Emergence of aggregation in the swarm sphere model with adaptive coupling laws

We present aggregation estimates for the swarm sphere model equipped with the adaptive coupling laws on a sphere. The temporal evolution of coupling strength is determined by a feedback rule incorporating the balance between relative spatial variations and linear damping. For the analytical treatment, we employ two adaptive feedback laws, namely anti-Hebbian and Hebbian laws. For the anti-Hebbian law, we provide a sufficient framework leading to the complete aggregation in which all particles aggregate to the same position and behave like one big point cluster asymptotically. Our frameworks are given in terms of the initial positions and the coupling strengths. For the Hebbian law, we provide proper subsets of the basin of attractions for the complete aggregation and bi-polar aggregation where particles aggregate to the north pole and south pole simultaneously. We also present a uniform \begin{document}$\ell_p$\end{document} -stability of the swarm sphere model with an adaptive coupling with respect to the initial data when the complete aggregation occurs exponentially fast.

can design a particle system exhibiting aggregation behaviors on the unit d-sphere S d embedded in R d+1 as a multi-dimensional generalization of the Kuramoto model. In this paper, we are interested in the swarm sphere model proposed in [24,25,28] generalizing the Kuramoto model as a special case. Let x i and κ ij be the position of the i-th particle on a sphere and the coupling strength between i and j-th particles, respectively. Then, the dynamics of (x i , κ ij ) is governed by a Cauchy problem for the swarm sphere model [28]: Here Ω i and ·, · are real (d + 1) × (d + 1) skew-symmetric matrix and the standard inner product in R d+1 , respectively. The nonnegative constants µ and γ are proportional to learning enhancement rate and friction, respectively, and |x| denotes the standard 2 -norm of x in R d+1 . It is easy to see that as long as particles stay on a sphere initially, particles stay on the same sphere. Thus, system (1) is well-posed as long as all particles stays away from zero. From the modeling viewpoint, there might be many different choices for the adaptive coupling law which reflects physical reality. In neuroscience, Hebbian rule [20] roughly says that the growth rate for the coupling strength is maximized when interacting neurons are in phase and is decreasing in the phase differences, e.g., Γ(θ) = cos θ. In contrast, anti-Hebbian rule refers to the opposite situation. In analogy with this, we consider the following two adaptive coupling laws (anti-Hebbian and Hebbian laws): Anti-Hebbian : Γ ah (x, y) = |x − y| 2 , Hebbian : Γ h,g (x, y) = 1 − |x − y| 2 2 and Γ h,p . ( System (1) can be reduced to the Kuramoto model with adaptive couplings for d = 1, which has been extensively studied in [2,10,14,18,19,27,29,31,32,33,35]. For the uniform all-to-all coupling in which κ ij is a uniform constant, i.e., κ ij = κ > 0, emergent dynamics of (1) has been extensively studied in [5,7,17] in which several analytic frameworks for the complete aggregation and practical aggregation (see Definition 2.1) have been investigated in terms of initial data and parameters. The purpose of this paper is to study the emergent dynamics of (1) under the feedback laws (2) and Ω i = Ω, i = 1, · · · , N , and we present sufficient frameworks leading to the complete aggregation and bi-polar aggregation. More precisely, our main results in this paper are three-fold. First, we consider anti-Hebian rule Γ ah (s) in (2) 1 . In this case, Γ(0) = 0, and this degeneracy causes lots of difficulty in the aggregation analysis. For this, we use the Lyapunov functional approach together with the Barbalat's lemma. For a given solution (X, K) to (1) and i, j, we introduce Lyapunov functionals L ij = L ij (X, K): Second, we consider the Hebbian feedback law (2) 2 . In this case, Γ h,p has a positive value at the origin and is decreasing until its first positive zero. Unlike the anti-Hebbian case, Γ ah can take positive and negative values so that diverse asymptotic patterns can emerge from the initial configurations. In particular, we are interested in the basin of attractions for the complete aggregation and bi-polar aggregation.
