Journal of Statistical Physics https://doi.org/10.1007/s10955-018-2091-0

On the Relaxation Dynamics of Lohe Oscillators on Some Riemannian Manifolds Seung-Yeal Ha1,2 · Dongnam Ko3

· Sang Woo Ryoo3

Received: 4 February 2018 / Accepted: 14 June 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018

Abstract We study the collective relaxation dynamics appearing in weakly coupled Lohe oscillators in a large coupling regime. The Lohe models on the unit sphere and unitary group were proposed as a nonabelian generalization of the Kuramoto model on the unit circle and their emergent dynamics has been extensively studied in previous literature for some restricted class of initial data based on the Lyapunov functional approach and order parameter approach. In this paper, we extend the previous partial results to cover a generic initial configuration via the detailed analysis on the order parameter measuring the modulus of the centroid. In particular, we present a detailed relaxation dynamics and structure of the resulting asymptotic states for the Lohe sphere model. We also present new gradient flow formulations for the Lohe matrix models with the same one-body Hamiltonians on some group manifolds. As a direct application of this new formulation, we show that every bounded Lohe flow which originated from any initial configuration converges asymptotically. Keywords Complete synchronization · The Kuramoto model · Order parameter · Phase-locked state Mathematics Subject Classification 15B48 · 92D25

B

Dongnam Ko [email protected] Seung-Yeal Ha [email protected] Sang Woo Ryoo [email protected]

1

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Gwanakro 1, Gwanak-gu, Seoul 08826, Republic of Korea

2

Korea Institute for Advanced Study, Hoegiro 85, Dongdaemun-gu, Seoul 02245, Republic of Korea

3

Department of Mathematical Sciences, Seoul National University, Gwanakro 1, Gwanak-gu, Seoul 08826, Republic of Korea

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1 Introduction Collective emergent dynamics often appears in many-body systems in our nature [1,4,31– 33,39]; to name a few, synchronous firing of fireflies, applause in concert halls, Josephson junction arrays in solid mechanics, etc. However, systematic research on such collective phenomena was made only in the past five decades by the two pioneers A. Winfree and Y. Kuramoto in [22,38]. To model synchronization of weakly coupled limit-cycle oscillators, they introduced first-order continuous systems for the oscillators’ phases, and showed that synchronous dynamics can emerge in the competition between the randomness due to distributed natural frequency and sinusoidal couplings between oscillators in a strong coupling regime. Although there were extensive research on the Kuramoto model in last four decades, there are still lots of mysterious phenomena waiting for scientific treatment. In this paper, we are interested in the relaxation dynamics of Lohe oscillators. In two recent papers [24,25], Lohe introduced a first-order consensus model for matrix oscillators and discussed several dynamic features of the proposed model based on analogy with the Kuramoto model and some numerical simulations. Let Ui , Ui† and Hi be a unitary time-evolution operator, its Hermitian conjugate (transpose conjugate) and Hamiltonian associated with the i-th Lohe matrix oscillator, respectively. Then, the first-order evolution law for Ui is governed by the following system: ⎧  N  ⎪ ⎨iU˙ U † = H + iκ  U U † − U U † , t > 0, i = 1, . . . , N , i i i j i i j 2N (1.1) j=1 ⎪ ⎩U (0) = U 0 , U 0 (U 0 )† = I , i d i i i where κ is a nonnegative coupling strength. Note that the unitarity of Ui is propagated along the Lohe flow (1.1). In two dimensions d = 2, we can expand Ui and Hi using Pauli’s matrices as a basis and rewrite the system (1.1) in terms of coefficients xi ∈ S3 ⊂ R4 as follows. x˙i = i xi +

N κ (xk − xi , xk xi ), i = 1, . . . , N . N

(1.2)

k=1

Throughout the paper, we call oscillators whose dynamics are governed by (1.1) and (1.2) as Lohe matrix oscillators and Lohe sphere oscillators, respectively. In a series of papers [6– 10,17], the first author and his collaborators studied extensively the emergent dynamics of both models (1.1) and (1.2) using the Lyapunov functional approach and the order parameter approach, and established the existence of phase-locked states and their stability emerging from some restricted class of initial configurations. However, in analogy with the Kuramoto model, the emergent dynamics can be generic in a large coupling regime. This generality of emergent dynamics has not yet been fully established in previous literature. Numerical simulations support the emergence of phase-locking from generic initial configurations as long as the coupling strength is large enough. This issue motivates the current study of this paper. In this paper, we are interested in the relaxation dynamics of the ensemble of Lohe oscillators toward a position-locked state via the extension of the order parameter approach and gradient flow approach in [6,19]. Before we present our main results, we introduce two functionals measuring the degree of position concentration: for a given Lohe sphere configuration X (t) = (x1 (t), . . . , x N (t)), we set N

1



D(X (t)) := max xi (t) − x j (t), ρ(t) :=

xi (t) , t ≥ 0. 1≤i, j≤N N i=1

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Note that lim D(X ) = 0

t→∞

⇐⇒

lim ρ(t) = 1.

t→∞

Next, we briefly discuss our main results. The first four results deal with the Lohe sphere oscillators on Sd and the last fifth result deals with the Lohe matrix oscillators on three group manifolds. First, we present a detailed relaxation dynamics of the identical Lohe sphere flow with Hi = H , i = 1, . . . , N from generic initial configurations with a positive order parameter. Our first result is about the a priori exponential synchronization (see Theorem 3.1): lim D(X ) = 0

t→∞

⇒

D(X ) ≈ e−κt as t → ∞.

Of course the zero convergence of D(X ) is not completely resolved yet. In fact, our second result deals with the asymptotic behavior of the order parameter ρ. Second, we show that the asymptotic order parameter converges to some value ρ ∞ , for the identical Lohe flow; more precisely, in any positive coupling regime, an initial configuration with non-overlapping positions tends either to a completely synchronized state, where every oscillator tends to a single common position, or to a (N − 1, 1)-type bi-polar state, where all oscillators except one approach the same position. In terms of the order parameter, we have a dichotomy expressed by ρ ∞ = 1, NN−2 . Like the Kuramoto flow in [19], we believe that for generic initial configuration, ρ ∞ will be unity (we refer to Theorem 3.2 for a detailed discussion). Our third result asserts that for a generic initial configuration, there exists a positive coupling strength κ∞ such that if the coupling strength κ is above the value κ∞ , then the Lohe sphere flow of nonidentical oscillators tends to the practical NN−1 -entrainment state asymptotically (see Theorem 4.1). The fourth result deals with practical partial entrainment. More precisely,   0 we show that a generic flow tends to a practical 21 + ρ6 -entrainment state asymptotically.   0 By reducing the required order level to 21 + ρ6 , the dependence of N vanishes in the sufficient condition of the coupling strength κ. Our last result shows that Lohe matrix models on three distinct group manifolds U(d), SU(d), R× SU(d) can be rewritten as gradient flows with analytical potentials. As noted in Sect. 5, the candidates for analytic potentials coincides with the potential for the Kuramoto model in the uni-dimensional case. Then, thanks to a new gradient flow formulation, we can see that all bounded flows are convergent. Of course, our general convergence results are restricted to the identical Hamiltonians Hi = H in (1.1) which generalizes the analogous results for the Kuramoto flow in [19]. The rest of the paper is organized as follows. In Sect. 2, we briefly review two Lohe models, namely, the Lohe matrix model and Lohe sphere model, and we study their basic properties which will be crucially used in later sections. We also provide a brief summary for the previous results. In Sect. 3, we deal with Lohe sphere oscillators on Sd which is a straightforward generalization of (1.2), and provide two estimates on the exponential synchronization and asymptotic values of the order parameter in the relaxation dynamics. In Sect. 4, we introduce two refined concepts, namely “asymptotic p-entrainment” and “practical p-entrainment”, and present two results on the formation of asymptotic entrainment from a generic initial configuration. In Sect. 5, we briefly discuss the convergence of bounded flows whose dynamics is governed by a gradient flow with analytical potential in a Riemannian setting, and we present new gradient flow formulations of the Lohe sphere model and the Lohe matrix models on group manifolds such as U(d), SU(d), and R × SU(d). Finally, Sect. 6 is devoted to the brief summary of our main results. In Appendix A, we provide supplementary calculations in relation with the proof of Theorem 4.2.

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2 Preliminaries In this section, we briefly introduce two Lohe models, namely the Lohe matrix and sphere models which exhibit emergent behaviors, and study their basic properties to be used in later sections.

2.1 The Lohe Matrix Model Let Ui and Ui† be the d × d unitary matrix and the corresponding Hermitian conjugate corresponding to the ith Lohe oscillator, respectively, and let Hi be a given d × d Hermitian matrix whose eigenvalues correspond to the natural frequencies of the ith Lohe oscillator. In [24,25], Max Lohe proposed a non-abelian matrix evolution model for Ui whose dynamics is given by the first-order matrix-valued ODE system:    † † iκ  N iU˙ i Ui† = Hi + 2N j=1 U j Ui − Ui U j , t > 0, i = 1, . . . , N , (2.1) Ui (0) = Ui0 , where the initial matrix Ui0 is in the unitary group U(d) and κ is the nonnegative uniform coupling strength. In the next lemma, we show that system (2.1) exhibits one conservation law and a kind of rotational symmetry. Lemma 2.1 [24,25] (Conservation law and symmetry) (1) Let {Ui } be a solution to (2.1) with initial data {Ui0 }. Then the quadratic quantity Ui Ui† is conserved along the Lohe flow (2.1): Ui (t)Ui† (t) = Ui0 Ui0† , t ≥ 0, 1 ≤ i ≤ N . (2) The Lohe system (2.1) is invariant under right-translation by a unitary matrix in the sense that if L ∈ U(d) and Vi = Ui L, then Vi satisfies    † † iκ  N V V − V V i V˙i Vi† = Hi + 2N j i j=1 i j , t > 0, i = 1, . . . , N , Vi (0) = Ui0 L. Next, we study the connection between the Lohe matrix model and the Kuramoto model. As a matter of fact, the Lohe matrix model has been introduced as a nonabelian counterpart of the Kuramoto model (2.3) on U(d). To see this connection, we consider the uni-dimensional case and ansatzs for Ui and Hi : Ui := e−iθi ,

Hi := νi ∈ R.

(2.2)

Then, we substitute (2.2) into (2.1) to recover the Kuramoto model [1,22,23]: θ˙i = νi +

N κ sin(θ j − θi ), t > 0, N

(2.3)

j=1

System (2.3) has been extensively studied in [2,3,11–16,18,21,27–29,35–37]. On the other hand, if we set κ = 0 and multiply the unitary matrix Ui to both sides of (2.1), we obtain the Schrödinger equation in finite-dimensional form: iU˙ i Ui† = Hi , or iU˙ i = Hi Ui .

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In this case, Ui and Hi can be regarded as the unitary time-evolution operator and Hamiltonian, respectively. Thus, we can regard the Lohe matrix model (2.1) as a generalized model for the Kuramoto model and the Schrödinger equation simultaneously. In general, like the Kuramoto model, the Lohe matrix model does not admit the equilibrium solution. Thus, we need to study a weaker concept of equilibria, namely, relative equilibria or phase-locked states. Now, we recall the definition of phase-locked states to the Lohe matrix model as follows. Definition 2.1 [17] Let {Ui (t)} be a solution to (2.1). (1) {Ui (t)} is a phase-locked state if and only if Ui (t)U j (t)† is a constant of motion for all t ≥ 0 and each pair i, j. (2) The Lohe flow {Ui (t)} achieves asymptotic phase-locking if and only if the limit of Ui U †j exists, as t tends to ∞, for each pair i, j. Remark 2.1 For d = 1, via the ansatz (2.2), it is easy to see that Ui (t)U j (t)† = e−i(θi −θ j ) : constant, i.e., θi − θ j : constant which coincides with the usual definition of phase-locked state for the Kuramoto model. Next, we recall the structure of the phase-locked state for (2.1) as follows. Proposition 2.1 [24,25] The phase-locked states {Ui } of (2.1) are of the form Ui = Ui∞ e−it , where Ui∞ ∈ U(d) and  is the constant d × d Hermitian matrix satisfying Ui∞ Ui∞† = Hi +

N  iκ  ∞ ∞† Uk Ui − Ui∞ Uk∞† . 2N k=1

2.2 The Lohe Sphere Model In this subsection, we discuss the Lohe sphere model (or the swarm model [30] on the dsphere Sd in the control theory community for identical particles) that can be derived from the Lohe matrix model for d = 2. Next, we provide a heuristic argument for the derivation of the Lohe sphere model following [6,24,25]. Consider a two-dimensional case with d = 2 in (2.1). In this case, we can use Pauli’s matrices {I2 , σ1 , σ2 , σ3 } as a basis of the unitary group U(2):

⎧   ⎪ xi4 + ixi1 xi2 + ixi3 ⎨U := e−iθi i 3 x k σ + x 4 I = e−iθi , i k=1 i k i 2 (2.4) −xi2 + ixi3 xi4 − ixi1 ⎪ 3 ⎩ k Hi = k=1 ωi σk + νi I2 , where I2 and σi are the identity matrix and Pauli matrices, respectively, defined by         10 1 0 0 −i 01 I2 := , σ1 := , σ2 := , σ3 := , 01 0 −1 i 0 10 and ωi = (ωi1 , ωi2 , ωi3 ) is a real three-vector, and the natural frequency νi is associated with the U(1) component of Ui . After some algebraic manipulations, we obtain 5N equations for the angles θi and the four-vectors xi ∈ S3 ⊂ R4 :

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θ˙i = νi +

N κ sin(θk − θi )xi , xk , t ∈ R, 1 ≤ i ≤ N , N k=1

(2.5)

N κ cos(θk − θi )(xk − xi , xk xi ), x˙i = i xi + N k=1

where ·, · is the standard inner product on R4 and i is a real 4×4 skew-symmetric matrix: ⎞ ⎛ 0 −ωi3 ωi2 −ωi1 ⎜ ω3 0 −ω1 −ω2 ⎟ i i i ⎟ i := ⎜ ⎝ −ω2 ω1 0 −ω3 ⎠ . i i i ωi1 ωi2 ωi3 0 Here, we concern the equation on S3 . We take θi = 0 and νi = 0 in (2.4) to obtain the swarming sphere model in (2.5) with xi ∈ S3 : x˙i = i xi +

N κ (xk − xi , xk xi ), t > 0, i = 1, . . . , N . N

(2.6)

k=1

It is easy to see that the unit modulus of xi is invariant along the dynamics (2.6). Now, we formally extend system (2.6) as a dynamical system on the unit sphere Sd to obtain the Lohe sphere model on Sd : x˙i = i xi +

N κ (xk − xi , xk xi ), i = 1, . . . , N , N

(2.7)

k=1

where i is a skew-symmetric matrix in Rd+1 , called the natural frequency. Below we introduce the emergent dynamics of (2.7). For this, we first introduce several functionals: For t ≥ 0 and X = (x1 , . . . , x N ), xc :=

N 1 xi , ρ := xc , A(t) := min xi (t), xc (t), B(t) := min xi , x j , 1≤i≤N 1≤i, j≤N N i=1

D() := max i −  j ∞ ,

D2 () := max i −  j op ,

1≤i, j≤N

1≤i, j≤N

D(X (t)) := max xi (t) − x j (t). 1≤i, j≤N

(2.8) Here the norm  · op is the operator norm as a linear operator of to see that

Rd+1 .

