Collective behaviors of the Lohe hermitian sphere model with inertia
aa r X i v : . [ m a t h - ph ] S e p COLLECTIVE BEHAVIORS OF THE LOHE HERMITIAN SPHEREMODEL WITH INERTIA
SEUNG-YEAL HA, MYEONGJU KANG, AND HANSOL PARK
Dedicated to the celebration of the 80th birthday of Prof. Shuxing Chen
Abstract.
We present a second-order extension of the first-order Lohe hermitian sphere(LHS)model and study its emergent asymptotic dynamics. Our proposed model incorporates aninertial effect as a second-order extension. The inertia term can generate an oscillatory be-havior of particle trajectory in a small time interval(initial layer) which causes a technicaldifficulty for the application of monotonicity-based arguments. For emergent estimates,we employ two-point correlation function which is defined as an inner product betweenpositions of particles. For a homogeneous ensemble with the same frequency matrix, weprovide two sufficient frameworks in terms of system parameters and initial data to showthat two-point correlation functions tend to the unity which is exactly the same as thecomplete aggregation. In contrast, for a heterogeneous ensemble with distinct frequencymatrices, we provide a sufficient framework in terms of system parameters and initial data,which makes two-point correlation functions close to unity by increasing the principal cou-pling strength. Introduction
Collective behaviors of a many-body system are often observed in biological complexnetworks, to name a few, flocking of birds, swarming of fish, herding of sheep, synchronousfiring of fireflies, neurons and pacemaker cells [1, 3, 9, 15, 19, 20, 26, 27, 28, 30] etc. In thispaper, we are interested in the aggregation phenomena of particles on a Hermitian sphere.To motivate our discussion, we begin with the first-order LHS model.Let z j = z j ( t ) be the position of the j -th particle on a Hermitian sphere at time t . Inorder to fix the idea, we begin with the first-order Lohe hermitian sphere model [16, 17, 18]on a Hermitian sphere HS dr := { z ∈ C d +1 : k z k = r } :(1.1) ˙ z j = Ω j z j + κ ( h z j , z j i z c − h z c , z j i z j ) + κ ( h z j , z c i − h z c , z j i ) z j , j = 1 , · · · , N, where h z , z i := P d +1 α =1 z α z α which is conjugate linear in the first argument and linear inthe second argument, and where Ω j is the skew-symmetric ( d + 1) × ( d + 1) matrix suchthat Ω † j = − Ω j and κ , κ are nonnegative coupling strengths. The emergent dynamics ofthe HLS model (1.1) has been extensively studied in [16, 17, 18] (see Section 2.1). Date : September 21, 2020.
Key words and phrases.
Emergence, Kuramoto model, Lohe sphere model, phase locked states.
Acknowledgment.
The work of S.-Y.Ha is supported by NRF-2020R1A2C3A01003881, the work ofM. Kang was supported by the National Research Foundation of Korea(NRF) grant funded by the Koreagovernment(MSIP)(2016K2A9A2A13003815), and the work of H. Park was supported by Basic ScienceResearch Program through the National Research Foundation of Korea(NRF) funded by the Ministry ofEducation (2019R1I1A1A01059585).
In this paper, we are interested in the large-time dynamics of the Cauchy problem to thesecond-order extension of (1.2) incorporating inertial effect to (1.1):(1.2) m (cid:16) ˙ v j − Ω j γ v j (cid:17) + γv j = κ ( h z j , z j i z c − h z c , z j i z j ) + κ ( h z j , z c i − h z c , z j i ) z j − m k v j k k z j k z j , z j ∈ HS dr , t > ,v j = ˙ z j − Ω j γ z j , z j (0) = z inj , v j (0) = v inj , h z inj , v inj i + h v inj , z inj i = 0 , where m is the strength of inertia which is nonnegative.It is easy to see that for zero inertia and unit friction constant, system (1.2) reduces tothe LHS model (1.1). The basic conservation law and solution splitting property will begiven in Lemma 2.1 and Lemma 2.2, respectively. Before we present our main results, wefirst recall the concepts of complete aggregation and practical aggregation as follows. Definition 1.1.
Let Z := { z j } be a solution to (1.2) .(1) The solution Z exhibits (asymptotic) complete aggregation if the following estimateholds. lim t →∞ max i,j k z i ( t ) − z j ( t ) k = 0 . (2) The solution Z exhibits (asymptotic) practical aggregation if the following estimateholds. lim κ →∞ lim sup ≤ t< ∞ max i,j k z i ( t ) − z j ( t ) k = 0 . Next, we briefly present our two main results on the large-time emergent dynamics of(1.2).First, we present a sufficient framework for the complete aggregation for the homogeneousensemble with Ω j = Ω. In this case, we may assume that Ω = 0 and z j satisfies m ¨ z j = − γ ˙ z j + κ (cid:0) z c − h z c , z j i z j (cid:1) + κ (cid:0) h z j , z c i − h z c , z j i (cid:1) z j − m k ˙ z j k z j . Under the following conditions on system parameters and initial data: γ ≫ m, G (0) ≪ , | ˙ G (0) | + G (0) ≪ . For the detailed conditions, we refer to frameworks ( F A F A
2) and ( F B F B
2) in Section4.1. Our first main result is concerned with the complete aggregation (see Theorem 4.1):lim t →∞ max i,j | z i ( t ) − z j ( t ) | = 0 . For this, we introduce two-point correlation functions: h ij = h z i , z j i , g ij := 1 − h ij , ≤ i, j ≤ N, G := 1 N N X i,j =1 | g ij | , and then, we also derive differential inequality for G : m ¨ G + γ ˙ G + 4 κ δ G ≤ f ( t ) , f ( t ) → t → ∞ . Then, via Gronwall’s differential inequality, we can derive the zero convergence of G :lim t →∞ G ( t ) = 0 , i.e., lim t →∞ h z i ( t ) , z j ( t ) i = 1 , ∀ i, j = 1 , · · · , N. SECOND-ORDER LOHE HERMITIAN MODEL 3
This clearly implies the complete aggregation in the sense of Definition 1.1.Second, we deal with a heterogeneous ensemble with distinct natural frequency matricesΩ i . In this situation, we derive a rather weak aggregation, namely practical aggregation.For this, we propose a framework on the system parameters and initial data: γ ≫ m, G (0) ≪ , | ˙ G (0) | + G (0) ≪ . For the detailed conditions, we refer to frameworks ( F C F C
2) in Section 4.2. As in theaggregation estimate to the homogeneous ensemble, we derive a second-order Gronwall’sinequality: m ¨ G + γ ˙ G + 4 κ δ G ≤ ∞ + 8 κ + 16 mγ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) , t > . Then, via the second-order Gronwall’s lemma (Lemma 5.4) and a suitable ansatz for m = m κ η , one can show G ( t ) . max n κ , κ η o , for t ≫ . This clearly implies the practical aggregation in Definition 1.1. We refer to Theorem 4.2for a detailed discussion.The rest of this paper is organized as follows. In Section 2, we briefly introduce thesecond-order LHS model and basic properties of the proposed model and discuss it withother previous models such as the first-order LHS model and the Kuramoto model, andreview the previous result on the first-order Lohe Hermitian model. In Section 3, we studythe characterization and instability of some distinguished states. In Section 4, we summarizeour main results on the emergent dynamics of the second-order LHS model. In Section 5and Section 6, we provide proofs of Theorem 4.1 and Theorem 4.2. Finally, Section 7 isdevoted to a brief summary of our main results and discussion on some remaining problemsfor a future work.
Notation : For a vector z = ( z , · · · , z d +1 ) ∈ C d +1 and w = ( w , · · · , w d +1 ) ∈ C d +1 , we setthe inner product h· , ·i and its corresponding ℓ -nrom: h z, w i := d +1 X i =1 z i w i , k z k := p h z, z i , HS d = HS d . For a given configuration C := { ( z j , w j := ˙ z j ) } , we set state and velocity diameters asfollows. D ( Z ) := max i,j | z i − z j | , D ( W ) := max i,j | w i − w j | . Preliminaries
In this section, we briefly introduce a second-order LHS model (1.2) and its basic prop-erties, and discuss its relations with other aggregation models such as the first-order LHSmodel and the Kuramoto model.
HA, KANG, AND PARK
The second-order LHS model.
In this subsection, we study basic properties of theseocnd-order LHS model. To factor out the rotational motion, we introduce an auxiliaryvariable u j on HS dR :(2.1) z j := e tγ Ω j u j , j = 1 , · · · , N. By direct calculations, one has(2.2) u j = e − tγ Ω j z j , ˙ u j = e − tγ Ω j v j , ¨ u j = e − tγ Ω j (cid:16) ˙ v j − γ Ω j v j (cid:17) , j = 1 , · · · , N. We substitute (2.2) into (1.2) and use the fact that Ω j is skew-Hermitian to derive theequations for u j : m ¨ u j + γ ˙ u j = κ N N X k =1 (cid:18) k u j k e Ω k − Ω jγ t u k − (cid:28) e Ω k − Ω jγ t u k , u j (cid:29) u j (cid:19) + κ N N X k =1 (cid:18) (cid:28) u j , e Ω k − Ω jγ t u k (cid:29) − (cid:28) e Ω k − Ω jγ t u k , u j (cid:29) (cid:19) u j − m k ˙ u j k k u j k u j ,u j (0) = u inj = z inj , ˙ u j (0) = ˙ u inj = v inj , h u inj , ˙ u inj i + h ˙ u inj , u inj i = 0 . (2.3)In the following lemma, we study the conservation of k z j k and k u j k . Lemma 2.1. (Conservation laws)
Let { z j } and { u j } be global solutions of (1.2) and (2.3) ,respectively. Then, ℓ -norms k z j k and k u j k are conserved quantity: ddt k z j ( t ) k = 0 , ddt k u j ( t ) k = 0 , ∀ t ≥ , j = 1 , · · · , N. Proof.