For the former situation, if the initial position of the particles is on the same hemisphere and the ratio between the maximum and minimum of coupling strength is sufficiently small, the complete aggregation can occur exponentially fast (Theorem 4.2): max where λ is a positive constant independent of N and t. On the other hand, the bi-polar aggregation can also emerge from the well-prepared initial configurations. More precisely, consider a configuration consisting of two approximate clusters scattered around the north and south poles, respectively. Furthermore, the intra coupling strengths between the particles belonging to the same subgroup are assumed to be positive(attraction), whereas the inter coupling strengths between the particles belonging to different groups are assumed to be negative(repulsion). Under this well-prepared situation, the particles belonging to the same group are likely to aggregate each other. In contrast, the particles belonging to different clusters are likely to depart each other. In fact, this plausible scenario can be made rigorous (see Theorem 6.3). Third, as a direct application of an exponential aggregation estimate, we present a uniform stability estimate of (1) with respect to the initial data in the sense of the Definition 6.1: For two pair of solutions (X, K) and (X,K) to (1), we have sup 0≤t<∞ The rest of this paper is organized as follows. In Section 2, we review basic properties of (1) and discuss its relation with the Kuramoto model, and recall previous results on the emergent dynamics of the Kuramoto model with the adaptive coupling laws. In Section 3, we consider the anti-Hebbian adaptive law and present the formation of complete aggregation from some admissible class of initial data using the Lyapunov functional approach. In Section 4, we consider the two Hebbian adaptive laws: one is a positive and non-increasing law which generalizes a cosine-type function and the other is also decreasing but it can be both positive and negative in their domain. For these Hebbian adaptive laws, we show that the complete aggregation can emerge from some class of initial configurations. In Section 5, we consider the second adaptive law and show that the bi-polar aggregation can emerge from the well-prepared initial data. In Section 6, we present the uniform aggregation estimate. Finally, Section 7 is devoted to a brief summary of our main result and future directions. Notation: For (d + 1) × (d + 1) matrix Ω and y ∈ R d+1 , we denote Ω := max 1≤l,m≤d+1 |Ω l,m | and |y| := y 2 1 + · · · + y 2 d+1 .
We set where M N (R + ) is the set of all N × N real matrices, and define a p -norm of the vector X:

Preliminaries.
In this section, we present elementary estimates for the swarm sphere model to be used in later sections, and briefly recall previous results on the emergent dynamics of the model with the adaptive coupling laws.
2.1. A swarm sphere model. First, we recall two concepts of emergent phenomena such as the complete aggregation and bi-polar aggregation in the following definition.
2. System (1) exhibits the bi-polar aggregation if there exists a non-empty subset N 0 of N := {1, · · · , N } such that Next, we present two elementary properties of the swarm sphere model (1) with adaptive couplings. Lemma 2.2. Let (X, K) be a solution to system (1) with initial data (X 0 , K 0 ) satisfying the symmetry relations: Then, the coupling matrix K = [κ ij ] is symmetric and its entries are nonnegative: Proof. We can obtain the following integro-differential equation from (1) 2 : Then, conditions (3) and (4) yield desired estimates.
Below, we show that the unit sphere S d is a positively invariant set for (1). Lemma 2.3. Let (X, K) be a solution to system (1) with initial data (X 0 , K 0 ) satisfying the relations: Then, x i stays on the sphere ∂B r (0) along the flow (1): Proof. Note that the pairwise coupling strengths satisfy the symmetry condition (4) due to Lemma 2.2. Now, we take the standard inner product of x i and (1) and use the symmetry of κ ij to get where the last equality is due to skew-symmetry of Ω i : For identical particles with Ω i = Ω, 1 ≤ i ≤ N, system (1) restricted on the set (∂B r (0)) N with r > 0 has a solution splitting property as a composition of the rotation and Lohe coupling. More precisely, we consider the following two sub-systems on x i = r: First, we introduce two solution operators R Ω (t) and L(t) for subsystems in (5). For u ∈ R d+1 and (d + 1) × (d + 1) skew-symmetric matrix Ω, R Ω (t) is a solution operator corresponding to (5) 1 : for u ∈ R d+1 , R Ω (t)u := e Ωt r 2 u, and L(t) is a solution operator corresponding to (5) 2 : for a solution x i (t) to (5) 2 with initial data u ∈ R d , x i (t) := L(t)x 0 i . Any solution to (1) can be written as a composite of two solution operators R Ω (t) and L(t) as follows.