Then, it is easy

0 ≤ ρ ≤ 1, − 1 ≤ A ≤ 1. and we use the definition of A to see A = min xi (t), x c (t) ≤ 1≤i≤N

  N N 1 1 xi , xc  = xi , xc = xc 2 = ρ 2 . N N i=1

(2.9)

i=1

Lemma 2.2 [6] Let X = (x1 , . . . , x N ) be the solution to system (2.7) satisfying the unit modulus conditions: xi  = 1, i = 1, . . . , N .

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Then, we have the following assertions: (1) The order parameter ρ satisfies the following differential equation:   N N dρ 2 2 1 = xc , i xi  + 2κ ρ 2 − xi , xc 2 , t > 0. dt N N i=1

(2.10)

i=1

Then, we also have   N dρ 2 1 xi , xc 2 , t > 0. ≥ −2D2 ()ρ + 2κ ρ 2 − dt N

(2.11)

i=1

(2) As long as A(t) > 0, the functional A satisfies dA ≥ −D2 () + κ A(1 − A). dt

(2.12)

Proof The detailed derivations for (2.10) and (2.12) can be found in Lemmas 3.4 and 3.6 of [6], respectively.   Lemma 2.3 [6] Suppose that the coupling strength and initial position satisfy κ > 0, xi0  = 1, xi0  = x 0j if i  = j, ρ 0 := ρ(0) > 0, and let X = (x1 , . . . , x N ) be a solution to (2.7). Then, we have d xi , x j  = (i −  j )xi , x j  + κ(1 − xi , x j )xc , xi + x j , t > 0, dt or, equivalently, d (1 − xi , x j ) = ( j − i )xi , x j  − κ(1 − xi , x j )xc , xi + x j  t > 0. dt Proof For the ensemble of identical oscillators with i = 0, a detailed proof can be found in Lemma 4.5 in [6]. For the readers’ convenience, we briefly discuss its proof. We take the inner product of (2.7) and x j to obtain x˙i , x j  = i xi , x j  +

N  κ  xk , x j  − xi , xk xi , x j  . N

(2.13)

k=1

Similarly, we have x˙ j , xi  = − j xi , x j  +

N  κ  xk , xi  − x j , xk xi , x j  , N

(2.14)

k=1

where we used the skew-symmetry of i :  j x j , xi  = −x j ,  j xi  = − j xi , x j . Finally, we add (2.13) and (2.14) to obtain

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 κ  d xi , x j  = (i −  j )xi , x j  + xk , xi + x j  − xi + x j , xk xi , x j  dt N k=1   = (i −  j )xi , x j  + κ xc , xi + x j  − xi + x j , xc xi , x j  N

= (i −  j )xi , x j  + κ(1 − xi , x j )xc , xi + x j .   Theorem 2.1 [5,6] (Identical oscillators) Suppose that the coupling strength and initial data satisfy κ > 0, i = 0, xi0  = 1, 1 ≤ i ≤ N , and let X = (x1 , . . . , x N ) be a solution to (2.7). Then, the following assertions hold. (1) If the initial data X 0 satisfies an extra condition 0 < D(X 0 ) := D(X (0)) <

1 , 2

(2.15)

then the diameter D(X ) decays exponentially: D(X (t))  D(X 0 )e−κt , t ≥ 0. (2) If the initial data satisfies an extra condition A0 := A(0) > 0,

(2.16)

then order parameter ρ approaches the unity: lim ρ(t) = 1.

t→∞

Proof (1) The exponential decay of D(X ) follows from a Gronwall type differential inequality for D(X )(in Lemma 3.1 of [5]): d D(X ) ≤ −κ D(X )(1 − 2D(X )). dt (2) For a detailed proof, we refer to Theorem 4.4 in [6]. In the course of the proof, we used the fact that ρ is bounded and non-decreasing so that it has a limit as t → ∞ and a contradiction argument.   Next, we state the emergent dynamics of the ensemble of nonidentical Lohe sphere oscillators as follows. Theorem 2.2 [6] (Nonidentical oscillators) Suppose that the coupling strength and initial data satisfy κ > 4N D2 (), xi0  = 1,    κ − κ 2 − 4N D ()κ 2 0 0 0 . min ρ , min xi , xc  > 1≤i≤N 2κ Then we have practical synchronization: lim lim inf ρ(t) = 1.

κ→∞ t→∞

Remark 2.2 In the proof of Theorem 2.2, we use a differential inequality:

123

(2.17)

On the Relaxation Dynamics of Lohe Oscillators...

κρ(1 − ρ) dρ ≥ , dt N which depends on the number of oscillators N . This is why N appears in the conditions (2.17). In the following two sections, we will generalize or strengthen the results in Theorems 2.1 and 2.2 to cover generic initial configurations.

3 Identical Lohe Sphere Oscillators In this section, we study the asymptotic dynamics of identical Lohe sphere oscillators with i =  under generic initial configurations via the order parameter ρ. Note that for identical oscillators with i = , system (2.7) becomes x˙i = xi +

N κ (xk − xi , xk xi ), i = 1, . . . , N . N k=1

Then, it follows from the operator splitting argument in [6] that we can assume that  = 0 without loss of generality: x˙i =

N κ (xk − xi , xk xi ), i = 1, . . . , N . N

(3.1)

k=1

In the following two subsections, we are interested in whether the restrictive conditions (2.15) and (2.16) in Theorem 2.1 can be replaced by generic conditions or not.

3.1 An Exponential Decay of D(X) In this subsection, we provide an exponential synchronization estimate under the a priori assumption “complete positional synchronization". More precisely, our first result can be summarized as follows. Theorem 3.1 (Exponential synchronization) Suppose that the coupling strength, i and initial data satisfy κ > 0, i = 0, xi0  = 1, i = 1, . . . , N . If X is a solution to (3.1) with complete positional synchronization: lim D(X (t)) = 0,

t→∞

then, there exist constants C1 , C2 > 0 which depend on initial data such that C1 e−κt ≤ D(X (t)) ≤ C2 e−κt , ∀ t > 0. Proof We briefly outline the proof as follows. • Step A For a lower bound estimate, we will prove the following estimates sequentially. ρ ≤ 1 + O(e−2κt )

⇒

D(X ) ≥ C1 e−κt .

• Step B For an upper bound estimate, we will prove the following estimates sequentially.

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lim D(X ) = 0 ⇒ lim B(t) = 1

t→∞

t→∞

⇒ B(t) ≥ 1 − |O(1)|e−2κt

⇒

D(X ) ≤ C2 e−κt .

The detailed proof will be provided after a series of lemmas in the sequel.

 

3.1.1 Basic Lemmas We assume that complete synchronization occurs asymptotically: lim D(X (t)) = 0.

t→∞

(3.2)

Under this a priori estimate (3.2), we will provide estimates for the order parameter ρ, the angle functional A defined in (2.8) and a supplementary angle functional B in three lemmas. Lemma 3.1 Let X = (x1 , . . . , x N ) be a solution to (3.1). Then, we have the following assertions. (1) As long as A > 0, we have A(t) ≥ 1 −



1 A0

 − 1 e−κt + O(e−2κt ), as t → ∞.

(2) As long as A > 0, the order parameter ρ satisfies differential inequalities: κρ(1 − ρ) ≤

dρ ≤ κρ(1 − ρ 2 ), t > 0. dt

Proof (1) It follows from (2.12) with D() = 0 that as long as A > 0, we have dA ≥ κ A(1 − A), t > 0. dt Hence, we have A(t) ≥

1+



1 1 A0

 , t ≥ 0, − 1 e−κt

which yields the desired estimate. (2) We use (2.10) in Lemma 2.2 to find N   dρ 2 1 xi , xc 2 . = 2κ ρ 2 − dt N

(3.3)

i=1

N To relate the term N1 i=1 xi , xc 2 with ρ, we take two steps. Note that there exists an orthogonal matrix M(t) ∈ O(d + 1) that varies smoothly in time such that M(t)xc (t) = (0, . . . , 0, ρ(t))T . We set x¯i (t) := M(t)xi (t), βi (t) := (x¯i (t))d+1 , i = 1, . . . , N . Then, since

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On the Relaxation Dynamics of Lohe Oscillators... N 1 x¯i (t) = M(t)xc (t) = (0, . . . , 0, ρ(t))T , N i=1

we have N N N d+1 1 1 1  M(t)xi (t) βi (t) = (x¯i (t))d+1 = = (M(t)xc (t))d+1 = ρ(t). N N N i=1

i=1

i=1

(3.4) On the other hand, note that xi , xc  = M(t)xi , M(t)xc  = ρβi .

(3.5)

Thus, we use (3.3) and (3.5) to obtain ρβi =

N 1 x j , xi  ≥ A > 0 N

⇒

βi > 0,

j=1

and N  1 2 2 dρ 2 = 2κ ρ 2 − ρ βi . dt N i=1

This again yields N  dρ 1 2 βi . = κρ 1 − dt N

(3.6)

i=1

Now, we claim: ρ2 ≤

N 1 2 βi ≤ ρ. N

(3.7)

i=1

Clearly, (3.6) and (3.7) yield the desired differential inequalities. • (Left-hand inequality in (3.7)) We use the Cauchy–Schwarz inequality and (3.4) to obtain

2 N 1 1 2 βi ≥ βi = ρ2. N N i

i=1

• (Right-hand inequality in (3.7)) Consider the following optimization problem:   maximize N1 i βˆi2  subject to N1 i βˆi = ρ, βˆi ∈ [0, 1].

(3.8)

The object function in (3.8) is a continuous function defined on a compact domain and thus attains a maximum. Since the map x  → x 2 is convex, a maximum point must have the form (βˆ1 , . . . , βˆ N ) = (0, . . . , 0, λ, 1, . . . , 1), λ ∈ [0, 1), up to a permutation.    m times

(3.9)

Here, m ∈ Z is determined by the following relationship with N ρ,

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m ≤ Nρ =

N

βˆi = m + λ < m + 1, 0 ≤ m ≤ N .

i=1

Then, λ = N ρ − m ∈ [0, 1). Hence 1 2 1 1 βˆi ≤ (12 + · · · + 12 +λ2 ) ≤ (m + λ) = ρ.   N N  N i m times  

This proves the claim (3.7), which along with (3.6), completes the proof. Remark 3.1 Below, we provide several implications of the results in Lemma 3.1. 1. The first estimate limt→∞ A(t) = 1 together with the relation A ≤ ρ 2 in (2.9) yield lim ρ(t) = 1.

t→∞

2. As a direct application of differential inequalities (2) in Lemma 3.1, for A > 0, 1+



1 1 ρ0



− 1 e−κt

1 ≤ ρ(t) ≤    1 + ρ12 − 1 e−2κt

1/2

, t ≥ 0.

0

Lemma 3.2 Let X = (x1 , . . . , x N ) be a given vector in (Sd ) N . Then, we have  √ D(X ) ≥ 2 − 2ρ 2 and D(X ) = 2 − 2B. Proof (i) By definition of D(X ) and xi − x j 2 = 2 − 2xi , x j , we have  D(X ) ≥

1/2 1/2  N N 1 1 2 x − x  = (2 − 2x , x ) i j i j N2 N2 i, j=1

 =

1 (2N 2 − N N2

i, j=1

N

1/2  ρβi ) = 2 − 2ρ 2 .

i=1

(ii) By definition of D(X ) and B, we have D(X ) = =

max xi − x j  = max

1≤i, j≤N

√

1≤i, j≤N



2 − 2xi , x j  =

! 2 − 2 min xi , x j  1≤i, j≤N

2 − 2B .  

Lemma 3.3 Let X = (x1 , . . . , x N ) be a solution to (3.1). Then, the functional B defined in (2.8) satisfies dB ≥ 2κ B(1 − B), t > 0. dt Proof (i) For a given t > 0, let i t , jt be indices such that B(t) = xit , x jt .

Then we have, by Lemma 2.3,

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(3.10)

On the Relaxation Dynamics of Lohe Oscillators...

dB d = xit , x jt  = κxc , xit + x jt (1 − xit , x jt ) = κxc , xit + x jt (1 − B). dt dt (3.11) On the other hand, we use the definition of B to see xc , xit  =

N N 1 1 xk , xit  ≥ B = B, N N k=1

(3.12)

k=1

and similarly for x jt . We combine (3.11) and (3.12), together with B ≤ 1, to obtain the desired estimate.   Remark 3.2 It follows from the differential inequality in (3.10) that if B0 := B(0) > 0, then we have 1   B(t) ≥ , t ≥ 0. 1 1 + B0 − 1 e−2κt We are now ready to provide a proof of Theorem 3.1.

3.1.2 Proof of Theorem 3.1 Below, we provide its proof into two steps. • Step A (LHS inequality): It follows from the estimates in Remark 3.1 that we have   1 1 ρ(t) ≤ 1 − − 1 e−2κt + O(e−4κt ), 2 |ρ 0 |2 We apply the above estimate and Lemma 3.2 to find     1 2 D(X ) ≥ 2 − 2ρ ≥ 2 − 1 e−2κt + O(e−4κt ) |ρ 0 |2 "   1 = 2 − 1 · e−κt + O(e−3κt ). |ρ 0 |2

1/2

(3.13)

• Step B (RHS inequality): By the assumption lim D(X (t)) = 0 and Lemma 3.2 we have t→∞

lim B(t) = 1.

t→∞

Thus we may change the starting time so that B0 > 0. This is possible since (3.1) is an autonomous system. Thus, we can use the result in Remark 3.2 to derive   1 B(t) ≥ 1 − − 1 e−2κt + O(e−4κt ). 0 B

Then, we substitute the above estimate into Lemma 3.2 to obtain "   √ 1 − 1 e−2κt + O(e−4κt ) D(X ) = 2 − 2B ≤ 2 0 B

"   1 − 1 · e−κt + O(e−3κt ). = 2 0

(3.14)

B

Finally, we combine (3.13) and (3.14) to derive the desired estimates. This completes the proof.

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3.2 Asymptotics of the Order Parameter In this subsection, we study possible asymptotic values of ρ as t → ∞. From the definition of ρ and the relation (2.11) in Lemma 2.2, we can see that ρ is bounded above by 1 and non-decreasing. Thus, we have ∃ lim ρ(t) = ρ ∞ .

