Emergent dynamics of the Lohe Hermitian sphere model with frustration
EEMERGENT DYNAMICS OF THE LOHE HERMITIAN SPHEREMODEL WITH FRUSTRATION
SEUNG-YEAL HA, MYEONGJU KANG, AND HANSOL PARK
Abstract.
We study emergent dynamics of the Lohe hermitian sphere(LHS) model whichcan be derived from the Lohe tensor model [24] as a complex counterpart of the Lohesphere(LS) model. The Lohe hermitian sphere model describes aggregate dynamics ofpoint particles on the hermitian sphere HS d lying in C d +1 , and the coupling terms in theLHS model consist of two coupling terms. For identical ensemble with the same free flowdynamics, we provide a sufficient framework leading to the complete aggregation in whichall point particles form a giant one-point cluster asymptotically. In contrast, for non-identical ensemble, we also provide a sufficient framework for the practical aggregation.Our sufficient framework is formulated in terms of coupling strengths and initial data. Wealso provide several numerical examples and compare them with our analytical results. Introduction
Emergent behaviors of complex systems are ubiquitous in nature, for example, flockingof birds, swarming of fish, flashing of fireflies and herding of sheep, etc [1, 2, 3, 4, 5, 6, 7, 22,34, 35, 36, 38, 39, 40, 41]. Several jargons such as aggregation, flocking, synchronization andherding are often used to describe such collective behaviors. Before we go into our topics, webriefly review several basic terminologies and concepts to be used in this paper. C denotesthe complex field and let C d be the cartesian product of d copies of C for a positive integer d . Thus, the points of C d are ordered d -tuples z = ( z , · · · , z d ) where z i ∈ C . Algebraically, C d is a d -dimensional vector space over C , and topologically it is the euclidean space R d of real dimension 2 d , so we may call it complex euclidean space. For z = ( z , · · · , z d ) and w = ( w , · · · , w d ) in C d , we define the inner product (cid:104)· , ·(cid:105) and the associated norm (cid:107) · (cid:107) : (cid:104) z, w (cid:105) := d (cid:88) i =1 ¯ z i w i , (cid:107) z (cid:107) := (cid:104) z, z (cid:105) , where we used physicist’s notation by conjugating the first argument in (cid:104)· , ·(cid:105) . Let v ∈ C d +1 and W ∈ C ( d +1) × ( d +1) be a complex vector and complex matrix, respectively. Then, wedenote i th -component and ( i, j )-component of the real vector v and real matrix A by [ v ] i Date : August 10, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Complete aggregation, collective behavior, emergence, Lohe hermitian spheremodel, practical aggregation.
Acknowledgment.
The work of S.-Y. Ha was supported by National Research Foundation of Korea(NRF-2020R1A2C3A01003881), the work of M. Kang was supported by the National Research Foundationof Korea(NRF) grant funded by the Korea government(MSIP)(2016K2A9A2A13003815), and the work ofH. Park was supported by Basic Science Research Program through the National Research Foundation ofKorea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585). a r X i v : . [ m a t h - ph ] A ug HA, KANG, AND PARK and [ A ] ij , respectively. Moreover, W † ∈ C d × d and (cid:107) W (cid:107) F are the hermitian conjugate andnorm of W : [ W † ] ij = [ W ] ji , ≤ i, j ≤ d + 1 , (cid:107) W (cid:107) F := Tr( W † W ) . In this paper, we are interested in an aggregate phenomenon of a particle ensemble on theunit (hermitian) sphere HS d in C d +1 under the effect of frustration: HS d := { z ∈ C d +1 : (cid:107) z (cid:107) = 1 } . Here we used the adjective “ hermitian ” to distinguish the unit sphere in C d +1 and the unitsphere in R d +1 .For phase-coupled limit-cycle oscillator models such as the Kuramoto model and theWinfree model, (interaction) frustration often appear as a form of phase shift, and it gener-ates diverse asymptotic patterns through the competitions between synchronizing enforcingterms and periodic enforcing terms. This is why the study of frustrated systems is so inter-esting from the viewpoint of nonlinear dynamics. For details, we refer to [20, 21]. Recently,aggregation modelings for a particle ensemble on the unit sphere S d in R d +1 has been ex-tensively studied in literature [8, 9, 10, 11, 12, 25, 29, 30, 31, 37, 32, 43] in the absence offrustration.To put our discussion in a proper setting, we begin with “ the Lohe sphere(LS) modelwith frustration ” introduced in [17] . Let x j = x j ( t ) ∈ S d be the position of the j -th Loheparticle. Then, the LS model in the presence of frustration reads as follows.(1.1) ˙ x j = Ω j x j + κN N (cid:88) k =1 (cid:16) V x k − (cid:104) x j , V x k (cid:105) x j (cid:17) , j = 1 , · · · , N, where Ω j ∈ R ( d +1) × ( d +1) is the natural frequency matrix of the j -th particle which is skew-symmetric (Ω t = − Ω), and V ∈ R ( d +1) × ( d +1) is the frustration matrix consisting of the sumof the identity matrix and skew-symmetric matrix W :(1.2) V = I d +1 + W and W t = − W. Then, system (1.1) - (1.2) can be rewritten as follows:(1.3) ˙ x j = Ω j x j + κN N (cid:88) k =1 (cid:16) x k − (cid:104) x j , x k (cid:105) x j (cid:17) + κN N (cid:88) k =1 (cid:16) W x k − (cid:104) x j , W x k (cid:105) x j (cid:17) . For W = 0, system (1.3) reduces to the Lohe sphere model on the complete graph, and itsemergent dynamics has been studied in [8, 9, 10, 11, 12, 31, 32, 43].In this paper, we are interested in the following two questions: • (Q1): What is the complex analogue of (1.1)? • (Q2): Can we rigorously verify emergent dynamics of the proposed complex coun-terpart?In the absence of frustration, i.e., V ≡ I d +1 , the complex analogue of the Lohe spheremodel has been proposed in [17] and its emergent dynamics was also studied for identicalensemble. In what follows, we briefly summarize our main results on (Q1) - (Q2). HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 3
First, we present the complex analog of system (1.1). Let z j = z j ( t ) be the position ofthe j -th particle on the unit sphere in HS d . Then, the proposed Lohe hermitian spheremodel with frustration reads as follows:(1.4) ˙ z j = Ω j z j + κ N N (cid:88) k =1 (cid:16) (cid:104) z j , z j (cid:105) V z k − (cid:104) V z k , z j (cid:105) z j (cid:17) + κ N N (cid:88) k =1 (cid:16) (cid:104) z j , V z k (cid:105) − (cid:104) V z k , z j (cid:105) (cid:17) z j . Here κ and κ are nonnegative constants, and the frustration matrices V and V take thesame form as in (1.2):(1.5) V = I d +1 + W , V = I d +1 + W , where Ω j , W and W are skew-hermitian matrices (see Section 2.1 for details):Ω † j = − Ω j , j = 1 , · · · , N, W † = − W , W † = − W . Note that for real vector case, the term (cid:104) z j , V z k (cid:105) − (cid:104) V z k , z j (cid:105) vanishes, and system (1.4)reduces to system (1.1). Hence, our proposed system (1.4) can be called a complex coun-terpart of (1.1). Moreover, it can be rewritten as a mean-field form:(1.6) ˙ z j = Ω j z j + κ (cid:16) (cid:104) z j , z j (cid:105) V z c − (cid:104) z c , z j (cid:105) z j (cid:17) + κ (cid:16) (cid:104) z j , V z c (cid:105) − (cid:104) V z c , z j (cid:105) (cid:17) z j , where z c = N (cid:80) Ni =1 z i .Second, we return to system (1.1), and study emergent dynamics of the Lohe spheremodel for non-identical ensemble in mean-field form:(1.7) ˙ x j = Ω j x j + κ (cid:16) (cid:104) x j , x j (cid:105) V x c − (cid:104) V x c , x j (cid:105) x j (cid:17) , x c := 1 N N (cid:88) j =1 x j . For the identical ensemble with Ω j = Ω, emergent dynamics for (1.7) has been alreadystudied in [17] in which exponential aggregation was achieved using a position diameter asa suitable Lyapunov functional. Hence, the previous approach in aforementioned literatureis global. In Section 3, we revisit complete aggregation problem using a local approach. Forthis, we introduce an inter-particle angle θ ij as follows. θ ij := cos − ( (cid:104) x i , x j (cid:105) ) , ≤ i, j ≤ N. For an identical ensemble, if the initial data Θ in satisfies θ inij < cot − (cid:18) (cid:107) W (cid:107) F √ (cid:19) , ≤ i, j ≤ N. then there exists a positive constant Λ ij = Λ ij ( N, κ, W, Θ in ) such that θ ij ( t ) ≤ θ inij exp ( − Λ ij t ) , t ≥ , which improves the earlier result in [17] (see Proposition 3.1). In contrast, for non-identicalensemble, if the initial data { x i } Ni =1 satisfymax i,j (cid:16) sin θ inij (cid:17) <
12 + √ (cid:107) W (cid:107) F and max i,j θ inij < π , practical aggregation emerges asymptotically (see Theorem 3.1):sup t ≥ (cid:16) max i,j θ ij (cid:17) ≤ π κ →∞ lim sup t →∞ max i,j (cid:16) sin θ ij ( t ) (cid:17) = 0 . Note that in [17], emergent dynamics for non-identical ensemble has not been studied.
