Asymptotic interplay of states and adapted coupling gains in the Lohe hermitian sphere model
aa r X i v : . [ m a t h - ph ] J a n ASYMPTOTIC INTERPLAY OF STATES AND ADAPTED COUPLINGGAINS IN THE LOHE HERMITIAN SPHERE MODEL
JUNHYEOK BYEON, SEUNG-YEAL HA, AND HANSOL PARK
Abstract.
We study emergent dynamics of the Lohe hermitian sphere (LHS) model withthe same free flows under the dynamic interplay between state evolution and adaptivecouplings. The LHS model is a complex counterpart of the Lohe sphere (LS) model onthe unit sphere in Euclidean space, and when particles lie in the Euclidean unit sphereembedded in C d +1 , it reduces to the Lohe sphere model. In the absence of interactionsbetween states and coupling gains, emergent dynamics have been addressed in [22]. In thispaper, we further extend earlier results in the aforementioned work to the setting in whichthe state and coupling gains are dynamically interrelated via two types of coupling laws,namely anti-Hebbian and Hebbian coupling laws. In each case, we present two sufficientframeworks leading to complete aggregation depending on the coupling laws, when thecorresponding free flow is the same for all particles. Introduction
Collective behaviors of classical and quantum systems are ubiquitous, e.g., aggregation ofbacteria, schooling of fishes, flocking of birds and synchronous firing of fireflies and neurons,etc [1, 2, 4, 7, 25, 30, 31, 32, 34, 35, 36, 37, 38]. These coherent phenomena were first modeledby two pioneers, Arthur Winfree [37] and Yoshiki Kuramoto [25] in almost half-century ago,and after their pioneering works, several mathematical models were proposed and studiedfrom the viewpoint of collective behaviors. Among them, our main interest in this paperlies in the LHS model [22] which corresponds to the special case of the Lohe tensor model[21]. The Lohe tensor model is a natural higher-dimensional extension of low-dimensionalaggregation models such as the Kuramoto model [1, 5, 9, 10, 13, 14, 15, 17], sphere models[8, 16, 23, 27, 28, 29, 33, 39] and matrix models [6, 11, 12, 24, 26]. Before we move onto thedescription of the LHS model, we first set the hermitian unit sphere HS d which is the unitsphere in C d +1 centered at the origin: z = ([ z ] , · · · , [ z ] d +1 ) ∈ C d +1 , ˜ z = ([˜ z ] , · · · , [˜ z ] d +1 ) ∈ C d +1 , h z, ˜ z i := d +1 X α =1 [ z ] α [˜ z ] α , k z k := p h z, z i , HS d := { z ∈ C d +1 | k z k = 1 } , Date : January 12, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Emergence, Lohe hermitian sphere model, synchronization, complex vector,tensor.
Acknowledgment.
The work of S.-Y. Ha was supported by National Research Foundation ofKorea(NRF-2020R1A2C3A01003881). The work of H. Park was supported by Basic Science Research Pro-gram through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2019R1I1A1A01059585) . where [ z ] α is the complex conjugate of [ z ] α . Equipped with these notation, the LHS modelwith the same free flow reads as follows:(1.1) ˙ z j = Ω z j + 1 N N X k =1 κ jk (cid:0) h z j , z j i z k − h z k , z j i z j (cid:1) + 1 N N X k =1 λ jk (cid:0) h z j , z k i − h z k , z j i (cid:1) z j , where κ jk and λ jk are constant coupling gains coined as “ Lohe sphere coupling gain ” and“ rotational coupling gain ” respectively. Here Ω is a ( d + 1) × ( d + 1) skew-hermitian matrix:Ω † = − Ω , j ∈ N := { , · · · , N } , where Ω † is the Hermitian conjugate of Ω.In this paper, we are interested in the following simple question:“What if dynamics of coupling gains interacts with the dynamics of states?i.e., dynamic interplay between coupling gains and state evolution. In thiscase, under what conditions, can a coupled system exhibit emergent dynam-ics?”The above question has been addressed in other aggregation models, e.g., the Kuramotomodel with adaptive couplings [19, 20], the Lohe sphere model with adaptive couplings[16], the Lohe matrix model with adaptive couplings [24]. Then the coupled dynamics for { ( z j , κ jk , λ jk ) is governed by the Cauchy problem to the LHS-AC model:˙ z j = Ω z j + 1 N N X k =1 κ jk (cid:16) h z j , z j i z k − h z k , z j i z j (cid:17) + 1 N N X k =1 λ jk (cid:16) h z j , z k i − h z k , z j i (cid:17) z j , ˙ κ jk = − γ κ jk + µ Γ ( z j , z k ) , ˙ λ jk = − γ λ jk + µ Γ ( z j , z k ) , t > , ( z j , κ jk , λ jk )(0) = ( z j , κ jk , λ jk ) ∈ HS d × R + × R , j, k ∈ N , (1.2)where µ k and γ k are positive constants.Throughout the paper, we use the following handy notation: Z := ( z , · · · , z N ) ∈ ( HS d ) N , K := [ κ ij ] , Λ := [ λ ij ] , max i,j := max ≤ i,j ≤ N , min i,j := min ≤ i,j ≤ N . Before we discuss our main results, we recall the concept of “ complete aggregation ” for theCauchy problem (1.2) as follows.
Definition 1.1.
Let ( Z, K, Λ) be a solution to (1.2) . Then, complete aggregation occursasymptotically if and only if following relations hold. lim t →∞ max ≤ i,j ≤ N k z i ( t ) − z j ( t ) k = 0 . Recall that the primary purpose of this paper is to provide sufficient frameworks leadingto complete aggregation for system (1.2). In general, there will be no functional dependencebetween κ jk and λ jk . From now on, we assume that the system parameters satisfy thefollowing relations:(1.3) Ω = 0 , γ = γ = γ, µ = µ = µ. HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 3
Motivated by the reduction from the Stuart-Landau(SL) model to the LHS model in Section2.3, we call the following relation as the SL coupling gain pair:(1.4) κ jk > , λ jk = − κ jk , j, k ∈ N . Under the setting (1.4), due to Lemma 2.2 and Lemma 2.3, system (1.2) becomes(1.5) ˙ z j = 1 N N X k =1 κ jk (cid:20) z k − (cid:16) h z j , z k i + h z k , z j i (cid:17) z j (cid:21) , t > , ˙ κ jk = − γκ jk + µ Γ ( z j , z k ) , ( z j , κ ij )(0) = ( z j , κ ij ) ∈ HS d × R + , i, j ∈ N . At the end of Section 2, we will see that system (1.5) on C ( d +1) N can be rewritten as theLohe sphere model on R d +1) N for a special case. Now, we set˜ λ jk := 12 κ jk + λ jk , j, k ∈ N , ˜Γ( z, w ) := 12 Γ ( z, w ) + Γ ( z, w ) , z, w ∈ HS d . Then, under the setting (1.3), system (1.2) can be rewritten as a perturbed system of (1.5):(1.6) ˙ z j = 1 N N X k =1 κ jk (cid:20) z k − (cid:16) h z j , z k i + h z k , z j i (cid:17) z j (cid:21) + 1 N N X k =1 ˜ λ jk ( h z j , z k i − h z k , z j i ) z j , ˙ κ jk = − γκ jk + µ Γ ( z j , z k ) , ˙˜ λ jk = − γ ˜ λ jk + µ ˜Γ( z j , z k ) , t > , ( z j , κ jk , λ jk )(0) = ( z j , κ jk , λ jk ) ∈ HS d × R + × R , j, k ∈ N . and we take the following ansatz for the coupling law Γ :(1.7) Γ ( w, z ) = k w − z k : Anti-Hebbian coupling law , − k w − z k : Hebbian coupling law . The choice and meaning of Anti-Hebbian and Hebbian coupling laws will be elaborated inSection 3. When the coupling gains κ jk and λ jk are simply positive constants and uniformlyindependent of j and k , emergent dynamics of (1.2) has been extensively studied in [22].However, for the coupled system (1.4), we will see that coupling gains tend to zero asymptot-ically. Hence, our presented results do not overlap with the results in aforementioned work.As complete aggregation occurs asymptotically, the vanishing of coupling gains is naturalin some sense, because the coupling gain will not be needed, once complete aggregation isachieved.In what follows, we briefly discuss main results of this paper. First, we study emergentbehaviors of (1.5) under (1.7) for Γ . For the anti-Hebbian coupling law, we use the followingLyapunov functional measuring the degree of aggregation:(1.8) L ij = 12 k z i − z j k + 14 µN N X k =1 ( κ ik − κ jk ) . BYEON, HA, AND PARK
Our first result deals with (1.5) with anti-Hebbian coupling law (1.7) . When initial datasatisfy following relation: max i,j L ij < , complete aggregation emerges and mutual coupling gains tend to zero asymptotically (seeTheorem 3.1): lim t →∞ k z i ( t ) − z j ( t ) k = 0 and lim t →∞ κ ij ( t ) = 0 , i, j ∈ N . Our second result is conserved with (1.5) incorporated by Hebbian coupling function (1.7) .In this case, instead of (1.8), we introduce another functional: D ( Z ) := 12 max i,j k z i − z j k , which is the half of square of state diameter.If there exist a constant κ satisfying the following relations0 < κ < min (cid:26) µγ , min i,j κ ij (cid:27) , max (cid:26) max i,j κ ij , µγ (cid:27) ≤ µκ µ − γκ , D ( Z ) < − γµ κ, then, there exist positive constants C > C > D ( Z ( t )) ≤ C e − C t , t > . We refer to Theorem 3.2 and Section 4 for details.Secondly, we study emergent behaviors of (1.6) with a general coupling gain pair: κ ij > , λ ij ∈ R , ∀ i, j ∈ N . Our third result can be stated as follows. Suppose that system parameters and initial datasatisfy ˜ λ ij = ˜ λ , i, j ∈ N , ˜Γ( t ) = 0 , ∀ t > , max i,j | ˜ λ | κ ij + max k,l L kl < , where L ij := L ij ( Z , K ) and ˜ λ ij := κ ij + λ ij .Then under anti-Hebbian coupling law (1.7) , we have complete aggregation and vanish-ing of coupling gains (see Theorem 3.3):lim t →∞ k z i ( t ) − z j ( t ) k = 0 and lim t →∞ κ ij ( t ) = 0 . Finally, suppose there exist a constant κ such that2 | ˜ λ | < κ < min (cid:26) µγ , min i,j κ ij (cid:27) , max (cid:26) max i,j κ ij , µγ (cid:27) ≤ µ ( κ − | ˜ λ | )2 µ − γκ , D ( Z ) < − γµ κ, and let ( Z, K ) be a solution to (1.6). Then under Hebbian coupling law (1.7) , there existpositive constants C > C > D ( Z ( t )) ≤ C e − C t , t > . See Theorem 3.4 for details.
HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 5
The rest of the paper is organized as follows. In Section 2, we present basic propertiesof the LHS-AC model (1.2), its relations with previous aggregation models and a reductionfrom the generalized Stuart-Landau model to the LHS model with special coupling pair(1.4). In Section 3, we briefly summarize our main results on the emergent collective behav-iors of (1.2). In Section 4, we study the emergent dynamics of (1.5). In Section 5, we studyemergent dynamics of (1.6). Finally, Section 6 is devoted to a brief summary of our mainresults and some remaining issues to be addressed in a future work.2.