(3.15)

t→∞

In this sequel, we study possible values of ρ ∞ . In Theorem 2.1, we have shown that if > 0, then we have ρ ∞ = 1. Thus, what matters is what will happen to ρ ∞ if A0 < 0. To motivate the asymptotics of ρ, we first consider the simplest system. A0

Example 3.1 Consider a two-particle system: x˙1 = Note that

κ κ (x2 − x2 , x1 x1 ), x˙2 = (x1 − x2 , x1 x2 ). 2 2

(3.16)

x + x 2 1 + x1 , x2  2

1 ρ 2 = xc 2 =

,

= 2 2   1 1 + x1 , x2  = ρ 2 . xc , x1  = xc , x2  = 2

(3.17)

By Lemma 2.3, we have d x1 , x2  = κ(1 − x1 , x2 2 ). dt Then, by direct calculation, we have x1 , x2 (t) =

(1 + x10 , x20 )e2κt − (1 − x10 , x20 ) (1 + x10 , x20 )e2κt + (1 − x10 , x20 )

, t ≥ 0.

This yields  1, x10 , x20   = − 1, lim x1 , x2 (t) = t→∞ − 1, x10 , x20  = − 1. Finally, we combine (3.17), (3.18) and the increasing property of ρ to find   1, ρ 0  = 0, 1, ρ 0  = 0, ∞ ρ = and lim xc , x1  = lim xc , x2  = 0 t→∞ t→∞ 0, ρ = 0. 0, ρ 0 = 0.

(3.18)

(3.19)

Note that for generic initial data X 0 , we have ρ ∞ = 1. As we can see in Example 3.1, it is important to see the asymptotic phase distribution of oscillators xi if we want to know ρ ∞ . In fact, we can show that all the xi tend toward the direction of xc or −xc , in the following sense: Lemma 3.4 Let X = (x1 , . . . , x N ) be a solution to (3.1) with κ > 0 and i = 0 for all i. Then, lim xc (t), x j (t) = ±ρ ∞ , for all j.

t→∞

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Proof We will use Barbalat’s lemma to prove the existence of the limit value. We already know that ρ has a limit value ρ ∞ . From Lemma 2.2,   N dρ 2 1 xi , xc 2 . (3.20) = 2κ ρ 2 − dt N i=1

d 2ρ

dρ 2 goes to zero, and From Barbalat’s lemma, if we already know 2 is bounded, then dt dt then (ρ ∞ )2 −

N 1 lim xi , xc 2 = 0, t→∞ N i=1

d 2ρ which proves our claim. It is quite straightforward to show that 2 is bounded, which looks dt reasonable from the analyticity of the dynamics (2.7). First, from (3.20), # # # dρ # # # ≤ ρκ. # dt # On the other hand, from Lemma 2.3, d xi , x j  = κ(1 − xi , x j )xc , xi + x j , dt hence we have

# # # # # # N # #1 # #d d # ≤ 4ρκ. # xi , xc # ≤ # , x  x i j # #N # # dt # j=1 dt #

Finally, using (3.20) again,   N   dρ 2 d dρ 2 1 d = 2κ − xi , xc 2 , dt dt dt N dt i=1

so that we have # # # 2 # # #2 # # N #d ρ # # dρ # # dρ # #d # 2 2ρ ## 2 ## + 2 ## ## ≤ 2κ 2ρ ## ## + |xi , xc | ## xi , xc ## ≤ 20ρ 2 κ 2 , dt dt dt N dt i=1

d 2ρ is bounded, and the conclusion holds.   dt 2 Remark 3.3 An argument similar to that of Lemma 3.4 will be employed in Remark 5.2, to obtain an analogous result for the Lohe model on U(d).

hence the second derivative

Below, we will show that for a many-body system with N ≥ 3, there is a dichotomy: either all particles aggregate to the same position asymptotically, or to a bi-polar state with the property that N − 1 particles aggregate to the same position x∗ ∈ Sd and the remaining one oscillator approaches the antipodal position −x∗ . Theorem 3.2 Suppose that the coupling strength, i and initial velocities satisfy κ > 0, i = 0, xi0  = 1, xi0  = x 0j if i  = j, ρ 0 > 0, and let X = (x1 , . . . , x N ) be a solution to (3.1). Then, we have the following assertions:

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(1) There exists at most one i such that lim xc (t), xi (t) = −ρ ∞ ,

t→∞

and then others j  = i satisfy lim xc (t), x j (t) = ρ ∞ .

t→∞

(2) ρ ∞ takes at most two values: ρ∞ ∈

N −2  ,1 . N

Proof (1) First, from Lemma 3.4, lim xc (t), x j (t) = ±ρ ∞ , for any j.

t→∞

Suppose that there are two distinct indices i 0  = j0 such that lim xc (t), xi0 (t) = lim xc (t), x j0 (t) = −ρ ∞ .

t→∞

t→∞

(3.21)

Then, for such indices i 0 and j0 , we have lim sup xi0 (t) − x j0 (t) t→∞





1 1





≤ lim sup xi0 (t) + xc (t) + lim sup − xc (t) − x j0 (t)

ρ(t) ρ(t) t→∞ t→∞   1/2 2 2 = lim sup 2 + + lim sup 2 + xi0 (t), xc (t) x j0 (t), xc (t) ρ(t) ρ(t) t→∞ t→∞

1/2

= 0. Hence, we have lim xi0 (t), x j0 (t) = 1.

t→∞

(3.22)

On the other hand, it follows from (3.21) and non-decreasing ρ that we have lim xc (t), xi0 (t) = lim xc (t), x j0 (t) = −ρ ∞ ≤ −ρ 0 .

t→∞

t→∞

Thus, there exists a positive constant t0 ≥ 0 such that xc (t), xi0 (t), xc (t), x j0 (t) ≤ −

ρ0 , ∀ t ≥ t0 . 2

This and Lemma 2.3 give d xi , x j  = κ(xc , xi0  + xc , x j0 )(1 − xi0 , x j0 ) dt 0 0 ≤ −κρ 0 (1 − xi0 , x j0 ) for t > t0 . Again this yields d (1 − xi , x j ) ≥ κρ 0 (1 − xi , x j ) for t > t0 . dt Since xi0  = x 0j , by the uniqueness of solutions to (3.1), we must have xi (t)  = x j (t) for all t, i.e., we have

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(3.23)

On the Relaxation Dynamics of Lohe Oscillators...

1 − xi (t), x j (t) > 0 for all t.

(3.24)

We combine (3.23) and (3.24) to get 1 − xi , x j (t) ≥ eκρ t (1 − xi0 , x 0j ), t ≥ t0 . 0

This contradicts (3.22). Thus we have the desired result. (2) From the first estimate (1), the following dichotomy holds. • Case A (complete positional synchronization): Suppose that lim xc (t), xi (t) = ρ ∞ , ∀ i = 1, . . . , N ,

t→∞

i.e., xi will be aligned in the direction of xc asymptotically. Since this is for every i, we have N 1 lim xc (t), xi (t) = lim xc (t), xc (t) = (ρ ∞ )2 . t→∞ t→∞ N

ρ ∞ = lim xc (t), xi (t) = t→∞

i=1

(3.25) Therefore, we have ρ ∞ = 1. • Case B (bi-polar positional synchronization): It follows from the result (i) of Lemma 3.2 that lim xc (t), xi (t) = ρ ∞ , i = 1, . . . , N − 1,

t→∞

lim xc (t), x N (t) = −ρ ∞ ,

t→∞

up to a permutation of the indices. We use the argument in (3.25) to derive ρ∞ =

N −2 . N

On the other hand, it is easy to see that lim xi (t), x j (t) = 1, 1 ≤ i, j ≤ N − 1,

t→∞

(3.26)

lim xi (t), x N (t) = − 1, i = 1, . . . , N − 1.

t→∞

Finally, we combine the results in Case A and Case B to obtain the desired result.

 

Remark 3.4 It follows from the differential inequalities of ρ that if ρ 0 = 0, then ρ(t) = 0, t ≥ 0, i.e., ρ ∞ = 0. In the course of the proof of Theorem 3.2, we have seen that a dichotomy occurs. Either all particles aggregate to their center of mass xc , or N −1 points aggregate to one position and the remaining one particle approaches the opposite position. In Theorem 3.1, we have shown that the first complete positional synchronization occurs exponentially fast. Next, we show that for the latter case, the N − 1 particles also aggregate to the same position exponentially fast. To prove this, we introduce functionals with the first N −1 partial configuration {x 1 , . . . , x N −1 }: $ := B

min

xi , x j ,

1≤i, j≤N −1

$ ) := D(X

max

1≤i, j≤N −1

xi − x j .

(3.27)

Lemma 3.5 Suppose that the coupling strength, i and initial configuration satisfy

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N ≥ 3, κ > 0, i = 0, xi0  = 1, $0 > xi0  = x 0j if i  = j, ρ 0 > 0, B

1 , N −1

and let X = (x1 , . . . , x N ) be a solution to (3.1) satisfying the a priori estimate: lim xc (t), x N (t) = −ρ ∞ .

t→∞

Then, we have

  $ 2κ(N − 1) dB 1 $− $), t > 0. B ≥ (1 − B dt N N −1

Proof Let i t and jt be indices depending on time t such that $(t). xit , x jt  = B Then, we have 1 xc , xit  = N Similarly, we have 1 xc , x jt  = N

N −1

xk , xit  + x N , xit  ≥

k=1

N −1

xk , x jt  + x N , x jt  ≥

k=1

& 1 % $− 1 . (N − 1)B N

(3.28)

& 1 % $− 1 . (N − 1)B N

(3.29)

Thus, we use (3.28), (3.29) and Lemma 2.3 to obtain

  $ dB 2κ(N − 1) 1 d $ $ $), B− = xit , x jt  = κxc , xit + x jt (1 − B) ≥ (1 − B dt dt N N −1

which yields the desired differential inequality. ˜ ) as follows. We are now ready to provide the exponential decay of D(X Proposition 3.1 Suppose that the coupling strength, i and initial velocities satisfy N ≥ 3, κ > 0, i = 0, xi0  = 1, xi0  = x 0j if i  = j, ρ 0 > 0, and let X = (x1 , . . . , x N ) be a solution to (3.1) satisfying the a priori estimate: lim xc (t), x N (t) = −ρ ∞ .

t→∞

Then, there exists a nonnegative constant t0 ≥ 0 such that    1 κ(N − 1) $ (t0 )), t > t0 . $ $ B(t0 ) − D(X (t)) ≤ exp − (t − t0 ) D(X N N −1 Proof It follows from Theorem 3.1 that there exists t0 ≥ 0 such that $(t0 ) > B

1 . N −1

By Lemma 3.5, we see that $(t) ≥ B $(t0 ), t ≥ t0 . B On the other hand, an argument similar to that given in Lemma 3.3 gives

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$ )= D(X and



$ 2 − 2B

  $ d $ κ(N − 1) 1 −1 d B −1 $ $) · B− D(X ) =  ≤  (2 − 2B dt N N −1 $ dt $ 2 − 2B 2 − 2B   κ(N − 1) 1 $ ), t ≥ t0 . $(t0 ) − ≤− B D(X N N −1

Then this Gronwall type inequality completes the proof.

 

Remark 3.5 As noted in the two-particle system, we conjecture that for a generic initial configuration X 0 with κ > 0, we have ρ ∞ = 1. This is because the bi-polar states are linearly unstable, as demonstrated in Appendix in Theorem B.1.

4 Nonidentical Lohe Sphere Oscillators In this section, we study emergent dynamics of the ensemble of the Lohe sphere oscillators for a generic initial configuration by improving the result of Theorem 4.9 of [6] (see Theorem 2.2). First, we introduce new concepts such as asymptotic p-entrainment and practical p-entrainment as follows. N be an ensemble of initial data on the Lohe sphere model Definition 4.1 Let X 0 = {xi0 }i=1 (2.7).

(1) The ensemble X 0 exhibits asymptotic p-entrainment for some p ∈ (0, 1) if there exist S ⊂ {1, . . . , N } and κ0 > 0 such that |S| ≥ p and lim sup max xi (t) − x j (t) = 0, ∀ κ > κ0 . N t→∞ i, j∈S (2) The ensemble X 0 exhibits practical p-entrainment for some p ∈ (0, 1) if there exists S ⊂ {1, . . . , N } such that for any > 0, there exist κ( ) and T ( ) satisfying |S| ≥ p and N

sup max xi (t) − x j (t) ≤ , ∀ κ > κ( ).

t>T ( ) i, j∈S

(4.1)

Remark 4.1 Note that Theorem 3.2 states that identical Lohe sphere oscillators with a generic initial configuration achieves asymptotic NN−1 -entrainment under any positive coupling κ > 0.

4.1 Emergence of Practical Entrainment In this subsection, we present practical entrainment for (2.7) introduced in Definition 4.1. For this, we first present several basic lemmas. Consider a first-order differential inequality: y  ≥ f (y), t > 0,

y(0) = y 0 ,

(4.2)

where f has two distinct positive real roots y− < y+ satisfying

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⎧ ⎨ < 0, y ∈ [0, y− ), f (y± ) = 0 and f (y) > 0, y ∈ (y− , y+ ), ⎩ < 0, y ∈ (y+ , ∞). Lemma 4.1 [6] Suppose that the differentiable function y = y(t) satisfies (4.2) with y 0 ∈ (y− , ∞). Then, we have lim inf y(t) ≥ y+ , and y(t) ≥ y− , ∀ t > 0, t→∞

Proof The proof can be found in Lemma 3.7 [6].

 

Lemma 4.2 Suppose that the coupling strength, natural frequencies, and initial data satisfy κ>

2 p D2 () , D() > 0, xi0  = 1, (2 p − 1)2

and let X (t) = (x1 , . . . , x N ) be a solution to (2.7) and let S ⊂ {1, . . . , N } be a set with the following a priori estimate: ' ( !  1 D2 ()  |S| 1 p := > and B S (0) > 1 − 1 − 4p 1 − p + , (4.3) N 2 2p 2κ where B S (t) := min xi (t), x j (t). i, j∈S

Then, we have

' ( !  1 D2 ()  lim inf B S (t) ≥ 1 + 1 − 4p 1 − p + . t→∞ 2p 2κ

Thus, the ensembles satisfying (4.3) exhibit the practical p-entrainment since ' ( !  D2 ()  1 1 + 1 − 4p 1 − p + → 1 as κ → ∞. 2p 2κ Proof We may find i t , jt ∈ S such that B S (t) = xit , x jt .