Since e − tγ Ω j is unitary, one can see k u j k = h u j , u j i = D e − tγ Ω j z j , e − tγ Ω j z j E = D e − tγ Ω j (cid:16) e − tγ Ω j (cid:17) † z j , z j E = h z j , z j i = k z j k . Hence, we only verify the conservation of the norm k u j k . Now we claim: ddt k u j k = h u j , ˙ u j i + h ˙ u j , u j i = 0 . Simple calculation yields(2.4) m ddt (cid:16) h u j , ˙ u j i + h ˙ u j , u j i (cid:17) = 2 m k ˙ u j k + h u j , m ¨ u j i + h m ¨ u j , u j i . Here, we use (2.3) and (2.4) to obtain h u j , m ¨ u j i = − γ h u j , ˙ u j i + κ N N X k =1 (cid:18) (cid:28) u j , e Ω k − Ω jγ t u k (cid:29) − (cid:28) e Ω k − Ω jγ t u k , u j (cid:29) (cid:19) k u j k + κ N N X k =1 (cid:18) (cid:28) u j , e Ω k − Ω jγ t u k (cid:29) − (cid:28) e Ω k − Ω jγ t u k , u j (cid:29) (cid:19) k u j k − m k ˙ u j k . This yields h u j , m ¨ u j i + h m ¨ u j , u j i = h u j , m ¨ u j i + h u j , m ¨ u j i = − γ (cid:16) h u j , ˙ u j i + h ˙ u j , u j i (cid:17) − m k ˙ u j k . SECOND-ORDER LOHE HERMITIAN MODEL 5
Now, we derive Gronwall’s inequality for h u j , ˙ u j i + h ˙ u j , u j i : m ddt ( h u j , ˙ u j i + h ˙ u j , u j i ) = 2 m k ˙ u j k + h u j , m ¨ u j i + h m ¨ u j , u j i = − γ ( h u j , ˙ u j i + h ˙ u j , u j i ) . Gronwall’s lemma and initial conditions imply ddt k u j k = h u j ( t ) , ˙ u j ( t ) i + h ˙ u j ( t ) , u j ( t ) i = e − γm t ( h u inj , ˙ u inj i + h ˙ u inj , u inj i ) = 0 , ∀ t > . (cid:3) Lemma 2.2.
Suppose Ω j satisfies (2.5) Ω † = − Ω , Ω j ≡ Ω for all j = 1 , · · · , N , where † denotes the Hermitian conjugate, and let { z j } be a solution to (1.2) . Then, u j defined in (2.1) satisfies m ¨ u j + γ ˙ u j = κ (cid:0) k u j k u c − h u c , u j i u j (cid:1) + κ (cid:0) h u j , u c i − h u c , u j i (cid:1) u j − m k ˙ u j k k u j k u j , ( u j (0) , ˙ u j (0)) = ( u inj , ˙ u inj ) , h u inj , ˙ u inj i + h ˙ u inj , u inj i = 0 , where u c := N P Nk =1 u k .Proof. We substitute the relation (2.5) into (2.3) to get the desired estimate. (cid:3)
Remark 2.1.
For a homogeneous ensemble with the common natural frequency Ω , one canalso see that z j satisfies m (cid:16) ˙ v j − γ Ω v j (cid:17) + γv j = κ ( h z j , z j i z c − h z c , z j i z j ) + κ ( h z j , z c i − h z c , z j i ) z j − m k v j k k z j k z j ,v j = ˙ z j − γ Ω z j , z j (0) = z inj , v j (0) = v inj , h z inj , v inj i + h v inj , z inj i = 0 . Relation with other aggregation models.
In this subsection, we briefly discussrelations with other aggregation models with (1.2). For a zero inertia and unit frictionconstant case: m = 0 and γ = 1system (1.2) becomes the first-order LHS model [17, 18]:(2.6) ˙ z j = Ω j z j + κ ( h z j , z j i z c − h z c , z j i z j ) + κ ( h z j , z c i − h z c , z j i ) z j . Moreover, for the special case with z j = x j ∈ S d ⊂ R d +1 and κ = 0, system (2.6) alsoreduces to the Lohe sphere model:(2.7) ˙ x j = ˆΩ j x j + κ ( h x j , x j i x c − h x c , x j i x j ) , ˆΩ ⊤ = − ˆΩ ∈ R ( d +1) × ( d +1) . The emergent dynamics of (2.7) has been extensively studied in [4, 5, 6, 7, 11, 13, 14, 15, 21,22, 23, 24, 25, 29]. In the sequel, we mainly discuss emergent behavior of the complex swarmsphere model (2.6). For this, we introduce new dependent variables: for state configuration { z j } ,(2.8) h ij := h z i , z j i , R ij := Re( h ij ) , I ij := Im( h ij ) ∀ i, j = 1 , , · · · , N. Note that h ij = h z i , z j i = 1 ⇐⇒ R ij = 1 and I ij = 0 , ∀ i, j = 1 , · · · , N. HA, KANG, AND PARK
For a homogeneous ensemble with Ω j = Ω, we expect the formation of complete aggregationwhich means lim t →∞ h ij = 1 . Hence, it is natural to introduce a Lyapunov functional depending on the quantities: | − h ij | = q (1 − R ij ) + I ij . and we set J ij := q (1 − R ij ) + I ij , ∀ i, j ∈ { , , · · · , N } , J M := max i,j J ij and D (Ω) := max i,j k Ω i − Ω j k F . Now, we briefly summarize emergent behaviors of (2.6) without proofs.
Theorem 2.1. [16]
The following assertions hold.(1) (A homogeneous ensemble): Suppose system parameters and initial data satisfy κ > κ ≥ , D (Ω) = 0 , J M (0) < r − κ κ , and let { z j } be the solution of (1.2) with initial data { z inj } . Then, there exists apositive constant ˜Λ such that J M ( t ) ≤ J M (0) exp (cid:16) − ˜Λ t (cid:17) , t > . (2) (A heterogeneous ensemble): Suppose system parameters and initial data satisfy κ ≥ , D (Ω) > , and let { z j } be a solution of (1.2) with the initial data { z inj } . Then, one has apractical aggregation: lim κ →∞ lim sup t →∞ J M ( t ) = 0 . Before we close this section, we recall that how (2.6) can be further reduced to the theKuramoto model which is one of prototype examples for synchronization. We assume thatthe second coupling is absent and dimension in unit-dimensional: κ = 0 , d = 1 . In this case, we take the ansatz:(2.9) z j := (cid:20) cos θ j sin θ j (cid:21) , Ω j := (cid:20) − ν j ν j (cid:21) , j = 1 , · · · , N. SECOND-ORDER LOHE HERMITIAN MODEL 7
We substitute (2.9) into (2.6) to obtain˙ θ i (cid:20) − sin θ j cos θ j (cid:21) = (cid:20) − ν i ν i (cid:21) (cid:20) cos θ j sin θ j (cid:21) + κN N X k =1 (cid:18)(cid:20) cos θ k sin θ k (cid:21) − (cid:28)(cid:20) cos θ j sin θ j (cid:21) , (cid:20) cos θ k sin θ k (cid:21)(cid:29) (cid:20) cos θ j sin θ j (cid:21)(cid:19) = ν i (cid:20) − sin θ j cos θ j (cid:21) + κN N X k =1 (cid:20) cos θ k − sin θ k − (cos θ i cos θ k + sin θ i sin θ k ) cos θ j cos θ k + sin θ k − (cos θ i cos θ k + sin θ i sin θ k ) sin θ j (cid:21) = ν j (cid:20) − sin θ j cos θ j (cid:21) + κN N X k =1 (cid:20) cos θ k − cos( θ j − θ k ) cos θ j sin θ k − cos( θ j − θ k ) sin θ j (cid:21) = ν j (cid:20) − sin θ j cos θ j (cid:21) + κN N X k =1 (cid:20) − sin θ j cos θ j (cid:21) sin( θ k − θ j ) . We take an inner product the above relation with ( − sin θ j , cos θ j ) ⊤ , we obtain the Kuramotomodel: ˙ θ j = ν j + κN N X k =1 sin( θ k − θ j ) , j = 1 , · · · , N. In summary, the LHS model generalizes the Lohe sphere model and Kuramoto model thatwere extensively studied in literature.3.
Characterization and instability of two distinguished states
In this section, we discuss the characterization and instability of two distinguished statesfor system (1.2) with zero frequency matrix and unit Hermitian sphere HS d :Ω j ≡ , k z j k = 1 , j = 1 , · · · , N. In this case, system (1.2) takes a much simpler form:(3.1) m ¨ z j + γ ˙ z j = κ (cid:0) z c − h z c , z j i z j (cid:1) + κ (cid:0) h z j , z c i − h z c , z j i (cid:1) z j − m k ˙ z j k z j . This can also be rewritten as a first-order system by introducing an auxiliary variable w j = ˙ z j : ˙ z j = w j , ˙ w j = − γm w j + κ m (cid:0) z c − h z c , z j i z j (cid:1) + κ m (cid:0) h z j , z c i − h z c , z j i (cid:1) z j − k w j k z j . (3.2)3.1. Characterization of equilibria.
Note that the algebraic equilibrium system associ-ated with (3.2):(3.3) ( w j = 0 , − γw j + κ (cid:0) z c − h z c , z j i z j (cid:1) + κ (cid:0) h z j , z c i − h z c , z j i (cid:1) z j − m k w j k z j = 0 . Proposition 3.1.