Preliminaries
In this section, we study several basic properties of the LHS-AC model (1.1) and itsrelations with other first-order aggregation models with emergent property. We also providea reduction from a generalized Stuart-Landau model to the LHS model.2.1.
Basic estimates.
In this subsection, we study basic properties of system (1.2) suchas the positivities of coupling gains, conservation of modulus of z i and solution splittingproperty. Lemma 2.1. (Positivity and symmetry of coupling gains)
Suppose that the coupling laws Γ and Γ take nonnegative values: Γ ( z, ˜ z ) ≥ , Γ ( z, ˜ z ) ≥ , z, ˜ z ∈ HS d , and let ( Z, K, Λ) be a solution to (1.2) . Then, we have the following assertions:(1) If initial coupling gains satisfy κ ij > , λ ij > , ∀ i, j ∈ N , then one has positivities of coupling gains: κ ij ( t ) > , λ ij ( t ) > , ∀ t > , ∈ i, j N . (2) If initial coupling gains satisfy κ ij = κ ji , λ ij = λ ji , ∀ i, j ∈ N , then symmetries of the coupling gains are preserved: κ ij ( t ) = κ ji ( t ) , λ ij ( t ) = λ ji ( t ) , ∀ t > , i, j ∈ N . Proof. (i) For the first assertion, we use (1.2) and Duhamel’s principle to find the followingrepresentations: for t ≥ κ ij ( t ) = e − γ t (cid:18) κ ij + Z t µ e γ s Γ ( z i ( s ) , z j ( s )) ds (cid:19) ,λ ij ( t ) = e − γ t (cid:18) λ ij + Z t µ e γ s Γ ( z i ( s ) , z j ( s )) ds (cid:19) . (2.1)Since system parameters µ k and γ k are nonnegative, it follows from (2.1) that κ ij ( t ) > , λ ij ( t ) > , t > . (ii) For the second assertion, we use the symmetry of κ ij , λ ij , Γ and Γ in the indexexchange i ←→ j to find the desired results. (cid:3) BYEON, HA, AND PARK
Lemma 2.2. (Conservation of modulus)
Let ( Z, K, Λ) be a solution to the Cauchy problem (1.2) . Then, the modulus k z i k is a conserved quantity: for i ∈ N , k z i k = 1 = ⇒ k z i ( t ) k = 1 , t > . Proof.
It follows from the symmetry of coupling strengths that h z i , ˙ z i i = h z i , Ω z i i + 1 N N X k =1 κ ik (cid:16) h z i , z k i − h z k , z i i (cid:17) h z i , z i i + 1 N N X k =1 λ ik (cid:16) h z i , z k i − h z k , z i i (cid:17) h z i , z i i , h ˙ z i , z i i = h Ω z i , z i i + 1 N N X k =1 κ ik (cid:16) h z k , z i i − h z i , z k i (cid:17) h z i , z i i + 1 N N X k =1 λ ik (cid:16) h z k , z i i − h z i , z k i (cid:17) h z i , z i i . (2.2)Since Ω i is skew-hermitian, we have(2.3) h z i , Ω z i i + h Ω z i , z i i = 0 . Finally, we combine (2.2) and (2.3) to obtain the desired estimate: ddt k z i k = ddt h z i , z i i = h z i , ˙ z i i + h ˙ z i , z i i = 0 . (cid:3) Now, we consider corresponding linear and nonlinear flows:(2.4) ( ˙ f j = Ω f j , t > , ∀ j ∈ N ,f j (0) = f j , and(2.5) ˙ w j = 1 N N X k =1 κ jk (cid:0) w k − h w k , w j i w j (cid:1) + 1 N N X k =1 λ jk (cid:0) h w j , w k i − h w k , w j i (cid:1) w j , ˙ κ jk = − γ κ jk + µ Γ ( w j , w k ) , ˙ λ jk = − γ λ jk + µ Γ ( w j , w k ) , ( w j (0) , κ jk (0) , λ jk (0)) = ( w j , κ jk , λ jk ) ∈ HS d × R + × R , i, j ∈ N . Let R and L j be solution operators to (2.4) and (2.5), respectively. Then, solutions to(2.4) and w j in (2.5) can be represented as follows. f j ( t ) = R ( t ) f j =: e Ω t f j , w j ( t ) := L j ( t )( W , K , Λ ) , ∀ j ∈ N . In next lemma, we show that the full solution operator to (1.2) can be expressed as acomposition of R and L j . Lemma 2.3. (Solution splitting property)
Let ( Z, K, Λ) be a solution to system (1.2) withinitial data ( Z , K , Λ ) satisfying Ω j ≡ Ω , j ∈ N . Then, z j can be decomposed as a composition of f j and w j : z j ( t ) = R ( t ) ◦ L j ( t )( Z , K , Λ ) . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 7
Proof.
We substitute z j = e Ω t w j into (1.1) to obtain e Ω t ˙ w j + Ω e Ω t w j = Ω e Ω t w j + 1 N N X k =1 κ jk (cid:0) e Ω t w k − h e Ω t w k , e Ω t w j i e Ω t w j (cid:1) + 1 N N X k =1 λ jk (cid:0) h e Ω t w j , e Ω t w k i − h e Ω t w k , e Ω t w j i (cid:1) e Ω t w j . (2.6)On the other hand, we use the skew-hermitian property of Ω to find(2.7) ( e Ω t ) † = ( e Ω t ) − . Finally, we combine (2.6) and (2.7) to get˙ w j = 1 N N X k =1 κ jk (cid:0) w k − h w k , w j i w j (cid:1) + 1 N N X k =1 λ jk (cid:0) h w j , w k i − h w k , w j i (cid:1) w j , so that w j ( t ) = L j ( t )( W , K , Λ ). This gives a desired result. (cid:3) Note that the LHS model (1.2) contains two terms involving with κ jk and λ jk . To seethe role of each coupling gain separately, we consider the following subsystems: • (Subsystem A): If we impose the following conditions on (1.2): λ jk = 0 for all j, k ∈ N , and µ = 0 , then we have λ jk ( t ) = 0 , ∀ t > , j, k ∈ N . In this case, we have Subsystem A: ˙ z j = Ω z j + 1 N N X k =1 κ jk (cid:0) h z j , z j i z k − h z k , z j i z j (cid:1) , t > , ˙ κ jk = − γ κ jk + µ Γ ( z j , z k ) , j, k ∈ N , ( z j (0) , κ jk (0)) = ( z j , κ jk ) ∈ HS d × R + . (2.8) • (Subsystem B): If we impose the following condition on (1.2): κ jk = 0 for all j, k ∈ N , and µ = 0 , then we have κ jk ( t ) = 0 ∀ t > , j, k ∈ N . In this case, one has Subsystem B: ˙ z j = Ω z j + 1 N N X k =1 λ jk (cid:0) h z j , z k i − h z k , z j i (cid:1) z j , t > , ˙ λ jk = − γ λ jk + µ Γ ( z j , z k ) , i, j ∈ N , ( z j (0) , λ jk (0)) = ( z j , λ jk ) ∈ HS d × R + . (2.9) BYEON, HA, AND PARK
Reductions to other aggregation models.
In this subsection, we study the rela-tions between (1.2) and other first-order aggregation models with adaptive couplings.For a real vector-valued state { z , z , · · · , z N } ⊂ R d +1 , the interaction terms in SubsystemB become zero: h z j , z k i − h z k , z j i = 0 , j, k ∈ N so that Subsystem B reduces to the free flow: ( ˙ z j = Ω z j , t > , j, k ∈ N , ˙ λ jk = − γ λ jk + µ Γ ( z j , z k ) . In next lemma, we show that the real-valuedness of the components of z j and κ ij arepropagated along system (2.8). Lemma 2.4.
Let ( Z, K, Λ) be a solution to (2.8) with initial data satisfying following con-ditions: z j ∈ R d +1 , Ω ∈ R ( d +1) × ( d +1) , κ jk = κ kj , ∀ j, k ∈ N . Then, one has z j ( t ) ∈ R d +1 , t ≥ , j ∈ N . Proof.
Since the R.H.S. of system (2.8) is Lipschitz continuous with respect to state vari-ables and uniformly bounded, global well-posedness of classical solutions are guaranteed bythe standard Cauchy-Lipschitz theory. Meanwhile, governing system (2.8) coincides withthe LS model on the sphere in R d +1 . On the other hand, the LS model has a unique solutionwhich is bounded in R d +1 . Thus, we have a desired result. (cid:3) By Lemma 2.4, if we assume that (
Z, K,
Λ) is a solution to (2.8) with following conditions: z j ∈ R d +1 , Ω ∈ R ( d +1) × ( d +1) , j ∈ N , then ( Z, K,
Λ) is a solution to system (2.10) with following conditions: γ = γ , µ = µ , Γ = Γ , κ ij = κ ji . This implies that system (2.8) can be reduced to (2.10) with real natural frequency matrices.Then, Subsystem A (2.8) reduces to the LS model with adaptive couplings: ˙ x j = Ω x j + 1 N N X k =1 κ jk (cid:16) h x j , x j i x k − h x k , x j i x j (cid:17) , t > , j ∈ N , ˙ κ jk = − γκ jk + µ Γ( x j , x k ) , ( x i (0) , κ jk (0)) = ( x i , κ jk ) ∈ S d × R + . (2.10)Next, we show that Subsystem A and Subsystem B can be reduced to the Kuramotomodel with adaptive couplings in three different ways: ˙ θ j = ν j + 1 N N X l =1 κ jl sin( θ l − θ j ) , ˙ κ jk = − γκ jk + µ Γ( θ k − θ j ) , t > , j, k ∈ N , (2.11) HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 9 where Γ satisfies Γ( − θ ) = Γ( θ ) , Γ( θ + 2 π ) = Γ( θ ) . First, let { x j } be a solution of the LS model (2.10) with adaptive couplings. We set d = 1 , x j = (cid:20) cos θ j sin θ j (cid:21) , Ω j = (cid:20) − ν j ν j (cid:21) . Then, system (2.10) can be converted to˙ θ j (cid:20) − sin θ j cos θ j (cid:21) = ν j (cid:20) − sin θ j cos θ j (cid:21) + 1 N N X k =1 κ jk (cid:18)(cid:20) cos θ k sin θ k (cid:21) − cos( θ j − θ i ) (cid:20) cos θ j sin θ j (cid:21)(cid:19) . This yields(2.12) ˙ θ j = ν j + 1 N N X k =1 κ jk sin( θ k − θ j )and we can obtain (2.11) .Note that the dynamics (2.10) of κ jk can also be expressed as˙ κ jk = − γκ jk + µ Γ (cid:18)(cid:20) cos θ j sin θ j (cid:21) , (cid:20) cos θ k sin θ k (cid:21)(cid:19) . From the simple assumption Γ( x, y ) = ˜Γ( k x − y k ), there is a proper function ˆΓ with thefollowing properties:Γ (cid:18)(cid:20) cos θ i sin θ i (cid:21) , (cid:20) cos θ j sin θ j (cid:21)(cid:19) = ˜Γ (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) θ i − θ j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) = ˆΓ( θ i − θ j ) . (2.13)It is easy to check that ˆΓ satisfiesˆΓ( − θ ) = ˆΓ( θ ) , ˆΓ( θ + 2 π ) = ˆΓ( θ ) . Thus, system (2.11) becomes(2.14) ˙ κ jk = − γκ jk + µ ˆΓ( θ k − θ j ) . Finally, we combine (2.12) and (2.14) to derive the Kuramoto model with adaptive cou-plings (2.11).Second, we consider Subsystem A (2.8). Let (
Z, K ) be a solution to (2.8) with d = 0 , z j = e i θ j , Ω j = i ν j . (2.15)Then we substitute (2.15) into (2.8) to geti ˙ θ j e i θ j = i ν j e i θ j + 1 N N X k =1 κ jk (cid:16) e i θ k − e i(2 θ j − θ k ) (cid:17) which can be simplified as ˙ θ j = ν j + 2 N N X k =1 κ jk sin( θ k − θ j ) . By the same arguments as in (2.13), we can also reduce (2.8) to˙ κ jk = − γ κ jk + µ ˆΓ ( θ k − θ j ) . Again, Subsystem A can be reduced to the Kuramoto model with adaptive couplings.Third, we consider Subsystem B (2.9). Let (
Z, K ) be a solution to (2.9) with (2.15).Then, by the same argument as in Subsystem A, we can convert (2.9) as ˙ θ j = ν j + 2 N N X k =1 κ jk sin( θ k − θ j ) , ˙ κ jk = − γκ jk + µ ˆΓ( θ k − θ j ) . This implies that Subsystem B can be reduced to the Kuramoto model with adaptivecoupling gains. To sum up, we can visualize aforementioned discussions in the followingdiagram. Subsystem A → Lohe sphere model(2.8) with adaptive couplings(2.10) ր ց ↓
Lohe hermitian sphere model Kuramoto modelwith adaptive couplings with adaptive couplings(1.1) (2.11) ց ր
Subsystem B(2.9)The LHS model with adaptive coupling gains can be reduced to Subsystem A and Sub-system B by setting λ jk ≡ κ jk ≡
0, respectively. Each subsystem can also be reducedto the Kuramoto model with adaptive couplings. This implies that each coupling term ofthe LHS model with adaptive couplings can be reduced to the Kuramoto model with adap-tive couplings. So we can conclude that the LHS model (1.1) with adaptive couplings iswell-defined.2.3.