Then, we use Lemma 2.3 to obtain N  1 d d BS = xit , x jt  = (it −  jt )xit , x jt  + κ 1 − xit , x jt  xk , xit + x jt  dt dt N k=1

1 = (it −  jt )xit , x jt  + κ(1 − B S ) xk , xit + x jt  + xk , xit + x jt  N k∈S k ∈S /  1 ≥ −(it −  jt )xit  + κ(1 − B S ) |S| · 2B S + (N − |S|) · (−2) N ≥ −D2 () + κ(1 − B S ) ( p · 2B S + (1 − p) · (−2))   ) D2 () * = 2κ − p B2S + B S − 1 − p + , 2κ

(4.4) where we have used the fact

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xi , x j  ≥ B S for i, j ∈ S, xi , x j  ≥ −1, for i, j ∈ S c . Finally, we apply Lemma 4.1 with y = BS ,

 D2 () * f (y) = 2κ − py + y − 1 − p + 2κ )



2

 

to find the desired estimate.

Next, we study the deviation of the Lohe flow for identical and nonidentical particles. For this, consider the Lohe sphere systems: for i = 1, . . . , N , ⎧ % & κ N ⎪ ⎨ y˙i = N j=1 y j − y%i , y j yi , t >&0, N (4.5) x˙i = i xi + Nκ j=1 x j − x i , x j x i , ⎪ ⎩ 0 0 yi (0) = xi (0) = yi , yi  = 1. We define ∞ := max i ∞ , X − Y ∞ := max xi − yi . 1≤i≤N

1≤i≤N

Lemma 4.3 Let X and Y be solutions to (5.14) with the same initial data, respectively. Then we have & op % 4κt e − 1 , t ≥ 0. X (t) − Y (t)∞ ≤ 4κ Proof Note that i xi  ≤ op and

% & % &

x j − xi , x j xi − y j − yi , y j yi











≤ x j − y j + yi , y j (yi − xi ) + yi − xi , y j xi + xi , y j − x j xi



≤ 2 x j − y j + 2 xi − yi  ≤ 4X − Y ∞ .

We choose i t so that X − Y ∞ = xit − yit . Then, we have d d X − Y ∞ = xit − yit  ≤ op + 4κX − Y ∞ . dt dt This yields the desired estimate.

 

Below, we present an improved result on the emergence of practical synchronization in Theorem 2.2 where the N -dependent differential inequality was employed: dρ κρ(1 − ρ) ≥ . dt N This is why N appears in (2.17). Now we replace this inequality by the L.H.S. of Lemma 3.1, which does not depend on N : dρ ≥ κρ(1 − ρ), dt and emulate the proof of Theorem 2.2 to remove N -dependence in the following corollary.

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Corollary 4.1 Suppose that the coupling strength and initial data satisfy  κ − κ 2 − 4D2 ()κ 0 0 0 0 κ > 4D2 (), xi  = 1, min{ρ , min xi , xc } > , 1≤i≤N 2κ and let X be a solution to (2.7). Then, we have lim lim inf ρ(t) = 1.

κ→∞ t→∞

Proof It follows from the same argument in Lemma 3.1 that we have dρ ≥ −D2 () + κρ(1 − ρ). dt The rest of the proof follows from Lemma 4.1.

 

In the following theorem, we show that in a large coupling regime, the Lohe sphere flow evolves toward the NN−1 -entrainment state asymptotically. Theorem 4.1 Suppose that D() and the initial data satisfy D() > 0, xi0  = 1, xi0  = x 0j if i  = j, ρ 0 > 0. Then, there exists a positive coupling strength κ∞ depending on the initial data such that if κ > κ∞ , then for the solution X to (2.7), X evolves toward a practical NN−1 -entrainment state. Proof For a two-particle system it is easy to see the emergence of practical entrainment, as demonstrated in Example 3.1. Thus, we consider N ≥ 3. As noted in Theorem 3.2 and Proposition 3.1, the dynamics of identical Lohe sphere oscillators are well understood in any positive coupling regime. Note that xi satisfies N & 1 d xi i 1 % = xi + x j − xi , x j xi . κ dt κ N

(4.6)

j=1

Then, in terms of a rescaled time-variable τ = κt, system (4.6) becomes N & 1 1 % d xi = i xi + x j − xi , x j xi . dτ κ N

(4.7)

j=1

Thus as κ → ∞, the dynamics (4.7) becomes close to the identical Lohe sphere flow with i = 0. Hence, we might be able to use the results for identical Lohe sphere flow in a large coupling regime to study the asymptotics of nonidentical Lohe sphere oscillators. Consider the identical Lohe sphere flow Y defined in Lemma 4.3, and introduce a slow time scale τ = κt along with a flow Y˜ : τ  y˜i (τ ) = yi . κ Then y˜i satisfies

⎧ N ⎪ % & ⎪ ⎨ d y˜i = 1 y˜ j −  y˜i , y˜ j  y˜i , τ > 0, dτ N j=1 ⎪ ⎪ ⎩ y˜ (0) = x 0 . i

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Then, it follows from Theorem 3.2 that y˜ exhibits NN−1 -entrainment and thus there exist τ0 > 0 and S ⊂ {1, . . . , N } with |S| ≥ N − 1 so that $S (τ0 ) := min  y˜i (τ0 ), y˜ j (τ0 ) > B i, j∈S

1 . N −1

We use Lemma 4.3 to see that # # ## # τ  + τ + τ τ0 , τ0 ,## 0 0 0 # # # ˜ − B S (τ0 )# = # min xi ( ), x j ( ) − min yi ( ), y j ( ) # #B S i, j∈S i, j∈S κ κ κ κ κ

τ   τ 

0 0

−Y ≤ 2 X

κ κ ∞ op 4τ0 (e − 1). ≤ 2κ Let p =

N −1 N .

Then p > 21 , and since 1 − [1 − 4 p(1 − p + D2 ()/κ)]1/2 1 = , κ→∞ 2p N −1 lim

we may find κ∞ such that κ > κ∞ implies $S (τ0 ) − B

op 4τ0 1 − [1 − 4 p(1 − p + D2 ()/κ)]1/2 (e − 1) > 2κ 2p

and κ>

2p D2 (). (2 p − 1)2

The assumptions of Lemma 4.3 are satisfied for κ > κ∞ at T0 =

τ0 κ .

 

4.2 Emergence of Practical Partial Entrainment Recall that in Theorem 4.1, we have shown that NN−1 -entrainment is possible from any generic initial configuration. However, we had no upper bound for the sufficient coupling strength κ(ε) - in particular, κ(ε) may be dependent on N - and similarly for the time T (ε). Thus, in this part, we describe a scheme for p-entrainment, where p may be less than NN−2 but κ(ε) and T (ε) are completely determined by ε, ρ 0 , D() and d. Next, we state our main results in this subsection as follows. Theorem 4.2 Suppose that the coupling strength, D() and initial data X 0 satisfy κ > 0, D() > 0, xi0  = 1, ρ 0 > 0, and let X be a solution to (2.7). Then, this ensemble exhibits practical with κ(ε) and T (ε) depending only on ε,

ρ0,

(4.8) 

1 2

+

 0

ρ 6

-entrainment,

D(), and d.

Proof The proof needs several technical lemmas, which include lemmas in Appendix A. We will come back to its detailed proof at the end of this subsection.   Next, we study several technical lemmas for the proof of Theorem 4.2.

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4.2.1 Technical Lemmas We begin with some estimates for the behavior of the order parameter ρ. Recall that the order parameter ρ satisfies the differential inequality (2.11):  . N  dρ 2 1 xi , xc  2 2D2 ()ρ + 2 1 − ≥ κρ 2 − , t > 0. (4.9) dt κρ 2 N ρ i=1

In the following lemma, we show that there exists a time when the R.H.S. of (4.9) becomes less than a positive value, 2D2 ()ρ. Lemma 4.4 Suppose that the coupling strength, D() and initial data X 0 satisfy N ≥ 3, κ > 0, D() > 0, xi0  = 1, ρ 0 > 0, and let X be a solution to (2.7). Then, there exists a time t0 ≥ 0 for which  N  1 xi , xc  2 ## 2D2 () ## ρ(t0 ) ≥ ρ 0 and 1 − ≤ # # . t=t0 N ρ(t) κρ(t) t=t0

(4.10)

(4.11)

i=1

Proof If the second inequality in (4.11) holds for t0 = 0, we are done. Thus, we assume that the second inequality in (4.11) does not hold at t = 0. By continuity, there exists δ > 0 such that  N  1 xi , xc  2 2D2 () 1− > , t ∈ [0, δ). (4.12) N ρ(t) κρ(t) i=1

Then, from the inequality (4.9), we have dρ dρ 2 ≥ 2D2 ()ρ, i.e., ≥ D2 (), ∀ t ∈ [0, δ). dt dt Note that as long as the condition (4.12) holds, ρ is strictly increasing and hence the value of ρ is larger than ρ 0 . Next, we argue that the relation (4.12) cannot hold for all time, i.e., the relation (4.11)2 should hold for some finite time. For this, we consider the set: #   # T := t1 > 0#(4.11)2 does not hold for t ∈ (0, t1 ) . Then, the set T is nonempty by assumption. Note that if the second inequality of (4.11) does not hold, then we have dρ 2 ≥ 2D2 ()ρ. dt We denote T ∗ := sup T , and we will show that T ∗ < ∞. Suppose not, i.e., T ∗ = ∞, then we have dρ 2 ≥ D2 ()ρ, t ∈ (0, ∞), dt or equivalently dρ ≥ D2 (), t ∈ (0, ∞). dt This yields that for t ≥ 0, we have

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ρ(t) ≥ ρ 0 and ρ(t) − (ρ 0 ) ≥ D2 ()t.

(4.13)

Note that the R.H.S. of the second inequality tends to ∞ as t → ∞, there exists t∗ > 0 such that ρ(t∗ ) > ρ 0 + 1, which is contradictory to the boundedness of ρ ≤ 1. Therefore, T ∗ < ∞ and there exists t0 > T ∗ satisfying (4.11).   Remark 4.2 The upper bound for t0 can be estimated as follows. It follows from (4.13)2 that we have 1 ≥ ρ(t0 ) ≥ ρ 0 + D2 ()t0 , i.e., t0 ≤

(1 − ρ 0 ) . D2 ()

Lemma 4.5 Suppose that in addition to (4.10), there exist constants α0 ∈ (0, 1), n 0 ∈ N and κ0 > 0 satisfying the relations: n0 2p 1 1 , D (), 1 − 2α0 > 2 2 2 N 2 p (2 p − 1) / 0 ρ 0 + (1 − α0 ) 1 N 2D2 () 1 ρ0 n0 − 1 · , − . p> + > 2 6 1 + (1 − α0 ) N α0 κ0 ρ 0 N

(4.14)

Then, for κ > κ0 , p and t0 a time for which (4.11) holds , there exists a set S ⊂ {1, . . . , N } such that |S| ≥ n 0 and

min xi , x j  >

i, j∈S

1 − [1 − 4 p(1 − p + D2 ()/κ)]1/2 , t ≥ t0 . (4.15) 2p

Proof For notational simplicity, we set βi (t0 ) =

 xi (t0 ), xc (t0 ) , i = 1, . . . , N and β0 := 1 − α0 , ρ(t0 )

and suppress t0 dependence in βi (t0 ), i.e., βi = βi (t0 ), i = 1, . . . , N , and define index sets: S+ := {i |βi > β0 }, S0 := {i | − β0 ≤ βi ≤ β0 }, S− := {i |βi < −β0 }. Then, clearly we have S+ ∪ S0 ∪ S− = {1, . . . , N }. Next, we claim that the set S+ plays the role of S in the statement of Lemma 4.5, i.e., S+ satisfies the conditions (4.15). This will be verified in two steps. • Step A (S+ satisfies (4.15)1 ): By condition (4.11)1 and the relation ρ = xc , we have |S0 | 1 |S+ | |S− | ρ 0 ≤ ρ(t0 ) = + (−β0 ) + β0 βi < N N N N i (4.16) |S0 | |S+ | = (1 + β0 ) + 2β0 − β0 , N N where we used the relations:

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|S+ | |S0 | |S− | =1− − . N N N

|βi | ≤ 1,

(4.17)

On the other hand, we use the relation (4.11)2 and (4.17) to see 1−

N |S0 | 2D2 () 2D2 () 1 2 |S+ | |S− | ≤1− βi ≤ ≤ + + β02 κρ0 κρ(t0 ) N N N N i=1

|S0 | |S0 | = 1 − (1 − β02 ) = 1 − α0 . N N This yields N 2D2 () · , i.e., |S0 | ≤ |S0 | ≤ α0 κ0 ρ0

/

0 N 2D2 () · . α0 κ0 ρ0

(4.18)

Then, it follows from (4.16) and (4.18) that we have ρ 0 + β02 2β0 |S0 | 1 |S+ | ρ 0 + β0 − · −1· > ≥ N 1 + β0 1 + β0 N N 1 + β02 n0 − 1 > . N

/

N 2D2 () · α0 κ0 ρ0

0 (4.19)

Here in the first inequality, we used ρ 0 + β02 ρ 0 + β0 ≥ 1 + β0 1 + β02

⇐⇒

1 − β0 ≥ ρ 0 (1 − β0 );

2β0 <1 1 + β0

and in the last inequality of (4.19), we used the last inequality of (4.14). Hence |S+ | ≥ n 0 . • Step B (S+ satisfies (4.15)2 ): Let i, j ∈ S+ . Then, we use xi (t0 ), xc (t0 ) = ρ(t0 )βi > ρ(t0 )β0 , to get

2



xi (t0 ) − β0 xc (t0 ) | = 1+β 2 − 2 β0 xi (t0 ), xc (t0 ) < 1−β 2 = α0 . (4.20) 0 0



ρ(t0 ) ρ(t0 ) Similarly, we have





x j (t0 ) − β0 xc (t0 ) < √α0 ,



ρ(t0 )

(4.21)

Then, we use (4.20) and (4.21) to get



x (t ) − x j (t0 ) 2



xi (t0 ) − x j (t0 ) < 2√α0 , i.e., xi (t0 ), x j (t0 ) = 1 − i 0 > 1 − 2α0 . 2 Hence, we have =

1 1 − [1 − 4 p(1 − p + D2 ()/κ)]1/2 > , 2p 2p

where we used the third relation in (4.14).

 

min xi (t0 ), x j (t0 ) > 1 − 2α0 >

i, j∈S

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1 2 nN0

On the Relaxation Dynamics of Lohe Oscillators...

4.2.2 Proof of Theorem 4.2 Suppose that N ≥ 3, κ > 0,

D() > 0, xi0  = 1, ρ 0 > 0.

We split its proof into two steps as follows. • Step A: We claim that for N ≥ 3 and ρ 0 > 0, there exists 0 < α0 < 1, n 0 ∈ N, κ0 > 0 satisfying (4.14). • Case A (N even): We choose m 0 ∈ Z such that 2(m 0 + 1) 2m 0 < ρ0 ≤ , N N and for such m 0 , we set n0 =

1m 2 0

2

 N 1152 D2 () ρ 0 /8, m 0 ≤ 1, + + 1, κ0 = , α = 0 m0 2 7 (ρ 0 )3 m 0 ≥ 2. 4N ,

(4.22)

• Case B (N odd): We choose m 0 ∈ Z such that 2m 0 − 1 2m 0 + 1 ≤ ρ0 < , N N and for such m 0 , we set n0 =

1m 2 0

2

N +1 640 D2 () + , κ0 = , α0 = 2 3 (ρ 0 )3



1 4N , m0 8N ,

m 0 ≤ 1, m 0 ≥ 2.

