations [12,13].

3.
A necessary condition for a mono-cluster flocking. In this section, we provide a framework for the non-existence of mono-cluster flocking and state a necessary condition for the emergence of a mono-cluster flocking.

3.1.
A framework and main result. In this subsection, we will introduce a framework for the non-existence of mono-cluster flocking.
be an initial non-flocking configuration of the ensemble of C-S particles. Then, we set sub-ensembles G 1 , · · · , G n of the total ensemble G according to initial velocity: for α = 1, · · · , n, Since we assume the initial configuration is not in the mono-cluster flocking state, we have n ≥ 2, and the original system (1.1) can be rewritten as: Here we assume the short-range communication weight such that For conveniences, we introduce local averages:

SEUNG-YEAL HA, DONGNAM KO AND YINGLONG ZHANG
Now, we describe the geometry of initial separation between sub-ensembles. For a given initial configuration (x 0 , v 0 ), we set For notational simplicity, we suppress (x 0 , v 0 ) dependence in T 0 , κ 0 in the following: Remark 3.1. We can easily see that θ 0 ∈ (0, π]. We next introduce a coupling strength κ 0 (x 0 , v 0 ) depending on the geometry of the initial configuration (x 0 , v 0 ).
• If initial configuration satisfies • If initial configuration satisfies min β =α,i,k then, we set Now we are ready to state our main result as follows.
Theorem 3.2. Let (x, v) be a global solution to (1.1) with initial data satisfying i.e., mono-cluster flocking does not occur asymptotically. Moreover, each groups are separating.

3.2.
Dynamics of local averages and fluctuations. In this subsection, we provide estimates on the local averages and fluctuations. In particular, we introduce a useful function which is crucial for the study of the Cucker-Smale model with unit speed: v m α (t) := min where e represents a unit constant vector, which will be replaced later by a fixed vector e α (T 0 ) depending on initial data. Then, we have the following proposition with respect to function v m α (t) defined in (3.4) as follows.

3.3.
Non-existence of mono-cluster flocking. In this subsection, we will provide the proof of Theorem 3.2. We first briefly outline our strategy as follows.
The proof of our main results can be split into three stages. For a given initial • Initial stage (from mixed configuration to segregated configuration): there exists a T 0 ≥ 0 such that, for any i, k, and β = α, where e α (T 0 ) is the unit vector in the direction of v c α (T 0 ).

SEUNG-YEAL HA, DONGNAM KO AND YINGLONG ZHANG
• Final stage (emergence of non-mono cluster configuration): finally we show that T * 0 = ∞ and obtain the non-existence of mono-cluster flocking.
3.3.1. Emergence of segregated configurations. In this subsection, we will show that the configuration at time T 0 is well segregated: Recall that In the sequel, we assume without loss of generality that αi,βk (0) < 0 so that T 0 > 0. Otherwise, T 0 = 0 and the desired estimate (3.5) holds trivially, and all the lemmas from Lemma 3.4 to Lemma 3.7 can be proved with better estimates. We stated this argument in the proof of Theorem 3.2, at the end of this section. As in the definition of e α (T 0 ), we set .
then the following estimates hold: for t ∈ [0, T 0 ] and β = α, Now, we use system (3.1), the upper bound of ψ in (1.2), and the above relation to get for any α ∈ {1, · · · , n}, Thus, we combine estimates (3.6) and (3.7) to obtain Then, we use estimate (3.8) and the assumption of κ to get For the second estimate, we use the estimate of v c α (t) in (i) for all α ∈ {1, · · · , n} to obtain that for any β = α Thus, we obtain the following from the assumption of κ.
Here the last inequality is from properties of cosine functions, (ii) By using a similar analysis as in the second estimate of (i), we can derive the first estimate in (ii). For the last inequality, we use (i) and the definition of e α (t) to see that for any α ∈ {1, · · · , n}, Now we combine relation (3.9) and the previous estimates to get Hence, we obtain Proof. For the desired estimate, we claim: Proof of claim (3.10). For all t ∈ [0, T 0 ], α = β and i, k, By Lemma 3.4 and the assumption of κ, we obtain We now integrate relation (3.10) to obtain We now take an minimum over α, β, i and k to obtain the desired result.