From the Stuart-Landau model to the LHS model.
In this subsection, we ex-plain how the special coupling gain relation λ jk = − κ jk can arise in the reduction fromthe generalized Stuart-Landau model to the LHS model.Consider a generalized Stuart-Landau model on C d +1 :(2.16) dz j dt = (cid:0) (1 − k z j k ) I d +1 + Ω (cid:1) z j + κN N X k =1 ( z k − z j ) , where z j ∈ C d +1 for all j ∈ N , Ω is a skew-hermitian matrix with the size ( d + 1) × ( d + 1)and I d +1 is the identity matrix with the size ( d + 1) × ( d + 1). HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 11
We substitute the ansatz: z j = r j w j , r j = k z j k and w j = z j k z j k , ∀ j ∈ N into (2.16) to see(2.17) ˙ r j w j + r j ˙ w j = (1 − r j ) r j w j + r j Ω w j + κN N X k =1 ( r k w k − r j w j ) . Then, h w j , (2.17) i implies˙ r j + r j h w j , ˙ w j i = (1 − r j ) r j + r j h w j , Ω w j i + κN N X k =1 ( r k h w j , w k i − r j ) . (2.18)If we take the real part of (2.18), one has˙ r j = (1 − r j ) r j + κN N X k =1 ( r k Re( h w j , w k i ) − r j ) . (2.19)Here we used the relations: h w j , ˙ w j i + h ˙ w j , w j i = h w j , ˙ w j i + h w j , ˙ w j i = 2Re h w j , ˙ w j i , h w j , Ω w j i = 0 . Now, we combine (2.18) and (2.19) to get˙ w j = Ω w j + κN N X k =1 r k r j (cid:16) w k − Re( h w k , w j i ) w j (cid:17) . Similarly, we impose r j ≡ w j = Ω w j + κN N X k =1 h w k − (cid:16) h w k , w j i + h w j , w k i (cid:17) w j i = Ω w j + κN N X k =1 ( w k − h w k , w j i w j ) − κ N N X k =1 ( h w j , w k i − h w k , w j i ) w j = Ω w j + κN N X k =1 h w k − (cid:16) h w k , w j i + h w j , w k i (cid:17) w j i . (2.20)Note that this is the special case of the LHS model (1.1) with κ = − κ .Next, we show that system (2.20) can be embedded as a system on the Euclidean spaceby extending ( d + 1)-dimensional complex-valued vector w ∈ C d +1 to 2( d + 1)-dimensionalreal-valued vector ˜ w ∈ R d +1) with the following map: w = ( w , · · · , w d +1 ) ˜ w = (cid:0) Re( w ) , · · · , Re( w d +1 ) , Im( w ) , · · · , Im( w d +1 ) (cid:1) . Now we will rewrite (2.20) in terms of { ˜ w j } . First, it is easy to see that(2.21) ˙˜ w j = ˜˙ w j . By simple calculation, we haveΩ w j = (cid:0) Re(Ω) + iIm(Ω) (cid:1)(cid:0)
Re( w j ) + iIm( w j ) (cid:1) = (cid:0) Re(Ω)Re( w j ) − Im(Ω)Im( w j ) (cid:1) + i (cid:0) Im(Ω)Re( w j ) + Re(Ω)Im( w j ) (cid:1) . This yields Re(Ω w j ) = Re(Ω)Re( w j ) − Im(Ω)Im( w j ) , Im(Ω w j ) = Im(Ω)Re( w j ) + Re(Ω)Im( w j ) . (2.22)Since Ω is a ( d + 1) × ( d + 1) complex skew-hermitian matrix, we know that Re(Ω) andIm(Ω) are symmetric. From this, we can define 2( d + 1) × d + 1) skew-symmetric matrix˜Ω as follows: ˜Ω = (cid:20) Re(Ω) − Im(Ω)Im(Ω) Re(Ω) (cid:21) . Then we have(2.23) g Ω w j = ˜Ω ˜ w j . Next, we rewrite h w k , w j i + h w j , w k i in terns of ˜ w k and ˜ w j as follows. By definition of thecomplex inner-product, we have h w k , w j i + h w j , w k i = w † k w j + w † j w k = 2Re( w k ) T Re( w j ) + 2Im( w k ) T Im( w j )= 2 ˜ w Tk ˜ w j = 2 h ˜ w k , ˜ w j i . (2.24)Finally we can express system (2.20) with { ˜ w j } and Ω using (2.21), (2.22), (2.23) and (2.24)to get ˙˜ w j = ˜Ω ˜ w j + κN N X k =1 ( ˜ w k − h ˜ w k , ˜ w j i ˜ w j )which is exactly the Lohe sphere model. In summary, from the proper map between C d +1 and R d +1) , we can transform the special case of the LHS model (1.1) with λ jk = − κ jk tothe LS model. Thus, we can see that system (2.21) is a gradient flow as in the LS model(see Proposition 5.1 in [18]).3. Frameworks for complete aggregation and main results
In this section, we briefly present our main results and sufficient frameworks leading tocomplete aggregation in the sense of Definition 1.1. As noted in the previous section, weconsider four different cases depending on the relations between coupling gains κ jk , λ jk andcoupling law Γ (anti-Hebbian or Hebbian law). ⋄ (Coupling gain pair): Depending on the relation between κ jk and λ jk , we consider thefollowing two cases: • Stuart-Landau coupling gain pair ( κ jk , λ jk ): κ jk > , λ jk = − κ jk , j, k ∈ N . • General coupling gain pair ( κ jk , λ jk ): κ jk > λ jk ∈ R , j, k ∈ N . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 13 ⋄ (Coupling law Γ ): Consider anti-Hebbian and Hebbian laws:(3.1) Γ ( w, z ) = k w − z k , Anti-Hebbian law1 − k w − z k , Hebbian lawThe motivation for (3.1) can be explained as follows. In literature [16, 19, 20] on thesynchronization with an adaptive coupling law, the following coupling law(3.2) Γ( θ, ˜ θ ) = cos( θ − ˜ θ )was often employed. Note that when the phase difference between the interactiong oscillatorsis small, it increases the mutual coupling strength. Thus, it is called Hebbian coupling law. Incontrast, when the phase difference is small, there is a case in which the rate of increment incoupling strength becomes small. This is called “ anti-Hebbian coupling law ” and the ansatz:(3.3) Γ( θ, ˜ θ ) = | sin( θ − ˜ θ ) | was used in aforementioned literature. Note that on HS d ,Γ ( z, ˜ z ) = k z − ˜ z k = 2(1 − Re h z, ˜ z i ) = 2 (cid:0) − Re (cid:0) cos θ ( z, ˜ z ) (cid:1)(cid:1) , where θ ( z, ˜ z ) is the angle between z and ˜ z , and Γ becomes smaller when the angle is small.In this sense, it plays the same role as anti-Hebbian law (3.3). In contrast, real part ofΓ ( z, ˜ z ) = 1 − k z − ˜ z k exhibits the same dynamics as (3.2).3.1. SL coupling gain pair.
Consider the Stuart-Landau coupling gain pair:(3.4) κ jk > , λ jk = − κ jk , ∀ t ≥ , j, k ∈ N . In fact, one can show that once initial gain pair satisfy (3.4), then the relation (3.4) will bepropagated along (1.2) under suitable conditions on system parameters and coupling laws(see Lemma 4.1).3.1.1.
Anti-Hebbian coupling law.
Consider the anti-Hebbian coupling law:(3.5) Γ ( z, ˜ z ) = k z − ˜ z k . Under the setting (3.4) and (3.5), system (1.2) becomes(3.6) ˙ z j = 1 N N X k =1 κ jk (cid:20) z k − (cid:16) h z j , z k i + h z k , z j i (cid:17) z j (cid:21) , t > , ˙ κ jk = − γκ jk + µ k z j − z k k , ( z j , κ jk )(0) = ( z j , κ jk ) ∈ HS d × R + , j, k ∈ N . For the emergent estimate, we use a Lyapunov functional approach: for i, j ∈ N ,(3.7) L ij = 12 k z i − z j k + 14 µN N X k =1 ( κ ik − κ jk ) . Note that at the completely aggregated state z i = z, κ ij = κ, i, j ∈ N the functional L ij is exactly zero. Thus, we can see that the functional L ij can measure howa state configuration and coupling gains are close to complete aggregated state.Now, we state our first result on the emergent dynamics for (1.1). Theorem 3.1.
Suppose initial data ( Z , K ) satisfy (3.8) max i,j L ij < , and let ( Z, K ) be a solution to (3.6) . Then, one has lim t →∞ k z i ( t ) − z j ( t ) k = 0 and lim t →∞ κ ij ( t ) = 0 , i, j ∈ N . Proof.
We leave its proof in Section 4. (cid:3)
Hebbian coupling law.