Lemma 2.4. Suppose that the initial positions and Ω i satisfy |x 0 i | = r, Ω i ≡ Ω, for all i = 1, · · · , N . Then, for any solution (X, K) to (1), we have  Then, we multiply e − Ωt r 2 to (5) and use the above relations to obtain Thus, we have We use the change of variables y i := e − Ωt Remark 1. 1. By Lemma 2.4, it suffices to consider the following Lohe system with Ω i = 0, 1 ≤ i ≤ N, r = 1: for the complete aggregation analysis. 2. For d = 1, system (1) can be reduced to the Kuramoto model under the following setting: for i, j = 1, · · · , N , Then, we use the relationsẋ i =θ i x ⊥ i , |x i | = 1 to see that system (7) becomeṡ We take the inner product of (8) and x ⊥ i to find Now, we use (9) and the relations: to get the Kuramoto model:θ 2.2. Previous results. As noticed in Remark 1, the Kuramoto model (10) can be understood as a special case of the swarm sphere model on the unit sphere. Next, we briefly review previous results on the Kurmoto model with adaptive couplings.
Consider the generalized Kuramoto model with the adaptive couplings [19]: where µ and γ are nonnegative constants proportional to the learning enhancement rate and friction, respectively and Γ is a feedback law satisfying the parity and periodicity conditions: Note that for µ = γ = 0 and κ 0 ij = κ N , system (11) reduces to the Kuramoto model [22,23]:θ In this case, the emergent dynamics of the Kuramoto model [22,23] has been extensively studied in [1,4,8,9,11,12,13,17,15,21,35,36] and we refer the reader to recent survey articles [12,16] in relation with the complete synchronization.
Then, the emergent dynamics for (12) can be summarized as follows.
Theorem 2.5. [19] (Identical particles) Suppose that the natural frequency vector and the initial data Θ 0 satisfy Then, for any solution Θ = Θ(t) to (12), there exists a positive number t 1 > 0 such that Theorem 2.6. [19] (Nonidentical particles) Suppose that the parameters µ, γ, the initial phase configuration and the coupling strength satisfy the following conditions:
Then, we have an asymptotic complete synchronization: Theorem 2.8. [19] (Nonidentical oscillators) Let Θ = Θ(t) be a solution to (13) with the a priori assumption: Then, we have an asymptotic complete synchronization: 3. Emegence of complete aggregation: Anti-Hebbian rule. In this section, we study an emergent dynamics of (1) with an anti-Hebbian rule Γ ah (x, y) = |x−y| 2 from well-prepared initial data. As long as there is no confusion, from now on we assume that x i stays on the unit sphere: Consider the swarm sphere model on the unit sphere for identical particles: Note that (14) 1 can also be rewritten aṡ For i, j = 1, · · · , N and (X, K), we introduce the Lyapunov functionals L ij : Note that the functional L ij is always nonnegative. Using the angle functional can be rewritten as follows.
Next, we study the rate of changes for L ij (t).
Lemma 3.1. Let (X, K) be a solution to (14). Then, L ij satisfies Proof. Note that the unit modulus of x i and x j yields We use the above relations and (15) to get d dt and d dt Now, we use (16) and combine (19) and (20) to yield the desired estimate.
Remark 2. Note thatL ij does not have definite sign due to the indefinite sign of h ij .