HA, KANG, AND PARK
Third, we consider system (1.6) with κ :˙ z j = Ω j z j + κ (cid:16) (cid:104) z j , z j (cid:105) V z c − (cid:104) z c , z j (cid:105) z j (cid:17) . Next, we introduce real and imaginary parts of the two-point correlation function (cid:104) z i , z j (cid:105) : R ij := Re( (cid:104) z i , z j (cid:105) ) , I ij := Im( (cid:104) z i , z j (cid:105) ) J ij := (cid:113) (1 − R ij ) + I ij , for i, j ∈ { , , · · · , N } . For identical ensemble, if the coupling strength and initial datasatisfy κ > i,j J inij < √ (cid:113) √ (cid:107) W (cid:107) F + 8 + √ (cid:107) W (cid:107) F , then J ij decays to zero exponentially fast, which illustrates the emergence of completeaggregation (see Theorem 4.1). In contrast, for non-identical ensemble, if initial data { z inj } satisfy max i,j J inij < √ (cid:113) √ (cid:107) W (cid:107) F + 8 + √ (cid:107) W (cid:107) F , then practical aggregation emerges, i.e.,lim κ →∞ lim sup t →∞ max i,j J ij ( t ) = 0 , (See Theorem 4.2).Lastly, we deal with the full dynamics (1.4) with κ > κ >
0. For identicalensemble, if coupling strengths, frustration matrix W and initial data satisfy κ > κ ≥ , W ≡ , max i,j J inij < √ (cid:16) − κ κ (cid:17)(cid:114) √ (cid:107) W (cid:107) F + 8 (cid:16) − κ κ (cid:17) + √ (cid:107) W (cid:107) F , then, the complete aggregation emerges exponentially fast (see Theorem 5.1). In contrast,for non-identical ensemble, if coupling strength κ is fixed and initial data satisfymax i,j J inij < √ (cid:113) √ (cid:107) W (cid:107) F + 8 + √ (cid:107) W (cid:107) F , then practical aggregation emerges (see Theorem 5.2):lim κ →∞ lim sup t →∞ max i,j J ij ( t ) = 0 . The rest of this paper is organized as follows. In Section 2, we briefly review basicproperties of the Lohe sphere and Lohe hermitian sphere models with frustrations, and recallprevious results on the emergent dynamics of the aforementioned models with frustration.In Section 3, we study emergent dynamics of the Lohe sphere model. In particular, ourpractical aggregation estimate improves earlier results. In Section 4, we present emergentdynamics of the Lohe hermitian sphere model with κ = 0. This is exactly complex analogueof the Lohe sphere model. The complex nature of ambient space will appear in conditions forcomplete and practical aggregation estimates. In Section 5, we study emergent dynamicsof the full dynamics (1.4) and provide sufficient conditions leading to the complete and HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 5 practical aggregations. In Section 6, we provide several numerical examples and comparethem with analytical results in previous sections. Finally, Section 7 is devoted to a briefsummary of our main results and some remaining issues which were not discussed in thiswork. 2.
Preliminaries
In this section, we study basic properties of the Lohe hermitian sphere model with frus-tration, and briefly review earlier results on the emergent dynamics of Lohe type modelswith frustration such as “ the Kuramoto model ” on the unit circle and “ the Lohe spheremodel ” on the unit sphere.2.1.
The LHS model with frustration.
Consider the LHS model on HS d with frustra-tion: for j = 1 , · · · , N ,(2.1) ˙ z j = Ω j z j + κ N N (cid:88) k =1 ( (cid:104) z j , z j (cid:105) V z k − (cid:104) V z k , z j (cid:105) z j ) + κ N N (cid:88) k =1 ( (cid:104) z j , V z k (cid:105) − (cid:104) V z k , z j (cid:105) ) z j . In order to rewrite system (2.1) into a mean-field form, we introduce a centroid z c := N (cid:80) Nk =1 z k . Then, the LHS model (2.1) can be rewritten as a mean-field form:(2.2) ˙ z j = Ω z j + κ (cid:16) (cid:104) z j , z j (cid:105) V z c − (cid:104) V z c , z j (cid:105) z j (cid:17) + κ (cid:16) (cid:104) z j , V z c (cid:105) − (cid:104) V z c , z j (cid:105) (cid:17) z j . Lemma 2.1.
Let { z j } be a solution to (2.2) . Then (cid:107) z j (cid:107) is a conserved quantity: ddt (cid:107) z j (cid:107) = 0 , for all t > , j = 1 , · · · , N. Proof.
Note that(2.3) ddt (cid:107) z j (cid:107) = (cid:104) ˙ z j , z j (cid:105) + (cid:104) z j , ˙ z j (cid:105) . We use (2.2) to estimate the second term in (2.3): (cid:104) z j , ˙ z j (cid:105) = (cid:68) z j , Ω j z j + κ (cid:16) (cid:104) z j , z j (cid:105) V z c − (cid:104) V z c , z j (cid:105) z j (cid:17) + κ (cid:16) (cid:104) z j , V z c (cid:105) − (cid:104) V z c , z j (cid:105) (cid:17) z j (cid:69) = (cid:104) z j , Ω z j (cid:105) + κ (cid:16) − (cid:104) V z c , z j (cid:105) + (cid:104) z j , V z c (cid:105) (cid:17) (cid:107) z j (cid:107) . (2.4)Now, we use the relation (cid:104) ˙ z j , z j (cid:105) = (cid:104) z j , ˙ z j (cid:105) to see(2.5) (cid:104) ˙ z j , z j (cid:105) = (cid:104) z j , ˙ z j (cid:105) = (cid:104) Ω z j , z j (cid:105) + κ (cid:16) (cid:104) z j , V z c (cid:105) − (cid:104) V z c , z j (cid:105) (cid:17) (cid:107) z j (cid:107) . In (2.3), we combine estimates (2.4) and (2.5) to obtain ddt (cid:107) z j (cid:107) = (cid:104) z j , ˙ z j (cid:105) + (cid:104) ˙ z j , z j (cid:105) = (cid:104) z j , Ω z j (cid:105) + (cid:104) Ω z j , z j (cid:105) = (cid:104) (Ω † + Ω) z j , z j (cid:105) = 0 , which yields the desired estimate. (cid:3) Next, we study solution splitting property, when the system has the same free flowΩ j = Ω:(2.6) ˙ z j = Ω z j + κ N N (cid:88) k =1 ( (cid:104) z j , z j (cid:105) V z k − (cid:104) V z k , z j (cid:105) z j ) + κ N N (cid:88) k =1 ( (cid:104) z j , V z k (cid:105) − (cid:104) V z k , z j (cid:105) ) z j . HA, KANG, AND PARK
Now, we consider the associated linear and nonlinear flows:(2.7) ˙ f j = Ω f j , ˙ w j = κ N N (cid:88) k =1 ( (cid:104) w j , w j (cid:105) ˜ V w k − (cid:104) ˜ V w k , w j (cid:105) w j ) + κ N N (cid:88) k =1 ( (cid:104) w j , ˜ V w k (cid:105) − (cid:104) ˜ V w k , w j (cid:105) ) w j , where(2.8) ˜ V ( t ) = e − Ω t V e Ω t and ˜ V ( t ) = e − Ω t V e Ω t . Then, it is easy to see that f j ( t ) = e Ω t f inj , and let N = N ( t ) be solution operators to (2.7)such that W ( t ) = N ( t ) W in , W ( t ) = ( w ( t ) , · · · , w N ( t )) . In next proposition, we study a solution splitting property of (2.6).
Proposition 2.1.
Let { z j } be a solution to (2.6) with initial data z in . Then, one has z j ( t ) = e Ω t ( N ( t ) z in ) j , t ≥ . Proof.
We substitute the ansatz z j ( t ) = e Ω t w j ( t ) , into system (2.2) to get e Ω t ( ˙ w j + Ω w j ) =Ω e Ω t w j + κ ( (cid:104) e Ω t w j , e Ω t w j (cid:105) V e Ω t w c − (cid:104) V e Ω t w c , e Ω t w j (cid:105) e Ω t w j )+ κ ( (cid:104) e Ω t w j , V e Ω t w c (cid:105) − (cid:104) V e Ω t w c , e Ω t w j (cid:105) ) e Ω t w j . This leads ˙ w j = κ (cid:16) (cid:104) w j , w j (cid:105) e − Ω t V e Ω t w c − (cid:68) e − Ω t V e Ω t w c , w j (cid:69) w j (cid:17) + κ (cid:16)(cid:68) w j , e − Ω t V e Ω t w c (cid:69) − (cid:68) e − Ω t V e Ω t w c , w j (cid:69)(cid:17) w j . (2.9)Now we use (2.8) to simplify (2.9) to find˙ w j = κ ( (cid:104) w j , w j (cid:105) ˜ V w c − (cid:104) ˜ V w c , w j (cid:105) w j ) + κ ( (cid:104) w j , ˜ V w c (cid:105) − (cid:104) ˜ V w c , w j (cid:105) ) w j . This complete the desired proof. (cid:3)
Remark 2.1.