In this part, we consider the Hebbian law:(3.9) Γ ( z, ˜ z ) = 1 − k z − ˜ z k . Under the setting (3.4) and (3.9), system (1.1) becomes ˙ z j = 1 N N X k =1 κ jk h z k − (cid:16) h z j , z k i + h z k , z j i (cid:1) z j i , t > , ˙ κ jk = − γκ jk + µ (cid:18) − k z k − z j k (cid:19) , ( z j , κ jk )(0) = ( z j , κ jk ) ∈ HS d × R + , j, k ∈ N . (3.10)For the emergent dynamics of (3.10), we introduce a Lyapunov function: D ij ( Z ) := 12 k z i − z j k , D ( Z ) := max i,j D ij ( Z ) . Note that D is the half of the square of state diameter, and complete aggregation occurs iflim t →∞ D ( Z ( t )) = 0 . The maximum of differentiable functions does not need to be differentiable, hence we cannotguarantee differentiability of D ( Z ). However, it follows from the analyticity of each D ij ( Z ), D ( Z ) is differentiable almost everywhere and we can regard ˙ D ( Z ( t )) as a weak derivativeof D ( Z ). By the continuity and estimate for ˙ D ( Z ) a.e. is enough to derive the estimate for D ( Z ) via direct integration.Now, we present our second result as follows. Theorem 3.2.
Suppose there exist a constant κ such that (3.11) 0 < κ < min (cid:26) µγ , min i,j κ ij (cid:27) , max (cid:26) max i,j κ ij , µγ (cid:27) ≤ µκ µ − γκ , D ( Z ) < − γµ κ. and let ( Z, K ) be a solution to (3.10) . Then, there exist positive constants C > and C > satisfying D ( Z ( t )) ≤ C e − C t , t > . Proof.
We leave its proof in Section 4.2. (cid:3)
HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 15
Asymptotically SL coupling gain pair.
In this subsection, we consider a couplinggain pair: κ jk > , λ jk ∈ R , ∀ j, k ∈ N . Note that unlike the previous subsection, we do not assume any functional relation between κ jk and λ jk . Nevertheless, we can still rewrite (1.2) as a perturbation of (1.5). To see this,we recall the dynamics of z j :(3.12) ˙ z j = 1 N N X k =1 κ jk (cid:0) h z j , z j i z k − h z k , z j i z j (cid:1) + 1 N N X k =1 λ jk (cid:0) h z j , z k i − h z k , z j i (cid:1) z j . To use the result in Section 3.1, we set(3.13) ˜ λ jk := 12 κ jk + λ jk , ∀ j, k ∈ N . Then, ˜ λ jk = 0 corresponds to exactly the same situation in Section 3.1. Now we can rewrite(3.12) using (3.13) into˙ z j = 1 N N X k =1 κ jk ( z k − R kj z j ) + 1 N N X k =1 ˜ λ jk ( h jk − h kj ) z j , and the dynamics of ˜ λ jk can also be expressed as follows:(3.14) ˙˜ λ jk = 12 ˙ κ jk + ˙ λ jk = 12 (cid:16) − γ κ jk + µ Γ ( z j , z k ) (cid:17) + (cid:16) − γ λ jk + µ Γ ( z j , z k ) (cid:17) . In what follows, we set γ = γ = γ, µ = µ = µ. (3.15)We combine (3.14) and (3.15) to get˙˜ λ jk = − γ (cid:18) κ jk + λ jk (cid:19) + µ (cid:18)
12 Γ ( z j , z k ) + Γ ( z j , z k ) (cid:19) . Now, we set ˜Γ( z, w ) := 12 Γ ( z, w ) + Γ ( z, w )to rewrite(3.16) ˙˜ λ jk = − γ ˜ λ jk + µ ˜Γ( z j , z k ) . Finally we combine (3.12) and (3.16) to get(3.17) ˙ z j = 1 N N X k =1 κ jk h z k − (cid:16) h z j , z k i + h z k , z j i (cid:1) z j i + 1 N N X k =1 ˜ λ jk ( h z j , z k i − h z k , z j i ) z j , t > , ˙ κ jk = − γκ jk + µ Γ ( z j , z k ) , ˙˜ λ jk = − γ ˜ λ jk + µ ˜Γ( z j , z k ) , j, k ∈ N , ( z j , κ jk , ˜ λ jk )(0) = ( z j , κ jk , ˜ λ jk ) ∈ HS d × R + × R . In what follows, we consider only following cases:˜ λ ij is independent of i and j , but is a function of t , and ˜Γ ≡ . Anti-Hebbian coupling law.
In this part, we study emergent dynamics of (3.17) withanti-Hebbian coupling law:(3.18) ˙ z j = 1 N N X k =1 κ jk h z k − (cid:16) h z j , z k i + h z k , z j i (cid:1) z j i + 1 N N X k =1 ˜ λ jk ( h z j , z k i − h z k , z j i ) z j , ˙ κ jk = − γκ jk + µ k z j − z k k , ˙˜ λ jk = − γ ˜ λ jk + µ ˜Γ( z j , z k ) , ( z j , κ jk , ˜ λ jk )(0) = ( z j , κ jk , ˜ λ jk ) ∈ HS d × R + × R , j, k ∈ N . Our third main result is concerned with a sufficient framework leading to complete aggre-gation.
Theorem 3.3.
Suppose that the following relations hold ˜ λ ij = ˜ λ , ∀ i, j ∈ N and ˜Γ( t ) ≡ , t > , for some constant ˜ λ , and let ( Z, K, ˜Λ) be a solution to (3.18) with initial data satisfyingthe following conditions: > max i,j | ˜ λ | κ ij + max k,l L kl , then we have lim t →∞ k z i ( t ) − z j ( t ) k = 0 and lim t →∞ κ ij ( t ) = 0 . Proof.
We leave its proof in Section 5.1. (cid:3)
Remark 3.1.
Since the coupling gains tend to zero asymptotically, the presented result iscompletely different from the previous result in [22] in which the coupling gains take thesame positive constant: κ ij ( t ) = κ > , ∀ i, j ∈ N . Hebbian coupling law.
Consider system (3.17) with a Hebbian coupling law: ˙ z j = 1 N N X k =1 κ jk h z k − (cid:16) h z j , z k i + h z k , z j i (cid:1) z j i + 1 N N X k =1 ˜ λ jk ( h z j , z k i − h z k , z j i ) z j , ˙ κ jk = − γκ jk + µ (cid:18) − k z j − z k k (cid:19) , ˙˜ λ jk = − γ ˜ λ jk + µ ˜Γ( z j , z k ) , ( z j , κ jk , ˜ λ jk )(0) = ( z j , κ jk , ˜ λ jk ) ∈ HS d × R + × R , j, k ∈ N , (3.19)Similar to Section 3.1.2, one has the same emergent dynamics. Theorem 3.4.
Suppose there exist a constant κ such that (3.20)2 | ˜ λ | < κ < min (cid:26) µγ , min i,j κ ij (cid:27) , max (cid:26) max i,j κ ij , µγ (cid:27) ≤ µ ( κ − | ˜ λ | )2 µ − γκ , D ( Z ) < − γµ κ. and let ( Z, K ) be a solution to (3.19) . Then, there exist positive constants C > and C > satisfying D ( Z ( t )) ≤ C e − C t , t > . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 17
Proof.
We leave its proof in Section 5.2. (cid:3) Collective dynamics under Stuart-Landau coupling gain pair
In this section, we study emergent dynamics of system (1.2) with the initial Stuart-Landaucoupling gain pair:(4.1) κ ij > , λ ij = − κ ij , ∀ i, j ∈ N . First, we show that if the initial SL coupling gain pair satisfies (4.1), then it is propagatedalong the dynamics (1.2).
Lemma 4.1.
Suppose that system parameters and initial coupling strengths satisfy γ = γ = γ, µ = µ = µ, Γ + 2Γ = 0 , λ ij = − κ ij , (4.2) and let ( Z, K, Λ) be a solution of (1.1) . Then we have (4.3) λ ij ( t ) = − κ ij ( t ) , ∀ t ≥ , i, j ∈ N . Proof.
It follows from (1.1) and (4.2) that˙ κ ij = − γκ ij + µ Γ ( z i , z j ) , ˙ λ ij = − γλ ij − µ ( z i , z j ) , t > . This yields ddt ( κ ij + 2 λ ij ) = − γ ( κ ij + 2 λ ij ) . By integrating the above relation, one has the desired estimate:( κ ij + 2 λ ij )( t ) = e − γt ( κ ij + 2 λ ij ) = 0 , t ≥ . (cid:3) Next, we substitute (4.3) into (1.1) to get the dynamics for (
Z, K ):(4.4) ˙ z j = 1 N N X k =1 κ jk (cid:20) z k − (cid:16) h z j , z k i + h z k , z j i (cid:17) z j (cid:21) , t > , ˙ κ jk = − γκ jk + µ Γ ( z j , z k ) , j, k ∈ N , ( z j , κ jk )(0) = ( z j , κ jk ) ∈ HS d × R + . In what follows, we consider two coupling laws for Γ as prototype examples for the anti-Hebbian and the Hebbian couplings between the coupling gain and state:(4.5) Γ ( z, ˜ z ) = k z − ˜ z k and Γ ( z, ˜ z ) = 1 − k z − ˜ z k . Anti-Hebbian coupling law.
Consider system (4.4) with anti-Hebbian coupling law(4.5) :(4.6) ˙ z j = 1 N N X k =1 κ jk (cid:20) z k − (cid:16) h z j , z k i + h z k , z j i (cid:17) z j (cid:21) , t > , ˙ κ jk = − γκ jk + µ k z j − z k k , j, k ∈ N , ( z j , κ jk )(0) = ( z j , κ jk ) ∈ HS d × R + . To study emergent dynamics of (4.6), we recall a Lyapunov functional L ij in (3.7): L ij = 12 k z i − z j k + 14 µN N X k =1 ( κ ik − κ jk ) . On HS d , the functional L ij can be rewritten as follows:(4.7) L ij = 1 − Re h z i , z j i + 14 µN N X k =1 ( κ ik − κ jk ) . Thus, it is natural to study the time-evolution of h z i , z j i . For notational simplicity, we use(4.8) h ij := h z i , z j i , R ij := Re h ij = 12 ( h ij + h ji ) , I ij := Im h ij = 12i ( h ij − h ji ) . Then, it is easy to see R ii = 1 , I ii = 0 , | R ij | ≤ | h ij | ≤ , R ij = R ji and I ij = − I ji , i, j ∈ N . We can rewrite (4.6) and a Lyapunov functional in (4.7):˙ z j = 1 N N X k =1 κ jk ( z k − R jk z j ) , L ij = 1 − R ij + 14 µN N X k =1 ( κ ik − κ jk ) . To sum up, system (4.6) on HS d becomes ˙ z j = 1 N N X k =1 κ jk ( z k − R jk z j ) , t > , ˙ κ jk = − γκ jk + µ k z j − z k k , j, k ∈ N , ( z j , κ jk )(0) = ( z j , κ jk ) ∈ HS d × R + . (4.9)Next, we study the time-evolution of L ij in a series of lemmas. Lemma 4.2.
Let ( Z, K ) be a solution to (4.9) . Then L ij satisfies ddt L ij = − N N X k =1 ( κ ik R ik + κ jk R jk )(1 − R ij ) − γ µN N X k =1 ( κ ik − κ jk ) , t > . Proof.