Let { ( z ej , w ej ) } be an equilibrium solution of (3.2) if and only if ( z ej , w ej ) is a constant state satisfying w ej = 0 , z ec = h z ec , z ej i z ej , ∀ j = 1 , · · · , N, where z ec = N P Nj =1 z ej . HA, KANG, AND PARK
Proof. (= ⇒ part): Suppose { ( z ej , w ej ) } is an equilibrium state. Then, it satisfies(3.4) w ej = 0 , κ ( z ec − h z ec , z ej i z ej ) + κ ( h z ej , z ec i − h z ec , z ej i ) z ej . Now, we use the relation k z ej k = 1 to see0 = h z ej , (3.4) i = ( κ + κ ) (cid:16) h z ej , z ec i − h z ec , z ej i (cid:17) . Since κ > κ ≥
0, one has 0 = h z ej , z ec i − h z ec , z ej i . (3.5)Then, we substitute (3.5) into (3.4) and use κ > z ec = h z ec , z ej i z ej . ( ⇐ = part): Suppose that a constant state ( z ej , w ej ) satisfies relations:(3.6) z ec = h z ec , z ej i z ej ∀ t ≥ w ej = 0 , j = 1 , · · · , N. We use the relation (3.6) and k z ej k = 1 to find (cid:10) z ej , z ec (cid:11) − (cid:10) z ec , z ej (cid:11) = D z ej , h z ec , z ej i z ej E − D h z ec , z ej i z ej , z ej E = h z ec , z ej ih z ej , z ej i − h z ec , z ej ih z ej , z ej i = h z ec , z ej ih z ej , z ej i − h z ej , z ec ih z ej , z ej i = − (cid:16) (cid:10) z ej , z ec (cid:11) − (cid:10) z ec , z ej (cid:11) (cid:17) . This yields(3.7) (cid:10) z ej , z ec (cid:11) − (cid:10) z ec , z ej (cid:11) = 0 , ∀ t ≥ . Finally, the relations (3.6) and (3.7) satisfy the equilibrium system (3.3). (cid:3)
Next, we introduce an order parameter ρ which measures the degree of aggregation. Fora given configuration { ( z j , w j ) } , we set(3.8) ρ := (cid:13)(cid:13)(cid:13) N X j z j (cid:13)(cid:13)(cid:13) , ρ ∞ := lim t →∞ ρ ( t ) if it does exist . Then, ρ = 0 , ρ is eithercompletely aggregated state or a bi-polar state. Corollary 3.1.
For d = 0 , let ( z ej , w ej ) be an equilibrium solution with ρ > and | z ej | = 1 .Then, one has z ej z ec ∈ R . Proof.
Let { ( z ej , w ej ) } be an equilibrium state with ρ > | z ej | = 1. Then, by Proposition3.1, one has(3.9) z ec = h z ec , z ej i z ej , j = 1 , · · · , N. On the other hand, since ρ = | z ec | > z c and z j as polar forms:(3.10) z ej = e i θ j and z ec = ρe i φ , ρ > . SECOND-ORDER LOHE HERMITIAN MODEL 9
Now, we substitute (3.10) into (3.9) to get ρe i φ = ρe i(2 θ j − φ ) . This yields e φ = e θ j , j = 1 , · · · , N. Hence, one has either θ j = φ or θ j = φ + π, j = 1 , · · · , N. Thus, z ej z ec = 1 ρ e i( θ j − φ ) ∈ n ρ , − ρ o . (cid:3) Instability of two distinguished states.
In this subsection, we study linear insta-bilities of two distinguished state “ bi-polar state and incoherence state ( ρ = 0)”. ( ˙ z j = w j , ˙ w j = − γm w j + κ m (cid:0) z c − h z c , z j i z j (cid:1) + κ m (cid:0) h z j , z c i − h z c , z j i (cid:1) z j − k w j k z j . (3.11)In the sequel, we consider z j and w j as real vectors in R d +2 . In other words, let x αj , y αj , a αj , b αj ∈ R be given as follows: z αj = x αj + i y αj , w αj = a αj + i b αj , j = 1 , · · · , N, α = 1 , · · · , d + 1 , (3.12)where z αj and w αj are a -th component of z j and w j , respectively. We rewrite (3.11) using(3.12): ˙ x j = a j , ˙ y j = b j , ˙ a j = − γm a j + κ m (cid:2) x c − (cid:0) h x c , x j i + h y c , y j i (cid:1) x j + (cid:0) h x c , y j i − h y c , x j i (cid:1) y j (cid:3) − κ m (cid:0) h y c , x j i − h x c , y j i (cid:1) y j − (cid:0) k a j k + k b j k (cid:1) x j , ˙ b j = − γm b j + κ m (cid:2) y c − (cid:0) h x c , x j i + h y c , y j i (cid:1) y j − (cid:0) h x c , y j i − h y c , x j i (cid:1) x j (cid:3) + 2 κ m (cid:0) h y c , x j i − h x c , y j i (cid:1) x j − (cid:0) k a j k + k b j k (cid:1) y j , For stability analysis, we also define I := ( x , · · · , x N , y , · · · , y N , a , · · · , a N , b , · · · , b N ) = ( c , · · · , c N ) ∈ R d +1) N , and consider the following Jacobian matrix at equilibrium I e : M := ∂ ˙ I ∂ I (cid:12)(cid:12)(cid:12)(cid:12) I = I e = ( M ij ) ≤ i,j ≤ , M ij := ∂ ( ˙ c ( i − N +1 , · · · , ˙ c ( i − N + N ) ∂ ( c ( j − N +1 , · · · , c ( j − N + N ) (cid:12)(cid:12)(cid:12)(cid:12) I = I e . By direct calculations, one has M = M = M = M = M = M = M = M = O ( d +1) N , M = M = I ( d +1) N , M = M = − γm I ( d +1) N , where we used w j = 0 at equilibrium to calculate M and M . Hence, M has followingform: M := ∂ ˙ I ∂ I = O d +1) N I d +1) N M s − γm I d +1) N ! , M s = (cid:18) M M M M (cid:19) , We use the fact that (cid:12)(cid:12)(cid:12)(cid:12)
A BC D (cid:12)(cid:12)(cid:12)(cid:12) = det( A − BD − C ) det( D )to observe the relation between eigenvalues of M and M s :det (cid:0) M − λI d +1) N (cid:1) = (cid:12)(cid:12)(cid:12)(cid:12) − λI d +1) N I d +1) N M s − (cid:0) γm + λ (cid:1) I d +1) N (cid:12)(cid:12)(cid:12)(cid:12) = det (cid:16) λ (cid:16) γm + λ (cid:17) I d +1) N − M s (cid:17) . It follows from the above equation that if λ is the eigenvalue of M s , then λ satisfying λ = λ (cid:18) γm + λ (cid:19) is also an eigenvalue of M .Suppose that M s has an eigenvalue λ p , whose real part is positive. Then, one can seeRe λ p = Re λ (cid:18) γm + Re λ (cid:19) − (Im λ ) ⇐⇒ Re λ = − γ ± q γ + 4 m (cid:0) Re λ p + (Im λ ) (cid:1) m , which implies that M has an eigenvalue which has positive real part. More precisely, onehas − γ + q γ + 4 m (cid:0) Re λ p + (Im λ ) (cid:1) m > . Hence, we need further estimates on M s . We calculate components of M s one by one. M = ( A jk ) j,k , M = ( A jk ) j,k , M = ( A jk ) j,k , M = ( A jk ) j,k , ≤ j, k ≤ N,A jk := ∂ ˙ a j ∂x k = (cid:18) ∂ ˙ a αj ∂x βk (cid:19) α,β , A jk := ∂ ˙ a j ∂y k = (cid:18) ∂ ˙ a αj ∂y βk (cid:19) α,β ,A jk := ∂ ˙ b j ∂x k = (cid:18) ∂ ˙ b αj ∂x βk (cid:19) α,β , A jk := ∂ ˙ b j ∂y k = (cid:18) ∂ ˙ b αj ∂y βk (cid:19) α,β , ≤ α, β ≤ d + 1 . More precisely, we have ∂ ˙ a αj ∂x βk = κ m ∂∂x βk (cid:16) x αc − (cid:0) h x c , x j i + h y c , y j i (cid:1) x αj + (cid:0) h x c , y j i − h y c , x j i (cid:1) y αj (cid:17) − κ m ∂∂x βk (cid:16)(cid:0) h y c , x j i − h x c , y j i (cid:1) y αj (cid:17) − ∂∂x βk (cid:16)(cid:0) k a j k + k b j k (cid:1) x αj (cid:17) = κ m (cid:18) δ αβ N − (cid:18)(cid:28) e β N , x j (cid:29) + h x c , δ jk e β i (cid:19) x αj − ( h x c , x j i + h y c , y j i ) ∂x αj ∂x βk + (cid:18)(cid:28) e β N , y j (cid:29) − h y c , δ jk e β i (cid:19) y αj (cid:19) − κ m (cid:18)(cid:18) h y c , δ jk e β i − (cid:28) e β N , y j (cid:29)(cid:19) y αj (cid:19) SECOND-ORDER LOHE HERMITIAN MODEL 11 = κ m (cid:18) δ αβ N − (cid:18) x βj N + δ jk x βc (cid:19) x αj − ( h x c , x j i + h y c , y j i ) ∂x αj ∂x βk + (cid:18) y βj N − δ jk y βc (cid:19) y αj (cid:19) − κ m (cid:18)(cid:18) δ jk y βc − y βj N (cid:19) y αj (cid:19) . Similarly, one can see ∂ ˙ a αj ∂y βk = κ m ∂∂y βk (cid:16) x αc − (cid:0) h x c , x j i + h y c , y j i (cid:1) x αj + (cid:0) h x c , y j i − h y c , x j i (cid:1) y αj (cid:17) − κ m ∂∂y βk (cid:16)(cid:0) h y c , x j i − h x c , y j i (cid:1) y αj (cid:17) − ∂∂y βk (cid:16)(cid:0) k a