Suppose that W and W satisfy [ W , Ω] = 0 and [ W , Ω] = 0 . Then, we have ˜ V = e − Ω t V e Ω t = V and ˜ V = e − Ω t V e Ω t = V , which implies the solution splitting property. Now we will study about relations between the Lohe hermitian sphere model with frus-tration and other synchronization models with frustration. First, if we set initial data Z in = { z inj } Nj =1 as set of real unit vectors and Ω j , W as real skew-symmetric matrix, thenwe can easily obtain the Lohe sphere model with frustration. We introduce how we canreduce the Lohe hermitian sphere model with frustration to Kuramoto model with frus-tration. The diagram below shows relations among the Lohe hermitian sphere model with HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 7 frustration, the Lohe sphere model with frustration, and the Kuramoto model with frustra-tion. Since we can obtain the Lohe sphere model with frustration from the Lohe hermitiansphere model with frustration by letting κ = 0, we denoted it in the diagram. Lohe Hermitian sphere modelwith frustration Lohe sphere modelwith frustrationKuramoto modelwith frustration κ =0 , d =0 κ =0 d =1 Next, we show how the LS model with frustration can be reduced to the Kuramoto modelwith frustration [15, 20, 21, 27, 42]. In what follows, we consider two subsystems of (1.4)separately. • Subsystem A ( κ > κ = 0): Consider the LHS model restricted on S d with auniform frustration:(2.10) ˙ z i = Ω i z i + κN N (cid:88) k =1 (cid:16) V z k − (cid:104) z i , V z k (cid:105) z i (cid:17) , where V is the frustration matrix of the form (1.5). For d = 1, We set(2.11) z i := (cid:20) cos θ i sin θ i (cid:21) , Ω i := (cid:20) − ν i ν i (cid:21) , V := (cid:20) cos α − sin α sin α cos α (cid:21) . We substitute (2.11) into (2.10) to obtain˙ θ i (cid:20) − sin θ i cos θ i (cid:21) = (cid:20) − ν i ν i (cid:21) (cid:20) cos θ i sin θ i (cid:21) + κN N (cid:88) k =1 (cid:18)(cid:20) cos α − sin α sin α cos α (cid:21) (cid:20) cos θ k sin θ k (cid:21) − (cid:28)(cid:20) cos θ i sin θ i (cid:21) , (cid:20) cos α − sin α sin α cos α (cid:21) (cid:20) cos θ k sin θ k (cid:21)(cid:29) (cid:20) cos θ i sin θ i (cid:21)(cid:19) = ν i (cid:20) − sin θ i cos θ i (cid:21) + κN N (cid:88) k =1 (cid:20) cos α cos θ k − sin α sin θ k − (cos θ i cos( θ k + α ) + sin θ i sin( θ k + α )) cos θ i sin α cos θ k + cos α sin θ k − (cos θ i cos( θ k + α ) + sin θ i sin( θ k + α )) sin θ i (cid:21) = ν i (cid:20) − sin θ i cos θ i (cid:21) + κN N (cid:88) k =1 (cid:20) cos( θ k + α ) − cos( θ i − θ k − α ) cos θ i sin( θ k + α ) − cos( θ i − θ k − α ) sin θ i (cid:21) = ν i (cid:20) − sin θ i cos θ i (cid:21) + κN N (cid:88) k =1 (cid:20) − sin θ i sin( θ k − θ i + α )cos θ i sin( θ k − θ i + α ) (cid:21) . This leads to the Kuramoto model with frustration:˙ θ i = ν i + κN N (cid:88) k =1 sin( θ k − θ i + α ) . For α = 0, the above system is exactly the Kuramoto model [13, 14, 16, 19, 26]. HA, KANG, AND PARK • Subsystem B ( κ = 0 and κ > z j = Ω j z j + κ N N (cid:88) k =1 ( (cid:104) z j , V z k (cid:105) − (cid:104) V z k , z j (cid:105) ) z j . We set d = 0 , Ω j = ν j i , z j = e i θ j , V = e i α , κ = κ . We substitute the above ansatz into (2.12) to findi e i θ j ˙ θ j = ν j i e i θ j + κ N N (cid:88) k =1 (cid:16) e i( θ k + α − θ j ) − e i( θ j − θ k − α ) (cid:17) e i θ j = i e i θ j (cid:32) ν j + κN N (cid:88) k =1 sin( θ k − θ j + α ) (cid:33) . After simplification, we obtain the Kuramoto model with frustration:˙ θ j = ν j + κN N (cid:88) k =1 sin( θ k − θ j + α ) . Previous results.
In this subsection, we briefly recall previous results on the emergentdynamics of two aforementioned particle models with frustration.2.2.1.
The Kuramoto model.
Let θ j = θ j ( t ) be the phase of the j -th oscillator whose dy-namics is governed by the Kuramoto model with frustration:(2.13) ˙ θ j = ν j + κN N (cid:88) k =1 sin( θ k − θ j + α ) , | α | < π , where α is the uniform size of frustration.For a brief description of [20], we set D (Θ) := max i,j | θ i − θ j | , D ( ˙Θ) := max i,j | ˙ θ i − ˙ θ j | , D ( ν ) := max i,j | ν i − ν j | . Note that D (Θ( t )) is Lipschitz continuous, hence it is differentiable except at times of col-lision between the extremal phases and their neighboring phases.Let κ, D ( ν ) and α be positive constants satisfying D ( ν ) + κ sin | α | < κ. Then we set D ∞ < D ∞ be two roots of the following trigonometric equation:sin x = D ( ν ) + κ sin | α | κ , x ∈ (0 , π ) . Proposition 2.2. [20]
Suppose system parameters D ( ν ) , κ and α satisfy D ( ν ) > , κ ≥ D ( ν )1 − sin | α | , < D (Θ ) < D ∞ − | α | , and let Θ = Θ( t ) be a solution to (2.13) . Then there exists t > such that D ( ˙Θ( t )) e − κ ( t − t ) ≤ D ( ˙Θ( t )) ≤ D ( ˙Θ( t )) e − κ cos( D ∞ + ε )( t − t ) , t ≥ t , HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 9 where ε (cid:28) is a positive constant satisfying D ∞ + ε < π . Remark 2.2.
For the Kuramto model with heterogeneous frustrations with α kj , its emergentdynamics has been studied in [18] . The LS model with the same free flow.
Let x j = x j ( t ) be a position of the j -th particleon the unit sphere S d whose dynamics is governed by the following system with frustrationand the same free flow:(2.14) ˙ x j = Ω x j + κN N (cid:88) k =1 ( V x k − (cid:104) x j , V x k (cid:105) x j ) , where the frustration matrix V is given by the following form:(2.15) V = I d +1 + W, where I d +1 is the ( d +1) × ( d +1) identity matrix and W is a ( d +1) × ( d +1) skew-symmetricmatrix.We further substitute (2.15) into (2.14) to get(2.16) ˙ x j = Ω x j + κN N (cid:88) k =1 ( x k − (cid:104) x j , x k (cid:105) x j ) (cid:124) (cid:123)(cid:122) (cid:125) synchronous motion + κN N (cid:88) k =1 ( W x k − (cid:104) x j , W x k (cid:105) x j ) . (cid:124) (cid:123)(cid:122) (cid:125) periodic motion Note that the presence of frustration matrix W induces a competition between ‘synchroniza-tion’ and ‘periodic motion’ , in the following sense. The second term on the RHS of (2.16)tends to bring the oscillators together. On the other hand, since W is a ( d + 1) × ( d + 1)skew-symmetric matrix, all eigenvalues of W are zero or purely imaginary. Hence, we caninterpret the last term on the RHS of (2.16), together with Ω x j , tries to pull the dynamicsinto a periodic motion. Theorem 2.1.
Suppose natural frequency matrix, frustration matrix and initial data satisfy (cid:107) W (cid:107) F < , max i,j (cid:16) − (cid:104) x ini , x inj (cid:105) (cid:17) < − (cid:107) W (cid:107) F , where W is a ( d + 1) × ( d + 1) skew-symmetric matrix, and let { x i } Ni =1 be a solution to (2.14) . Then, one has lim κ →∞ lim sup t →∞ max i,j (cid:107) x i ( t ) − x j ( t ) (cid:107) = 0 . In next three sections, we study the emergent dynamics of (1.4) with heterogeneousfrustrations into two cases. First, we study the emergence of practical aggregation for thecase with κ = 0. Second, we study the practical aggregation of (1.4) with κ > κ >
0. More precisely, we study the LH model with frustration, i.e., either κ > , κ = 0 , or κ > , κ > . Consider the LHS model with κ = 0:Subsystem A: ˙ z j = Ω j z j + κ (cid:16) (cid:104) z j , z j (cid:105) V z c − (cid:104) V z c , z j (cid:105) z j (cid:17) andSubsystem B: ˙ z j = Ω j z j + κ (cid:16) (cid:104) z j , z j (cid:105) V z c − (cid:104) V z c , z j (cid:105) z j (cid:17) + κ (cid:16) (cid:104) z j , V z c (cid:105) − (cid:104) V z c , z j (cid:105) (cid:17) z j . For Subsystem A, it is easy to see that z ini ∈ R d +1 = ⇒ z j ( t ) ∈ R d +1 , t > . Emergent dynamics of the LS model: real vector case
In this section, we provide improved aggregation estimate for the LHS model with frus-tration which can be obtained from the LHS model restricted on the unit sphere S d . In thiscase, the state z j = x j ∈ R d . Then, the state x j satisfies(3.1) System A: ˙ x j = Ω j x j + κ (cid:16) (cid:104) x j , x j (cid:105) V x c − (cid:104) V x c , x j (cid:105) x j (cid:17) , where V = I d +1 + W .Before we discuss the aggregation estimate, we recall the concept of “ complete and prac-tical aggregations ”. Definition 3.1.
Let X = { x j } be a dynamic ensemble whose evolution is governed bysystem (3.1) .(1) The ensemble X exhibits complete aggregation, if the following estimate holds. lim t →∞ D ( X ( t )) = 0 , where D ( X ) := max i,j (cid:107) x i − x j (cid:107) .(2) The ensemble X exhibits practical aggregation, if the following estimate holds. lim κ →∞ lim sup t →∞ D ( X ( t )) = 0 . For later practical aggregation estimate, the following lemma will be used crucially.