Next, we show that L ij satisfies the non-increasing property under some restricted initial data. Proposition 1. Suppose that the initial data (X 0 , K 0 ) satisfy the following relations: for i, j = 1, · · · , N , (21) Then, for any solution (X, K) to (14) and i, j ∈ {1, · · · , N }, we have the following two assertions: 1. h ij and κ ij are strictly positive for all t: 2. L ij is nonincreasing along the flow (14).
Proof. We will use the continuity argument. First, we define a set T : Due to the assumption (21), and the continuity of the solution, we can see that the set T is nonempty, and we set T * := sup T . We claim: Since h ij > 0, t ∈ (0, T * ), it follows from Lemma 3.1 that we have h ij (t) > 0, for all i, j = 1, · · · , N and t > 0.
On the other hand, since κ 0 ij > 0 and 1 − h ij ≥ 0, we have which also yield the nonpositivity of the R.H.S. of (18). Hence, we can conclude that L ij (t) is non-increasing.
Next, we recall Barbalat's lemma to be used throughout the paper. Then, f tends to zero as t → ∞: As a direct corollary of Proposition 1 and Lemma 3.2, we next show that the coupling matrix K becomes a constant multiple of the matrix J N with entries 1: Corollary 1. Let (X, K) be a solution to (14) with the initial data satisfying Then, we have lim Proof. Basically, we will use Barbalat's lemma in (3.2) and estimate (18).
For given i, j, it follows from Lemma 3.1 that we have This implies the desired estimate (24). • Step B. In order to derive (23) from the integrability estimate (24), we use Barbalat's lemma. For this, we set It suffices to check that f ij,k is uniformly continuous. For the uniform continuity of f ij,k (t), we need to check the uniform boundedness ofḟ ij,k = 2(κ ik (t) − κ jk (t))(κ ik (t) −κ jk (t)) which is implied by the uniform boundedness of κ ik (t) and (14) 2 . For this, we use (14) 2 to get By Gronwall's lemma, we have the uniform boundedness of κ ij .
Thus, κ ij is uniformly bounded and so isκ ij . Hence,ḟ ij,k is uniformly bounded. Now, we apply Lemma 3.2 to see Next, we recall the Gronwall type lemma to be used later.
be a C 1 -function satisfying the following differential inequality: where α is a positive constant and f : R + ∪ {0} → R is a continuous function decaying to zero as its argument goes to infinity. Then y satisfies Proof. The proof can be found in Appendix A of [6].
Finally, we arrive at our first main result on the emergence of the complete aggregation.  (14) with initial data (X 0 , K 0 ) satisfying Then for every i, j = 1, · · · , N , we have Proof.
• Step A (Derivation of Gronwall's inequality for L ij ): It follows from Proposition 1 that we have . Then, the relation (18) and κ ij > 0 imply We set We use (16) and combine (27) with (28) to obtain This yields • Step B (Derivation of estimates (26)): Depending on the integrability of Λ m , consider the following two cases: Case B.1: Suppose that Λ m is not integrable: In this case, the relation (29) yields By definition of L ij and (30), we have Next, we also show that κ ij tends to zero asymptotically. For this, it follows from the relation (14) 2 that we havė Then, we apply Lemma 3.3 using the relations (31) and (32) In this case, we can apply Barbalat's lemma to get We use (25) to see thatḣ ij andκ ij are uniformly bounded: Then, we use (33) and differentiate the dynamics of κ ij in (14) 1 to yield thatκ ij is also uniformly bounded: Hence we apply Barbalat's lemma to conclude that lim t→∞κ ik (t) = lim t→∞κ jk (t) = 0.
Finally we use the dynamics of κ ij in (14) 1 to conclude that Remark 3. Note that our convergence estimates does not yield explicit convergence rate due to the use of Barbalat's lemma. 4. Emergence of complete aggregation: Hebbian rule. In this section, we consider two Hebbian rules denoted by Γ h,p and Γ h,g . The former case is called the "positive Hebbian rule" in the sense that Γ h,p takes positive values in its domain, and the latter case Γ h,g is called the "general Hebbian rule" where adaptive law can attain both positive and negative values in its domain. For a positive Hebbian rule, we motivate its structure based on the cosine function, and for a general Hebbain rule, we take the following explicit ansatz: In the following two subsections, we will show that the complete aggregation can occur for both positive and general Hebbian rules.