By direct calculations, one has(4.10) ddt L ij = −
12 ( ˙ h ij + ˙ h ji ) + 12 µN N X k =1 ( κ ik − κ jk )( ˙ κ ik − ˙ κ jk ) . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 19
Note that the terms in the R.H.S. of (4.10) can be estimated as follows. • (Estimate of the first term in (4.10)): We use (4.8) and (4.9) to find˙ h ij + ˙ h ji = h z i , ˙ z j i + h ˙ z i , z j i + h z j , ˙ z i i + h ˙ z j , z i i = 2 N N X k =1 Re( h z i , ˙ z j i + h ˙ z i , z j i )= 2 N N X k =1 Re (cid:16) κ jk ( h ik − R jk h ij ) + κ ik ( h kj − R ik h ij ) (cid:17) = 2 N N X k =1 (cid:16) κ jk ( R ik − R jk R ij ) + κ ik ( R jk − R ik R ji ) (cid:17) = 2 N N X k =1 (cid:16) ( κ ik − κ jk )( R jk − R ik ) + ( R ik κ ik + R jk κ jk )(1 − R ij ) (cid:17) . (4.11) • (Estimate of the second term in (4.10)): Similar to the first term, one has12 µN N X k =1 ( κ ik − κ jk )( ˙ κ ik − ˙ κ jk )= 12 µN N X k =1 ( κ ik − κ jk )( − γκ ik − µh ik − µh ki + γκ jk + µh jk + µh kj )= − γ µN N X k =1 ( κ ik − κ jk ) − N N X k =1 ( κ ik − κ jk )( h ik + h ki − h jk − h kj )= − γ µN N X k =1 ( κ ik − κ jk ) + 1 N N X k =1 ( κ ik − κ jk )( R jk − R ik ) . (4.12)In (4.10), we combine (4.11) and (4.12) to find the desired result. (cid:3) Lemma 4.3.
Let ( Z, K ) be a solution to (4.9) with the initial data ( Z , K ) satisfying thefollowing relations: max i,j L ij < . Then, we have the following assertions:(1) R ij and κ ij are strictly positive: R ij ( t ) > , κ ij ( t ) > , t ≥ , i, j ∈ N . (2) L ij is non-increasing function: L ij ( t ) ≤ L ij , t ≥ , i, j ∈ N . Proof.
Let ( i, j ) ∈ N be fixed. Since κ ij >
0, by Lemma 2.1, one has κ ij ( t ) > , t > . Now, it follows from L ij < R ij > µN N X k =1 ( κ ik − κ jk ) ≥ . We claim: R ij ( t ) > , t ≥ . For this, we introduce a set T ij : T ij := { τ ∈ [0 , ∞ ) : R ij ( t ) > , t ∈ [0 , τ ) } . Then, by (4.13) and continuity of R ij , one has T ij = ∅ . Suppose that t ∗ ij := sup T ij < ∞ . Then, one has lim t ր t ∗ ij R ij ( t ) = 0 . We choose the index ( i , j ) by ( i , j ) := arg min ( k,l ) t ∗ kl . By the minimality of t ∗ i j and Lemma 4.2, we have1 − R i j ( t ) ≤ L i j ( t ) < L i j , so that R i j ( t ) > − L i j > , t ∈ (0 , t ∗ i j ) . We take t ր t ∗ i j to derive a contradiction:0 = lim t ր t ∗ i j R i j ( t ) ≥ − L i j > , Hence, we verified the claim: t ∗ i j = ∞ and R ij ( t ) > , t ∈ (0 , ∞ ) . By minimality, we have t ∗ ij = ∞ for each index ( i, j ) . On the other hand, we have 1 − R ij ≥ . Therefore, it follows from Lemma 4.2 that the derivative of L ij is not positive for every t ∈ [0 , ∞ ) which yields the desired result. (cid:3) Before we provide a proof of Theorem 3.1, we state Barbalat’s lemma and Gr¨onwall typelemma without proofs.
Lemma 4.4. (Barbalat’s Lemma [3])
Suppose f : [0 , ∞ ) → R is uniformly continuous andsatisfies ∃ lim t →∞ Z t f ( s ) ds < ∞ . Then, one has lim t →∞ f ( t ) = 0 . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 21
Lemma 4.5. [16]
Let y : [0 , ∞ ) → [0 , ∞ ) be a C function satisfying y ′ ≤ − αy + f, t > , y (0) = y , where α is a positive constant and f : [0 , ∞ ) → R is a continuous function satisfying lim t →∞ f ( t ) = 0 . Then y satisfies y ( t ) ≤ α (cid:16) max s ∈ [ t/ ,t ] | f ( s ) | (cid:17) + y e − αt + k f k L ∞ α e − αt , t ≥ . Proof.
For a proof, we refer to Appendix A of [16]. (cid:3)
Now we are ready to provide a proof of Theorem 3.1.
Proof of Theorem 3.1 : Let (
Z, K ) be a solution to (3.6) with the initial data ( Z , K )satisfying max i,j L ij < , min i,j κ ij > , and we choose an index ( I, J ) := arg max ( i,j ) ∈N L ij . To apply Lemma 4.4 to κ ij , we will show uniform boundedness of ˙ κ ij in order to verifyuniform continuity of κ ij . Note that(4.14) ˙ κ ij = − γκ ij + µ k z i − z j k ≤ − γκ ij + 4 µ, ˙ κ ij = − γκ ij + µ k z i − z j k ≥ − γκ ij and from (4.14) we obtain unfirom upper bound of κ ij as κ ij ( t ) ≤ κ ij e − γt + 4 µγ (1 − e − γt ) ≤ κ ij + 4 µγ ≤ max k,l κ kl + 4 µγ . (4.15)From Lemma 2.1, κ ij is uniformly bounded below by 0. Therefore, κ ij is uniformly bounded.On the other hand, from (4.14) ˙ κ ij is also uniformly bounded, therefore κ ij is uniformlycontinuous.It follows from Lemma 4.3 that1 − R ij ( t ) < L ij , therefore R ij ( t ) > − L ij > − L IJ > , t > . Therefore we have˙ L ij = − (cid:18) − R ij N (cid:19) N X k =1 ( κ ik R ik + κ jk R jk ) − γ µN N X k =1 ( κ ik − κ jk ) ( ∵ Lemma 4.2) ≤ − (cid:18) − R ij N (cid:19) N X k =1 (cid:0) (1 − L IJ )( κ ik + κ jk ) (cid:1) − γ µN N X k =1 ( κ ik − κ jk ) ( ∵ R ij ( t ) > − L IJ ) ≤ − min ( N N X k =1 (cid:0) (1 − L IJ )( κ ik + κ jk ) (cid:1) , γ )| {z } =: K ij − R ij + 14 µN N X k =1 ( κ ik − κ jk ) ! = −K ij L ij , for any index i and j . This implies(4.16) L ij ( t ) ≤ L ij exp (cid:18) − Z t K ij ( s ) ds (cid:19) . Since K ij ≥
0, we have only two possible cases:either Z ∞ K ij dt = ∞ or Z ∞ K ij dt < ∞ . • Case A (cid:0)R ∞ K ij dt = ∞ (cid:1) : We first assume that Z ∞ K ij ( s ) ds = ∞ . As L ij ≥
0, we use (4.16) to find lim t →∞ L ij ( t ) = 0 . In particular, we have lim t →∞ k z i ( t ) − z j ( t ) k = 0 . Now we recall that dynamics of κ ij is defined by(4.17) ˙ κ ij = − γκ ij + µ k z i − z j k , t > . By κ ij >
0, the assumption, Lemma 2.1, one has κ ij ( t ) > , t ≥ . Therefore we can apply Lemma 4.5 to (4.17) to findlim t →∞ κ ij ( t ) = 0 , verifying the desired result. • Case B (cid:0)R ∞ K ij dt < ∞ (cid:1) : Now assume that Z ∞ K ij dt < ∞ . and consider the set A := ( t ∈ (0 , ∞ ) : 1 N N X k =1 (cid:0) (1 − L IJ )( κ ik + κ jk ) (cid:1) ≥ γ ) . Then for the Lebesgue measure m , we have following relation2 γm ( A ) = Z A γ = Z A K ij dt ≤ Z ∞ K ij dt < ∞ , HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 23 therefore m ( A ) < ∞ . This yields, Z ∞ N N X k =1 (cid:0) (1 − L IJ )( κ ik + κ jk ) (cid:1) dt = Z A N N X k =1 (cid:0) (1 − L IJ )( κ ik + κ jk ) (cid:1) dt + Z R + \ A N N X k =1 (cid:0) (1 − L IJ )( κ ik + κ jk ) (cid:1) dt = Z A N N X k =1 (cid:0) (1 − L IJ )( κ ik + κ jk ) (cid:1) dt + Z R + \ A K ij dt ( ∵ Definition of A ) ≤ Z A N N X k =1 (cid:20)(cid:0) − L IJ (cid:1) (cid:18) max i,j κ ij + 4 µγ (cid:19)(cid:21) dt + Z R + \ A K ij dt ( ∵ (4.15))= 2 m ( A ) (cid:20)(cid:0) − L IJ (cid:1) (cid:18) max i,j κ ij + 4 µγ (cid:19)(cid:21) dt + Z R + \ A K ij dt ≤ m ( A ) (cid:20)(cid:0) − L IJ (cid:1) (cid:18) max i,j κ ij + 4 µγ (cid:19)(cid:21) dt + Z ∞ K ij dt < ∞ . (4.18)Since L ij < Z ∞ ( κ ik + κ jk ) dt < ∞ , ∀ i, j, k ∈ N . Now we use the uniform continuity of κ ij . By lemma 4.4, we obtainlim t →∞ κ ik ( t ) = lim t →∞ κ jk ( t ) = 0 . (4.19)On the other hand, we recall the result of (4.11):˙ R ij = 12 ( ˙ h ij + ˙ h ji ) = 1 N N X k =1 (cid:16) ( κ ik − κ jk )( R jk − R ik ) + ( R ik κ ik + R jk κ jk )(1 − R ij ) (cid:17) . From uniform boundedness of κ ij and R ij , we have uniform boundedness of ˙ R ij . Combiningthese all together with uniform boundedness of ˙ κ ij , we obtain uniform boundedness of¨ κ ij = − γ ˙ κ ij − µ ˙ R ij , which leads to uniform continuity of ˙ κ ij . As integration of ˙ κ ij is finite from (4.19): Z ∞ ˙ κ ij ( s ) ds = − κ ij , again from Lemma 4.4, we can conclude thatlim t →∞ ˙ κ ik ( t ) = lim t →∞ ˙ κ jk ( t ) = 0 . (4.20)Therefore, taking limit t → ∞ to the dynamics˙ κ ij = − γκ ij + µ k z i − z j k , with (4.19) and (5.21) yield the desired resultlim t →∞ k z i ( t ) − z j ( t ) k = 0 . (cid:3) Hebbian coupling law.