Lemma 3.1.
Let W ∈ R d × d be a real skew-symmetric matrix and x, y ∈ R d be vectors.Then we have (cid:12)(cid:12)(cid:12) (cid:104) x, W y (cid:105) (cid:12)(cid:12)(cid:12) ≤ √ (cid:107) W (cid:107) F (cid:112) (cid:107) x (cid:107) (cid:107) y (cid:107) − (cid:104) x, y (cid:105) , where (cid:104)· , ·(cid:105) is the inner product in R d .Proof. By direct calculations, one has (cid:104) x, W y (cid:105) = (cid:88) i,j [ x ] i [ W ] ij [ y ] j = 12 (cid:88) i,j ([ x ] i [ W ] ij [ y ] j + [ x ] j [ W ] ji [ y ] i )= 12 (cid:88) i,j ([ x ] i [ W ] ij [ y ] j − [ x ] j [ W ] ij [ y ] i ) = 12 (cid:88) i,j [ W ] ij ([ x ] i [ y ] j − [ y ] i [ x ] j ) , where we used dummy variable exchange and skew-symmetry of matrix W . HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 11
Then, it follows from the Cauchy-Schwarz inequality that |(cid:104) x, W y (cid:105)| = 14 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i,j [ W ] ij ([ x ] i [ y ] j − [ y ] i [ x ] j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) i,j [ W ] ij · (cid:88) i,j ([ x ] i [ y ] j − [ y ] i [ x ] j ) = 14 (cid:107) W (cid:107) F (cid:88) i,j ([ x ] i [ y ] j + [ y ] i [ x ] j − x ] i [ y ] j [ y ] i [ x ] j )= 12 (cid:107) W (cid:107) F · ( (cid:107) x (cid:107) (cid:107) y (cid:107) − (cid:104) x, y (cid:105) ) . This yields the desired estimate. (cid:3)
For emergent dynamics, we introduce θ ij measuring the angle between x i and x j :(3.2) θ ij := cos − ( (cid:104) x i , x j (cid:105) ) , ≤ i, j ≤ N. Lemma 3.2.
Let { x j } be a solution to (3.1) . Then, θ ij satisfies ˙ θ ij ≤ (cid:107) Ω i − Ω j (cid:107) F − κN tan (cid:18) θ ij (cid:19) N (cid:88) k =1 (cid:104) cos θ ik + cos θ jk − (cid:107) W (cid:107) F √ (cid:16) sin θ ik + sin θ jk (cid:17)(cid:105) . Proof.
It follows from (3.1) that ddt (cid:104) x i , x j (cid:105) = (cid:42) Ω i x i + κN N (cid:88) k =1 ( V x k − (cid:104) V x k , x i (cid:105) x i ) , x j (cid:43) + (cid:42) x i , Ω j x j + κN N (cid:88) k =1 ( V x k − (cid:104) V x k , x j (cid:105) x j ) (cid:43) = (cid:104) Ω i x i , x j (cid:105) + (cid:104) x i , Ω j x j (cid:105) + κN N (cid:88) k =1 (cid:0) (cid:104) V x k , x j (cid:105) + (cid:104) x i , V x k (cid:105) (cid:1) (1 − (cid:104) x i , x j (cid:105) ) . Now, we use the skew-symmetry of Ω i , Ω j and W to find ddt (cid:104) x i , x j (cid:105) = (cid:104) (Ω i − Ω j ) x i , x j (cid:105) + (1 − (cid:104) x i , x j (cid:105) ) · κN N (cid:88) k =1 (cid:104) x k , x i + x j (cid:105) + (1 − (cid:104) x i , x j (cid:105) ) · κN N (cid:88) k =1 (cid:104) W x k , x i + x j (cid:105) . We use the defining relation (3.2) for θ ij to obtain ddt cos θ ij = (cid:68) (Ω i − Ω j ) x i , x j (cid:69) + κN (1 − cos θ ij ) N (cid:88) k =1 (cos θ ik + cos θ jk )+ κN (1 − cos θ ij ) N (cid:88) k =1 (cid:16) (cid:104) W x k , x i (cid:105) + (cid:104) W x k , x j (cid:105) (cid:17) , or equivalently˙ θ ij = − (cid:104) (Ω i − Ω j ) x i , x j (cid:105) sin θ ij (cid:124) (cid:123)(cid:122) (cid:125) =: I − κN − cos θ ij sin θ ij N (cid:88) k =1 (cos θ ik + cos θ jk ) (cid:124) (cid:123)(cid:122) (cid:125) =: I − κN − cos θ ij sin θ ij N (cid:88) k =1 ( (cid:104) W x k , x i (cid:105) + (cid:104) W x k , x j (cid:105) ) (cid:124) (cid:123)(cid:122) (cid:125) =: I . (3.3) • (Estimate of I ): We use Lemma 3.1 with A := Ω i − Ω j which is a skew-symmeric toobtain |(cid:104) (Ω i − Ω j ) x i , x j (cid:105)| ≤ √ (cid:107) Ω i − Ω j (cid:107) F (cid:113) (cid:107) x i (cid:107) (cid:107) x j (cid:107) − (cid:104) x i , x j (cid:105) = 1 √ (cid:107) Ω i − Ω j (cid:107) F sin θ ij . Then we use the above estimate to derive(3.4) |I | ≤ √ (cid:107) Ω i − Ω j (cid:107) F . • (Estimate of I and I ): Similarly, one has |I | ≤ κN tan (cid:18) θ ij (cid:19) N (cid:88) k =1 (cos θ ik + cos θ jk ) , |I | ≤ κ (cid:107) W (cid:107) F √ N tan (cid:18) θ ij (cid:19) N (cid:88) k =1 (sin θ ik + sin θ jk ) . (3.5)In (3.3), we combine all the estimates (3.4) and (3.5) to get˙ θ ij ≤ (cid:107) Ω i − Ω j (cid:107) F − κN tan (cid:18) θ ij (cid:19) N (cid:88) k =1 (cos θ ik + cos θ jk )+ κ (cid:107) W (cid:107) F √ N tan (cid:18) θ ij (cid:19) N (cid:88) k =1 (sin θ ik + sin θ jk )= 12 (cid:107) Ω i − Ω j (cid:107) F − κN tan (cid:18) θ ij (cid:19) N (cid:88) k =1 (cid:104) cos θ ik + cos θ jk − (cid:107) W (cid:107) F √ (cid:16) sin θ ik + sin θ jk (cid:17)(cid:105) . (3.6) (cid:3) Now, we are ready to provide our first main result on the aggregation of pairwise particles.
Proposition 3.1. (Complete aggregation)
Suppose the initial data Θ in satisfy θ inij < cot − (cid:18) (cid:107) W (cid:107) F √ (cid:19) , ∀ i, j ∈ { , , · · · , N } , (3.7) HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 13 and let { x i } be the solution of (3.3) with the same free flow Ω j ≡ Ω . Then there exists apositive constant Λ ij = Λ ij ( N, κ, W, Θ in ) such that θ ij ( t ) ≤ θ inij exp ( − Λ ij t ) , t ≥ . Proof.
We use Ω j = Ω and Lemma 3.2 to find˙ θ ij ≤ − κN tan (cid:18) θ ij (cid:19) N (cid:88) k =1 (cid:18) cos θ ik + cos θ jk − (cid:107) W (cid:107) F √ θ ik + sin θ jk ) (cid:19) . On the other hand, it follows from (3.7) that θ inij < cot − (cid:18) (cid:107) W (cid:107) F √ (cid:19) = ⇒ cos θ inij − (cid:107) W (cid:107) F √ θ inij > . This implies ˙ θ ij (cid:12)(cid:12)(cid:12) t =0+ ≤ , ∀ i, j ∈ { , , · · · , N } . Note that f ( θ ) = cos θ − (cid:107) W (cid:107) F √ θ is a decreasing function for θ ∈ [0 , π ]. Thus, it is easyto see(3.8) ˙ θ ij ≤ − κN tan (cid:18) θ ij (cid:19) N (cid:88) k =1 (cid:18) cos θ inik + cos θ injk − (cid:107) W (cid:107) F √ θ inik + sin θ injk ) (cid:19) . Then, we use (3.8) and the relation tan θ ≥ θ, θ ∈ [0 , π/
2) to find˙ θ ij ≤ − κ N θ ij N (cid:88) k =1 (cid:18) cos θ inik + cos θ injk − (cid:107) W (cid:107) F √ θ inik + sin θ injk ) (cid:19) =: − Λ ij θ ij , which implies the desired estimate. (cid:3) Next, we consider the heterogenous ensemble in the sense that there exists i (cid:54) = j suchthat Ω i (cid:54) = Ω j . In the following theorem, we consider the second main result on the emergence of practicalaggregation.
Theorem 3.1. (Practical aggregation)
Let { x i } be the solution of (3.3) with initial data { x ini } : max i,j (cid:16) sin θ inij (cid:17) <
12 + √ (cid:107) W (cid:107) F and max i,j θ inij < π . Then, we have sup t ≥ (cid:16) max i,j θ ij (cid:17) ≤ π and lim κ →∞ lim sup t →∞ max i,j (cid:16) sin θ ij ( t ) (cid:17) = 0 . Proof.