(4.23)

Then, for both cases, the relation N2 < n 0 ≤ N is easily verified. The other conditions will be checked in Appendix A. • Step B: We use Lemma 4.2 to conclude the p-entrainment. We set . 1152 D2 () 0 κ(ε) := max , κ (ε, ρ ) , 1 7 (ρ 0 )3 (4.24) 1 T (ε) := t0 + T1 (ε) ≤ + T1 (ε). 2D2 ()ρ 0 Here κ1 and t0 are the coupling strength and time appearing in Lemma 4.5, and T1 is the time we should wait for sup max xi (t) − x j (t) ≤ ε.

t>T (ε) i, j∈S

Note that κ(ε) and T1 (ε) are functions of ε, ρ 0 , D2 (), and d. This is because the function    D2 () 2 f (y) = 2κ − py + y − 1 − p + 2κ used in Lemma 4.2 via Lemma 4.1 depends only on p, d, ρ 0 and κ. In detail, for a proper value of κ(ε), we can set 1 + [1 − 4 p(1 − p + D2 ()/κ)]1/2 1 > 1 − cos ε. 2p 2 On the other hand, from Step A,

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min xi (t0 ), x j (t0 ) >

i, j∈S

1 − [1 − 4 p(1 − p + D2 ()/κ)]1/2 . 2p

Hence for κ(ε), min xi (t0 + T1 (ε)), x j (t0 + T1 (ε)) > 1 − cos ε,

i, j∈S

for a finite time T1 (ε) which may depend on κ(ε), ε, ρ 0 , D2 (), and d. Therefore, sup max xi (t) − x j (t) ≤ ε.

t>T (ε) i, j∈S

5 Gradient Flow Formulation of the Lohe Models In this section, we present gradient flow formulations of the Lohe matrix models for i = , and using this new gradient flow formulation, we show that all initial configurations tend to the complete entrainment state asymptotically.

5.1 A Gradient Flow on Riemannian Manifolds Recall that for a finite-dimensional gradient flow with an analytical potential on the Euclidean space Rd , it is well known that a uniformly bounded flow converges. Below, we extend this result to a Riemannian setting. First, we recall the Łojasiewicz inequality in a Riemannian framework where the proof of Łojasiewicz inequality in Rd can be generalized in a straightforward manner. Theorem 5.1 Let (M, g) be a Riemannian C ω −manifold with Riemannian metric g. Then, for an open subset U ⊂ M, a point p ∈ U , and a real analytic function f : U → R, there exist constants γ ∈ [ 21 , 1), C L and an open neighborhood V satisfying p ∈ V ⊂ U such that | f (z) − f ( p)|γ ≤ C L ∇ f (z), ∀z ∈ V .

(5.1)

Proof For the Euclidean space M = Rd , the Łojasiewicz inequality (5.1) has been proved in [26]. Since the inequality (5.1) has a local nature, the proof for the Euclidean space can be trivially extended to the Riemannian manifold M.   We now apply Theorem 5.1 to study the asymptotic behavior of gradient flows with analytical potentials on Riemannian manifolds. The following theorem is an extended version of the arguments in [13], which was used for the Kuramoto model in R N . Theorem 5.2 Let f : U → R be a real analytic function defined on an open subset U of a Riemannian manifold (M, g), and suppose the smooth path θ : [0, ∞) → U is an integral curve of −∇ f with starting point θ 0 ∈ U , i.e.,   dθ (t) d = θ (0) = θ 0 , θ∗ = −∇ f (θ (t)). dt dt (Here, the gradient vector −∇ f is obtained from the covariant vector −d f under the natural identification induced by the Riemannian metric g on M. The pushforward θ∗ is the

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On the Relaxation Dynamics of Lohe Oscillators...

corresponding derivative as well.) If the image of θ is contained in a compact subset K of U , then

 

d

= 0. θ∗ ∃ lim θ (t) = θ∞ ∈ K and lim

t→∞ t→∞ dt

Proof Without loss of generality, we may assume the initial condition θ 0 is not an equilibrium. Then, θ (t) can never be an equilibrium point in finite time since f is analytic. Since the image of θ is contained in the compact metric space K , we may find a sequence {tn }∞ n=1 and a point θ∞ ∈ M such that t1 ≤ t2 ≤ · · · ≤ tn → ∞,

lim θ (tn ) = θ∞ .

n→∞

On the other hand, f (θ (t)) satisfies d f (θ (t)) = −∇ f (θ (t))2 , dt and since f is bounded on K , there exists a limit limt→∞ f (θ (t)) = f ∞ and we must have f (θ∞ ) = f ∞ by continuity of f . Then, we apply Theorem 5.1 at θ∞ to obtain constants γ ∈ [ 21 , 1), C L and an open neighborhood θ∞ ∈ V ⊂ U such that | f (z) − f ∞ |γ ≤ C L ∇ f (z), ∀z ∈ V . Now consider the metric d induced on M through the Riemannian metric, and choose r > 0 such that Br (θ∞ ) ⊂ V . Define the function h : [0, ∞) → [0, ∞) by h(t) = ( f (θ (t)) − f ∞ )1−γ . Clearly, as t → ∞, h ↓ 0. So for any ε ∈ (0, ε0 ) there exists some T0 ∈ {tn } such that |h(t) − h(T0 )| ≤

ε(1 − γ ) ε , ∀t ≥ T0 and θ (T0 ) − θ∞  < . 3C L 3

We now intend to show that θ (t) ∈ Bε (θ∞ ) for t ≥ T0 . The following set T = {t1 > T0 : θ (t) ∈ B (θ∞ ) for all t ∈ (T0 , t1 )}

is nonempty by continuity of θ (t). Also, for any t1 ∈ T , d dh(t) = (1 − γ )( f (θ (t)) − f ∞ )−γ f (θ (t)) = −(1 − γ )( f (θ (t)) − f ∞ )−γ ∇ f (θ (t))2 dt dt 1−γ ≤− ∇ f (θ (t)), t ∈ (T0 , t1 ). CL Hence

3

t1 T0

 

3



θ∗ d dτ =

dτ

t1

∇ f (θ (τ ))dτ ≤ −

T0

C L (h(t1 ) − h(T0 )) ε ≤ . 1−γ 3

(5.2)

Now, we set T1 = sup T . If T1 < ∞ then clearly T1 ∈ T . Then by inequality (5.2), 3 T1 d d(θ (T1 ), θ∞ ) ≤ d(θ (T1 ), θ (T0 )) + d(θ (T0 ), θ∞ ) ≤ θ∗ ( )dτ + θ (T0 ) − θ∞  dτ T0 2 ≤ . 3

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This contradicts the definition of T1 since we may choose an element of T greater than T1 by continuity of θ (t). Hence T1 = ∞, i.e., θ (t) ∈ B (θ∞ ), t ≥ T0 . By taking the limit t1 → ∞ in (5.2), we obtain 3 ∞ ε θ˙ (τ )dτ ≤ , 3 T0 which implies the existance of the limit lim θ (t) = θ∞ .

t→∞

On the other hand, by the continuity of ∇ f , the limit

 

d

= lim  − ∇ f (θ (t)) =  − ∇ f (θ∞ ), lim θ∗ t→∞

dt t→∞ must be 0 since the limit limt→∞ θ (t) exists.

 

Corollary 5.1 Let f : M → R be a real analytic function defined on a compact Riemannian manifold M, and suppose that the smooth path θ : [0, ∞) → M is an integral curve of −∇ f with starting point θ 0 ∈ M, i.e.   d 0 θ (0) = θ , θ∗ = −∇ f (θ (t)). dt Then the limit lim θ (t) = θ∞ ∈ M

t→∞

exists and

 

d

= 0. θ∗ lim

t→∞

dt

Proof We set M = U = K in Theorem 5.2 to get the proof.

 

5.2 The Kuramoto Model is a Gradient Flow We briefly discuss the gradient flow formulation of the Kuramoto model as a motivation to the Riemannian manifolds: θ˙i = νi +

N κ sin(θ j − θi ), i = 1, . . . , N . N

(5.3)

j=1

Recall that the Lohe matrix model in one-dimension d = 1 is reduced to the Kuramoto model via the correspondence relation Ui = e−iθi ,

Hi = νi , i = 1, . . . , N .

Then, we define an order parameter (Rk , φk ) and potential Vk as follows:

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(5.4)

On the Relaxation Dynamics of Lohe Oscillators...

Rk eiφk :=

N 1 iθ j e ,  := (θ1 , . . . , θ N ), N j=1

ν := (ν1 , . . . , ν N ), Vk () := −ν ·  −

(5.5)

κ cos(θ j − θi ). N i, j

We use (5.4) to rewrite the potential Vk in terms of R: Vk () = −ν ·  − κ N Rk2 .

(5.6)

Then it is easy to see that the Kuramoto flow (5.3) is a gradient flow: ˙ = −∇ Vk ().  This fact is used in [13] to prove the emergence of synchronization in the Kuramoto model whose domain is R N . Hence it is natural to ask whether the Lohe matrix and sphere models can be written as a gradient flow or not. This will be the main focus of the next subsections.

5.3 Gradient Flow Formulations of the Lohe Models In this subsection, we present gradient flow formulations for generalized Lohe models on four Riemannian manifolds: M : Sd , U(d),

SU(d), R × SU(d).

In the following, we consider Lohe flows on the above Riemannian manifolds and present a gradient flow formulation.

5.3.1 A Lohe Flow on Sd Consider the Lohe sphere model for identical oscillators: x˙i =  +

  N N 1 κ (xk − xi , xk xi ) =  + κP⊥ x k , i = 1, . . . , N . xi N N k=1

(5.7)

k=1

where  is a skew-symmtric matrix and P⊥ xi is the orthogonal projection onto the tangent plane perpendicular to xi : P⊥ xi y = y − x i , yx i . Without loss of generality, we assume  = 0. Now, we introduce a potential function V0 : V0 (X ) := −

N κ xi , xk . 2N

(5.8)

i,k=1

Note that the potential function V0 is analytic, and recall that the surface gradient is defined as the projection of the gradient onto the tangent plane. Proposition 5.1 The Lohe sphere model (5.7) with  = 0 is a gradient flow: # # x˙i = −∇xi V0 # d , Txi S

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# # where ∇xi V0 #

Txi Sd

denotes the orthogonal projection of the gradient vector ∇xi V0 ∈ Rd+1

onto the tangent plane Txi Sd at xi , which is the induced gradient vector on the manifold Sd . Proof It follows from (5.8) that for each i = 1, . . . , N , ∇ x i V0 = −

N κ xk . N k=1

This yields # # ∇ x i V0 #

Txi Sd

# N κ ## =− xk # # N k=1

=− Txi Sd

N κ (xk − xk , xi xi ) . N k=1

  As a direct application of Theorem 5.2 and Proposition 5.1, we obtain the following convergence result for the Lohe flow from any initial data. Corollary 5.2 Let X = X (t) be a Lohe flow whose dynamics is governed by the system (5.7) with  = 0. Then, there exists an equilibrium X ∞ such that X ∞ = lim X (t). t→∞

Proof Since (Sd ) N is a compact manifold, the Lohe flow X is always bounded. Thus it follows from Theorem 5.2 and Proposition 5.1 that any Lohe flow converges.   Next, we briefly argue that why we need to restrict i = 0. One natural question is the existence of potential functions when the natural frequencies are not identical. For simplicity, let κ = 0 so that the interaction terms disappear: x˙i = i xi , i = 1, . . . , N , where the i ’s are skew-symmetric matrices. In this case, a possible ansatz on the nontrivial potential function might be V (x) = −

N

x tj  j x j ,

j=1

but then the derivatives of V are zero due to skew-symmetry of i : −

∂V = (it + i )xi = 0, i.e., V = constant. ∂ xi

On the other hand, for the Kuramoto model, the potential for nonidentical oscillators still exists on the covering space (R1 ) N of (S1 ) N as we can see in (5.5), which is not well-defined on (S1 ) N . For the Lohe sphere model, let us consider d > 1, since d = 1 is exactly the same as the Kuramoto model. The sphere Sd for d > 1 is a simply connected manifold, which has no proper covering space. Therefore, the potential function should be on (Sd ) N if it exists. However, it cannot exist on (Sd ) N since this model has a periodic solution. We can easily see this from the following example. Example 5.1 Let κ = 0 and

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⎛ 0 ⎜0 ⎜ ⎜ 1 = ⎜ ... ⎜ ⎝0 1

0 0 .. . 0 0

··· ··· .. . ··· ···

0 0 .. . 0 0

⎞ −1 0 ⎟ ⎟ .. ⎟ ,  = 0 for any k  = 1. k . ⎟ ⎟ ⎠ 0 0

Then the i ’s are skew-symmetric matrices. With initial data x10 = (1, 0, . . . , 0)t , it has a periodic solution x1 (t) = exp(1 t)x10 = (cos(t), 0, . . . , 0, sin(t))t , with xk (t) = xk0 for k  = 1. Remark 5.1 The Lohe sphere model on Sd for nonidentical oscillators does not have a potential function on Sd . However, for d = 1, it has a potential on its covering space (R1 ) N , as same as the Kuramoto model.

5.3.2 A Lohe Flow on U(d) In this part, we consider the Lohe matrix model on the unitary Hamiltonians Hi = H : ⎧   † † κ N ⎪ ˙ † ⎪ j=1 U j Ui − Ui U j , ⎨ Ui Ui = −iH + 2N  N  † † ˙ i = −iHUi + κ U , U U U − U U U j i i i j=1 i j 2N ⎪ ⎪ ⎩ Ui (0) = Ui0 ,

group U(d) with identical or equivalently t > 0,

(5.9)

i = 1, . . . , N .