4.1.
A positive Hebbian rule. In this subsection, we present aggregation estimate for positive Hebbian rule Γ h,p satisfying the following structural conditions: We first present basic estimates for the coupling matrix.
Then, Gronwall's lemma implies Now, we choose a positive time t * such that Finally, we set and let (X, K) be a solution to (1) with (34). Then, there exist positive constants C and λ such that Proof. First, note that Thus, it suffices to show the exponential zero convergence of 1 − h ij as t → ∞. For this, we set ∆(H) := max Since h ij ≤ 1, the functional ∆(H) is always nonnegative, and we will show that ∆(H) decays to zero exponentially fast. Since it is Lipschitz continuous, for a given t ≥ 0, we set indices i, j depending on t such that Then, for such i, j, we use (15) to get After tedious algebraic manipulation, we have where Below, we estimate the terms I 1i separately.
• (Estimate of I 11 ): We use Lemma 4.1 to find • (Estimate of I 12 ): We again use Lemma 4.1 to get

This yields
• (Estimate of I 13 ): Since we have Finally, we combine all estimates (36), (37) and (38) to obtain a Riccati differential inequaltiy: Then, by the comparison principle of ODEs and explicit representation of solutions to the Riccati equation, we have where and the two terms inside the parentheses in the denominator are positive due to the assumptions (35). Finally, we take the limit t → ∞ to the both sides of (39) to obtain the desired result.

Remark 4.
We recall the dynamics of κ ij with (34): and this can be written as It follows from Theorem 4.2 that the second term in the right-hand side converges to zero. Then, Lemma 3.3 yields lim t→∞ κ ij (t) = µ γΓ (0), for all i, j = 1, · · · , N .

4.2.
A general Hebbian rule. In this subsection, we present the complete aggregation estimate for an adaptive rule: Note that compared to the adaptive rule (34) which always attain positive values, this type (40) can be both positive and negative. Then, the system reads aṡ We define the functionals for (41) as SinceL ij (t) is defined as the difference of two positive functionals, the sign ofL ij (t) is indefinite unlike that of L ij . Next, we study the rate of change ofL ij .
Proof. We estimate the terms in the R.H.S. of (42) separately below. Note that d dt and 1 2µN We use (43) − (44) to obtain d dt For the adaptive rule Γ ah (x, y) = |x − y| 2 , we can associate the corresponding Lyapunov functionals as the sum of two positive functionals (see (18).) However, for the general Hebbian law (40), the sign of the functionals in (42) is indefinite. Hence, it is difficult to apply the Lyapunov functional method as it is.
Next, we introduce two frameworks (F A ) and (F B ) in which the complete aggregation will be verified.
• (F A 1): The initial pairwise coupling strengths are moderately strong in the sense that Next, we introduce another framework for the complete aggregation.
• (F B 1): The initial pairwise coupling strengths are moderately strong in the sense that • (F B 2): The initial configuration X 0 satisfy

SEUNG-YEAL HA, DOHYUN KIM, JAESEUNG LEE AND SE EUN NOH
Next, we introduce the extremal critical coupling strengths: Lemma 4.4. Suppose that one of frameworks (F A ) and (F B ) hold, and let (X, K) be a solution to (41). Then, we have Proof. We use the relation (45) 2 to see the equivalence: (i) (Framework (F A )): Suppose that the framework (F A ) holds.
(Derivation of (46) 1 ): Since γ > 3 2 , we have and from (F A 1) 1 , we see that We combine (48) with (49) to conclude that From the assumption (F A 1) which holds for all i, j and (50), we show that κ M satisfies relation (47).