For a distributed set { Ω j } of natural frequency matrices, we set D (Ω) := max i,j (cid:107) Ω i − Ω j (cid:107) F . Then, it follows from Lemma 3.2 that˙ θ ij ≤ D (Ω) − κN tan (cid:16) θ ij (cid:17) N (cid:88) k =1 (cid:104) cos θ ik + cos θ jk − (cid:107) W (cid:107) F √ (cid:16) sin θ ik + sin θ jk (cid:17)(cid:105) . Now we set θ M ( t ) := max i,j θ ij ( t ) , T := { t : max i,j θ ij ( t ) < π , ∀ t ∈ [0 , t ) } , T ∞ := sup T . Then we have˙ θ M ≤ D (Ω) − κN tan (cid:18) θ M (cid:19) N (cid:88) k =1 (cid:104) cos θ ik + cos θ jk − (cid:107) W (cid:107) F √ (cid:16) sin θ ik + sin θ jk (cid:17)(cid:105) ≤ D (Ω) − κ tan (cid:18) θ M (cid:19) (cid:16) θ M − √ (cid:107) W (cid:107) F sin θ M (cid:17) , a.e. t ∈ [0 , T ∞ ) . This yields,12 cos (cid:18) θ M (cid:19) ˙ θ M ≤ D (Ω) cos (cid:18) θ M (cid:19) − κ sin (cid:18) θ M (cid:19) (cid:104) (cid:18) − (cid:18) θ M (cid:19)(cid:19) − √ (cid:107) W (cid:107) F sin (cid:18) θ M (cid:19) cos (cid:18) θ M (cid:19) (cid:105) ≤ D (Ω) − κ sin (cid:18) θ M (cid:19) (cid:104) (cid:18) − (cid:18) θ M (cid:19)(cid:19) − √ (cid:107) W (cid:107) F sin (cid:18) θ M (cid:19) (cid:105) ≤ D (Ω) − κ sin (cid:18) θ M (cid:19) (cid:104) − (4 + 2 √ (cid:107) W (cid:107) F ) sin (cid:18) θ M (cid:19) (cid:105) , a.e. t ∈ [0 , T ∞ ) . (3.9)To simplify (3.9) further, we set s := sin (cid:18) θ M (cid:19) . Then, we have ˙ s ≤ D (Ω) − κs (cid:16) − (4 + 2 √ (cid:107) W (cid:107) F ) s (cid:17) , ≤ t < T ∞ . (3.10)For a practical aggregation estimate, we uses a similar argument in [23]. We define thefollowing quadratic polynomial: p ( s ) = 14 D (Ω) − κs (2 − (4 + 2 √ (cid:107) W (cid:107) F ) s ) = (4 + 2 √ (cid:107) W (cid:107) F ) κs − κs + 14 D (Ω) . Then the discriminant D of above quadratic polynomial is D = 4 κ − D (Ω) κ (4 + 2 √ (cid:107) W (cid:107) F ) = κ (4 κ − D (Ω)(4 + 2 √ (cid:107) W (cid:107) F )) . If κ > D (Ω)(2 + 2 √ (cid:107) W (cid:107) F )4 , we have two distinct roots s < s : s = 2 κ − (cid:113) κ − κ (4 + 2 √ (cid:107) W (cid:107) F ) D (Ω)2(4 + 2 √ (cid:107) W (cid:107) F ) κ ,s = 2 κ + (cid:113) κ − κ (4 + 2 √ (cid:107) W (cid:107) F ) D (Ω)2(4 + 2 √ (cid:107) W (cid:107) F ) κ . Moreover, since the coefficient of s and p (0) are positive, one has0 < s . HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 15 If s ( t ) is the solution of (3.10) with initial data s (0) < s , we have T ∞ = ∞ and lim sup t →∞ s ( t ) ≤ s . Explicit formula of s and s provide uslim κ →∞ s = 0 and lim κ →∞ s = 12 + √ (cid:107) W (cid:107) F , so that we can conclude that if s (0) <
12 + √ (cid:107) W (cid:107) F we havelim κ →∞ lim sup t →∞ s ( t ) = 0 . (cid:3) Emergent dynamics of the LS model: complex vector case
In this section, we study emergent dynamics of the complex LS model. First, we considerthe case in which the second coupling is not present, i.e., κ = 0, i.e.,˙ z j = Ω j z j + κ (cid:16) (cid:104) z j , z j (cid:105) V z c − (cid:104) V z c , z j (cid:105) z j (cid:17) . For this, we first generalize the result of Lemma 3.1 as follows.
Lemma 4.1.
Let x, y ∈ C d be complex vectors, and W ∈ C d × d be a skew-hermitian matrixi.e., W † = − W . Then, one has (cid:12)(cid:12)(cid:12) (cid:104) W x, y (cid:105) + (cid:104) y, W x (cid:105) (cid:12)(cid:12)(cid:12) ≤ √ (cid:107) W (cid:107) F (cid:112) (cid:107) x (cid:107) (cid:107) y (cid:107) − Re( (cid:104) x, y (cid:105) ) . Proof.
We basically use the same argument as in Lemma 3.1. (cid:104)
W x, y (cid:105) + (cid:104) y, W x (cid:105) = (cid:88) i,j (cid:16) [ W ] ij [ x ] j [ y ] i + [ y ] i [ W ] ij [ x ] j (cid:17) = (cid:88) i,j (cid:16) − [ x ] j [ W ] ji [ y ] i + [ y ] i [ W ] ij [ x ] j (cid:17) = (cid:88) i,j (cid:16) − [ x ] i [ W ] ij [ y ] j + [ y ] i [ W ] ij [ x ] j (cid:17) = (cid:88) i,j [ W ] ij ([ y ] i [ x ] j − [ x ] i [ y ] j ) . This and the Cauchy-Schwarz inequality yield |(cid:104)
W x, y (cid:105) + (cid:104) y, W x (cid:105)| = (cid:88) i,j [ W ] ij ([ y ] i [ x ] j − [ x ] i [ y ] j ) ≤ (cid:88) i,j | [ W ] ij | (cid:88) i,j (cid:12)(cid:12)(cid:12) [ y ] i [ x ] j − [ x ] i [ y ] j (cid:12)(cid:12)(cid:12) = (cid:107) W (cid:107) F (cid:16) (cid:107) x (cid:107) (cid:107) y (cid:107) − (cid:104) x, y (cid:105) − (cid:104) y, x (cid:105) (cid:17) = 2 (cid:107) W (cid:107) F · (cid:0) (cid:107) x (cid:107) (cid:107) y (cid:107) − Re( (cid:104) x, y (cid:105) ) (cid:1) . This implies the desired estimate. (cid:3)
Remark 4.1.
Let z i and z j be complex vectors with (cid:107) z i (cid:107) = (cid:107) z j (cid:107) = 1 . Then, we use Lemma4.1 to get (cid:107)(cid:104) W z i , z j (cid:105) + (cid:104) z j , W z i (cid:105)| ≤ √ (cid:107) W (cid:107) F · (cid:113) − Re( (cid:104) z i , z j (cid:105) ) . In the following subsections, we study emergent dynamics of (1.4) with κ = 0 and thefull system separately.Consider the system which can be obtained from the full system (1.4) with κ = 0:(4.1) (cid:40) ˙ z j = Ω j z j + κ ( (cid:104) z j , z j (cid:105) V z c − (cid:104) V z c , z j (cid:105) z j ) ,V = I d +1 + W . or equivalently˙ z j = Ω j z j + κ (cid:16) (cid:104) z j , z j (cid:105) z c − (cid:104) z c , z j (cid:105) z j (cid:17) + κ (cid:16) (cid:104) z j , z j (cid:105) W z c − (cid:104) W z c , z j (cid:105) z j (cid:17) . Next we consider the temporal evolution of (cid:104) z i , z j (cid:105) . Lemma 4.2.
Let { z j } be the solution of the system (4.1) . Then, one has ddt (cid:104) z i , z j (cid:105) = (cid:104) (Ω i − Ω j ) z i , z j (cid:105) + κ (cid:16) − (cid:104) z i , z j (cid:105) )( (cid:104) V z c , z j (cid:105) + (cid:104) z i , V z c (cid:105) (cid:17) . Proof.
Note that(4.2) ddt (cid:104) z i , z j (cid:105) = (cid:104) ˙ z i , z j (cid:105) + (cid:104) z i , ˙ z j (cid:105) . For each term in the R.H.S. of (4.2), we use (4.1) to see (cid:104) ˙ z i , z j (cid:105) = (cid:104) Ω i z i , z j (cid:105) + κ ( (cid:104) V z c , z j (cid:105) − (cid:104) z i , V z c (cid:105)(cid:104) z i , z j (cid:105) ) , (cid:104) z i , ˙ z j (cid:105) = (cid:104) z i , Ω j z j (cid:105) + κ ( (cid:104) z i , V z c (cid:105) − (cid:104) V z c , z j (cid:105)(cid:104) z i , z j (cid:105) ) , (4.3)Now, we combine (4.2) and (4.3) to get ddt (cid:104) z i , z j (cid:105) = (cid:104) ˙ z i , z j (cid:105) + (cid:104) z i , ˙ z j (cid:105) = (cid:104) Ω i z i , z j (cid:105) + (cid:104) z i , Ω j z j (cid:105) + κ (cid:16) (cid:104) V z c , z j (cid:105) − (cid:104) z i , V z c (cid:105)(cid:104) z i , z j (cid:105) + (cid:104) z i , V z c (cid:105) − (cid:104) V z c , z j (cid:105)(cid:104) z i , z j (cid:105) (cid:17) = (cid:104) (Ω i − Ω j ) z i , z j (cid:105) + κ (1 − (cid:104) z i , z j (cid:105) )( (cid:104) V z c , z j (cid:105) + (cid:104) z i , V z c (cid:105) ) . (cid:3) Identical ensemble.