j k + k b j k (cid:1) x αj (cid:17) = − κ m (cid:18)(cid:18)(cid:28) e β N , y j (cid:29) + h y c , δ jk e β i (cid:19) x αj − (cid:18) h x c , δ jk e β i − (cid:28) e β N , x j (cid:29)(cid:19) y αj − (cid:0) h x c , y j i − h y c , x j i (cid:1) ∂y αj ∂y βk (cid:19) − κ m (cid:18)(cid:18)(cid:28) e β N , x j (cid:29) − h x c , δ jk e β i (cid:19) y αj + (cid:0) h y c , x j i − h x c , y j i (cid:1) ∂y αj ∂y βk (cid:19) = − κ m (cid:18)(cid:18) y βj N + δ jk y βc (cid:19) x αj − (cid:18) δ jk x βc − x βj N (cid:19) y αj − (cid:0) h x c , y j i − h y c , x j i (cid:1) ∂y αj ∂y βk (cid:19) − κ m (cid:18)(cid:18) x βj N − δ jk x βc (cid:19) y αj + (cid:0) h y c , x j i − h x c , y j i (cid:1) ∂y αj ∂y βk (cid:19) . In the same way, we can observe ∂ ˙ b αj ∂x βk = − κ m (cid:18)(cid:18) x βj N + δ jk x βc (cid:19) y αj − (cid:18) δ jk y βc − y βj N (cid:19) x αj − (cid:0) h y c , x j i − h x c , y j i (cid:1) ∂x αj ∂x βk (cid:19) − κ m (cid:18)(cid:18) y βj N − δ jk y βc (cid:19) x αj + (cid:0) h x c , y j i − h y c , x j i (cid:1) ∂x αj ∂x βk (cid:19) ,∂ ˙ b αj ∂y βk = κ m (cid:18) δ αβ N − (cid:18) y βj N + δ jk y βc (cid:19) y αj − ( h x c , x j i + h y c , y j i ) ∂y αj ∂y βk + (cid:18) x βj N − δ jk x βc (cid:19) x αj (cid:19) − κ m (cid:18)(cid:18) δ jk x βc − x βj N (cid:19) x αj (cid:19) . In what follows, we study stability of two distinguished states. • (Instability of an incoherence state): Since the trace of a matrix is equal to the sum of itseigenvalues, we observeTr M s = Tr M + Tr M = N X j =1 Tr A jj + N X j =1 Tr A jj = N X j =1 d +1 X α =1 ∂ ˙ a αj ∂x αj + N X j =1 d +1 X α =1 ∂ ˙ b αj ∂y αj = κ m N X j =1 d +1 X α =1 (cid:18) N − (cid:18) x αj N + x αc (cid:19) x αj + h x c , x j i + (cid:18) y αj N − y αc (cid:19) y αj (cid:19) − κ m N X j =1 d +1 X α =1 (cid:18)(cid:18) y αc − y αj N (cid:19) y αj (cid:19) + κ m N X j =1 d +1 X α =1 (cid:18) N − (cid:18) y αj N + y αc (cid:19) y αj + h y c , y j i + (cid:18) x αj N − x αc (cid:19) x αj (cid:19) − κ m N X j =1 d +1 X α =1 (cid:18)(cid:18) x αc − x αj N (cid:19) x αj (cid:19) = κ m N X j =1 d +1 X α =1 (cid:18) N − (cid:0) x αj (cid:1) N + (cid:0) y αj (cid:1) N (cid:19) + 2 κ m N X j =1 d +1 X α =1 (cid:0) y αj (cid:1) N + κ m N X j =1 d +1 X α =1 (cid:18) N − (cid:0) y αj (cid:1) N + (cid:0) x αj (cid:1) N (cid:19) + 2 κ m N X j =1 d +1 X α =1 (cid:0) x αj (cid:1) N = 2( d + 1) κ m + 2 κ m > , where we used x c = y c = 0 . Hence, M s has at least one eigenvalue whose real part is positive and so does M . Therefore,we can conclude that the incoherence state is unstable. • (Instability of bi-polar state): Suppose that there exists a point z and integer n such that k z k = 1 , ≤ n < (cid:22) N (cid:23) , z i = − z, z j = z, ≤ i ≤ n, n + 1 ≤ j ≤ N. Without loss of generality, we can assume that z = z ∞ = (0 , · · · ,
1) using the rotationalsymmetry of the LHS with inertia. Then, we have z c = (cid:18) , · · · , , N − nN (cid:19) . Then, further calculation yields ∂ ˙ a αj ∂x βk = κ m (cid:18) δ αβ N − (cid:18) x βj N + δ jk x βc (cid:19) x αj − h x c , x j i ∂x αj ∂x βk (cid:19) and ∂ ˙ b αj ∂x βk = 0 . We observe ( n + 1)( d + 1)-th column of M s : M s ˜ e ( n +1)( d +1) = (cid:16)(cid:0) A ,n +1 (cid:1) ,d +1 , · · · , (cid:0) A N,n +1 (cid:1) d +1 ,d +1 , , · · · , (cid:17) ⊤ = 2 κ ( N − n ) mN ˜ e ( n +1)( d +1) , where { ˜ e α } d +1) Nα =1 is a standard basis on R d +1) N . Since n is smaller than ⌊ N/ ⌋ , we havepositive eigenvalue. Therefore, we can conclude that the bipolar state is unstable. SECOND-ORDER LOHE HERMITIAN MODEL 13 Presentation of main results
In this section, we briefly summarize frameworks for the emergent dynamics of the second-order extension of the first-order LHS model.4.1.
Complete aggregation.
In this subsection, we present an emergent dynamics of thehomogeneous ensemble with the same natural frequency matrix Ω j = Ω. For this, we set h ij := h z i , z j i , g ij := 1 − h ij , ≤ i, j ≤ N, G := 1 N N X i,j =1 | g ij | , R ( ˙ Z ) := max j k ˙ z j k , R ( Z ) := max j |h z c , z j i − h z j , z c i| ,M := max (cid:26) k w in k , · · · , k w inN k , κ + κ ) γ (cid:27) , ν := γ + p γ − mκ δ m . Then, it is easy to see that | h ij | ≤ , | g ij | ≤ , h ij = h ji and g ij = g ji , ≤ i, j ≤ N. Now, we set up two sufficient frameworks for complete synchronization. For a fixed δ ∈ (0 , • ( F A m , γ , κ and δ satisfy γ − mκ δ > , m, γ, κ > , κ ≥ . • ( F A G (0) < κ + 16 mM κ δ < (1 − δ ) N , ˙ G (0) + ν G (0) < ν (8 κ + 16 mM )4 κ δ . And also, our second framework is given as follows. • ( F B m , γ , κ and δ satisfy γ − mκ δ < , m, γ > , κ ≥ . • ( F B G (0) < mγ (8 κ + 16 mM ) < (1 − δ ) N , ˙ G (0) + γ m G (0) < γ (8 κ + 16 mM ) . Our first main result is concerned with the complete aggregation of a homogeneous en-semble.
Theorem 4.1.
Suppose that the sufficient frameworks ( F A - ( F A or ( F B - ( F B hold.Moreover, assume that initial data and natural frequency satisfy k z inj k = 1 , Ω j = 0 , j = 1 , · · · , N. Let { z j } be the global solution of (1.2) . Then, we have lim t →∞ G ( t ) = 0 , i.e., lim t →∞ h ij ( t ) = 1 , ∀ ≤ i, j ≤ N. Proof.
Although the detailed proof can be found in Section 5, we briefly sketch some ingre-dients for reader’s convenience. Since g ij + g ji = 2 − h z i , z j i − h z j , z i i = k z i − z j k , one has lim t →∞ G ( t ) = 0 = ⇒ lim t →∞ D ( Z ( t )) = 0 . Thus, it suffices to verify(4.1) lim t →∞ G ( t ) = 0 . By straightforward calculation to be performed in next section, we can derive second-orderdifferential inequality for G in (3.1): m ¨ G + γ ˙ G + 4 κ G ≤ κ √ N G + 2 κ R ( Z ) + 16 m R ( ˙ Z ) . Next, we use sufficient frameworks to show the uniform boundedness of G , which yields(4.2) m ¨ G + γ ˙ G + 4 δκ G ≤ κ R ( Z ) + 16 m R ( ˙ Z ) . Then, we use the relations in Proposition 5.1 and (5.15):(4.3) lim t →∞ R ( Z ( t )) = 0 , lim t →∞ R ( ˙ Z ( t )) = 0 , and the second-order Gronwall’s inequality (4.2) together with (4.3) to derive (4.1). (cid:3) Practical aggregation.
In this subsection, we first list a framework ( F C ) formulatedin terms of system parameters and initial data for a practical synchronization.First, we introduce several notation: R ( V ) := max j k v j k , Ω ∞ := max j k Ω j k F ,U ( m, Ω ∞ , κ , κ , γ ) := 4Ω ∞ + 8 κ + 16 mγ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) . Now, we set up a sufficient framework for practical aggregation. For a fixed δ ∈ (0 , • ( F C m , γ , κ and δ satisfy γ − mκ δ > , m, γ > , κ ≥ . • ( F C R ( V in ) < γ ( κ + κ ) , G (0) < κ δ U ( m, Ω ∞ , κ , κ , γ ) < (1 − δ ) N , ˙ G (0) + ν G (0) < ν κ δ U ( m, Ω ∞ , κ , κ , γ ) . Since practical aggregation is discussed with sufficiently large κ , there is no state about κ in the framework F C .Under the above framework, our second result deals with the emergence of practicalaggregation for a heterogeneous ensemble. SECOND-ORDER LOHE HERMITIAN MODEL 15
Theorem 4.2.