(Derviation of (46) 2 ): We use the first relation of the framework (F A 1) 2 , (50) and the definition of κ m to see that (46) 2 holds. (ii) (Framework (F B )) Suppose that the framework (F B ) holds.
(Derivation of (46) 2 ) Since (46) 2 is equivalent to In the following theorem, we show that the exponential aggregation can occur under the above two frameworks. Theorem 4.5. Suppose that exactly one of frameworks (F A ) and (F B ) hold, and let (X, K) be a solution to (41). Then, the complete aggregation occurs exponentially, i.e., there exist positive constants C > 0 and λ > 0 such that Proof. Since the proof is rather lengthy, we split its proof into several steps below.
• Step A (Uniform upper and lower bounds for the coupling strength): We use (41) 2 and a rough estimate h ij ≤ 1 to seė This implies Thus, we have a uniform upper bound for κ ij : For a uniform lower bound for the coupling strengths, we define a set T and its supremum as follows: • Step B (T * = ∞): Next, we claim: We use (46) 2 to see that T = ∅ and hence T * is well-defined. Suppose not, i.e., T * < ∞. Then, we have lim For a given t > 0, we choose indices i, j depending on t such that Then, for such i and j, 1 − h ij satisfies In (52), we use the arguments similar to that in the proof of Theorem 4.2 using the relation: Note that the coefficient 2(2κ m − κ M ) of ∆(H) in the R.H.S. of (53) is strictly negative under our two frameworks (see (46)). Now we use the comparison principle and the initial condition Next, we use the dynamics of κ ij (t) and (55) to obtaiṅ We integrate the above relation using the integrating factor to get • Step C (Exponential decay estimate of ∆(H)): Note that the defining condition (45) yields 2µ γ Then, relation (56) becomes which is contradictory to (51). Thus T * = ∞ and the following Riccati type differential inequality holds for ∆(H): Now, consider the Riccati differential equation: which can be integrated to give On the other hand, it follows from the comparison principle that Finally, we combine (57) and (58) to obtain which yields the exponential decay of ∆(H). Furthermore, we check that the initial condition in (54) Remark 5. Note that in Theorem 3.4, we employed the adaptive law Γ(s) = |s| 2 . Thus, we do not know the exact decay rate of ∆(H), since we use Barbalat's lemma to obtain the desired limit. However in Theorems 4.2 and 4.5, we can guarantee that the decay rate is exponential due to the uniform positiveness of κ ij (t) for all time. This explicit exponential decay of ∆(H) will be crucially used in the uniform stability analysis in Section 6.

5.
Emergence of bi-polar aggregation. In this section, we study the emergence of bi-polar aggregation of the swarm sphere model with the Hebbian adaptive rule in Section 4.2: Note that Γ h,g does not have a definite sign, for example, when x i and x j are located on the different hemisphere, Γ h,g takes a negative value. Thus, in the following two subsections, we will show that under some well-controlled situations, the bi-polar aggregation can emerge from the well-prepared initial configuration.
5.1. A two-particle system. Consider the two-particle system: We assume κ 21 = κ 12 = κ and set h := x 1 , x 2 . Then, it follows from (59) thaṫ Note that system (60) has three equilibria (κ ∞ , h ∞ ): From the linear stability analysis, the first two equilibria are (asymptotically) stable and the last equilibrium is unstable. For the stability of the first two equilibria, see the vector field in Figure 1 below.

5.2.