Note that we may assume H = 0 in (5.9) without loss of generality, if necessary by considering a change of variable Ui → eiH t Ui , so that N  κ  U j Ui† Ui − Ui U †j Ui , t > 0. U˙ i = 2N

(5.10)

j=1

On the other hand, motivated by the Kuramoto model in Sect. 5.2, we introduce an order parameter R and a potential V1 for (5.9) with Hi = 0: R 2 :=

N    κ 1  N and V1 {Ui }i=1 := − N R 2 . tr Ui U †j 2 N 2

(5.11)

i, j=1

Here, the Riemannian metric of U(d) is induced by the natural inclusion U(d) → Md,d (C). For the one-dimensional case, the relations (5.11) are exactly the same as (5.5) and (5.6) via the relation (5.4). Proposition 5.2 The Lohe model (5.9) with Hi = 0 is a gradient flow with analytical potential

V1 in (5.11):

# ∂ V1 ## U˙ i = − ∂Ui #TU

i

U(d)

, i = 1, . . . , N , t > 0. 2

Proof The function V1 has an obvious polynomial extension to all of Md,d (C) N = R2d N 2 viewed as a real analytic manifold. Since each variable Ui is in Md,d (C) = R2d , the partial

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derivatives of a matrix can be calculated by the partial derivatives of each real and imaginary 2 kl component of Ui on R2d . Let u ikl = aikl + ) ibi be the (k, l)-element of matrix Ui*. First, we   use tr(Ui U †j ) = dk,l=1 u ikl u¯ klj = dk,l=1 (aikl a klj + bikl bklj ) + i(a klj bikl − aikl bklj ) to see V1 = −

d N N * κ κ ) kl kl (ai a j + bikl bklj ) + i(a klj bikl − aikl bklj ) tr(Ui U †j ) = − 2N 2N i, j=1

=−

κ 2N

N

i, j=1 k,l=1

d

) * aikl a klj + bikl bklj ,

i, j=1 k,l=1

where we cancel the imaginary term by symmetry of the indices i, j. This yields ∂ V1 ∂aikl

=−

N κ kl aj , N j=1

∂ V1 ∂bikl

=−

N κ kl bj , N j=1

2

and thus reverting back to the coordinates of Md,d (C) N = R2d N , we have ⎛ ⎞

# d d N N ∂ V1 ∂ V1 ## ∂ V1 2κ κ ⎝− = + i kl E kl = a klj − i bklj ⎠ E kl kl ∂Ui #TU Md,d (C) N N ∂a ∂b i i k,l=1 k,l=1 j=1 j=1 i =−

d d N N & κ % kl κ kl kl uj E a j + ibklj E kl = − N N j=1 k,l=1

=−

κ N

N

j=1 k,l=1

Uj,

j=1

where E kl denotes the d × d matrix whose (k, l)-coordinate is 1 and the other coordinates are 0. Now, we should project it on the tangent space TUi U(d) of U(d) at Ui . The orthogonal projection π : TId Md,d (C) → u(d) is given by A  → 21 (A − A† ) since u(d) = TId U(d) = {X ∈ Md,d (C) | X + X † = 0} is the set of skew Hermitian matricies (See Chapter 4 in [34]). Since TUi U(d) is the right translate u(d)Ui of u(d), we can see that the orthogonal projection πUi : TUi Md,d (C) → TUi U(d) is given by AUi  → π(A)Ui = 21 (A − A† )Ui for an element AUi ∈ TUi Md,d (C). Hence we may calculate



# # # ∂ V1 ## ∂ V1 ## ∂ V1 ## † = πUi U Ui =π ∂Ui #TU U(d) ∂Ui #TU Md,d (C) ∂Ui #TU Md,d (C) i i i i ⎞ ⎛ N N & κ κ % = π ⎝− U j Ui† ⎠ Ui = − U j Ui† − Ui U †j Ui . N 2N j=1

j=1

Therefore, we have −

# ∂ V1 ## ∂Ui #TU

i

U(d)

which yields the desired estimate.

123

Ui† =

N  κ  U j Ui† − Ui U †j , 2N j=1

 

On the Relaxation Dynamics of Lohe Oscillators...

As a direct application of Theorem 5.2 and Proposition 5.2, we have the convergence of the flow e−H t Ui as t → ∞. Corollary 5.3 Suppose that the Hamiltonians Hi are the same: Hi = H , i = 1, . . . , N , and let Ui = Ui (t) be a solution to the Cauchy problem (5.9). Then, the quantities e−iH t Ui converge for any initial configuration {Ui0 }. Proof Upon change of variables Ui  → eiH t Ui , we may assume H = 0. Then we are done by Theorem 5.2 and Proposition 5.2.   Remark 5.2 The above corollary implies that for the Lohe model (2.1) with positive coupling strength and zero natural frequencies, lim U˙i (t) = 0 for all i.

t→∞

This can be proved by more elementary means similar to those of Lemma 3.4, without employing the profound result of Łojasiewicz’s inequality and its implications on real analytic flows. However, this does not imply that the relative phases Ui U †j should converge, because the set of pairs of Ui U †j , (i, j = 1, . . . , N ) of equilibrium points is not in general discrete. This in turn highlights the significance of Łojasiewicz’s inequality in proving the convergence of Ui . Proof We first prove the existence of ρ ∞ where ρ is defined by the mean value, ρ = Uc , Uc =

N 1 Ui , N i=1

where we use the trace norm A =



tr(A A† ),

A ∈ Md,d (C).

Note that N  κ κ  U j Ui† − Ui U †j = (Uc Ui† − Ui Uc† ), U˙ i Ui† = 2N 2 j=1

so we may write κ U˙ i = (Uc − Ui Uc† Ui ). 2 From the definition of the norm, κ2 tr(Uc − Ui Uc† Ui )(Uc† − Ui† Uc Ui† ) 4 κ2 = tr(Uc Uc† − Ui Uc† Ui Uc† − Uc Ui† Uc Ui† + Ui Uc† Ui Ui† Uc Ui† ) 4 κ2 = tr(Uc Uc† − Ui Uc† Ui Uc† − Ui† Uc Ui† Uc + Uc† Uc ) 4 κ2 = Re tr(Uc Uc† − Ui Uc† Ui Uc† ). 2 On the other hand, U˙ i 2 =

123

S.-Y. Ha et al. N κ (Uc − Ui Uc† Ui ). U˙ c = 2N i=1

Thus, κ 2 ˙ 2 d 2 d (Uc Uc† − Ui Uc† Ui Uc† ) = Ui  , ρ = Uc 2 = 2Re tr(U˙ c Uc† ) = Re tr dt dt N κN i=1 i=1 √ hence ρ is non-decreasing and bounded above by d, which implies the existence of ρ ∞ . Note that

κ

κ κ√



d(1 + d), U˙ i  = (Ui − Uc Ui† Ui ) ≤ (Ui  + Uc Ui† Ui ) ≤ 2 2 2 and thus that



N N

1

1 ˙ κ√



Ui  ≤ d(1 + d). U˙ c  =

U˙ i ≤

N

N 2 N

i=1

N

i=1

Now, according to the previous calculations, d ˙ 2 d κ2 Ui  = Re tr(Uc Uc† − Ui Uc† Ui Uc† ) dt dt 2 κ2 = Re tr(U˙ c Uc† +Uc U˙ c† − U˙ i Uc† Ui Uc† −Ui U˙ c† Ui Uc† − Ui Uc† U˙ i Uc† −Ui Uc† Ui U˙ c† ) 2 = κ 2 Re tr(U˙ c Uc† − U˙ i Uc† Ui Uc† − Ui U˙ c† Ui Uc† ), and hence we conclude that # # #d # # U˙ i 2 # ≤ κ 2 (U˙ c U †  + U˙ i U † Ui U †  + Ui U˙ † Ui U † ) c c c c c # dt # ≤ Therefore,

κ3 d(1 + d)(1 + 2d). 2

d 2ρ is bounded, and by Barbalat’s lemma we obtain the conclusion of the remark. dt 2  

As we discussed on Sd , we can consider the existence of potential functions for nonidentical oscillators. In U(d), a possible potential function for κ = 0 is V (U ) = i

N

tr(U †j H j U j ).

j=1

However, this is a constant function on U(d) since all U j are unitary. We can rigorously prove the nonexistence of potential functions on U(d) by showing periodic solutions. Example 5.2 For the case of κ = 0 and H1 = h 1 Id with a nonzero real scalar h 1 , the oscillator U1 follows the equation U˙1 = −ih 1 U1 . Therefore, we have a periodic solution U1 (t) = exp(−ih 1 t)U10 . Since U(d) is not a simply connected manifold, we have the following potential function on the covering space R × SU(d) similar to the Kuramoto model. Remark 5.3 The universal covering space of U(d) is R × SU(d). A Lohe flow on U(d) with nonidentical oscillators can have a potential function if and only if the natural frequencies Hi are all scalars. We will see this and analyze a Lohe flow on R × SU(d) in a later subsection.

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On the Relaxation Dynamics of Lohe Oscillators...

5.3.3 A Lohe Flow on SU(d) Consider a Cauchy problem for the generalized Lohe model on SU(d), which was introduced in [20]:  *  )  † † † † κ N 1 U˙ i Ui† = −iHi + 2N j=1 U j Ui − Ui U j − d tr U j Ui − Ui U j Id , t > 0, (5.12) Ui (0) = Ui0 , i = 1, . . . , N , where Hi ∈ isu(d)(recall that su(d) consists of skew-hermitian matrices with zero trace). First, we show that system (5.12) preserves the structure of SU(d). Lemma 5.1 Special unitarity is preserved along the dynamics (5.12), i.e., Ui0 ∈ SU(d)

⇒ Ui (t) ∈ SU(d), t ≥ 0.

Proof Since the R.H.S. of (5.12)1 is anti-Hermitian, it is clear to see the unitarity of Ui . Thus, we only need to check whether detUi = 1 or not. Note that U˙ i Ui† has trace zero, and hence Ui−1 U˙ i = Ui−1 (U˙ i Ui† )Ui has trace zero. We use the Jacobi’s determinant formula (See Proposition 15.21 and Problem 15.7 in [34] for detail) to see that d detUi = detUi · tr(Ui−1 U˙ i ) = 0, dt so that detUi = 1 for t ≥ 0.

 

Proposition 5.3 Suppose that the constant hamiltonians Hi are the same, i.e., Hi = H , i = 1, . . . , N , and let Ui = Ui (t) be a solution to the Cauchy problem (5.12). Then, the Lohe model (5.12) can be rewritten as a gradient flow with analytical potential V1 in (5.11): # ∂ V1 ## ˙ , i = 1, . . . , N , t > 0. Ui = − ∂Ui #TU U(d) i

Here, the Riemannian metric on SU(d) is given by the inclusion SU(d) → Md,d (C). Proof The proof is almost the same with that of Proposition (5.2). Again the function V1 2 has an obvious polynomial extension to all of Md,d (C) N = R2d N viewed as a real analytic manifold, and has gradient # N κ ∂ V1 ## = − Uj. (5.13) ∂Ui #TU Md,d (C) N i

j=1

projection π : Now we want to project it on the tangent%space TUi SU(d). The orthogonal & TId Md,d (C) → su(d) is given by A  → 21 A − A† − d1 tr(A − A† )Id where su(d) = {X ∈ Md,d (C) | X + X † = 0, tr(X ) = 0}. Note that the derivative of (detX = 1) at Id is (tr(X ) = 0) because of Jacobi’s determinant formula. (See Chapter 4 in [34] for details) The projection π comes from the composition of the projection from Md,d (C) onto u(d) and the projection from u(d) onto su(d). Since the projection π1 : u(d) → su(d) is given by the projection into the traceless matrices, A  → A − d1 tr(A)Id , the projection π is obtained from the composition Md,d (C) → u(d) → su(d),   1 1 1 † † † A  → (A − A )  → A − A − tr(A − A )Id . 2 2 d Therefore, the orthogonal projection of (5.13) on TUi U(d) is given by AUi  → π(A)Ui ,

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# ∂ V1 ## − ∂Ui #TU

i

U(d)

Ui†

N  *  κ 1 ) † † † † U j Ui − Ui U j − tr U j Ui − Ui U j Id . = 2N d j=1

  Then, as a direct application of Proposition 5.3, we have the following corollary. Corollary 5.4 Suppose that Hamiltonians Hi are the same: Hi = H , i = 1, . . . , N , and let Ui = Ui (t) be a solution to the Cauchy problem (5.10) with H = 0. Then, the quantities e−i H t Ui converge for any initial configuration {Ui0 }. Remark 5.4 A Lohe flow on SU(d) with nonidentical oscillators does not have a potential function since SU(d) is simply connected and there is a periodic solution as in Example 5.2.

5.3.4 A Lohe Flow on R × SU(d) Consider the universal covering space of U(d): R × SU(d) → U(d), (θ, V )  → eiθ V . If we use this representation as a parametrization Ui = eiθi Vi of each particle described under the Lohe model, {θi , Vi } behaves as N * κ ) iθ˙i Id + V˙i Vi† = −iHi + V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) . 2N j=1

Note that Vi ∈ SU(d), so

V˙i Vi† ∈ su(d), hence tr(V˙i Vi† ) = 0.

We may take the trace on both sides and use the above relation in order to split the derivatives for the R × SU(d) model: N * 1 iκ ) θ˙i = − trHi − tr V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) , d 2d N j=1

N i κ ) V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) V˙i Vi† = −iHi + trHi Id + d 2N j=1 * * ) 1 − tr V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) Id . d As a special case, we choose Hi = −νi Id , νi ∈ R to rewrite (5.14) as follows. N * iκ ) tr V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) , θ˙i = νi − 2d N

(5.14)

j=1

κ V˙i = 2N

N )

V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j )

j=1

 * 1  − tr V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) Id Vi . d We set a real analytic potential on (R × SU(d)) N :

123

(5.15)

On the Relaxation Dynamics of Lohe Oscillators...



N V2 {θi , Vi }i=1

= −d

N i=1



νi θi −

N N  κ κ  i(θi −θ j ) dνi θi − tr e Vi V j† , N R2 = − 2 2N i=1

(5.16)

i, j=1

where R is the order parameter defined by the relation (5.11). Proposition 5.4 The Lohe model (5.15) is a gradient flow with analytical potential V2 in (5.16): # # ∂ V2 ## # θ˙i = −∇θi V2 # , V˙i = − , TR ∂ Vi #TV SU(d) i

where the Riemannian metric of the SU(d) components of (R × SU(d)) N are given by the inclusion SU (d) → Md,d (C), and those of the R components of (R × SU (d)) N are given as d times of the standard Riemannian metric of R. Proof Note that the function V2 given in (5.16) is defined on (R × Md,d (C)) N viewed as a real analytic manifold, and has gradient ⎞ ⎛ # N * ∂ V2 ## 1⎝ iκ ) − = tr V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) ⎠ , νi d − ∂θi #T R d 2N j=1

= νi −

iκ 2d N

N

* ) tr V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) ,

j=1

# N ∂ V2 ## κ − = V j ei(θ j −θi ) , ∂ Vi #T Md,d (C) N j=1

where we consider R N × Md,d (C) N with its ordinary metric times d on its R N component and with its ordinary metric on its Md,d (C) N component. We perform an orthogonal projection onto su(d) # to obtain ∂ V2 ## − Vi† ∂V # i TV SU (d) i

=

N  * κ 1 ) V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) − tr V j Vi† ei(θ j −θi ) − Vi V j† ei(θi −θ j ) Id . 2N d j=1

Therefore, we can conclude that the Lohe flow (5.16) is a gradient flow with analytic potential V2 .   As a direct corollary of Theorem 5.2 and Proposition 5.4, we have the following a priori convergence result. Corollary 5.5 Let (θi , Vi ) be a solution to system (5.15) such that the phase variables θi are uniformly bounded. Then, the quantities Ui = eiθi Vi converge.