Suppose that the sufficient framework ( F C - ( F C holds, and let { z j } bethe solution of (1.2) with k z inj k = 1 , j = 1 , · · · , N . Then, we have a practical aggregation: lim κ →∞ lim sup t →∞ G ( t ) = 0 . Proof.
We briefly sketch a key idea. Detailed argument can be found in Section 6. In thecourse of proof, we will derive the following differential inequality: m ¨ G + γ ˙ G + 4 κ δ G ≤ ∞ + 8 κ + 16 mγ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) , ∀ t ∈ (0 , T ∗ ) . Then, this yields G ( t ) < Ω ∞ + 2 κ κ δ + 4 mγ κ δ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) , ∀ t > . For a sufficiently large κ ≥ max (cid:8) Ω ∞ , κ (cid:9) and a suitable ansatz for m : m = m κ η , η > , m > , one has lim sup t →∞ G ( t ) < Ω ∞ + 2 κ κ δ + 64 γ δ · m κ η . This implies the desired result. (cid:3) Emergence of complete aggregation
In this section, we provide estimates on the complete aggregation to the second-orderLHS model with inertia for a homogeneous ensemble:Ω j = Ω , k z j k = 1 , j = 1 , · · · , N. Furthermore, by Lemma 2.2, without loss of generality, we may assume Ω = 0. In thissituation, z j satisfies(5.1) ( m ¨ z j = − γ ˙ z j + κ (cid:0) z c − h z c , z j i z j (cid:1) + κ (cid:0) h z j , z c i − h z c , z j i (cid:1) z j − m k ˙ z j k z j ,z j (0) = z inj , ˙ z j (0) = ˙ z inj , h z inj , ˙ z inj i + h ˙ z inj , z inj i = 0 , j = 1 , · · · , N. Since the proof of Theorem 4.1 is very lengthy, we briefly delineate a proof strategy in severalsteps. Recall that our main purpose in this section is to derive a sufficient frameworks(setting) leading to the complete aggregation:(5.2) lim t →∞ h z i , z j i = 1 , i.e., lim t →∞ D ( Z ( t )) = 0 . • Step A: We introduce an energy functional E and via a time-decay estimate of it,we show thatlim t →∞ k ˙ z j ( t ) k = 0 , lim t →∞ |h z c , z j i − h z j , z c i| = 0 , j = 1 , · · · , N. See Proposition 5.1 for details. • Step B: We derive a second-order differential inequality for G :(5.3) m ¨ G + γ ˙ G + 4 κ G ≤ κ √ N G + f ( t ) , f ( t ) → t → ∞ . • Step C: We use a second-order Gronwall’s lemma (Lemma 5.5) and the result ofStep A to derive a zero convergence of G :lim t →∞ G ( t ) = 0 , which implies (5.2).In the following two subsections, we perform the above three steps one by one.5.1. Zero convergence of energy functional.
For a solution { z j } to (1.2), we define anenergy functional: E := 1 N N X j =1 (cid:18) m k ˙ z j k − m κ + 2 κ κ + κ ) |h z j , ˙ z j i| + κ k z c − z j k (cid:19) = 1 N N X j =1 m (cid:18) k ˙ z j k − κ + 2 κ κ + κ ) |h z j , ˙ z j i| (cid:19) + κ (cid:0) − k z c k (cid:1) . In the following lemma, we check the following two properties of E :1. E ≥ E = 0 ⇐⇒ k z c k = 1 and k ˙ z j k = 0 , j = 1 , · · · , N . Lemma 5.1.
Suppose the coupling strengths κ and κ satisfy κ > and κ ≥ , and let { z j } be a solution to (5.1) . Then the following assertions hold.(1) The energy functional E is nonnegative: E ( t ) ≥ , ∀ t ≥ . (2) The energy functional E is zero if and only if k z c k = 1 and k ˙ z j k = 0 , j = 1 , · · · , N. Proof. (i) The first assertion follows from0 < κ + 2 κ κ + κ ) < , |h z j , ˙ z j i| ≤ k z j k · k ˙ z j k , κ > , k z c k ≤ . (ii) Note that E = 0 ⇐⇒ k ˙ z j k − κ + 2 κ κ + κ ) |h z j , ˙ z j i| = 0 , ∀ i = 1 , · · · , N, − k z c k = 0 ⇐⇒ ˙ z j = 0 , ∀ j = 1 , · · · , N, and k z c k = 1 . (cid:3) Next, we study a nonincreasing property of E along system (1.2). Lemma 5.2.
Let { z j } be the solution of (5.1) . Then, we have (5.4) d E dt = − γN N X j =1 (cid:18) k ˙ z j k − κ + 2 κ κ + κ ) |h z j , ˙ z j i| (cid:19) ≤ , t > . SECOND-ORDER LOHE HERMITIAN MODEL 17
Proof.
First, we use the relation k z j k = 1 to see(5.5) h z j , ˙ z j i + h ˙ z j , z j i = 0 . We use (5.5) to obtain m ddt k ˙ z j k = h ˙ z j , m ¨ z j i + h m ¨ z j , ˙ z j i = − γ k ˙ z j k + (cid:2) κ ( h ˙ z j , z c i − h z c , z j ih ˙ z j , z j i ) + ( c.c ) (cid:3) + (cid:2) κ ( h z j , z c i − h z c , z j i ) h ˙ z j , z j i + ( c.c ) (cid:3) − (cid:2) m k ˙ z j k h ˙ z j , z j i + ( c.c ) (cid:3) = − γ k ˙ z j k + κ ( h z c , ˙ z j i + h ˙ z j , z c i ) − κ ( h z c , z j i − h z j , z c i ) h ˙ z j , z j i + 2 κ ( h z j , z c i − h z c , z j i ) h ˙ z j , z j i = − γ k ˙ z j k + κ ( h z c , ˙ z j i + h ˙ z j , z c i ) + ( κ + 2 κ ) ( h z c , z j i − h z j , z c i ) h z j , ˙ z j i , (5.6)where ( c.c ) means the complex conjugate of the previous term.We take summation (5.6) over j and divide by N to obtain ddt N N X j =1 m k ˙ z j k = − γm N N X j =1 m k ˙ z j k + κ ddt k z c k + κ + 2 κ N N X j =1 (cid:16) h z c , z j i − h z j , z c i (cid:17) h z j , ˙ z j i , or equivalently ddt N N X j =1 m k ˙ z j k + κ (cid:0) − k z c k (cid:1) = − γm N N X j =1 m k ˙ z j k + κ + 2 κ N N X j =1 ( h z c , z j i − h z j , z c i ) h z j , ˙ z j i . (5.7)On the other hand, one has m ddt |h z j , ˙ z j i| = m ddt ( h z j , ˙ z j ih ˙ z j , z j i )= m (cid:2)(cid:0) k ˙ z j k + h z j , ¨ z j i (cid:1) h ˙ z j , z j i + h z j , ˙ z j i (cid:0) k ˙ z j k + h ¨ z j , z j i (cid:1)(cid:3) . (5.8)Then, we use (5.7), (5.8) and the following relation: m (cid:0) h z j , ¨ z j i + k ˙ z j k (cid:1) = − γ h z j , ˙ z j i + ( κ + κ ) ( h z j , z c i − h z c , z j i )to obtain m ddt |h z j , ˙ z j i| = − γ |h z j , ˙ z j i| + 2 ( κ + κ ) ( h z c , z j i − h z j , z c i ) h z j , ˙ z j i , or equivalently(5.9) ( h z c , z j i − h z j , z c i ) h z j , ˙ z j i = m κ + κ ) ddt |h z j , ˙ z j i| + γκ + κ |h z j , ˙ z j i| . Finally, we combine (5.7) and (5.9) to get the desired result. (cid:3)
Remark 5.1.
Note that the estimate (5.4) can be rewritten as (5.10) d E dt = − γm E + 2 κ γm (1 − k z c k ) , ∀ t > . As a corollary of Lemma 5.1 and Lemma 5.2, we obtain the following result.
Corollary 5.1.
Suppose system parameters satisfy m > , γ > , κ > and κ ≥ , and { z j } be the solution of (5.1) . Then, we have the following estimates: ( i ) ∃ E ∞ := lim t →∞ E ( t ) . ( ii ) max ≤ j ≤ N k ˙ z j k ≤ max (cid:26) k w in k , · · · , k w inN k , γ ( κ + κ ) (cid:27) =: M . ( iii ) lim t →∞ k z c k = 1 = ⇒ lim t →∞ E ( t ) = 0 . Proof. (i) Since E ( t ) ≥ , ˙ E ( t ) ≤ , ∀ t > , E converges as t → ∞ .(ii) It follows from (5.6) that if κ > κ ≥
0, we have m ddt k ˙ z j k ≤ − γ k ˙ z j k + 4( κ + κ ) k ˙ z j k , or equivalently, ddt k ˙ z j k ≤ − γm k ˙ z j k + 2 m ( κ + κ ) . This implies k ˙ z j k ≤ (cid:18) k w inj k − γ ( κ + κ ) (cid:19) e − γm t + 2 γ ( κ + κ ) , ∀ t > . Hence, for all j , we have k ˙ z j k ≤ max (cid:26) k w in k , · · · , k w inN k , γ ( κ + κ ) (cid:27) . (iii) It follows from (5.10) that(5.11) E ( t ) = E (0) e − γtm + 2 γκ m Z t e − γm ( t − s ) (1 − k z c ( s ) k ) ds, ∀ t ≥ . Suppose that lim t →∞ k z c ( t ) k = 1 . Then, for any positive small ε , there exists a positive time T = T ( ε ) > − k z c ( t ) k < ε, ∀ t > T ( ε ) . Then, (5.11) becomes E ( t ) = E (0) e − γtm + 2 γκ m Z T ( ε )0 e − γm ( t − s ) (1 − k z c ( s ) k ) ds SECOND-ORDER LOHE HERMITIAN MODEL 19 + 2 γκ m Z tT ( ε ) e − γm ( t − s ) (1 − k z c ( s ) k ) ds ≤ E (0) e − γtm + 2 γκ m Z T ( ε )0 e − γm ( t − s ) ds + 2 γκ εm Z tT ( ε ) e − γm ( t − s ) ds = E (0) e − γtm + κ e − γm t (cid:16) e γm T ( ε ) − (cid:17) + κ ε (cid:16) − e − γm ( t − T ( ε )) (cid:17) , ∀ t > T ( ε ) . This implies that for t ≫ E ( t ) ≤ κ ε. Since ε is arbitrary, we have desired zero convergence of E . (cid:3) Remark 5.2.