A many-body system. In this subsection, we study a bi-polar aggregation estimate for the many-body system (41) with N ≥ 3: As noticed in the two-particle system in the previous subsection, the bi-polar configuration can emerge from the well-prepared initial configuration. Throughout this subsection, the index letter i denotes an element of G 1 , i.e., i runs from 1 to N 1 and the index letter j denotes an element of G 2 , i.e., j runs from 1 to N 2 . For the coupling strength κ 1 ij and κ 2 ij , they are concerned with the same subgroups G 1 and G 2 , respectively and for the κ d ij , it is concerned with the other groups. For example, for the notation κ 1 ij and for j ∈ G 1 , then we automatically realize that i also belongs to G 1 . By the same way, for κ d ij and i ∈ G 1 , then j must belong to G 2 . In order to describe a sufficient framework leading to the bi-polar aggregation, we assume that the system is composed of two sub-ensemble G 1 and G 2 : We relabel the position of particles as {x 1i } N1 i=1 and {x 2j } N2 j=1 , respectively, and the coupling matrix K = (κ m ) is also represented as follows: x 2j , i = 1, · · · , N 1 , j = 1, · · · , N 2 . Note that system (61) can be rewritten as follows. For i = 1, · · · , N 1 and j = 1, · · · , N 2 , Since we are interested in the bi-polar aggregation, we look for a framework in which the particles in the same sub-ensemble G i , i = 1, 2 aggregate to the same position: whereas for the particles belonging to different sub-ensembles, To quantify (63) and (64), we introduce the angle functionals and maximal functionals: Note that ∆(H 1 ) and ∆(H 2 ) measure the maximal distances between the particles in the sub-ensembles G 1 and G 2 , respectively, and ∆(H d ) measures the maximal distance between the particles in G 1 and G 2 . In this regard, we will design our framework leading to the estimate: In the following lemma, we provide the rate of changes for the above functionals.
Now, we introduce a class of well-prepared initial framework which leads to the bi-polar aggregation state. (The framework F C ) Let µ and γ be positive constants.
• (F C 2): The initial position configuration satisfies (Framework F D ) Let µ and γ be positive constants.
Theorem 5.3. Suppose that exactly one of frameworks (F C ) and (F D ) hold, and let (X, K) be a solution to (61). Then, the complete aggregation occurs exponentially, i.e., there exist positive constantsC > 0 andλ > 0 such that Proof. We basically follow the strategy similar to that in the proof of Theorem 4.5. • Step A (One-sided boundedness of coupling strengths): We define a set T where δ a , a = 1, 2 is defined in the frameworks (F C ) and (F D ), respectively. We use either (F C 1) or (F D 1) and the continuity of the coupling strengths to see that the setT should contain a small time interval [0, ε), i.e., ε ∈T . Hence, we have T = ∅. andT * is well-defined. On the other hand, by the same argument in Step A of Theorem 4.5 and from (F C 1) and (F D 1), Suppose not, i.e.,T * < ∞. Then, at least one of the following relations hold: It follows from Lemma 5.1 that for t ∈ [0,T * ), We add the above relations to obtain the differential inequality for ∆(H 1 )+∆( Hence, we have T * = ∞. Finally, the differential inequality (73) holds for whole time t ∈ (0, ∞): κ M Now, consider the Riccati differential equation: which can be integrated to give Then, it follows from the comparison principle that Finally, we combine (81) with (82) to obtain that which yields the exponential decay of ∆(H 1 (t)), ∆(H 2 (t)) and ∆(H d (t)). Furthermore, we can check that the initial condition in (74) 20δ 2 − 12ρ 9ρ 2 + 4ρ , for the framework F D .
6. Uniform p -stability. In this section, we provide the uniform p -stability estimate of an adapative swarm sphere model (1). First, we recall definition of the stability with respect to the initial data as follows.
Definition 6.1. For p ∈ [1, ∞], system (1) is uniformly p -stable with respect to the initial data, if for two solutions X andX to (1) with initial data X 0 ,X 0 , respectively, there exists a uniform positive constant G independent of N and t such that sup 0≤t<∞ Remark 6.
1. Note that Definition 6.1 does not take into account the coupling strength function. In fact, in Proposition 2, we will see that |κ ij −κ ij | tends to 0 exponentially fast.