Proof of Theorem 3.2.
In this subsection, we provide the proof of Theorem 3.2. Recall that we defined a normal vector in the direction of v c α (T 0 ): .
Note that it is a well-defined since v c α (T 0 ) cannot be zero from previous lemmas. We define (3.11) Lemma 3.6. Let (x, v) be a global solution to (3.1) with non-flocking initial data (x 0 , v 0 ). If the coupling strength κ satisfies (ii) By the definition of T * 0 and estimate (i), assertion (ii) holds trivially.
Then, we have Hence, we have T * 0 > T 0 . (ii) We use Lemma 3.5 and Lemma 3.6 to obtain . Thus, by the non-increasing property of ψ(t), we have We are now ready to provide the proof of Theorem 3.2 as follows.
The proof of Theorem 3.2. Let (x, v) be a global solution to (1.1) with non-flocking initial data (x 0 , v 0 ). If the coupling strength κ satisfies Then, we claim: for t ∈ (T 0 , ∞), For the proof of the above claim, we consider two cases: • Case A. Suppose that we have Then, it follows from the arguments in Lemma 3.7 that On the other hand, we use Proposition 3.3, Lemma 3.4, Lemma 3.7 and the as- In particular, we have v αi (T * 0 ) · e α (T 0 ) > cos θ 0 4 This contradicts inequality (3.12). Thus, we have T * 0 = ∞. Therefore, the conclusion (ii) of Lemma 3.6 implies the conclusion of Theorem 3.2.
• Case B. Suppose that we have In this case, we do not need Lemma 3.4, Lemma 3.5 again, and cos θ0 4 is replaced by cos θ0 8 , e α (T 0 ) is replaced by e α (0) in our definition of T * 0 in (3.11). Then Lemma 3.6 and Lemma 3.7 hold with T 0 = 0,λ 0 = cos θ0 8 − cos 7θ0 8 and without smallness of κ. Recall that Then, for κ < κ 0 , we use the similar arguments in Case A to obtain Finally, it follows from Case A and Case B that we complete the proof of Theorem 3.2.
4. Emergence of multi-cluster flockings. In this section, we present an emergence of multi-cluster flocking to the Cucker-Smale model (1.1). In Section 3, we divided the particles into n sub-ensembles G 1 , · · · , G n according to their initial velocities, and showed that for a small coupling strength κ < κ 0 , any two different particles in different groups do not flock. Thus, it is natural to ask whether two different particles in the same group will flock or not in a small coupling regime.
In the sequel, we will concentrate this question by allowing the initial velocities of different particles in the same group to be slightly different.

4.1.
A framework and main result. As in Section 3, we define some parameters θ 0 , δ 0 and r 0 related to the separations of each sub-ensembles: Now, we introduce the local fluctuations and l 2 -type functionals that measure the total fluctuations of each group: We next state our framework (F) for a multi-cluster flocking as follows.
• (F1) (Initial configuration): Initial configuration is well-separated and initial fluctuations are sufficiently small in the sense that • (F2) (Coupling strength): The coupling strength takes an intermediate value and the initial distance is large such that where A and β α are positive constants defined by the following relations  Suppose that the framework (F) holds, and let (x αi , v αi ) be a solution to system (4.1) with initial configuration (x 0 αi , v 0 αi ). Then, we have the following estimates: i.e., the multi-cluster flocking emerges.
Remark 4.3. In Section 3, it follows from the classification that we assume v 0 αi = v c α (0) for any α ∈ {1, · · · , n}. Thus the initial assumption of δ 0 in the above theorem is satisfied naturally. Theorem 4.2 exhibits the stability of multi-cluster flocking in the following sense.
, v αi (t) (not necessary to be solutions of system (4.1)) be a configuration such that (ii) G tends to a multi-cluster flocking in the sense of Definition 2.2 (2).

Then there exists
then solution G(t) of system (4.1) with initial data G(0) tends to multi-cluster flocking in the sense of Definition 2.2 (2).