Uniform boundedness of ˙ z j provides us the uniform boundedness of ¨ z j since m k ¨ z j k = (cid:13)(cid:13) − γ ˙ z j + κ (cid:0) z c − h z c , z j i z j (cid:1) + κ (cid:0) h z j , z c i − h z c , z j i (cid:1) z j − m k ˙ z j k z j (cid:13)(cid:13) ≤ γ k ˙ z j k + 2( κ + κ ) + m k ˙ z j k . Similarly, one can also get the uniform boundedness of d z j dt . In next proposition, we study zero convergence of ˙ z j . Proposition 5.1.
Suppose system parameters and initial data satisfy m > , γ > , κ > , κ ≥ , k z inj k = 1 for all j = 1 , · · · , N , and let { z j } be the solution of (5.1) . Then, we have lim t →∞ k ˙ z j ( t ) k = 0 , j = 1 , · · · , N. Proof.
We integrate (5.4) to find E ( t ) + 2 γN N X j =1 Z t (cid:18) k ˙ z j ( s ) k − κ + 2 κ κ + κ ) |h z j ( s ) , ˙ z j ( s ) i| (cid:19) ds = E in < ∞ . This yields Z ∞ (cid:18) k ˙ z j ( t ) k − κ + 2 κ κ + κ ) |h z j ( t ) , ˙ z j ( t ) i| (cid:19) dt < ∞ . By straightforward calculation, one can show that time-derivative of the integrand is uni-formly bounded because z j , ˙ z j and ¨ z j are bounded, i.e., the integrand is uniformly contin-uous. Hence, we can apply Barbalat’s lamma to get(5.12) lim t →∞ (cid:18) k ˙ z j ( t ) k − κ + 2 κ κ + κ ) |h z j ( t ) , ˙ z j ( t ) i| (cid:19) = 0 . On the other hand, note that0 ≤ κ κ + κ ) k ˙ z j ( t ) k ≤ k ˙ z j ( t ) k − κ + 2 κ κ + κ ) |h z j ( t ) , ˙ z j ( t ) i| . (5.13)Finally, we combine estimates (5.12) and (5.13) to derive the desired zero convergence of˙ z j . (cid:3) Next, we study all possible equilibria in terms of order parameter ρ ∞ . Corollary 5.2.
Under the same assumptions of Theorem 4.1, let { z j } be the solution of (1.2) and let ρ ∞ be an asymptotic order parameter defined by (3.8) . Then, the followingtrichotomy holds: ρ ∞ = 0 , ρ ∞ = 1 , or there exists integer n such that ρ ∞ = N − nN , ≤ n < (cid:22) N (cid:23) .Proof. For the case ρ ∞ = 0 or 1, we are done. Hence, we consider only the case: ρ ∞ ∈ (0 , . In Remark 5.2, we noticed that d z j dt is uniformly bounded. Hence, we can apply Barbalat’sLemma [2] and Proposition 5.1 to havelim t →∞ ¨ z j ( t ) = 0 . Then, (5.1) becomeslim t →∞ ( κ ( z c − h z c , z j i z j ) + κ ( h z j , z c i − h z c , z j i ) z j ) = 0 , j = 1 , · · · , N. (5.14)Since z j is bounded, we can take h z j , · i to getlim t →∞ ( h z j , z c i − h z c , z j i ) = 0 , j = 1 , · · · , N. (5.15)We combine (5.14), (5.15) with the fact that k z j k = 1 to obtainlim t →∞ (cid:16) z c − h z c , z j i z j (cid:17) = 0 , j = 1 , · · · , N. Again, since z c is bounded, we can take h z c , · i to getlim t →∞ (cid:0) ρ − h z c , z j i (cid:1) = 0 , j = 1 , · · · , N, or equivalently,(5.16) lim t →∞ h z c , z j i = δ j ρ ∞ , δ j ∈ { , − } , j = 1 , · · · , N. Then, we sum (5.16) over j and divide by N to obtain (cid:0) ρ ∞ (cid:1) = lim t →∞ k z c k = ρ ∞ N N X j =1 δ j . This implies ρ ∞ = 1 N N X j =1 δ j . Since ρ ∞ >
0, there must be an integer n such that n = |{ j : δ j = − , j = 1 , · · · , N }| , ≤ n < (cid:22) N (cid:23) , which guarantees our desired result. (cid:3) Remark 5.3.
In the results of the above lemma, we call first two cases by1. ρ = 0 ⇐⇒ incoherence state,2. ρ = 1 ⇐⇒ complete aggregation. SECOND-ORDER LOHE HERMITIAN MODEL 21
On the other hand, the remaining case is called bi-polar state, which is defined as follows: lim t →∞ dist (cid:0) S, { z j } (cid:1) = 0 , S := (cid:8) { p j } ∈ ( HS d ) N : p j = ± a, ∃ a ∈ HS d (cid:9) . Proof of Theorem 4.1.
In this subsection, we provide a proof of our first main resultby analyzing asymptotic behaviors of angle parameter G and diameter functional R i ( Z ).First, we recall two-point correlation functions h ij and g ij : h ij = h z i , z j i , g ij = 1 − h ij , ∀ i, j = 1 , · · · , N. Next, we derive an evolution equation for | g ij | . Lemma 5.3.
Let { z j } be a solution of (5.1) with k z inj k = 1 , j = 1 , · · · , N . Then, | g ij | satisfies m d dt | g ij | + γ ddt | g ij | + 2 (cid:2) κ + m ( k ˙ z i k + k ˙ z j k ) (cid:3) | g ij | = κ N N X k =1 ( g ik + g ki + g kj + g jk ) | g ij | + κ ( h z c , z j i − h z j , z c i )( h z i , z j i − h z j , z i i )+ κ ( h z i , z c i − h z c , z i i )( h z i , z j i − h z j , z i i ) + 2 m ˙ g ij ˙ g ji + m ( k ˙ z j k − h ˙ z i , ˙ z j i + k ˙ z i k ) g ji + m ( k ˙ z i k − h ˙ z j , ˙ z i i + k ˙ z j k ) g ij . (5.17) Proof.
Recall that z j satisfies(5.18) m ¨ z j = − γ ˙ z j + κ (cid:0) z c − h z c , z j i z j (cid:1) + κ (cid:0) h z j , z c i − h z c , z j i (cid:1) z j − m k ˙ z j k z j . We use g ij = g ji to find m d dt | g ij | = m d dt ( g ij g ji ) = m ¨ g ij g ji + 2 m ˙ g ij ˙ g ji + m ¨ g ji g ij . On the other hand, it follows from g ij = 1 − h z i , z j i that(5.19) m ¨ g ij = −h m ¨ z i , z j i − m h ˙ z i , ˙ z j i − h z i , m ¨ z j i . Then, we use (5.18) to get h z i , m ¨ z j i = − γ h z i , ˙ z j i + κ h z i , z c i − κ h z c , z j ih z i , z j i + κ ( h z j , z c i − h z c , z j i ) h z i , z j i − m k ˙ z j k h z i , z j i . (5.20)We combine (5.19) and (5.20) to obtain m ¨ g ij = γ ( h z i , ˙ z j i + h ˙ z i , z j i ) − κ ( h z i , z c i + h z c , z j i ) + κ ( h z c , z j i + h z i , z c i ) h z i , z j i− κ ( h z j , z c i − h z c , z j i ) h z i , z j i − κ ( h z c , z i i − h z i , z c i ) h z i , z j i + m k ˙ z j k h z i , z j i + m k ˙ z i k h z i , z j i − m h ˙ z i , ˙ z j i = − γ ˙ g ij − κ ( h z i , z c i + h z c , z j i ) g ij − κ ( h z j , z c i − h z c , z j i ) h z i , z j i − κ ( h z c , z i i − h z i , z c i ) h z i , z j i− m ( k ˙ z i k + k ˙ z j k ) g ij + m ( k ˙ z j k − h ˙ z i , ˙ z j i + k ˙ z i k ) . Note that ( h z i , z c i + h z c , z j i ) g ij = 1 N N X k =1 (2 − g ik − g kj ) g ij . This yields m ¨ g ij = − γ ˙ g ij − κ g ij + κ N N X k =1 ( g ik + g kj ) g ij − κ ( h z j , z c i − h z c , z j i ) h z i , z j i − κ ( h z c , z i i − h z i , z c i ) h z i , z j i− m ( k ˙ z i k + k ˙ z j k ) g ij + m ( k ˙ z j k − h ˙ z i , ˙ z j i + k ˙ z i k ) . (5.21)We multiply (5.21) by g ji to find m ¨ g ij g ji + γ ˙ g ij g ji + (cid:2) κ + m ( k ˙ z i k + k ˙ z j k ) (cid:3) | g ij | = κ N N X k =1 ( g ik + g kj ) | g ij | − κ ( h z j , z c i − h z c , z j i ) h z i , z j i g ji − κ ( h z c , z i i − h z i , z c i ) h z i , z j i g ji + m ( k ˙ z j k − h ˙ z i , ˙ z j i + k ˙ z i k ) g ji . (5.22)We sum up (5.22) over all i, j and its complex conjugate to obtain the desired estimate. (cid:3) Next, we quote some useful Lemmas on the second-order Gronwall type differential in-equality from [8] and [10] without proofs.