2. Since our stability mainly relies on the exponential aggregation estimate, we use the following result proved in Theorem 4.2 and Theorem 4.5: there exist uniform positive constants C 0 and D 0 such that max 1≤i,j≤N Before we discuss the uniform stability, we first present several basic Gronwalltype lemmas. Lemma 6.2. Let y : R + → R + be a C 1 -function satisfying a differential inequality: Then we have the following assertions: 1. If α i and β i satisfy there exist uniform positive constants C 0 and D 0 > 0 such that 2. If α i and β i satisfy there exists a uniform constant C 1 such that 3. If α i and β i satisfy there exists a uniform constant C 2 such that Proof. (i) By the comparision principle of ODE and method of integrating factor, we have Hence, there exist uniform positive constants C 0 and D 0 > 0 such that (ii) We multiply (84) with the integrating factor to find y(t) ≤ y 0 e α 1 β 1 (1−e −β 1 t ) ≤ e α 1 β 1 y 0 =: C 1 y 0 , t ≥ 0. (iii) We also multiply the same integrating factor (85) to obtain This implies Thus we can find a uniform constant such that In next proposition, we show that |κ ij −κ ij | tends to zero exponentially fast and κ ij tends to the positive value exponentially fast. Proposition 2. Let (X, K) and (X,K) be two solutions to (1) with either (34) or (40). Then, there exist positive uniform constants (C i , D i ), i = 1, 2 such that Proof. • (Derivation of the first estimate): It follows from (1) 2 that we have We multiply κ ij −κ ij to the both sides of (87) to obtain Then, we apply (83) to obtain that We use Lemma 3.3 to conclude that there exist uniform positive constants C 1 , D 1 such that |κ ij (t) −κ ij (t)| ≤ C 1 e −D1t , t ≥ 0.
• (Derivation of the second estimate): We use (83), dynamics of κ ij and Lemma 6.2(i) to conclude that there exist uniform positive constants C 2 , D 2 such that |κ ij (t) − κ ∞ | ≤ C 2 e −D2t , κ ∞ := µ γ Γ(0). Now, we are ready to present a uniform p -stability of (1). Theorem 6.3. Let (X, K) and (X,K) be two solutions to (1) with either (34) or (40). Suppose a priori that exponential aggregation occurs. Then for p ∈ [1, ∞), there exists a uniform positive constant G > 0 such that Proof. It follows from (1) that We now take an inner product (88) with (x i −x i ) to obtain We divide the above inequality by |x i −x i | to find a differential inequality for |x i −x i |: Next, we multiply the above equation with p|x i −x i | p−1 and use the estimates (83) and (86) to obtain where κ M is an upper bound for κ ij . Again, we use (86) 2 to find where we used Hölder's inequality. We sum up (89) over the index i to have Finally, we apply Lemma 6.2 (iii) to find a uniform constant G > 0 such that X(t) −X(t) p ≤ G X −X 0 p , t ≥ 0.

7.
Conclusions. In this paper, we have studied the emergent dynamics of the swarm sphere model equipped with adaptive couplings whose adaptiveness is influenced by the closeness between the particles. The coupling strength in the swarm sphere model is designed to evolve via the competing mechanism between a linear damping and spatial variations. To make a realistic modeling for the real world phenomena, we basically employ the two adaptive feedback rules characterized by the function Γ = Γ(s). The first law is called Hebbian, and the second law is called anti-Hebbian. For the Hebbian case, we just determine that the adaptive function has a special form, say, Γ(s) = s 2 so that we can apply the Lyapunov functional method. By the decay estimate for the Lyapunov functional, we show that complete aggregation can occur under some restricted initial data. In contrast, for the anti-Hebbian case, we also specify the form of adaptive function, say Γ(s) = 1 − s 2 2 . In this case, this anti-Hebbian function can attain both positive and negative values in their domain. Thus, several interesting dynamic patterns other than the complete aggregation can emerge depending on the initial configuration and parameters. Finally, as a direct application of the (complete) aggregation estimate, we also show that swarm sphere model is uniformly p -stable with respect to the initial data. This generalizes the earlier result for the swarm sphere model with a uniform coupling constant.