6 Conclusion In this paper, we have studied the relaxation dynamics of the ensemble of Lohe sphere and matrix oscillators from generic initial configurations. For an ensemble of Lohe oscillators

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under the same Hamiltonians, we show that exponential position synchronization occurs asymptotically under the a priori assumption that complete position synchronization occurs. Moreover, we also showed that there are only two possible asymptotic states when the initial positions are all different and have a positive order parameter. This kind of conditions reminds us of the corresponding condition for the Kuramoto model in [19]. For distinct constant Hamiltonians, the relaxation dynamics are rather complicated. In this heterogeneous ensemble, we show that if the coupling strength is sufficiently large, then the heterogeneous ensemble tends to the practical NN−1 -entrained state asymptotically. However, in the intermediate coupling strength regime, we show that practical partial entrainment states will emerge from an initial configuration with positive order parameter. For the convergence result of the Lohe matrix oscillators, we employ a gradient flow approach. For the Kuramoto model, the gradient flow structure of the Kuramoto model plays a key role in the resolution of the complete synchronization in [19]. However, it is not known whether the Lohe matrix models admit gradient flow formulations with analytical potentials in a general setting. Fortunately, our finding in this paper says that at least for Lohe matrix models with the same Hamiltonian, the Lohe matrix flows can admit a gradient flow formulation so that the Lohe matrix flow on compact group manifolds such as U(d) and SU(d) always converge from any initial configuration. Of course, we cannot resolve the gradient flow formulation of the Lohe matrix model with heterogeneous Hamiltonians. We confirmed that it is no longer a gradient flow according to Remark 5.1, 5.3, and 5.4. Hence the asymptotic behaviors of nonidentical oscillators are still open problems for both sphere and matrix model. However, similar behaviors as in the Kuramoto model are observed through numerical simulations. We leave this issue for a future work. Acknowledgements The work of S.-Y. Ha was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03. The work of D. Ko was supported by the TJ Park fellowship from POSCO. The work of S. W. Ryoo was supported by the Student Directed Education program of Seoul National University.

Appendix A: Verification of Conditions (4.14) In this section, we show that the choices explicitly verify that the choices (4.22) and (4.23) satisfy the conditions (4.14), that is the conditions

2p (2 p − 1)2

D2 () < κ0 ,

1 − 2α0 >

1 , 2p

1 ρ0 + , 2 6 / 0 1 N 2D2 () n0 − 1 ρ 0 + (1 − α0 ) − . > · 1 + (1 − α0 ) N α0 κ0 ρ 0 N p>

A.1 N is even In this case, recall that we choose

123

(A.1) (A.2) (A.3) (A.4)

On the Relaxation Dynamics of Lohe Oscillators...

1m 2 N 2(m 0 + 1) 2m 0 0 < ρ0 ≤ , n0 = + + 1, N N 2 2  1152 D2 () ρ 0 /8, m 0 ≤ 1, , α0 = m 0 κ0 = 0 3 7 (ρ ) m 0 ≥ 2. 4N , m0 ≈ 21 + Here, n 0 was roughly chosen so that p = nN0 ≈ 21 + 2N would hold in most cases. On the other hand, (A.2) suggests that

α0 <

ρ0 4

(A.5)

and hence that (A.3)

1 1 1 1 1 1 1 p m0 − ≈ − ≈ − (1 − ( p − 1)) = = . 2 4p 2 4 1 + ( p − 1) 2 4 4 4N

+(1−α0 ) > n 0N−1 , hence we can finally choose κ0 large When m 0 ≥ 2, this happens to give ρ1+(1−α 0) enough so as to make (A.1) and (A.4) true. On the other hand, when m 0 ≤ 1, we need to choose a smaller α0 , because then n 0 = N2 + 1 and so 0

ρ 0 + (1 − α0 ) n0 − 1 1 > = 1 + (1 − α0 ) N 2

⇔

α0 < 2ρ 0 ,

m0 1 ≤ 4·2 = 18 . The while ρ0 could be very small. The denominator 8 was chosen because 4N funny coefficient 1152 7 in the choice for κ0 will be explained in the demonstration of (A.4) when m 0 ≥ 2; the condition (A.1) is weaker than (A.4).

• Case A (m 0 ≤ 1): We verify the relations (A.1)–(A.4) one by one. Note that by our choice, we have n0 =

N 4 ρ0 + 1, ρ 0 ≤ , α0 = . 2 N 8

• (Verification of (A.1)) We have p=

1 1 n0 = + , N 2 N

so 1 + (2/N ) N2 2p = = (2 p − 1)2 (2/N )2 16

  8 1 8 1152 4+ ≤ 0 2 (4 + 4) = 0 2 ≤ , N (ρ ) (ρ ) 7(ρ 0 )3

hence 2p (2 p − 1)2

1152 D2 () = κ0 . 7 (ρ 0 )3

D2 () <

• (Verification of (A.2)) We have (1 − 2α0 )

ρ0 2 1 2 2n 0 1 = (1 − )(1 + ) ≥ (1 − )(1 + ) > 1, i.e., 1 − 2α0 > , N 4 N N N 2p

where we use the fact that N ≥ 3. • (Verification of (A.3)) We use (A.5) and

4 m0 5 2

= 0 to find

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n0 1 1 1 ρ0 1 ρ0 = + ≥ + > + . N 2 N 2 4 2 6 • (Verification of (A.4)) By the choice of α0 and κ0 in (A.5), we have 0 / N 2D2 () N N 2D2 () 7(ρ 0 )2 7N ρ 0 28 · = · 0< = 0. · = ≤ < 1, i.e., α0 κ0 ρ 0 ρ 0 /8 576 72 72 α0 κ0 ρ 0 p=

Hence, we have 1 ρ 0 + (1 − α0 ) − 1 + (1 − α0 ) N

/

N 2D2 () · α0 κ0 ρ 0

0 =

1 n0 − 1 1 + 7ρ 0 /8 −0> = . 2 − ρ 0 /8 2 N

• Case B (m 0 ≥ 2): Similar to Case A, we check the relations (A.1)–(A.4) one by one. • (Verification of (A.1)): Since 1m 2 N m0 N 0 + n0 = +1> + , 2 2 2 2 we have m0 2n 0 −1> . (A.6) 2p − 1 = N N This yields     1 N N 1 2p + 1 < = + 1 (2 p − 1)2 2p − 1 2p − 1 m0 m0   N N 2(m 0 + 1) 2(m 0 + 1) = +1 · · 2(m 0 + 1) m0 2(m 0 + 1) m0     1 5 1152 1 4 ≤ 0 ·4 ≤ ·4+1 ≤ 0 , ρ ρ0 ρ ρ0 7(ρ 0 )3 Hence 2p (2 p − 1)2

D2 () <

1152 D2 () = κ0 . 7 (ρ 0 )3

m0 and (A.6) to obtain • (Verification of (A.2)) We use α0 = 4N  m 0  m0  m0  m0  2n 0 (1 − 2α0 ) p = (1 − 2α0 ) > 1− 1+ =1+ 1− > 1, N 2N N 2N N i.e., we have

1 − 2α0 >

1 . 2p

• (Verification of (A.3)) We again use the relation (A.6) to get p=

1 m0 1 2(m 0 + 1) m0 1 1 ρ0 2 n0 > + = + · ≥ + ρ0 · = + . N 2 2N 2 N 4(m 0 + 1) 2 4·3 2 6

• (Verification of (A.4)) By the choice of α0 =

m0 4N

and κ0 , we have

7N 2 (ρ 0 )2 4N 2 7(ρ 0 )2 7 · 4(m 0 + 1)2 N 2D2 () = · = · ≤ 0 α0 κ0 ρ m0 576 144m 0 144m 0   7m 0 1 2 = . 1+ 36 m0

123

(A.7)

On the Relaxation Dynamics of Lohe Oscillators...

On the other hand, we have 2m 0 m0 + 1 − 4N ρ 0 + (1 − α0 ) n 0 − 1 − > N m0 − 1 + (1 − α0 ) N 1 + 1 − 4N m 0 (7N + m 0 ) . = 2N (8N − m 0 )



m0 1 + 2N 2

 =

4N + 7m 0 N + m0 − 8N − m 0 2N

(A.8) Thus, we use m 0 ≥ 2, (A.7) and (A.8) to find / 0 ρ 0 + (1 − α0 ) n 0 − 1 1 N 2D2 () · − − 1 + (1 − α0 ) N N α0 K0ρ0 (m 0 + 1)2 m 0 (7N + m 0 ) −7 2N (8N − m 0 ) 36N m 0 14N (5m 20 − 8m 0 − 4) + 25m 30 + 14m 20 + 7m 0 = 36N m 0 (8N − m 0 ) 14N (m 0 − 2)(5m 0 + 2) + 25m 30 + 14m 20 + 7m 0 = > 0. 36N m 0 (8N − m 0 ) >

i.e. we have ρ 0 + (1 − α0 ) 1 − 1 + (1 − α0 ) N

/

N 2D2 () · α0 K0ρ0

0 >

(A.9)

n0 − 1 . N

The role of the coefficient 1152 7 appears in the last line of (A.9). For any coefficient ζ , the last line of (A.9) would appear as replacing 1152 7 N a(m 0 ) + b(m 0 ) , N m 0 (8N − m 0 ) where a(m 0 ) is a quadratic in m 0 and b(m 0 ) is a cubic in m 0 , depending on ζ . If a(2) < 0, then we may choose N sufficiently large so that this rational function takes a negative value. Hence we must have a(2) ≥ 0, which forces us to choose ζ ≥ 1152 7 . This choice of ζ happens to make the rational function positive for all valid values of m 0 and N .

A.2 N is odd Recall that we choose m 0 to satisfy 2m 0 − 1 2m 0 + 1 ≤ ρ0 < . N N Then, for such m 0 , we set n0 =

1m 2 0

2

N +1 640 D2 () + , α0 = , κ0 = 2 3 (ρ 0 )3

(A.10) 

1 4N , m0 8N ,

m 0 ≤ 1, m 0 ≥ 2.

(A.11)

We again claim that the choices (A.10)–(A.11) satisfy (A.1)–(A.4). The rationale for choosing n 0 , and α0 when m 0 ≥ 2, is similar to A.1. However, when m 0 ≤ 1, we have −1 n 0 = N 2+1 and n 0N−1 = N2N , which is less than 21 , so we can be a little more lenient with our choice of α0 so as to make

ρ 0 +(1−α0 ) 1+(1−α0 )

>

n 0 −1 N

even when ρ 0 ≈ 0. Again, the coefficient

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in the choice of κ0 is necessitated by (A.4) when m 0 ≥ 2; the condition (A.1) is weaker than (A.4). 640 3

• Case C (m 0 ≤ 1): Note that the relations (A.11) yield 1 N +1 3 , n0 = and ρ 0 < . 4N 2 N • (Verification of (A.1)) We use the relations (A.12) to obtain α0 =

(A.12)

2 nN0 2p 9(N + 1) N 2 640 1 = = N (N + 1) = . · < 12 · 0 2 < & % 2 0 )3 n0 (2 p − 1)2 N 9 (ρ ) 3(ρ 2N −1 Thus, we have 640 D2 () 2p D2 () < = κ0 . (2 p − 1)2 3 (ρ 0 )3 • (Verification of (A.2)) We again use (A.12) to get (1 − 2α0 )2 p = (1 − 2α0 )

2n 0 1 N +1 1 = (1 − )· =1+ N 2N N 2N

 1−

1 N

 > 1.

• (Verification of (A.3)) We use (A.12) to get p=

1 1 1 ρ0 n0 = + > + . N 2 2N 2 6

• (Verification of (A.4)) Note that 0<

3(ρ 0 )2 N 2D2 () 3N 2 (ρ 0 )2 27 · = 4N 2 · = < < 1, i.e., 0 α0 κ0 ρ 320 80 80 0 / N 2D2 () · = 0. α0 κ0 ρ 0

This yields 1 ρ 0 + (1 − α0 ) − 1 + (1 − α0 ) N ≥

1− 2−

1 4N 1 4N

−0−

/

0 N 2D2 () n0 − 1 · − α0 κ0 ρ 0 N

N −1 4N − 1 N −1 7N − 1 = − = > 0, 2N 8N − 1 2N 2N (8N − 1)

i.e., 1 ρ 0 + (1 − α0 ) − 1 + (1 − α0 ) N

/

N 2D2 () · α0 κ0 ρ 0

0 >

n0 − 1 > 0. N

• Case D (m 0 ≥ 2): By definition of n 0 in (A.11), we have 1m 2 N + 1 m0 N +1 m0 − 1 N 0 + > −1+ = + , n0 = 2 2 2 2 2 2 1m 2 N + 1 m0 N +1 0 + n0 = ≤ + . 2 2 2 2 These yield 2p =

123

m0 − 1 m0 + 1 2n 0 > + 1 and 2 p < + 1. N N N

(A.13)

(A.14)

On the Relaxation Dynamics of Lohe Oscillators...

• (Verification of (A.1)) It can easily be seen that x  →

2x (2x−1)2

=

1 2x−1

1 · ( 2x−1 + 1) is

decreasing for x > 21 . Thus we can use (A.14)1 to see We use (A.10) and (A.14) to see   m 0 −1 2m 0 + 1 1 2 N (m 0 − 1 + N ) N (m 0 − 1 + N ) 2p N +1 < < < · · (m 0 − 1)2 (m 0 − 1)2 N ρ0 (2 p − 1)2 ( m 0N−1 )2    >1

  2N 2 2(m 0 − 1) + 3 1 2 · ≤ · (m 0 − 1)2 N ρ0 2  2(m 0 − 1) + 3 1 =2· · 0 2 m0 − 1 (ρ ) 1 640 ≤ 2 · 52 · 0 2 < , (ρ ) 3(ρ 0 )3 where in the third inequality, we used (A.10) to find the relation: 2m 0 − 1 ≤ ρ0 ≤ 1 N

⇒

m 0 − 1 < 2m 0 − 1 ≤ N .

Thus, we have 2p (2 p − 1)

2

D2 () <

• (Verification of (A.2)) We use α0 =

m0 8N

640 D2 () = κ0 . 3 (ρ 0 )3

and (A.14) to obtain

m0 m0 − 1 3m 0 − 1 m 0 (m 0 − 1) 2n 0 > (1− )(1+ ) = 1+ − N 4N N  4N 4N 2 m 0 (m 0 − 1) m0 − 1 m0 m0 − 2− > 1, ≥1+ =1+ 2N 4N 2 4N N

(1 − 2α0 )2 p = (1−2α0 )

where in the last inequality, we also used the following relation from (A.14): 2 ≥ 2p >

m0 − 1 +1 N

⇒

N > m 0 − 1.