Lemma 5.4. [8]
Let y = y ( t ) be a nonnegative C − function satisfying the following differ-ential inequality: a ¨ y + b ˙ y + cy + d ≤ , t > , where a, b and c are positive constants. Then, we have the following assertions:(1) Suppose that b − ac > . Then, one has y ( t ) ≤ − dc + (cid:18) y (0) + dc (cid:19) e − ν t + a √ b − ac (cid:18) ˙ y (0) + ν y (0) + 2 db − √ b − ac (cid:19)(cid:0) e − ν t − e − ν t (cid:1) where ν and ν are given as follows: ν := b + √ b − ac a and ν := b − √ b − ac a . Moreover, if the following conditions hold: y (0) + dc < and y ′ (0) + ν y (0) + 2 db − √ b − ac < , (5.23) then, y ( t ) is uniformly bounded: y ( t ) < − dc . (2) Suppose that b − ac < . Then, one has y ( t ) ≤ − adb + e − b a t (cid:20) y (0) + 4 adb + (cid:18) b a y (0) + ˙ y (0) + 2 db (cid:19) t (cid:21) . SECOND-ORDER LOHE HERMITIAN MODEL 23
Moreover, if the following conditions hold: y (0) < − adb and b a y (0) + ˙ y (0) + 2 db < , (5.24) then, y ( t ) is uniformly bounded: y ( t ) < − adb . Lemma 5.5. [10]
Let y = y ( t ) be a nonnegative C -function satisfying the second-orderdifferential inequality: a ¨ y + b ˙ y + cy ≤ f, t > , where a, b, c and d are positive constants and f = f ( t ) is a nonnegative C -function whichconverges to zero as t → ∞ . Then, y vanishes asymptotically: lim t →∞ y ( t ) = 0 . Now, we are ready to provide a proof of our first main result on the complete aggregationof (5.1).
Proof of Theorem 4.1 : Suppose that the sufficient frameworks ( F A F A
2) or ( F B F B
2) hold. Moreover, assume that initial data and natural frequency satisfy k z inj k = 1 , Ω j = 0 , j = 1 , · · · , N. Let { z j } be the global solution of (1.2). Then, we claim:lim t →∞ G ( t ) = 0 . It follows from (5.17) that G satisfies m ¨ G + γ ˙ G + 4 κ G ≤ κ N N X i,j,k =1 ( | g ik | + | g jk | ) | g ij | + 2 κ N N X i =1 |h z i , z c i − h z c , z i i| + 2 mN N X i,j =1 | ˙ g ij | + 2 mN N X i,j =1 ( k ˙ z j k + 2 |h ˙ z i , ˙ z j i| + k ˙ z i k )=: I + I + I + I , where I := 2 κ N N X i,j,k =1 ( | g ik | + | g jk | ) | g ij | , I := 2 κ N N X i =1 |h z i , z c i − h z c , z i i| , I := 2 mN N X i,j =1 | ˙ g ij | , I := 2 mN N X i,j =1 ( k ˙ z j k + 2 |h ˙ z i , ˙ z j i| + k ˙ z i k ) . Below, we provide estimates for I i one by one. • Case A (Estimate of I ): We use the Cauchy-Swartz inequality to obtain2 κ N N X i,j,k =1 | g ik | · | g ij | ≤ κ √ N N N X k =1 vuut N X i =1 | g ik | ! · N N X j =1 vuut N X i =1 | g ij | ! ≤ κ √ N N vuut N N X k =1 N X i =1 | g ik | ! · N N X j =1 N X i =1 | g ij | ! ≤ κ √ N G . This implies(5.25) I ≤ κ √ N G . • Case B (Estimate of I ): By (5.25), we have(5.26) I ≤ κ R ( Z ) . • Case C (Estimate of I ): We use | ˙ g ij | = |h ˙ z i , z j i + h z i , ˙ z j i| ≤ k ˙ z i k + k ˙ z j k ) ≤ R ( ˙ Z )to find(5.27) I ≤ m R ( ˙ Z ) . • Case D (Estimate of I ): Similarly, we use k ˙ z j k + 2 |h ˙ z i , ˙ z j i| + k ˙ z i k ≤ k ˙ z i k + k ˙ z j k ) ≤ R ( ˙ Z ) , to find(5.28) I ≤ m R ( ˙ Z ) . We combine all the estimates (5.25), (5.26), (5.27), (5.28) of I k ’s to obtain(5.29) m ¨ G + γ ˙ G + 4 κ G ≤ κ √ N G + 2 κ R ( Z ) + 16 m R ( ˙ Z ) . Now, we derive a uniform bound for G using (5.29):sup ≤ t< ∞ G ( t ) < (1 − δ ) N .
We define a temporal set T for δ ∈ (0 , T := { T ∈ (0 , ∞ ) : G ( t ) < (1 − δ ) /N, ∀ t ∈ (0 , T ) } . By initial conditions, the set T is nonempty. Hence we can define T ∗ := sup T . Now we claim: T ∗ = ∞ . Suppose not, i.e., T ∗ < ∞ . SECOND-ORDER LOHE HERMITIAN MODEL 25
Then, we have lim t → T ∗ − G ( t ) = (1 − δ ) N . (5.30)On the other hand, it follows from (5.29) that, for t ∈ (0 , T ∗ ), we have m ¨ G + γ ˙ G + 4 κ δ G ≤ κ R ( Z ) + 16 m R ( ˙ Z ) . We use Corollary 5.1 to obtain2 κ R ( Z ) + 16 m R ( ˙ Z ) ≤ κ + 16 mM . Hence, one has m ¨ G + γ ˙ G + 4 κ δ G ≤ κ + 16 mM , t ∈ (0 , T ∗ ) . Note that ( F A
1) and ( F B
1) are the first and second case of Lemma 5.4, respectively. More-over, ( F A
2) and ( F B
2) satisfy the condition (5.23) and (5.24) of Lemma 5.4, respectively.So, we apply Lemma 5.4 to obtain( F A ) = ⇒ G ( t ) < κ + 16 mM κ δ < (1 − δ ) N , t ∈ (0 , T ∗ ) , ( F B ) = ⇒ G ( t ) < mγ (8 κ + 16 mM ) < (1 − δ ) N , t ∈ (0 , T ∗ ) , which contradicts to (5.30). Therefore, we have T ∗ = ∞ and so that m ¨ G + γ ˙ G + 4 κ δ G ≤ κ R ( Z ) + 16 m R ( ˙ Z ) , ∀ t > . We use Theorem 4.1 and (5.15) to seelim t →∞ (cid:2) κ R ( Z ) + 16 m R ( ˙ Z ) (cid:3) = 0 . Then, we can apply Lemma 5.5 to conclude that complete synchronization occurs. (cid:3) Emergence of practical aggregation
In this section, we study the emergent dynamics of (1.2) with distinct set of naturalfrequency matrices Ω j ’s. Unlike to the homogeneous ensemble in previous section, wecannot expect the emergence of complete aggregation in which all the states collapse to thesame state. Instead, we study a weaker concept of aggregation estimate, namely practicalaggregation introduced in Definition 1.1. The key ingredient as in homogeneous ensembleis to derive a suitable second-order differential inequality for G . In fact, similar to (5.3), wederive m ¨ G + γ ˙ G + 4 κ δ G ≤ ∞ + 8 κ + 16 mγ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) , ∀ t > . Then, via the second-order Gronwall’s lemma (Lemma 5.4) and a suitable ansatz for m = m κ η , one can show G ( t ) . max n κ , κ η o , for t ≫ . This clearly implies the desired practical aggregation estimate.
Derivation of Gronwall’s inequality for G . Recall that the LHS model on the unithermitian sphere k z j k = 1:(6.1) m ¨ z j = mγ Ω j ˙ z j + mγ Ω j v j − γ ˙ z j + Ω j z j + κ ( z c − h z c , z j i z j )+ κ ( h z j , z c i − h z c , z j i ) z j − m k v j k z j , ( z j , ˙ z j ) (cid:12)(cid:12)(cid:12) t =0+ = ( z inj , w inj ) , h z inj , w inj i + h w inj , z inj i − γ Ω j h z inj , z inj i = 0 , where v j = ˙ z j − γ Ω j z j .Parallel to Lemma 5.3, in the following lemma, we derive a dynamical system of g ij . Lemma 6.1.
Let { z j } be a global solution of (6.1) . Then, | g ij | satisfies m d dt | g ij | + γ ddt | g ij | + 2 (cid:2) κ + m ( k v i k + k v j k ) (cid:3) | g ij | = − mγ (cid:0) h z i , Ω j v j i + h Ω i v i , z j i + h z i , Ω j ˙ z j i + h Ω i ˙ z i , z j i (cid:1) g ji − ( h z i , Ω j z j i + h Ω i z i , z j i ) g ji − mγ (cid:0) h z j , Ω i v i i + h Ω j v j , z i i + h z j , Ω i ˙ z i i + h Ω j ˙ z j , z i i (cid:1) g ij − ( h z j , Ω i z i i + h Ω j z j , z i i ) g ij + κ N N X k =1 ( g ik + g ki + g kj + g jk ) | g ij | + κ ( h z c , z j i − h z j , z c i )( h z i , z j i − h z j , z i i )+ κ ( h z i , z c i − h z c , z i i )( h z i , z j i − h z j , z i i ) + 2 m ˙ g ij ˙ g ji − m ( h ˙ z i , ˙ z j i g ji + h ˙ z j , ˙ z i i g ij ) + m ( k v i k + k v j k )( g ji + g ij ) . (6.2) Proof.