Dynamics of local averages and fluctuations.
In this subsection, we study the time-evolution of local averages and fluctuations. We use the same definition as in (3.2).
Lemma 4.5. Let x αi , v αi be a solution to system (4.1). Then, local averages and fluctuations satisfy and Proof. (i) (Derivation of (4.3)): It follows from the definition of x c α that we havė Because of the skew-symmetric property of ψ( x αk − x αi ) v αk − v αi in the exchange of i ←→ k, the first term on the above equation becomes zero. And note that (ii) (Derivation of (4.4)): The first equation easily follows from the definitions of fluctuations. It follows from equality (3.1) 2 that we havė Then, we complete the proof of this Lemma.
In the following proposition, we derive estimates on the time-derivatives of X α and V α . Proposition 4.6. Let x αi , v αi , α = 1, · · · , n be a solution to system (3.1). Then we have, for any α, Proof. (i) We multiply equation (4.4) 1 by 2x αi (t), and add the results together over i = 1, · · · , N α yields Then, we divide (4.6) by 2|X α (t)| to obtain (ii) We multiply equation (4.4) 2 by 2v αi (t) and sum the resulting relation over i = 1, · · · , N α to obtain (4.7) • (Estimate on I 11 ): It is easy to see that • (Estimate on I 12 ): We exchange i ←→ k to get In the last inequality, we used Nα i=1v αi = 0 and get (4.5) to get that Thus, we use the upper bound of ψ and v αi to obtain where in the second equality we have used that Hence, we obtain

SEUNG-YEAL HA, DONGNAM KO AND YINGLONG ZHANG
In (4.7), we combine all estimates of I 1i , i = 1, · · · , 4 to obtain We now divide the above relation by 2V α to obtain the desired estimate.

Proof of Theorem 4.2.
In this subsection, we prove the emergence of multicluster flocking configurations for the Cucker-Smale dynamics.
Definition 4.7. Define where e α (0) is the unit vector in the direction of v c α (0) as before.

SEUNG-YEAL HA, DONGNAM KO AND YINGLONG ZHANG
Then, we apply the same arguments in Lemma 4.9 to derive the estimate: • (Estimate of estimate (ii)): Firstly, we claim that where A and β α are defined in (4.2).
(iii) For all t ∈ (0, +∞), we use assertions (4.14) to get that Thus we have V α (t) → 0, as t → +∞.  illustrates the conclusion of Lemma 3.7, which means that each group is separating at least after the time T 0 . Here Figure 1(b) tells us that our T 0 is quite bigger (nearly 10 times bigger) than the first separation time. However, this ratio is reasonable value according to the proof of Lemma 3.7. It is also clear that each group is separating at the instant T 0 . Figure 2 and 3 denote the temporal evolutions of D(x(t)) and θ(x(t), v(t)), which measure the diameters of positions and velocities of the whole ensemble, respectively. For a small time interval, D(x(t)) begins to decrease, since initially all the particles are gathering. Because κ is not big enough to bond the whole ensemble, they just pass each other and separate to the infinity. These mechanisms are well represented in Figure 2. Along this procedure, the velocities are hard to change its value, since κ is small. This is the basic idea of proofs on Lemma 3.4, Lemma 3.5, Lemma 3.6, and Lemma 3.7, since the whole separation is trivial for κ = 0. This also can be seen in Figure 3, which shows that the minimal difference of velocity angles θ(x(t), v(t)) does not change much. It begins to decrease, since initial effects of global attraction, but large separation of positions makes it increase since each group's internal attraction is bigger than inter-groups attraction.

5.2.
Total separation of particles. In Theorem 3.2, the definition of group was quite restricted, two particles are in the same group if they have the same velocity initially. This has enough implication due to two reasons. First, people usually assume general configuration of initial velocity. Second, our objective is to verify the non-existence of flocking. In this subsection, we choose initial data randomly to see the behavior of generic initial configurations. The initial positions and velocities are chosen uniformly in [−1, 1] 2 and S 1 , respectively. Figure 4(a) shows one simulation of initial data. In this case, the relative positions and velocities have no good relation, therefore the values of T 0 and κ 0 are as in the estimation. This typical example shows that the condition for Theorem 3.2 is as , v(t)) for time t ∈ (0, 2000) Figure 3. Temporal evolution of θ(x(t), v(t)) follows. κ 0 = 1.1658 × 10 −8 , T 0 = 18884. Figure 4(b) shows that total separation condition is satisfied near t = 30 for κ = 0.9 × κ 0 . The notable point is, however, that Theorem 3.2 is generally satisfied with respect to initial data if we give small κ. On the other hand, the result of Theorem 4.2 cannot be applied to general initial data, since the existence of local flocking itself is not guaranteed for every initial configuration.