Consider system (4.4) with the Hebbian coupling law (4.5) : ˙ z j = 1 N N X k =1 κ jk ( z k − R jk z j ) , t > , ˙ κ jk = − γκ jk + µ (cid:18) − k z j − z k k (cid:19) , j, k ∈ N , ( z j , κ jk )(0) = ( z j , κ jk ) ∈ HS d × R + . For a given system parameters γ and µ , we choose positive constants κ m and κ M such that(4.21) 12 κ M < κ m ≤ µγ (cid:18) − κ m κ M (cid:19) . Proposition 4.1.
For positive constants κ m and κ M satisfying (4.21) , suppose initial datasatisfy (4.22) D ( Z ) < κ m κ M − , κ m < min i,j κ ij , and let ( Z, K ) be a solution to system (3.10) satisfying a priori assumption: (4.23) sup ≤ t< ∞ max i,j κ ij ( t ) ≤ κ M . Then, there exist positive constants D and D such that D ( Z ( t )) ≤ D e − D t , t > . Proof.
By (4.22) and continuity of solution, the set˜ T ij := { τ ∈ (0 , ∞ ) : κ ij ( t ) > κ m , ∀ t ∈ (0 , τ ) } 6 = ∅ . Now, we set˜ T := \ i,j ˜ T ij = n τ ∈ (0 , ∞ ) : min i,j κ ij ( t ) > κ m , ∀ t ∈ (0 , τ ) o , ˜ t ∗ := sup ˜ T . In the course of proof of Lemma 4.2, we have˙ h ij + ˙ h ji = 2 N N X k =1 (cid:0) ( κ ik − κ jk )( R jk − R ik ) + ( R ik κ ik + R jk κ jk )(1 − R ij ) (cid:1) . In the sequel, for notational simplicity, we set D ij := D ij ( Z ) , D := D ( Z ) , D := D ( Z ) . This and defining relation (4.22) of D imply˙ D ij = 12 ddt (2 − h ij − h ji ) = −
12 ( ˙ h ij + ˙ h ji )= − N N X k =1 (cid:0) ( κ ik + κ jk ) D ij (cid:1) + 1 N N X k =1 (cid:0) κ ik D ik + κ jk D jk (cid:1) D ij + 1 N N X k =1 ( κ ik − κ jk )( D jk − D ik )=: J + J + J . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 25
From now on, we will regard i and j as a function of t . For each t , we assume that i, j areindices such that D = D ij . It follows from (4.22) and the definition of ˜ t ∗ that(4.24) J ≥ κ m D , J ≤ κ M D , J ≤ κ M − κ m ) D , t ∈ [0 , ˜ t ∗ ) . This leads to a differential inequality:(4.25) ˙ D < − κ m − κ M ) D + 2 κ M D , t ∈ [0 , ˜ t ∗ ) , where the factor (2 κ m − κ M ) is positive from the first inequality of (4.22).We apply the comparison principle to (4.25) to get D ( t ) ≤ (cid:16) D − κ m − κ M κ M (cid:17) e (2 κ m − κ M ) t + κ m − κ M κ M , t ∈ [0 , ˜ t ∗ ) , i.e. exponential decay occurs in t ∈ [0 , ˜ t ∗ ). Hence the proof is done if we verify ˜ t ∗ = ∞ .Now we use the initial condition D < κ m κ M − − κ m − κ M ) D + 2 κ M D < , whenever D ∈ (cid:18) , κ m κ M − (cid:19) . Therefore, D is decreasing in t ∈ [0 , ˜ t ∗ ). Hence we have˙ κ ij = − γκ ij + µ − µ D ij ≥ − γκ ij + µ − µ D ≥ − γκ ij + 2 µ κ M − κ m κ M , t ∈ [0 , ˜ t ∗ ) . By comparison principle, one has κ ij ≥ (cid:18) κ ij − µγ κ M − κ m κ M (cid:19) e − γt + 2 µγ (cid:18) κ M − κ m κ M (cid:19) ≥ (cid:0) κ ij − κ m (cid:1) e − γt + κ m , t ∈ [0 , ˜ t ∗ ) , where the last equality holds from the second inequality of (4.21). By definition of ˜ T , thereexist indices k and l such that κ m = lim t ր ˜ t ∗ κ kl . Therefore if ˜ t ∗ is finite, one has κ m = lim t ր ˜ t ∗ κ kl ≥ lim t ր ˜ t ∗ (cid:0) κ kl − κ m (cid:1) e − γt + κ m = (cid:0) κ kl − κ m (cid:1) e − γ ˜ t ∗ + κ m > κ m , which is contradictory, and we obtain our desired result. (cid:3) Now we are ready to provide a proof of our second main result.
Proof of Theorem 3.2.
Recall the conditions (3.11):(4.26) 0 < κ < min (cid:26) µγ , min i,j κ ij (cid:27) , max (cid:26) max i,j κ ij , µγ (cid:27) ≤ µκ µ − γκ , D ( Z ) < − γµ κ. Now, it suffices to show that the above conditions satisfy (4.22) and (4.23):(4.27) κ m < min i,j κ ij , D ( Z ) < κ m κ M − , sup ≤ t< ∞ max i,j κ ij ( t ) ≤ κ M . We first figure out κ m and κ M satisfying (4.21):(4.28) 12 κ M < κ m , κ m ≤ µγ (cid:18) − κ m κ M (cid:19) . Since κ is a candidate of κ m , we assume that κ m satisfies (4.26). Rewriting (4.28) , we have κ m ≤ µγ (cid:18) − κ m κ M (cid:19) ⇐⇒ µκ m µ − γκ m ≤ κ M . (4.29)Optimizing κ M under (4.29), we have 2 µκ m µ − γκ m = κ M . Therefore, as we set(4.30) κ m = κ and κ M = 2 µκ µ − γκ , (4.28) is achieved. In particular, as κ satisfies (4.26) , we have12 κ M < κ m ⇐⇒ κ < µγ , which is true from (4.26) . Hence (4.28) is achieved. • (Verification of (4.22)): Clearly, (4.26) implies (4.27) . By the setting (4.30), one has2 κ m κ M − κ µκ µ − γk − − γκµ . Hence (4.26) is equivalent to (4.27) under the setting (4.30). • (Verification of (4.23)): Note that κ ij ( t ) = e − γt (cid:18) κ ij + Z t µe γs R ij ds (cid:19) ≤ e − γt (cid:18) κ ij + Z t µe γs ds (cid:19) ≤ (cid:18) κ ij − µγ (cid:19) e − γt + µγ ≤ max (cid:26) max i,j κ ij , µγ (cid:27) ≤ κ M . Finally, we can apply Proposition 4.1 to derive the desired estimate. (cid:3) Collective dynamics under asymptotic SL coupling gain pair
In this section, we study the emergent dynamics of the system (1.1) for a general couplinggain pair ( κ ij , λ ij ): κ ij ( t ) > , λ ij ( t ) ∈ R , t ≥ , i, j ∈ N . Recall that our governing system is given as follows:(5.1) ˙ z j = 1 N N X k =1 κ jk ( z k − R jk z j ) + 1 N N X k =1 ˜ λ jk ( h z j , z k i − h z k , z j i ) z j , t > , ˙ κ ij = − γκ ij + µ Γ ( z i , z j ) , ˙˜ λ ij = − γ ˜ λ ij + µ ˜Γ( z i , z j ) , ( z j , κ ij , ˜ λ ij )(0) = ( z j , κ ij , ˜ λ ij ) ∈ HS d × R + × R , i, j ∈ N . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 27
Before we proceed analysis on the emergent behavior of (5.1), we observe that the ratio of κ ij and ˜ λ ij is bounded below by the ratio of Γ and ˜Γ in the following lemma. Lemma 5.1.
Suppose that coupling gains and coupling law satisfy cκ ij ≥ ˜ λ ij and c Γ ( z i , z j ) ≥ ˜Γ( z i , z j ) , i, j ∈ N for some constant c > , and let ( Z, K, ˜Λ) be a solution to (3.17) . Then, one has cκ ij ( t ) ≥ ˜ λ ij ( t ) , ∀ t > . Proof.
We use (3.17) to see ddt (cid:0) cκ ij − ˜ λ ij (cid:1) = − γ (cid:0) cκ ij − ˜ λ ij (cid:1) + µ (cid:0) c Γ ( z i , z j ) − ˜Γ( z i , z j ) (cid:1) ≥ − γ (cid:0) cκ ij − ˜ λ ij (cid:1) . Therefore, we have cκ ij ( t ) − ˜ λ ij ( t ) ≥ e − γt (cid:0) cκ ij − ˜ λ ij (cid:1) ≥ . and this is the desired result. (cid:3) Parallel to the presentation in Section 4, in what follows, we consider two type of couplinglaws for Γ as in Section 4: Γ ( z, ˜ z ) : k z − ˜ z k , − k z − ˜ z k . Anti-Hebbian coupling law.
In this subsection, we study emergent dynamics of(3.17) with the anti-Hebbian coupling law:(5.2) ˙ z j = 1 N N X k =1 κ jk (cid:16) z k − R jk z j (cid:17) + 1 N N X k =1 ˜ λ jk (cid:16) h z j , z k i − h z k , z j i (cid:17) z j , t > , ˙ κ jk = − γκ jk + µ k z j − z k k , ˙˜ λ jk = − γ ˜ λ jk + µ ˜Γ( z j , z k ) , ( z j , κ jk , ˜ λ jk )(0) = ( z j , κ jk , ˜ λ jk ) ∈ HS d × R + × R , j, k ∈ N . As in Section 4, we study the temporal evolution of the Lyapunov functional L ij introducedin (3.7) . Lemma 5.2.
Let ( Z, K, ˜Λ) be a solution to (5.2) . Then, the functional L ij satisfies ddt L ij = − (cid:18) − R ij N (cid:19) N X k =1 ( κ ik R ik + κ jk R jk ) − γ µN N X k =1 ( κ ik − κ jk ) − I ij N N X k =1 (˜ λ ik I ik − ˜ λ jk I jk ) . (5.3) Proof.
By definition of L ij , one has(5.4) ˙ L ij = −
12 ( ˙ h ij + ˙ h ji ) + 12 µN N X k =1 ( κ ik − κ jk )( ˙ κ ik − ˙ κ jk ) . Next, we estimate two terms in the R.H.S. of (5.4) separately. • (Estimate of the first term in (5.4)): By straightforward calculations, one has˙ h ij + ˙ h ji = h z i , ˙ z j i + h ˙ z i , z j i + h z j , ˙ z i i + h ˙ z j , z i i = 2 N N X k =1 Re( h z i , ˙ z j i + h ˙ z i , z j i )= 2 N N X k =1 Re h κ jk ( h ik − R jk h ij ) + κ ik ( h kj − R ik h ij ) + ˜ λ jk ( h jk − h kj ) h ij + ˜ λ ik ( h ki − h ik ) h ij ) i = 2 N N X k =1 h κ jk ( R ik − R jk R ij ) + κ ik ( R jk − R ik R ji ) (cid:1) − λ jk I jk I ij − λ ik I ki I ij ) i . (5.5) • (Estimate of the second term in (5.4)): Again, one has12 µN N X k =1 ( κ ik − κ jk )( ˙ κ ik − ˙ κ jk )= 12 µN N X k =1 ( κ ik − κ jk )( − γκ ik − µh ik − µh ki + γκ jk + µh jk + µh kj )= − γ µN N X k =1 ( κ ik − κ jk ) − N N X k =1 ( κ ik − κ jk )( h ik + h ki − h jk − h kj )= − γ µN N X k =1 ( κ ik − κ jk ) − N N X k =1 ( κ ik − κ jk )( R ik − R jk ) . (5.6)In (5.4), we combine (5.5) and (5.6) to obtain the desired estimate. (cid:3) Lemma 5.3.