In this subsection, we study emergent behaviors of identicalensemble. For four points ( z i , z j , z k , z l ) lying in a general position, we introduce a cross-ratio-like quantity [17, 28]: C ijkl [ Z ] := (1 − (cid:104) z i , z j (cid:105) )(1 − (cid:104) z k , z l (cid:105) )(1 − (cid:104) z i , z l (cid:105) )(1 − (cid:104) z k , z j (cid:105) ) . Next, we show that C ijkl is conserved along the dynamics (4.1) with Ω i = Ω. Proposition 4.1.
Suppose the natural frequency matrices are the same: Ω j = Ω , j = 1 , · · · , N, and let { z j } be a global solution of system (4.1) . Then, one has C ijkl [ Z ( t )] = C ijkl [ Z (0)] , t ≥ . HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 17
Proof.
Since Ω i = Ω j , we have(4.4) ddt (cid:104) z i , z j (cid:105) = κ (1 − (cid:104) z i , z j (cid:105) ) (cid:16) (cid:104) V z c , z j (cid:105) + (cid:104) z i , V z c (cid:105) (cid:17) . This implies(4.5) ddt (1 − (cid:104) z i , z j (cid:105) ) = − κ (1 − (cid:104) z i , z j (cid:105) )( (cid:104) V z c , z j (cid:105) + (cid:104) z i , V z c (cid:105) ) . Again this yields the desired estimate: ddt C ijkl = ddt (cid:18) (1 − (cid:104) z i , z j (cid:105) )(1 − (cid:104) z k , z l (cid:105) )(1 − (cid:104) z i , z l (cid:105) )(1 − (cid:104) z k , z j (cid:105) ) (cid:19) = − κ (cid:18) (1 − (cid:104) z i , z j (cid:105) )(1 − (cid:104) z k , z l (cid:105) )(1 − (cid:104) z i , z l (cid:105) )(1 − (cid:104) z k , z j (cid:105) ) (cid:19) × (cid:16) (cid:104) V z c , z j (cid:105) + (cid:104) z i , V z c (cid:105) + (cid:104) V z c , z l (cid:105) + (cid:104) z k , V z c (cid:105) − (cid:104) V z c , z l (cid:105)− (cid:104) z i , V z c (cid:105) − (cid:104) V z c , z j (cid:105) − (cid:104) z k , V z c (cid:105) (cid:17) = 0 . (cid:3) Next we use (4.5) to obtain ddt | − (cid:104) z i , z j (cid:105)| = ddt (cid:0) (1 − (cid:104) z i , z j (cid:105) )(1 − (cid:104) z j , z i (cid:105) ) (cid:1) = − κ (1 − (cid:104) z i , z j (cid:105) )( (cid:104) V z c , z j (cid:105) + (cid:104) z i , V z c (cid:105) )(1 − (cid:104) z j , z i (cid:105) ) + ( c.c. )= − κ | − (cid:104) z i , z j (cid:105)| ( (cid:104) V z c , z j (cid:105) + (cid:104) z i , V z c (cid:105) + (cid:104) V z c , z i (cid:105) + (cid:104) z j , V z c (cid:105) )= − κ | − (cid:104) z i , z j (cid:105)| (cid:16) (cid:104) V z c , z i + z j (cid:105) + (cid:104) z i + z j , V z c (cid:105) (cid:17) . (4.6)Now we introduce new variables defined as follows:(4.7) R ij := Re( (cid:104) z i , z j (cid:105) ) , I ij := Im( (cid:104) z i , z j (cid:105) ) ∀ i, j ∈ { , , · · · , N } . Since (cid:104) z i , z i (cid:105) = 1, it is easy to see that R ii = 1 and I ii = 0 , i = 1 , · · · , N. Lemma 4.3.
Let R ij and I ij be quantities defined by the relation (4.7) . Then, the followingassertions hold.(1) R ij and I ij satisfy symmetry-antisymmetry relations: R ij = R ji and I ij = − I ij , ≤ i, j ≤ N. (2) ( R ij , I ij ) lies on the unit ball: | R ij | + | I ij | ≤ , ≤ i, j ≤ N. Proof. (i) Note that (cid:104) z i , z j (cid:105) = (cid:104) z j , z i (cid:105) , i.e., R ij + i I ij = R ji + i I ji = R ji − i I ji . We compare the real and imaginary parts of the above relation to find the proof of the firstassertion.(ii) The second assertion follows from the relation |(cid:104) z i , z j (cid:105)| ≤ (cid:3) Now, we set J ij := (cid:113) (1 − R ij ) + I ij , ∀ i, j ∈ { , , · · · , N } . Then, note that the zero convergence of J ij yields the emergence of the complete aggrega-tion. Lemma 4.4.
Suppose the natural frequency matrices are the same: Ω j = Ω , j = 1 , · · · , N, and let { z j } be a global solution of system (4.1) . Then, the functional J ij satisfies d J ij dt ≤ − κ N J ij N (cid:88) k =1 (cid:32) − J ik − √ √ (cid:107) W (cid:107) F J ik + 1 − J jk − √ √ (cid:107) W (cid:107) F J jk (cid:33) . Proof.
From (4.6), we have ddt ((1 − R ij ) + I ij )= − κ N ((1 − R ij ) + I ij ) N (cid:88) k =1 (2 R ik + 2 R jk + (cid:104) W z k , z i + z j (cid:105) + (cid:104) z i + z j , W z k (cid:105) ) . Also, we can apply Lemma 4.1 to obtain |(cid:104) W z k , z i (cid:105) + (cid:104) z i , W z k (cid:105)| ≤ √ (cid:107) W (cid:107) F (cid:113) − R ik + I ik . Moreover, from the relation R ik + I ik ≤ − R ik + I ik = (1 + R ik )(1 − R ik ) + I ik ≤ − R ik ) + I ik ≤ (cid:114) (4 + 1) (cid:16) (1 − R ik ) + I ik (cid:17) = (cid:114) (cid:16) (1 − R ik ) + I ik (cid:17) . If we combine above calculations, we have ddt ((1 − R ij ) + I ij ) ≤ − κ N ((1 − R ij ) + I ij ) N (cid:88) k =1 (cid:18) R ik + 2 R jk − √ (cid:107) W (cid:107) F (cid:113) (cid:2) (1 − R ik ) + I ik (cid:3) − √ (cid:107) W (cid:107) F (cid:113) (cid:2) (1 − R jk ) + I jk (cid:3)(cid:19) . Then, we can rewrite above relation as d J ij dt ≤ − κ N J ij · N (cid:88) k =1 (cid:16) R ik + 2 R jk − √ √ (cid:107) W (cid:107) F ( J ik + J jk ) (cid:17) HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 19 and this implies d J ij dt ≤ − κ N J ij N (cid:88) k =1 (cid:32) R ik + R jk − √ √ (cid:107) W (cid:107) F J ik − √ √ (cid:107) W (cid:107) F J jk (cid:33) . Note that the definition of J ij yeilds R ij = 1 − (cid:113) J ij − I ij ≥ − J ij , from which we can get the desired result: d J ij dt ≤ − κ N J ij N (cid:88) k =1 (cid:32) − J ik − √ √ (cid:107) W (cid:107) F J ik + 1 − J jk − √ √ (cid:107) W (cid:107) F J jk (cid:33) . (4.8) (cid:3) Now, we are ready to present our third main result.
Theorem 4.1.
Suppose the natural frequency matrix and initial data satisfy (4.9) Ω j = Ω , J inij < √ (cid:113) √ (cid:107) W (cid:107) F + 8 + √ (cid:107) W (cid:107) F , i, j = 1 , · · · , N, and let { z j } be the solution for (4.1) with initial data { z inj } . Then, there exists a positiveconstant Λ ij such that |J ij ( t ) | (cid:46) exp ( − α ij t ) , (cid:107) z i ( t ) − z j ( t ) (cid:107) (cid:46) exp (cid:18) − Λ ij t (cid:19) , where Λ ij := κ N N (cid:88) k =1 (cid:32) − ( J inik ) − √ √ (cid:107) W (cid:107) F J inik + 1 − ( J injk ) − √ √ (cid:107) W (cid:107) F J injk (cid:33) . Proof.
From the assumption (4.9) and the inequality (4.8), we know that J ij always de-creases. This implies J ij ( t ) ≤ J inij , t ≥ , i, j ∈ { , , · · · , N } . Then we can also obtain d J ij dt ≤ − κ N J ij N (cid:88) k =1 (cid:32) − J ik − √ √ (cid:107) W (cid:107) F J ik + 1 − J jk − √ √ (cid:107) W (cid:107) F J jk (cid:33) ≤ − κ N J ij N (cid:88) k =1 (cid:32) − ( J inik ) − √ √ (cid:107) W (cid:107) F J inik + 1 − ( J injk ) − √ √ (cid:107) W (cid:107) F J injk (cid:33) . From the assumption (4.9), we haveΛ ij > i, j ∈ { , , · · · , N } . Then, we have d J ij dt ≤ − Λ ij J ij . This implies J ij ( t ) ≤ J inij exp( − Λ ij t ) and (cid:107) z i − z j (cid:107) = 2 − R ij ≤ J ij ≤ J inij exp( − Λ ij t ) . (cid:3) Non-identical ensemble.