We use (6.1) to obtain h z i , m ¨ z j i = mγ h z i , Ω j v j i + mγ h z i , Ω j ˙ z j i − γ h z i , ˙ z j i + h z i , Ω j z j i + κ h z i , z c i − κ h z c , z j ih z i , z j i + κ ( h z j , z c i − h z c , z j i ) h z i , z j i − m k v j k h z i , z j i . Then, we have m ¨ g ij = −h z i , m ¨ z j i − m h ˙ z i , ˙ z j i − h m ¨ z i , z j i = − mγ (cid:0) h z i , Ω j v j i + h Ω i v i , z j i + h z i , Ω j ˙ z j i + h Ω i ˙ z i , z j i (cid:1) − h z i , Ω j z j i − h Ω i z i , z j i − γ ˙ g ij − κ h z i , z c i g ij − κ h z c , z j i g ij − κ ( h z j , z c i − h z c , z j i ) h z i , z j i − κ ( h z c , z i i − h z i , z c i ) h z i , z j i− m h ˙ z i , ˙ z j i + m ( k v i k + k v j k ) − m ( k v i k + k v j k ) g ij . (6.3)We substitute h z i , z c i = 1 N N X k =1 (1 − g ik ) = 1 − N N X k =1 h ik , SECOND-ORDER LOHE HERMITIAN MODEL 27 into (6.3) and multiply both sides of (6.3) by h ji to get m ¨ g ij g ji + γ ˙ g ij g ji + (cid:2) κ + m ( k v i k + k v j k ) (cid:3) | g ij | = − mγ (cid:0) h z i , Ω j v j i + h Ω i v i , z j i + h z i , Ω j ˙ z j i + h Ω i ˙ z i , z j i (cid:1) g ji + κ N N X k =1 ( g ik + g kj ) | g ij | − ( h z i , Ω j z j i + h Ω i z i , z j i ) g ji − κ ( h z j , z c i − h z c , z j i ) h z i , z j i g ji − κ ( h z c , z i i − h z i , z c i ) h z i , z j i g ji − m h ˙ z i , ˙ z j i g ji + m ( k v i k + k v j k ) g ji . (6.4)We sum (6.4) and its complex conjugate to obtain the desired result. (cid:3) As in Corollary 5.1, we observe the uniform bound of k v j k . Since h v j , m ˙ v j i = mγ h v j , Ω j v j i − γ k v j k + κ h v j , z c i − κ h z c , z j ih v j , z j i + κ ( h z j , z c i − h z c , z j i ) h v j , z j i − m k v j k h v j , z j i , we have m ddt k v j k ≤ − γ k v j k + 4( κ + κ ) k v j k , (6.5)where we use h z j , v j i + h v j , z j i = 0. (6.5) implies k v j ( t ) k ≤ (cid:18) k v inj k − κ + κ ) γ (cid:19) e − γm t + 2 γ ( κ + κ ) . Hence, we can obtain the uniform bound of k v j k : k v j k ≤ max (cid:26) k v in k , · · · , k v inN k γ ( κ + κ ) (cid:27) , j = 1 , · · · , N. (6.6)Also, we have the uniform bound of k ˙ z j k : k ˙ z j k ≤ k v j k + k Ω j k F γ ≤ max (cid:26) k v in k , · · · , k v inN k , γ ( κ + κ ) (cid:27) + Ω γ ∞ , j = 1 , · · · , N, (6.7)where Ω ∞ := max j k Ω j k F .6.2. Proof of Theorem 4.2.
In this subsection, we provide a proof of our second mainresult on the emergence of practical aggregation. First, we begin with the derivation of auniform bound for G . • Step A (Derivation of uniform bound for G ): Suppose the framework ( F C F C
2) hold,and let Z = ( z , · · · , z N ) be a solution of (6.1). Then, one hassup ≤ t< ∞ G ( t ) < (1 − δ ) N .
It follows from (6.2) that m ¨ G + γ ˙ G + 4 κ G ≤ κ √ N G + 4 m Ω ∞ γ h R ( V ) + q R ( ˙ Z ) i + 12 m R ( ˙ z ) + 4 m R ( V ) + 4Ω ∞ + 8 κ . (6.8) We define a temporal set T for δ ∈ (0 , T := { T ∈ (0 , ∞ ) : G ( t ) < (1 − δ ) /N, ∀ t ∈ (0 , T ) } . By initial conditions, the set T is nonempty. Hence we can define T ∗ := sup T . Now we claim: T ∗ = ∞ . Suppose not, i.e., T ∗ < ∞ . Then, we have lim t → T ∗ − G ( t ) = (1 − δ ) N . (6.9)On the other hand, it follows from (6.8) that, for t ∈ (0 , T ∗ ), we have m ¨ G + γ ˙ G + 4 κ δ G ≤ m Ω ∞ γ h R ( V ) + p R ( ˙ z ) i + 12 m R ( ˙ z ) + 4 m R ( V ) + 4Ω ∞ + 8 κ . (6.10)We use (6.6), (6.7) and ( F C to obtain4 m Ω ∞ γ R ( V ) + 4 m R ( V ) ≤ m Ω ∞ ( κ + κ ) γ + 16 m ( κ + κ ) γ = 8 mγ ( κ + κ ) (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) , and Ω ∞ γ p R ( ˙ z ) + 12 m R ( ˙ z ) ≤ m Ω ∞ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) γ + 12 m (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) γ = 4 mγ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3)(cid:2) ∞ + 6( κ + κ ) (cid:3) . Hence, one has(6.11) 4 m Ω ∞ γ h R ( V ) + p R ( ˙ z ) i + 12 m R ( ˙ z ) + 4 m R ( V ) ≤ mγ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) . Now, we combine (6.10) and (6.11) to get m ¨ G + γ ˙ G + 4 κ δ G ≤ ∞ + 8 κ + 16 mγ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) , t ∈ (0 , T ∗ ) . Note that ( F C
1) is the first case of Lemma 5.4 and ( F C and ( F C satisfy the condition(5.23) of Lemma 5.4. So, we apply Lemma 5.4 to obtain(6.12) G ( t ) < κ δ ∞ + 8 κ + 16 mγ (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) ! < (1 − δ ) N , t ∈ (0 , T ∗ ) , which contradicts to (6.9). Therefore, we have T ∗ = ∞ .Now, we are ready to provide a proof of Theorem 4.2. SECOND-ORDER LOHE HERMITIAN MODEL 29 • Step B (Derivation of practical aggregation estimate): It follows from (6.12) that(6.13) G ( t ) < Ω ∞ + 2 κ κ δ + 4 m (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) γ κ δ , ∀ t > . For the case κ ≥ max (cid:8) Ω ∞ , κ (cid:9) , we have G ( t ) < Ω ∞ + 2 κ κ δ + 4 mκ h Ω ∞ κ + 2 + κ κ i γ δ ≤ Ω ∞ + 2 κ κ δ + 64 γ δ mκ , ∀ t > . Hence as(6.14) mκ → κ → ∞ , one has a practical synchronization. To satisfy the constraints (6.14), we assume that thereexist m > η > m = m κ η . Then, it follows from (6.13) and (6.15) that, for κ ≥ max (cid:8) Ω ∞ , κ (cid:9) , G ( t ) < Ω ∞ + 2 κ κ δ + 64 γ δ · m κ η , ∀ t > , which implies the desired result. (cid:3) Remark 6.1.
There could be a question about the possibility for second inequality of ( F C .We verify it holds for sufficiently large κ . We substitute (6.15) into the second inequalityof ( F C to obtain Ω ∞ + 2 κ κ δ + 4 m (cid:2) Ω ∞ + 2( κ + κ ) (cid:3) δγ κ η < (1 − δ ) N . (6.16)
One can observe that left hand side of (6.16) converges to zeros as κ goes to infinity. Conclusion
In this paper, we have studied emergent behaviors of the second-order LHS model whichcan be realized as a second-order extension of the first-order LHS model introduced inauthors’ earlier work [16, 17, 18]. For a homogeneous ensemble with the same naturalfrequency matrix Ω j = Ω, we provided emergence of complete aggregation in the sensethat all states aggregate to the same state asymptotically. For this, under a suitable setof system parameters and initial data with a finite energy, we show that the two-pointcorrelation functions between states tend to zero asymptotically, which denote the formationof complete aggregation. By linear stability analysis, we also showed that the incoherentstate and bi-polar state are linearly unstable. In contrast, for a heterogeneous ensemble,we provided a sufficient framework leading to practical aggregation which means that statediameter can be made small by increasing the principle coupling strength. Of course, thereare several issues to be discussed in a future work. For example, we only considered positivecoupling strengths (attractive couplings) in this work. However, the coupling strengths canbe negative, i.e., repulsive couplings or they can be time-dependent or state-dependentwhich make asymptotic dynamics more richer. We leave these interesting issues in a futurework. References [1] Acebron, J. A., Bonilla, L. L., P´erez Vicente, C. J. P., Ritort, F. and Spigler, R.:
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J. Theor.Biol. (1967), 15-42.(Seung-Yeal Ha) Department of Mathematical SciencesSeoul National University, Seoul 08826 andKorea Institute for Advanced Study, Hoegiro 85, 02455, Seoul, Republic of Korea
E-mail address : [email protected] (Myeongju Kang) Department of Mathematical SciencesSeoul National University, Seoul 08826, Republic of Korea
E-mail address : [email protected] (Hansol Park) Department of Mathematical SciencesSeoul National University, Seoul 08826, Republic of Korea
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