Suppose that the following relations hold: ˜ λ ij = ˜ λ , i, j ∈ N and ˜Γ( t ) ≡ , ∀ t > for some constant ˜ λ , and let ( Z, K, ˜Λ) be a solution to (5.2) . Then, the following assertionshold:(1) There exists a function ˜ λ = ˜ λ ( · ) such that ˜ λ ij ( t ) = ˜ λ ( t ) , t > , ∀ i, j ∈ N . (2) The functional L ij satisfies ˙ L ij ≤ − (cid:18) − R ij N (cid:19) N X k =1 (cid:16) κ ik R ik + κ jk R jk − | ˜ λ | (cid:17) − γ µN N X k =1 ( κ ik − κ jk ) , t > . Proof.
It follows from (5.2) and (5.7) that(5.8) ˙ z j = 1 N N X k =1 κ jk (cid:16) z k − R jk z j (cid:17) + 1 N N X k =1 ˜ λ jk (cid:16) h z j , z k i − h z k , z j i (cid:17) z j , t > , ˙ κ jk = − γκ jk + µ k z j − z k k , ˙˜ λ jk = − γ ˜ λ jk , ( z j , κ jk , ˜ λ jk )(0) = ( z j , κ jk , ˜ λ jk ) ∈ HS d × R + × R , j, k ∈ N . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 29 (i) By (5.8) , one has ˜ λ ij ( t ) = ˜ λ e − γt =: ˜ λ ( t ) , t > , which yields the first assertion.(ii) Now, we estimate the last term of (5.3) as follows.2 I ij N N X k =1 (˜ λ ik I ik − ˜ λ jk I jk ) = 2 I ij N N X k =1 (cid:0) ˜ λ ik ( I ik − I jk ) − I jk (˜ λ jk − ˜ λ ik ) (cid:1) = 2 I ij N N X k =1 ˜ λ ( I ik − I jk ) . We use the triangle inequality and the Cauchy-Schwarz inequality to find | I ij | = q − R ij = q (1 − R ij )(1 + R ij ) ≤ q − R ij ) , | I ik − I jk | = (cid:12)(cid:12)(cid:12)(cid:12) i ( h ik − h ki − h jk + h kj ) (cid:12)(cid:12)(cid:12)(cid:12) = 12 |h z i − z j , z k i + h z k , z j − z i i|≤ k z i − z j k = q − R ij ) . (5.9)Finally, we combine (5.3) and (5.9) to obtain˙ L ij = − (cid:18) − R ij N (cid:19) N X k =1 ( κ ik R ik + κ jk R jk ) − γ µN N X k =1 ( κ ik − κ jk ) − I ij N N X k =1 ˜ λ ik ( I ik − I jk ) ≤ − (cid:18) − R ij N (cid:19) N X k =1 ( κ ik R ik + κ jk R jk ) − γ µN N X k =1 ( κ ik − κ jk ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ij N N X k =1 ˜ λ ik ( I ik − I jk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − (cid:18) − R ij N (cid:19) N X k =1 ( κ ik R ik + κ jk R jk ) − γ µN N X k =1 ( κ ik − κ jk ) + 4 (cid:18) − R ij N (cid:19) N X k =1 | ˜ λ | = − (cid:18) − R ij N (cid:19) N X k =1 (cid:16) κ ik R ik + κ jk R jk − | ˜ λ | (cid:17) − γ µN N X k =1 ( κ ik − κ jk ) , and this is the desired result. (cid:3) Lemma 5.4.
Suppose that the following relations ˜ λ ij = ˜ λ , i, j ∈ N and ˜Γ( t ) = 0 , t > hold for some constant ˜ λ , and let ( Z, K, ˜Λ) be a solution to (5.2) with the initial datasatisfying > max i,j | ˜ λ | κ ij + max k,l L kl . Then, L ij is non-increasing.Proof. If ˜ λ = 0, by the same argument in a proof of Lemma 4.3, we are done. We choosea constant c satisfying cκ ij ≥ | ˜ λ | , (5.10)for any indices i and j . Since ˜Γ ≡
0, by Lemma 5.1 we have cκ ij ( t ) ≥ | ˜ λ ( t ) | , t > . Hence, one has˙ L ij ≤ − (cid:18) − R ij N (cid:19) N X k =1 ( κ ik R ik + κ jk R jk − | ˜ λ | ) − γ µN N X k =1 ( κ ik − κ jk ) ≤ − (cid:18) − R ij N (cid:19) N X k =1 (cid:0) κ ik ( R ik − c ) + κ jk ( R jk − c ) (cid:1) − γ µN N X k =1 ( κ ik − κ jk ) . (5.11)Now we recall that κ ij is positive by Lemma 2.1 because κ ij >
0. Therefore if we assumemin k,l R kl > c, (5.12)then ˜ T ∗ ij := { τ ∈ (0 , ∞ ) : R ij ( t ) > c, t ∈ (0 , τ ) } , is non-empty for each i, j and we can introduce˜ t ∗ ij := sup ˜ T ∗ ij , ( i , j ) := arg min k,l ˜ t ∗ kl . Therefore, it follows from the minimality of ˜ t ∗ i j that L i j ( t ) < L i j , t ∈ (0 , ˜ t ∗ i j ) . By definition (4.7), one has 1 − R i j ( t ) ≤ L i j ( t ) , t ≥ . Hence we have1 − R i j ( t ) ≤ L i j ( t ) < L i j , so that R i j ( t ) > − L i j > t ∈ (0 , ˜ t ∗ i j ) . Therefore by taking t ր ˜ t ∗ i j , from the continuity of R i j we obtain R i j (˜ t ∗ i j ) ≥ − L i j > . If we impose the relationship 1 − L ij > c, (5.13)one can obtain ˜ t ∗ i j < ∞ = ⇒ − L i j > c = R i j (˜ t ∗ i j ) ≥ − L i j , which is contradictory. Therefore ˜ t ∗ i j = ∞ , (5.14)so that ˜ t ∗ ij = ∞ and L ij is a non-increasing function of t . Now we choose optimal c satisfying(5.10). Namely, we set c = max i,j | ˜ λ | κ ij , and (5.12) and (5.13) can be specified asmin k,l R kl > i,j | ˜ λ | κ ij and 1 − L ij > i,j | ˜ λ | κ ij , (5.15)respectively. Since R ij > − L ij , (5.15) is achieved from a priori condition and we have adesired result. (cid:3) HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 31
As a consequence of continuity argument used in a proof of Lemma 5.4, we obtain thefollowing result.
Lemma 5.5.
Suppose that the following relations ˜ λ ij = ˜ λ , i, j ∈ N and ˜Γ( t ) = 0 , t > hold for some constant ˜ λ , and let ( Z, K, ˜Λ) be a solution to (5.2) with the initial datasatisfying > max i,j | ˜ λ | κ ij + max k,l L kl . (5.16) Then we have min
K,L R KL ( t ) > I,J | ˜ λ | κ IJ , t > . In particular, for any index k , we have κ ik R ik − max l,m | ˜ λ | κ lm ! > . Proof.
This is a direct consequence of (5.14). (cid:3)
Now we are ready to provide a proof of Theorem 3.3.
Proof of Theorem 3.3.
Since max i,j | ˜ λ | κ ij satisfies relationship (5.10), from (5.11) it followsthat˙ L ij ≤ − (1 − R ij ) 1 N N X k =1 κ ik R ik − max i,j | ˜ λ | κ ij ! + κ jk R jk − max i,j | ˜ λ | κ ij !!| {z } =: O ij − γ µN N X k =1 ( κ ik − κ jk ) ≤ − min {O ij , γ } | {z } =: M ij − R ij + 14 µN N X k =1 ( κ ik − κ jk ) ! =: −M ij L ij . This leads to(5.17) L ij ( t ) ≤ L ij exp (cid:18) − Z t M ij ( s ) ds (cid:19) . Since O ij is positive from Lemma 5.5, so is M ij . Hence we have only two possible cases:either Z ∞ M ij dt = ∞ or Z ∞ M ij dt < ∞ . • Case A ( R ∞ M ij dt = ∞ ): In this case, as we did in a proof of Theorem 3.1, we havelim t →∞ L ij ( t ) = 0 = ⇒ lim t →∞ k z i ( t ) − z j ( t ) k = 0 . Therefore, it follows from Lemma 4.5 that˙ κ ij = − γκ ij + µ k z i − z j k = ⇒ lim t →∞ κ ij ( t ) = 0 . • Case B ( R ∞ M ij dt < ∞ ): consider the set B := { t ∈ (0 , ∞ ) | O ij ≥ γ } . Then for the Lebesgue measure m , we have2 γm ( B ) = Z B γ = Z B M ij dt ≤ Z ∞ M ij dt < ∞ , therefore m ( B ) < ∞ . This yields, Z ∞ O ij dt = Z B O ij dt + Z R + \ B O ij d = Z B O ij dt + Z R + \ B M ij dt ( ∵ Definition of B ) ≤ Z B N N X k =1 " R ik + R jk − max i,j | ˜ λ | κ ij ! (cid:18) max i,j κ ij + 4 µγ (cid:19) dt + Z R + \ B M ij dt ( ∵ (4.15)) ≤ Z B N N X k =1 " − max i,j | ˜ λ | κ ij ! (cid:18) max i,j κ ij + 4 µγ (cid:19) dt + Z R + \ B M ij dt = 2 m ( B ) " − max i,j | ˜ λ | κ ij ! (cid:18) max i,j κ ij + 4 µγ (cid:19) + Z R + \ B M ij dt = 2 m ( B ) " − max i,j | ˜ λ | κ ij ! (cid:18) max i,j κ ij + 4 µγ (cid:19) + Z ∞ M ij dt < ∞ . (5.18)Since each summand of O ij is positive, (5.18) implies Z ∞ κ ik R ik − max i,j | ˜ λ | κ ij ! + κ jk R jk − max i,j | ˜ λ | κ ij !! dt < ∞ , ∀ i, j, k ∈ N . (5.19)Now, we use the relationship 1 − R ij ≤ L ij ≤ L ij and a priori condition (5.16) to get R IJ ≥ − max k,l L kl > max i,j | ˜ λ | κ ij , for any indices I and J . Therefore by (5.19) we have − max K,L L KL − max I,J | ˜ λ | κ IJ ! Z ∞ ( κ ik + κ jk ) dt < ∞ , ∀ i, j, k ∈ N . Again, by lemma 4.4, we obtain lim t →∞ κ ik = lim t →∞ κ jk = 0 . (5.20)On the other hand, we recall the result of 5.5:˙ R ij = 12 ( ˙ h ij + ˙ h ji )= 1 N N X k =1 h κ jk ( R ik − R jk R ij ) + κ ik ( R jk − R ik R ji ) (cid:1) − λ jk I jk I ij − λ ik I ki I ij ) i . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 33
From uniform boundedness of κ ij ,˜ λ ij , R ij and I ij , we have uniform boundness of ˙ R ij . Com-bining this together with uniform boundedness of ˙ κ ij , we obtain uniform boundedness of¨ κ ij = − γ ˙ κ ij − µ ˙ R ij , which leads to uniform continuity of ˙ κ ij . As integration of ˙ κ ij is finite from (5.20): Z ∞ ˙ κ ij ( s ) ds = − κ ij , again from by lemma 4.4, we concludelim t →∞ ˙ κ ik ( t ) = lim t →∞ ˙ κ jk ( t ) = 0 . (5.21)Therefore, we take limit t → ∞ to the dynamics˙ κ ij = − γκ ij + µ k z i − z j k to find the desired result lim t →∞ k z i ( t ) − z j ( t ) k = 0 . (cid:3) Hebbian coupling law.