In this subsection, we study practical aggregation of non-identical ensemble to (4.1).
Lemma 4.5.
Let { z j } be a global solution of system (4.1) . Then, the functional J ij satisfies d J ij dt ≤ √ J ij D (Ω) − κ N J ij N (cid:88) k =1 (cid:32) − J ik − √ √ (cid:107) W (cid:107) F J ik + 1 − J jk − √ √ (cid:107) W (cid:107) F J jk (cid:33) . Proof.
As in (4.6), one has ddt | − (cid:104) z i , z j (cid:105)| = (1 − (cid:104) z j , z i (cid:105) ) (cid:104) (Ω i − Ω j ) z i , z j (cid:105) + (1 − (cid:104) z i , z j (cid:105) ) (cid:104) (Ω j − Ω i ) z j , z i (cid:105)− κ | − (cid:104) z i , z j (cid:105)| ( (cid:104) V z c , z i + z j (cid:105) + (cid:104) z i + z j , V z c (cid:105) )Now, we use Lemma 4.1 to get | (1 − (cid:104) z j , z i (cid:105) ) (cid:104) (Ω i − Ω j ) z i , z j (cid:105) + (1 − (cid:104) z i , z j (cid:105) ) (cid:104) (Ω j − Ω i ) z j , z i (cid:105)| = |(cid:104) (Ω i − Ω j ) z i , (1 − (cid:104) z j , z i (cid:105) ) z j (cid:105) + (cid:104) (1 − (cid:104) z j , z i (cid:105) ) z j , (Ω i − Ω j ) z i (cid:105)|≤ √ (cid:107) Ω i − Ω j (cid:107) F (cid:113) (cid:107) z i (cid:107) (cid:107) (1 − (cid:104) z j , z i (cid:105) ) z j (cid:107) − Re (cid:2) (cid:104) z i , (1 − (cid:104) z j , z i (cid:105) ) z j (cid:105) (cid:3) = √ (cid:107) Ω i − Ω j (cid:107) F (cid:113) | − (cid:104) z j , z i (cid:105)| − Re (cid:2) (cid:104) z i , z j (cid:105) (1 − (cid:104) z j , z i (cid:105) ) (cid:3) = √ (cid:107) Ω i − Ω j (cid:107) F (cid:113) (1 − R ij ) + I ij − Re (cid:2) ( R ij + i I ij − R ij − I ij ) (cid:3) = √ (cid:107) Ω i − Ω j (cid:107) F (cid:113) (1 − R ij ) + I ij − ( R ij − R ij − I ij ) + I ij = √ (cid:107) Ω i − Ω j (cid:107) F (cid:113) (1 − R ij − I ij )((1 − R ij ) + I ij ) + 2 I ij ≤ √ (cid:107) Ω i − Ω j (cid:107) F (cid:113) J ij (3 − R ij − I ij ) ≤ √ D (Ω) J ij , where D (Ω) = max i,j (cid:107) Ω i − Ω j (cid:107) F . Finally we can obtain the desired estimate. d J ij dt ≤ √ J ij D (Ω) − κ N J ij N (cid:88) k =1 (cid:32) − J ik − √ √ (cid:107) W (cid:107) F J ik + 1 − J jk − √ √ (cid:107) W (cid:107) F J jk (cid:33) . (4.10) (cid:3) Theorem 4.2.
Suppose initial data { z inj } satisfy (4.9) and let { z j } be the solution of (4.1) .Then, we have following practical aggregation: lim κ →∞ lim sup t →∞ J ij = 0 . Proof.
We will use a similar argument with Theorem 3.1. First, consider the followingquartic polynomial: p ( J ) = κ J (cid:32) J + √ √ (cid:107) W (cid:107) F J − (cid:33) + 3 √ D (Ω) . HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 21
For a sufficiently large κ , the polynomial p has two positive solutions, called J ± , whichsatisfy lim κ →∞ J + = 2 √ (cid:113) √ (cid:107) W (cid:107) F + 8 − √ (cid:107) W (cid:107) F , lim κ →∞ J − = 0 . If we set J M ( t ) = max i,j J ij ( t ) , we have J M d J M dt ≤ √ D (Ω) + κ J M (cid:32) J M + √ √ (cid:107) W (cid:107) F J M − (cid:33) . Hence, we can obtain that if J M (0) < √ (cid:113) √ (cid:107) W (cid:107) F + 8 + √ (cid:107) W (cid:107) F , then we can obtain following practical aggregation:lim κ →∞ lim t →∞ J M = 0 . (cid:3) Remark 4.2.
Since the leading coefficient of p is O ( κ ) and p is quartic, for sufficientlylarge κ , one has J − = O (cid:16) κ − (cid:17) . Emergent dynamics of the full LHS model
In this section, we study emergent dynamics of the LHS model. In other words, we needto consider the effect of κ ( (cid:104) z j , V z c (cid:105) − (cid:104) V z c , z j (cid:105) ) z j , V = I d +1 + W . Lemma 5.1.
Let { z j } be a global solution of system (1.4) . Then, the functional J ij satisfies d J ij dt ≤ √ J ij D (Ω) − κ N J ij N (cid:88) k =1 (cid:32) − J ik − √ √ (cid:107) W (cid:107) F J ik + 1 − J jk − √ √ (cid:107) W (cid:107) F J jk (cid:33) + κ N J ij N (cid:88) k =1 ( J ik + J kj ) + √ κ (cid:107) W (cid:107) F . Proof.
It suffices to consider additional term including κ and then, we can use the calcu-lation in proofs of Lemma 4.2 and 4.4: κ ( (cid:104) z j , V z c (cid:105) − (cid:104) V z c , z j (cid:105) ) (cid:104) z i , z j (cid:105) + κ ( (cid:104) V z c , z i (cid:105) − (cid:104) z i , V z c (cid:105) ) (cid:104) z i , z j (cid:105) = κ (cid:104) z i , z j (cid:105) ( (cid:104) z j − z i , V z c (cid:105) − (cid:104) V z c , z j − z i (cid:105) ) . From this calculation, we have κ (cid:104) z i , z j (cid:105) (1 − (cid:104) z j , z i (cid:105) )( (cid:104) z j − z i , V z c (cid:105) − (cid:104) V z c , z j − z i (cid:105) )+ κ (cid:104) z j , z i (cid:105) (1 − (cid:104) z i , z j (cid:105) )( (cid:104) z i − z j , V z c (cid:105) − (cid:104) V z c , z i − z j (cid:105) )= κ ( (cid:104) z i , z j (cid:105) − (cid:104) z j , z i (cid:105) )( (cid:104) z j − z i , V z c (cid:105) − (cid:104) V z c , z j − z i (cid:105) )= − κ i I ij N N (cid:88) k =1 ( (cid:104) z i − z j , V z k (cid:105) − (cid:104) V z k , z i − z j (cid:105) ) . We also have (cid:104) z i − z j , V z k (cid:105) − (cid:104) V z k , z i − z j (cid:105) = (cid:104) z i , z k (cid:105) − (cid:104) z j , z k (cid:105) − (cid:104) z k , z i (cid:105) + (cid:104) z k , z j (cid:105) + (cid:104) z i − z j , W z k (cid:105) − (cid:104) W z k , z i − z j (cid:105) = 2i I ik + 2i I kj + (cid:104) z i − z j , W z k (cid:105) − (cid:104) W z k , z i − z j (cid:105) . We combine this identity with the previous results in proofs of Lemma 4.2 and 4.4 to have ddt | − (cid:104) z i , z j (cid:105)| = (1 − (cid:104) z j , z i (cid:105) ) (cid:104) (Ω i − Ω j ) z i , z j (cid:105) + (1 − (cid:104) z i , z j (cid:105) ) (cid:104) (Ω j − Ω i ) z j , z i (cid:105)− κ N ((1 − R ij ) + I ij ) N (cid:88) k =1 (cid:16) R ik + 2 R jk + (cid:104) W z k , z i + z j (cid:105) + (cid:104) z i + z j , W z k (cid:105) (cid:17) − κ I ij N N (cid:88) k =1 ( I ik + I kj ) + 2 κ i I ij N N (cid:88) k =1 (cid:16) (cid:104) z i − z j , W z k (cid:105) − (cid:104) W z k , z i − z j (cid:105) (cid:17) . (5.1)Finally, we combine the relations (4.10) and (5.1) to obtain following result: d J ij dt ≤ √ J ij D (Ω) − κ N J ij N (cid:88) k =1 (cid:32) − J ik − √ √ (cid:107) W (cid:107) F J ik + 1 − J jk − √ √ (cid:107) W (cid:107) F J jk (cid:33) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ I ij N J ij N (cid:88) k =1 ( I ik + I kj ) − κ i I ij N J ij N (cid:88) k =1 (cid:16) (cid:104) z i − z j , W z k (cid:105) − (cid:104) W z k , z i − z j (cid:105) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . From the definition of J ij , we have | I ij | ≤ J ij , | I ik | ≤ J ik , | I kj | ≤ J kj , and also we have |(cid:104) z i − z j , W z k (cid:105)| ≤ (cid:107) W (cid:107) F · (cid:107) z i − z j (cid:107) = (cid:107) W (cid:107) F (cid:113) − R ij ) ≤ √ (cid:107) W (cid:107) F J ij . Finally, we can obtain the desired estimate: d J ij dt ≤ √ J ij D (Ω) − κ N J ij N (cid:88) k =1 (cid:32) − J ik − √ √ (cid:107) W (cid:107) F J ik + 1 − J jk − √ √ (cid:107) W (cid:107) F J jk (cid:33) + κ N J ij N (cid:88) k =1 ( J ik + J kj ) + √ κ (cid:107) W (cid:107) F . (cid:3) HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 23
For a configuration z , we set J M ( t ) := max i,j J ij . Theorem 5.1.