5.3.
Emergence of multi-cluster flockings. In this subsection, we consider Theorem 4.2, whose conclusion is on the local flocking in each groups with global nonflocking between groups. The results are similar to the subsection 5.3, but the difference comes from the fluctuation of initial velocities. Since Theorem 4.2 has restriction on X 0 and V 0 , the L 2 -norm of fluctuation, its result depend on the number of particles N . Hence we set small number of particles as follows, for the  Initial configuration is also chosen in a similar way as in subsection 5.1. The reference positions and velocities of each group are fixed and we give small random fluctuation for each particle. In contrast to the previous setting, we set separating initial conditions to see the behavior after separation.
Hence all the restrictions for Theorem 4.2 are satisfied. Here δ 0 is chosen to be quite small value in order to satisfy the condition (F2). In Figure 5(b), we can observe the minimal difference of velocity angle is nearly constant, which implies κ is so small that the interaction between each groups are little. On the other hand, κ is large enough to make local flocking of each group, as we can see Figure 6. Figure 6(a) shows position fluctuation is bounded on each group and Figure 6(b) implies that the velocities are gathering on each group with algebraic decay rate. Therefore, we can see the strong evidence of the multi-cluster flocking in this simulation.  Figure 7 shows the behavior of solutions when we have smaller κ, namely, κ = 0.09 * κ 1 . The graphs are plotted with 10 times larger time axis, in order to investigate the trends of variables in a time scale t/κ. The system seems to exhibit three clusters although the coupling strength is small enough to violate the condition (F2) of Theorem 4.2. In this case, however, the motion of X (t) does not match with the conclusion of Theorem 4.2, as we easily can see X (t) > X (0) + A = 0.1579 + 0.1052 for some t.
Therefore, the conditions of Theorem 4.2 is not close from optimal to the multicluster flocking, but it will destroy the parameters we assumed for the bootstrapping arguments of local flocking phenomena. It suggests that we need different approaches to get a closer guess on the critical value of the coupling strength.
Much smaller value of κ = 0.009 * κ 1 seems to get a larger number of clusters as in Figure 8. The finite-time simulation can not prove the non-flocking, but X (t) 6. Conclusion. In this paper, we presented a quantitative estimate on the critical coupling strength for the transition from local flocking state to global flocking state, and we also provided a sufficient framework for the multi-cluster flocking. More precisely, we first provided a possible range of the critical coupling strength for the mono-cluster flocking. For a long-ranged communication weight, any positive coupling strength can push initial data to the corresponding mono-cluster flocking state asymptotically. Thus, in this case, the critical coupling strength can set as zero. However, this scenario is completely different for a short-ranged communication weight. As noticed in [27], even for two-dimensional setting and short-ranged communication weight, a sufficiently large coupling strength is required to guarantee the emergence of mono-cluster flocking. It seems that finding the exact form of the critical coupling strength looks a pretty challenging task, if possible, because it might depend on the delicate geometric information of initial configuration. In this paper, we instead tried to estimate the range of the critical coupling strength for a given initial data. In Theorem 3.2, we have shown that if κ < κ 0 , then mono-cluster flocking cannot happen. Thus, the critical coupling strength should be larger than κ 0 depending on the initial configurations, especially for the distance on positions and velocities of individual particles. Second, we generalized bi-cluster flocking result in [13], which treated the same unit-speed model in two-dimensional setting. More precisely, we provided a sufficient framework for the emergence of multi-cluster flocking in terms of κ. Unlike to the result of the Cucker-Smale model in [28], the bounds for κ both depend on the configurations of positions and velocities. The flow of the proof contains the same idea as in [28], i.e., we employed the Lyapunov functional approach with continuity arguments which describe expected particle formations after sufficiently long time. However, the interactions on unit-speed model is quite different from the Cucker-Smale model, for example, oppositely directed two particles do not have any interacting force. Motivated by this simple example, the condition on multi-cluster flocking is more restricted from the Cucker-Smale model. One of the problems is that the velocity cannot guarantee sufficient distance between groups since all the particles have unit-speed. This is why Lemma 3.7 has more rough estimate on relative velocities. In summary, we have provided the first analytic method on the emergent behaviors with respect to general initial data on the agent-based models with velocity restrictions. Of course, there are still many unresolved issues related to the topics treated in this paper. For example, as aforementioned, the possible range of the critical coupling strength for mono-cluster is not optimal, thus it will be interesting problem to optimize the critical coupling strength using optimization theory. This will be treated in future.