Hence, we have 1 − 2α0 > • (Verification of (A.3)) We use

m0 2 !

≥

m 0 −1 2

n0 ≥

1 . 2p

and (A.11) to find

m0 + N . 2

This yields p=

2 1 m0 1 2m 0 + 1 m0 1 1 ρ0 n0 ≥ + = + · > + ρ0 · > + . N 2 2N 2 N 2(2m 0 + 1) 2 2·5 2 6

• (Verification of (A.4)) We use α0 =

m0 8N

and κ0 =

640 D2 () 3 (ρ 0 )3

to obtain

N 2D2 () 8N 2 3(ρ 0 )2 3(2m 0 + 1)2 3N 2 (ρ 0 )2 · = · < . = α0 κ0 ρ 0 m0 320 40m 0 40m 0

(A.15)

On the other hand, we have

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m0 + 1 − 8N m0 − 1 + N − m0 1 + 1 − 8N 2N 8N + 15m 0 − 8 m 0 − 1 + N = − 16N − m 0 2N m 0 (15N + m 0 − 1) . = 2N (16N − m 0 )

ρ 0 + (1 − α0 ) n 0 − 1 − ≥ 1 + (1 − α0 ) N

(A.16)

Thus, it follows from (A.15), (A.16) and m 0 ≥ 2 that we have 0 / ρ0 + (1 − α0 ) n 0 − 1 1 N 2D2 () − − · 1 + (1 − α0 ) N N α0 κ0 ρ 0 m 0 (15N + m 0 − 1) 3(2m 0 + 1)2 − 2N (16N − m 0 ) 40N m 0 2 12N (9m 0 − 16m 0 − 4) + 32m 30 − 8m 20 + 3m 0 = 40m 0 N (16N − m 0 ) 12N (m 0 − 2)(9m 0 + 2) + 8m 20 (4m 0 − 1) + 3m 0 = 40m 0 N (16N − m 0 ) > 0. >

(A.17)

This yields ρ0 + (1 − α0 ) 1 − 1 + (1 − α0 ) N

/

N 2D2 () · α0 K 0 ρ0

0 >

n0 − 1 . N

The role of the coefficient 640 3 appears in the penultimate term in (A.17), and the reason for this choice is the same as Case B.

Appendix B: Linear Instability of the Bipolar State In this section, we discuss the linear stability of the bipolar state for the Lohe model with identical oscillators: x˙i = xi +

 N  κ xi , xk  xk − xi , i = 1, . . . , N . N xi , xi 

(B.1)

k=1

By replacing the variables xi by e−t xi , we may assume  = 0, so that x˙i =

 N  κ xi , xk  xi , i = 1, . . . , N . xk − N xi , xi 

(B.2)

k=1

By the O(d)-symmetry of the above model, i.e., by replacing yi by Ayi for some A ∈ O(d), and by permuting the indices, we may assume the bipolar state (y1 , . . . , y N ) in question consists of the following:  (0, . . . , 0, 1), 1 ≤ i ≤ r, yi = (0, . . . , 0, −1), r < i ≤ N ,

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On the Relaxation Dynamics of Lohe Oscillators...

where r = 0, . . . , N describes the number of oscillators on the north pole. We consider a perturbation {xi } of {yi }: ⎧ 6 ⎨(x 1 , . . . , x d , 1 − (x 1 )2 − . . . − (x d )2 ), 1 ≤ i ≤ r , i i i i 6 xi = ⎩(x 1 , . . . , x d , − 1 − (x 1 )2 . . . − (x d )2 ), r < i ≤ N , i

i

i

i

√ where |xia | " 1/ d for i = 1, . . . , N , a = 0, . . . , d − 1. Discarding quadratic terms, we can see that  1 + O(x2 ), 1 ≤ i, j ≤ r or r < i, j ≤ N , xi , x j  = −1 + O(x2 ), 1 ≤ i ≤ r < j ≤ N or 1 ≤ j ≤ r < i ≤ N ,  N d−1 a 2 where x2 := i=1 a=0 (x i ) . Hence, if we project (B.2) onto the first d variables, we obtain the system of equations

x˙ia =

N κ a κ(2r − N ) a xk − xi + O(x3 ), i = 1, . . . , r , a = 0, . . . , d − 1. (B.3) N N k=1

x˙ia =

N κ a κ(N − 2r ) a xk − xi + O(x3 ), i = r + 1, . . . , N , a = 0, . . . , d − 1. N N k=1

(B.4) Note that we do not need the (d + 1)th component, since it is determined by the first d components. Linearizing (B.3), (B.4) yields z˙ ia =

N κ a κ(2r − N ) a z i , i = 1, . . . , r , a = 0, . . . , d − 1. zk − N N

(B.5)

N κ a κ(N − 2r ) a z i , i = r + 1, . . . , N , a = 0, . . . , d − 1. zk − N N

(B.6)

k=1

z˙ ia =

k=1

Then our result on linear stability is as follows: Theorem B.1 Let (B.5), (B.6) be the linear approximation to (2.7) at the bipolar state (y1 , . . . , y N ). (1) When r = 0 or r = N , i.e., near the completely synchronized state, the linear system (B.5), (B.6) is diagonalizable with eigenvalue 0 with multiplicity d, and −1 with multiplicity d(N − 1). Hence the linear stability is indeterminate. (2) When 1 ≤ r ≤ N − 1, i.e., near a bipolar state that is not the completely synchronized state, the linear system (B.5), (B.6) is diagonalizable with eigenvalue 0 with multiplicity d, 1 with multiplicity d, 2r N−N with multiplicity d(N − r − 1), N −2r with multiplicity N d(r − 1); hence it is linearly unstable. Remark B.1 In both cases, 0 is an eigenvalue with multiplicity at least d. This follows from the fact that (2.7) possesses an O(d)-symmetry, and from the fact that the orbit of the corresponding action of O(d) on the bipolar states in (S d ) N has dimension d. Proof This system is just the linear ODE z = κz,

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where zT = (z 10 , . . . , z 0N , . . . , z 1d−1 , . . . , z d−1 N ), and  is the block matrix  = diag(0 , . . . , 0 ).    d copies

Here, 0 is defined as 1 2r − N N − 2r A− IN − 2 B N N N where A is the N × N -matrix all of whose entries are 1, and B is the N × N -matrix in block form   I 0 B= r 0 0 0 :=

where Ir is the r ×r unit matrix, and the 0’s represent zero matrices of the appropriate dimension. According to Corollary C.1 in the next section,  is diagonalizable, with eigenvalues ⎧ ⎪ if r = 0, N , ⎨0 with multiplicity d, − 1 with multiplicity d(N − 1) 0 with multiplicity d, 1 with multiplicity d, ⎪ ⎩ 2r −N N −2r N with multiplicity d(N − r − 1), N with multiplicity d(r − 1) if 1 ≤r ≤ N −1.  

Appendix C: Calculation of Some Eigenvalues Lemma C.1 Let k be a field of characteristic zero. Let N ∈ N , m ∈ Z such that 0 ≤ r ≤ N . Let A be the N × N -matrix all of whose entries are 1 ∈ k, and let B be the N × N -matrix in the block form   I 0 B= r 0 0 where Ir is the r ×r unit matrix over k, and the 0’s represent zero matrices of the appropriate dimension. Then for any μ ∈ k, the matrix A − μB has characteristic polynomial ⎧ N −1 (t − N ) ⎪ if r = 0, ⎨t φ(t) = t N −r −1 (t + μ)r −1 (t 2 + (μ − N )t + μ(r − N )) if 1 ≤ r ≤ N − 1, (C.1) ⎪ ⎩ if r = N , (t + μ) N −1 (t + (μ − N )) over k. Proof • Case A (r = 0): Then B = 0, so we are searching for the characteristic polynomial of A. If N = 1, then it is immediate that the characteristic polynomial is φ(t) = t − 1, as asserted above. So we assume N ≥ 2. Then it is easy to verify that the relation A2 − N A = 0, holds,as well as the relations A  = 0,

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A − N I N  = 0.

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Hence the minimal polynomial m(t) of A is m(t) = t 2 − N t = t(t − N ), and since the minimal polynomial and the characteristic polynomial must share the same irreducible factors, we conclude that φ(t) must be of the form φ(t) = t N −a (t − N )a , where a ∈ Z is an integer 1 ≤ a ≤ N − 1. This allows us to calculate, using the binomial theorem, (coefficient of t N −1 in φ(t)) = −a N , but we can also directly calculate (coefficient of t N −1 in φ(t)) = −tr(A) = −N . Hence we must have a = 1, i.e., φ(t) = t N −1 (t − N ), as asserted. • Case B (r = N ): Then B = I N , the N × N identity matrix, so φ(t) = det(t I N − (A − μB)) = det((t + μ)I N − A) = (t + μ) N −1 (t + μ − N ), by using the result of Case A. This agrees with our assertion (C.1). • Case C (1 ≤ r ≤ N − 1): Without loss of generality, we may consider μ to be a variable over k, i.e., that the field of rational functions k(μ) is a transcendental extension of k, and consider A − μB as a matrix over k(μ). Then we must verify the relation (C.1) over the polynomial ring k(μ)[t]. First, it is easy to verify the relations A2 = N A,

B 2 = B,

AB A = r A.

(C.2)

Using these relations, we proceed to verify the following by direct expansion: (A − μB)2 = N A + μ2 B − μAB − μB A, (A − μB)3 = (N 2 − μr )A − μ3 B + (μ2 − μN )AB + (μ2 − μN )B A + μ2 B AB, (A − μB)4 = (N 3 − 2N μr + r μ2 )A + μ4 B + (−μ3 + μ2 N − μN 2 + μ2 r )AB + (−μ3 + μ2 N − μN 2 + μ2 r )B A + (−2μ3 + μ2 N )B AB. This permits us to verify the relation (A − μB)4 + (2μ − N )(A − μB)3 + (μ2 − 2μN + μr )(A − μB)2 − μ2 (N − r )(A − μB) = 0. (The rationale for the preceding computation is that, the relations (C.2) force the powers of A − μB to be linear combinations of the five terms A, B, AB, B A, BAB only, and hence that five powers of A − μB suffice to give a linear relation. Luckily, we needed only four.) Hence the minimal polynomial m(t) of A − μB over k(μ) is a divisor of t 4 + (2μ − N )t 3 + (μ2 − 2μN + μr )t 2 − μ2 (N − r )t = t(t + μ)(t 2 + (μ − N )t + μ(r − N )). But t 2 + (μ − N )t + μ(r − N ) is irreducible over k(μ) if and only if it factors into linear factors, that is to say, if its discriminant (as a quadratic polynomial in t) (μ − N )2 − 4μ(r − N ) = μ2 − 2(2r − N )μ + N 2

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is a square in k(μ), which holds if and only if its discriminant (as a quadratic polynomial in μ) 4(2r − N )2 − 4N 2 = 16r (r − N ) = 0 is zero in k. Since k has characteristic zero, this is equivalent to r = 0 or r = N in Z. These cases were already excluded. Hence t 2 + (μ − N )t + μ(r − N ) is irreducible over k(μ). Since m(t) and φ(t) share the same irreducible factors, we conclude that φ(t) = t N −a−2b (t + μ)a (t 2 + (μ − N )t + μ(r − N ))b , where a, b are nonnegative integers such that a + 2b ≤ N . As in Case A, we use the binomial theorem to calculate (coefficient of t N −1 in φ(t)) = aμ + b(μ − N ), and compare this with (coefficient of t N −1 in φ(t)) = −tr(A) = r μ − N . Thus aμ + b(μ − N ) = r μ − N holds in k(μ), and since k has characteristic zero, we must have a = r − 1, b = 1 in Z, i.e., φ(t) = t N −r −1 (t + μ)r −1 (t 2 + (μ − N )t + μ(r − N )),  

as asserted.

Remark C.1 We will only be interested in the case μ = 4r − 2N . The reason for introducing μ as a variable is that, in the case 1 ≤ r ≤ N − 1, the calculation of the characteristic polynomial becomes easier once we consider μ as a transcendental element over k. Corollary C.1 The matrix 1 2r − N N − 2r A− IN − 2 B N N N over R is diagonalizable, with eigenvalues ⎧ ⎪ if r = 0, N , ⎨0 with multiplicity 1, − 1 with multiplicity N − 1 0 with multiplicity 1, 1 with multiplicity 1, ⎪ ⎩ 2r −N N −2r N with multiplicity N − r − 1, N with multiplicity r − 1 if 1 ≤ r ≤ N − 1. 0 :=

Proof Diagonalizability follows from the symmetry of 0 and the spectral theorem for real symmetric matrices. Let μ = 2(2r − N ). Then we can see that t 2 + (μ − N )t + μ(r − N ) = t 2 + (4r − 3N )t + 2(2r − N )(r − N ) = (t + 2r − N )(t + 2r − 2N ), so (C.1) becomes

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⎧ N −1 (t − N ) ⎪ if r = 0, ⎨t φ(t) = t N −r −1 (t + 4r − 2N )r −1 (t + 2r − N )(t + 2r − 2N ) if 1 ≤r ≤ N −1, (C.3) ⎪ ⎩ if r = N , (t + 2N ) N −1 (t + N ) (we used μ = 2N when r = N ). Thus, the desired characteristic polynomial is    1 2r − N N − 2r det t I N − A− IN − 2 B N N N 1 = N det [N t I N − (A − (N − 2r )I N − 2(2r − N )B)] N 1 = N det [(N t + N − 2r )I N − (A − μB)] N 1 = N φ(N t + N − 2r ) N ⎧ 1 (N t + N − 2 · 0) N −1 (N t + N − 2 · 0 − N ) if r = 0, ⎪ ⎪ NN ⎪ ⎨ 1 (N t + N − 2r ) N −r −1 ((N t + N − 2r ) + (4r − 2N ))r −1 = NN ⎪ ·((N t + N − 2r ) + 2r − N )((N t + N − 2r ) + (2r − 2N )) if 1 ≤ r ≤ N − 1, ⎪ ⎪ ⎩ 1 (N t + N − 2 · N + 2N ) N −1 ((N t + N − 2 · N ) + N ) if r = N , NN ⎧ N −1 ⎪ if r = 0, ⎨t(t + 1) 2r −N r −1 N −r −1 = t(t − 1)(t + N −2r ((t + N )) if 1 ≤ r ≤ N − 1, N ) ⎪ ⎩ if r = N . t(t + 1) N −1 This completes the proof. (The diagonalizability of 0 , which is equivalent to the diagonalizability of A − μB, can also be shown as follows. If r = 0 or N , we use the diagonalizability of A. If 1 ≤ r ≤ N − 1, we use the fact that the minimal polynomial of A − μB divides t(t + μ)(t 2 + (μ − N )t + μ(r − N )) = t(t + 4r − 2N )(t + 2r − N )(t + 2r − 2N ), which consists of distinct linear factors, and hence the minimal polynomial of A −μB must factor into distinct linear factors.)  

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