Suppose system parameters and initial data satisfy κ > κ ≥ , W ≡ , D (Ω) = 0 , J inij < √ (cid:16) − κ κ (cid:17)(cid:114) √ (cid:107) W (cid:107) F + 8 (cid:16) − κ κ (cid:17) + √ (cid:107) W (cid:107) F ∀ i, j = 1 , · · · , N, (5.2) and let { z j } be the solution of (1.4) with initial data { z inj } . Then, there exists a positiveconstant ˜Λ such that J M ( t ) ≤ J M (0) exp (cid:16) − ˜Λ t (cid:17) , t > , where ˜Λ := κ (cid:32) − κ κ − |J inM | − √ √ (cid:107) W (cid:107) F J inM (cid:33) . Proof.
Again, we use Lemma 5.1 to obtain d J M dt ≤ √ J M D (Ω) − κ J M (cid:32) − J M − √ √ (cid:107) W (cid:107) F J M (cid:33) + 2 κ J M + √ κ (cid:107) W (cid:107) F . (5.3)By the assumption (cid:107) W (cid:107) F = 0 and D (Ω) = 0, one has d J M dt ≤ − κ J M (cid:32) − κ κ − J M − √ √ (cid:107) W (cid:107) F J M (cid:33) . If we use the same argument with Theorem 4.1 and assumption on the initial data, weobtain J M ( t ) ≤ J M (0) exp (cid:16) − ˜Λ t (cid:17) . (cid:3) Theorem 5.2.
Let { z j } be the solution of (1.4) with the initial data { z inj } = satisfying (4.9) . Then, for fixed κ , we can obtain following practical aggregation lim κ →∞ lim sup t →∞ J M ( t ) = 0 . Proof.
We first rewrite (5.3) as d J M dt ≤ √ J M D (Ω) + √ κ (cid:107) W (cid:107) F − κ J M (cid:32) − κ κ − J M − √ √ (cid:107) W (cid:107) F J M (cid:33) . From the similar argument with Theorem 4.2, we can also define quartic polynomial asfollows: p ( J ) = 3 √ D (Ω) + √ κ (cid:107) W (cid:107) F J − κ J (cid:32) − κ κ − J − √ √ (cid:107) W (cid:107) F J (cid:33) = κ J (cid:32) J + √ √ (cid:107) W (cid:107) F J − (cid:33) + κ J (cid:16) J + √ (cid:107) W (cid:107) F (cid:17) + 3 √ D (Ω) . Then, for sufficiently large κ relative to κ , we can also set two positive solutions of p as J ± . Also we know thatlim κ →∞ J + = 2 √ (cid:113) √ (cid:107) W (cid:107) F + 8 + √ (cid:107) W (cid:107) F , lim κ →∞ J − = 0 . Hence, assumption on initial condition implies the desired result. (cid:3)
Remark 5.1.
As in Remark 4.2, for sufficiently large κ , one has J − = O (cid:16) κ − (cid:17) . Numerical simulation
In this section, we provide several numerical examples in order to confirm analyticalresults in Sections 4 and 5 on the asymptotic behavior of complex LS model and LHSmodel, respectively. In all simulations, we set N = 50 , ∆ t = 0 . , z i ∈ HS , i = 1 , · · · , , and used the fourth order Runge-Kutta method.6.1. The complex Lohe sphere model.
In this subsection, we observe analytical resultson the complex Lohe sphere model (4.1) in Section 4.6.1.1.
Identical case.
In this part, we observe the emergent behavior of (4.1) with all naturalfrequencies are equal. That is, there exists Ω ∈ C × such thatΩ † = − Ω , Ω i = Ω , i = 1 , · · · , . We perform a numerical simulation under plausible condition to observe the complete ag-gregation in Theorem 4.1. In all simulations, we chose coupling strength, natural frequencyΩ and frustration W satisfying κ = 1 , Re(Ω jk ) , Im(Ω jk ) ∈ [ − , , Re[( W ) jk ] , Im[( W ) jk ] ∈ [ − . , . , ≤ j, k ≤ , and take initial data satisfying (4.9). Under those settings, we observe the dynamics in timeinterval [0 , J M , from which one can observe the completeaggregation when it converges to zero. Furthermore, linearity of Figure 1 (b) exhibits theexponential decaying of J M .6.1.2. Nonidentical case.
In this part, we observe the emergent behavior of (4.1) with dis-tinct natural frequencies. We perform numerical simulation under plausible condition toobserve the practical aggregation in Theorem 4.2. In simulation, we chose natural frequen-cies Ω i and frustration W satisfyingRe[(Ω i ) jk ] , Im[(Ω i ) jk ] ∈ [ − , , Re[( W ) jk ] , Im[( W ) jk ] ∈ [ − . , . , ≤ j, k ≤ , take initial data satisfying (4.9) and observe the dynamics in time interval [0 , κ = 1 , , , . In Figure 2 (a), we plot the graph of J M for various κ . For each κ , we denote thecorresponding J M as J κ M . Then, one can observe that the asymptotic bound of J κ M gets HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 25 (a) Graph of J M (b) Graph of log J M Figure 1.
Emergence of complete aggregationcloser to zero as κ becomes larger, which implies the practical aggregation. Moreover, inFigure 2 (b), (c) and (d), we plot the graphs of J M , √ κ J κ M and κ J κ M for κ = 5 , , J M is bounded by the graphs of √ κ J κ M and κ J κ M . More precisely, asymptotic bound of J M is larger than that of √ κ J κ M , whileless than that of κ J κ M . And these observations support Remark 4.2.6.2. The LHS model.
In this subsection, we observe the analytical result on the LoheHermitian sphere models in Section 5.6.2.1.
Identical ensemble.
In this part, we observe the emergent behavior of (1.4) with allnatural frequencies are equal. We perform numerical simulation under plausible conditionto observe the complete aggregation in Theorem 5.1. In simulation, we chose couplingstrength, natural frequency Ω and frustrations W , W satisfying κ = 4 , κ = 1 , W = 0 , Re[( W ) jk ] , Im[( W ) jk ] ∈ [ − . , . , Re(Ω jk ) , Im(Ω jk ) ∈ [ − , , ≤ j, k ≤ , so that κ > κ , and take initial data satisfying (5.2) . Under those settings, we observethe dynamics in time interval [0 , J M , from which one can observe the completeaggregation when it converges to zero. Furthermore, linearity of Figure 3 (b) exhibits theexponential decaying of J M .6.2.2. Nonidentical ensemble.
In this part, we observe the emergent behavior of (1.4) withdistinct natural frequencies. We perform numerical simulation under plausible condition toobserve the practical aggregation in Theorem 5.2. In simulation, we chose natural frequen-cies Ω i and frustration W , W satisfyingRe[(Ω i ) jk ] , Im[(Ω i ) jk ] ∈ [ − , , Re[( W l ) jk ] , Im[( W l ) jk ] ∈ [ − . , . , ≤ i ≤ N, ≤ j, k ≤ d, l = 0 , (a) Graphs of J κ M for various κ (b) Graphs of J M , √ J M and 5 J M (c) Graphs of J M , √ J M and 10 J M (d) Graphs of J M , √ J M and 20 J M Figure 2.
Emergence of practical aggregationtake initial data satisfying (4.9) and observe the dynamics in time interval [0 , κ , κ ) = (1 , , (5 , , (10 , , (20 , . In Figure 4 (a), we plot the graph of J M for various κ , but κ fixed. For each κ , wedenote the corresponding J M as J κ M . Then, one can observe that the asymptotic bound of J κ M gets closer to zero as κ becomes larger, which implies the practical synchronization.Moreover, in Figure 4 (b), (c) and (d), we plot the graphs of J M , √ κ J κ M and κ J κ M for κ = 5 , ,
20, respectively. One can observe that the graph of J M is bounded by thegraphs of √ κ J κ M and κ J κ M . More precisely, asymptotic bound of J M is larger than thatof √ κ J κ M , while less than that of κ J κ M . And these observations support Remark 5.1. HE LOHE HERMITIAN SPHERE MODEL WITH FRUSTRATION 27 (a) Graph of J M (b) Graph of log J M Figure 3.
Emergence of complete aggregation7. conclusion
In this paper, we have studied emergent behaviors of the Lohe hermitian sphere modelwith frustration. Frustration can act as an anti-aggregation mechanism so that it pre-vents a formation of aggregate phenomenon. However, when coupling is strong enough andfrustration is small enough, the complete aggregation and practical aggregation can emergedepending on the nature of frequency matrices. For both cases, we provide explicit sufficientframeworks in terms of coupling strengths and initial data, and then within our proposedframework, we show that the LHS model can exhibit collective behaviors. For non-identicalparticles with different free flows, our estimates on the practical aggregation can be viewedas a weak aggregation estimate in the sense that our practical aggregation estimate doesnot give us on the formation of phase-locked states for the LHS model. Moreover, we do notknow how many phase-locked states with positive order parameter can exist in a genericsetting, and whether periodic orbits can exhibit in the LHS model is not known yet. Forthe identical ensemble, nontrivial periodic orbits will be excluded due to the constancy ofcross-ratio-like quantity. Of course, aforementioned questions on the phase-locked statesare also open, even for the Lohe sphere model. These interesting questions will be left fora future work.
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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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