In this subsection, we study emergent dynamics of (3.17)with Hebbian coupling law: ˙ z j = 1 N N X k =1 κ jk ( z k − R jk z j ) + 1 N N X k =1 ˜ λ jk ( h z j , z k i − h z k , z j i ) z j , t > , ˙ κ jk = − γκ jk + µ (cid:18) − k z j − z k k (cid:19) , ˙˜ λ jk = − γ ˜ λ jk + µ ˜Γ( z j , z k ) , ( z j , κ jk , ˜ λ jk )(0) = ( z j , κ jk , ˜ λ jk ) ∈ HS d × R + × R , j, k ∈ N . (5.22)Basically, we follow the same arguments in Section 4.2 to derive the emergent dynamics ofsystem (5.22). Proposition 5.1.
Suppose that the following relations hold ˜ λ ij = ˜ λ , ˜Γ( t ) = 0 , t > , i, j ∈ N , and that there exists a function ˜ λ , positive constants κ m , κ M such that ˜ λ ij ( t ) = ˜ λ ( t ) , i, j ∈ N , t > , κ M + 2 | ˜ λ | < κ m ≤ µγ · κ M − κ m + 2 | ˜ λ | κ M , (5.23) and let ( Z, K, ˜Λ) be a solution to (5.2) with initial data satisfying (5.24) min i,j κ ij > κ m , D < κ m − | ˜ λ | κ M − , and a priori condition max i,j sup ≤ t< ∞ κ ij ( t ) ≤ κ M . (5.25) Then, there exist positive constants D and D such that D ( Z ( t )) ≤ D e − D t , t > . Proof.
As in a proof of Proposition 4.1, by (5.24) the set¯ T := { τ ∈ (0 , ∞ ) : min i,j κ ij ( t ) > κ m , ∀ t ∈ (0 , τ ) } is nonempty. So ¯ t ∗ := sup ¯ T is well defined in (0 , ∞ ]. We use the same argument in a proofof Lemma 5.2 to find˙ h ij + ˙ h ji = 2 N N X k =1 (cid:16) κ jk ( R ik − R jk R ij ) + κ ik ( R jk − R ik R ji ) (cid:1) − λ jk I jk I ij − λ ik I ki I ij ) (cid:17) . Then, we use the above relation to find˙ D ij ( t ) = −
12 ( ˙ h ij + ˙ h ji )= − N N X k =1 (cid:16) ( κ ik + κ jk ) D ij (cid:17) + 1 N N X k =1 (cid:16) κ ik D ik + κ jk D jk (cid:17) D ij + 1 N N X k =1 ( κ ik − κ jk )( D jk − D ik ) + 1 N N X k =1 λ jk I jk I ij + ˜ λ ik I ki I ij )=: J + J + J + J . Below, we estimate the term J i separately. • (Estimate of J i , i = 1 , , J ≥ κ m D , J ≤ κ M D , J ≤ κ M − κ m ) D , t ∈ [0 , ¯ t ∗ ) . • (Estimate of J ): We use the estimate I ij ≤ | I ij | = q − R ij = q (1 − R ij )(1 + R ij ) ≤ q − R ij ) = p D ij ≤ √ D to find(5.27) J ≤ D| ˜ λ | = 8 D| e − γt ˜ λ | < D| ˜ λ | . As in a proof of Proposition 4.1, for each t , we assume that indices i, j are chosen to satisfy D = D ij . We now combine (5.26) and (5.27) to get the Riccati type differential inequality:(5.28) ˙ D < − κ m − κ M − | ˜ λ | ) D + 2 κ M D , t ∈ [0 , ¯ t ∗ ) , where we use the first inequality of (5.23) to see2 κ m − κ M − | ˜ λ | > . We apply the comparison principle to (5.28) to find(5.29) D ( t ) ≤ (cid:16) D − κ m − κ M − | ˜ λ | κ M (cid:17) e (2 κ m − κ M − | ˜ λ | ) t + κ m − κ M − | ˜ λ | κ M , t ∈ [0 , ¯ t ∗ ) . HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 35
Next, we will verify ¯ t ∗ = ∞ . Since we will use proof by contradiction, suppose that ¯ t ∗ is finite. We use the initial condition D < κ m − | ˜ λ | κ M − , to see that D is decreasing in t ∈ [0 , ¯ t ∗ ) from (5.28). Therefore, one has˙ κ ij = − γκ ij + µ − µ D ij ≥ − γκ ij + µ − µ D≥ − γκ ij + 2 µ κ M − κ m + 2 | ˜ λ | κ M , t ∈ [0 , ¯ t ∗ ) . By comparison principle, one obtains κ ij ≥ κ ij − µγ κ M − κ m + 2 | ˜ λ | κ M ! e − γt + 2 µγ κ M − κ m + 2 | ˜ λ | κ M ! ≥ (cid:0) κ ij − κ m (cid:1) e − γt + κ m , t ∈ [0 , ¯ t ∗ ) , where the last equality holds from the second inequality of (5.23). From definition of ¯ T ,there exist indices k and l such that κ m = lim t ր ¯ t ∗ κ kl . Since ¯ t ∗ is finite, we have following inequality: κ m = lim t ր ¯ t ∗ κ kl ≥ lim t ր ¯ t ∗ (cid:0) κ kl − κ m (cid:1) e − γt + κ m = (cid:0) κ kl − κ m (cid:1) e − γ ¯ t ∗ + κ m > κ m , which is a contradictory. Thus ¯ t ∗ = ∞ andmin i,j κ ij ( t ) > κ m , t ∈ [0 , ∞ ) . Then the relation (5.29) implies our desired estimate. (cid:3)
Proof of Theorem 3.4.
Recall the conditions (3.20):(5.30)2 | ˜ λ | < κ < min (cid:26) µγ , min i,j κ ij (cid:27) , max (cid:26) max i,j κ ij , µγ (cid:27) ≤ µ ( κ − | ˜ λ | )2 µ − γκ , D ( Z ) < − γµ κ. Now, it suffices to show that the above conditions satisfy (5.24) and (5.25), i.e., κ m < min i,j κ ij , D ( Z ) < κ m − | ˜ λ | κ M − , sup ≤ t< ∞ max i,j κ ij ( t ) ≤ κ M . (5.31)We first figure out κ m and κ M satisfying (5.23):(5.32) 12 κ M + 2 | ˜ λ | < κ m , κ m ≤ µγ · κ M − κ m + 2 | ˜ λ | κ M . Since κ is a candidate of κ m , we will assume that κ m satisfies (5.30). Rewriting (5.32) , wehave κ m ≤ µγ · κ M − κ m + 2 | ˜ λ | κ M ⇐⇒ µ ( κ m − | ˜ λ | )2 µ − γκ m ≤ κ M . (5.33) Optimizing κ M under (5.33), we have2 µ ( κ m − | ˜ λ | )2 µ − γκ m = κ M . Therefore, as we set(5.34) κ m = κ and κ M = 2 µ ( κ − | ˜ λ | )2 µ − γκ , (5.32) is achieved. In particular, as κ satisfies (5.30) , we have12 κ M + 2 | ˜ λ | < κ m ⇐⇒ κ < µγ , which is true from (5.30) . Hence (5.32) is achieved. • (Verification of (5.31)): Clearly, (5.30) implies (5.31) . By the setting (5.34), one has2 κ m − | ˜ λ | κ M − κ − | ˜ λ | µ ( κ − | ˜ λ | )2 µ − γκ − − γκµ . Hence (5.30) is equivalent to (5.31) under the setting (5.34). As we did in a proof ofTheorem 3.2, (5.31) is verified by the estimate κ ij ( t ) = e − γt (cid:18) κ ij + Z t µe γs R ij ds (cid:19) ≤ e − γt (cid:18) κ ij + Z t µe γs ds (cid:19) ≤ (cid:18) κ ij − µγ (cid:19) e − γt + µγ ≤ max (cid:26) max i,j κ ij , µγ (cid:27) ≤ κ M . Finally, we can apply the result of Proposition 5.1 to derive the desired estimate. (cid:3) Conclusion
In this paper, we have studied the emergent dynamics of the LHS model with adaptivecoupling gains. When the dynamics of coupling gains are decoupled from the dynamics ofstate, say, they are simply constants, in previous literature, several sufficient frameworkswere proposed for complete aggregation in which all states collapse to the same state.However, when coupling gains and state evolutions are intertwined via adaptive couplinglaws, emergent dynamics are more delicate and interesting. In order to couple the dynamicsof coupling gains and state, we employ two types of coupling laws, namely anti-Hebbianlaw and Hebbian law in analogy with the dynamics of brain neurons. The former causes theincrement of coupling gain, as the state differences become larger, whereas the latter causesthe opposite effect. In the case of the same free flow for all particles, states aggregate tothe same state asymptotically for some class of initial data and system parameters. Whenrotational coupling gain is the minus of the half of the sphere coupling gain, our first resultsays that the relative state tends to zero and coupling gains tend to zero asymptotically.Since the coupling gain becomes smaller over time, analysis of complete aggregation is highlynontrivial and difficult to analyze. Despite this apparent difficulty, we use the Lyapunovfunctional approach and Barbalat’s lemma to show that the relative states and couplinggains tend to zero for the anti-Hebbian case. For the Hebbian coupling case, we showthat the square of the state diameter tends to zero exponentially fast for some admissibleclass of initial data and initial system parameters. The same things can be done for an
HE LOHE HERMITIAN SPHERE MODEL WITH ADAPTED COUPLINGS 37 asymptotically SL coupling gain pair. All presented results in this paper deal with theensemble of particles with the same free flows. There are several issues that have not beenaddressed in this work. For example, for an ensemble of LHS particles with the same freeflow, a bi-polar state can emerge as one of the resulting asymptotic patterns. Then, isthis bi-polar configuration unstable as for the Lohe sphere model? For the ensemble ofparticles with heterogeneous free flows, emergent dynamics is completely unknown even fornonnegative coupling gains, not to mention adaptive coupling gains. These interesting issueswill be left for future work.
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Department of Mathematical SciencesSeoul National University, Seoul 08826, Republic of Korea
Email address : [email protected] (Seung-Yeal Ha) Department of Mathematical Sciences and Research Institute of MathematicsSeoul National University, Seoul 08826 andKorea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea
Email address : [email protected] (Hansol Park) Department of Mathematical SciencesSeoul National University, Seoul 08826, Republic of Korea
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