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Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration

We study phase concentration for the Kuramoto-Sakaguchi(K-S) equation with frustration via detailed estimates on the dynamics of order parameters. The Kuramoto order parameters measure the overall degree of phase concentrations. When the coupling strength is sufficiently large and the size of frustration parameter is sufficiently small, we show that the amplitude order parameter has a positive lower bound uniformly in time, and we also show that the total mass concentrates on the translated phase order parameter by a frustration parameter asymptotically, whereas the mass in the region around the antipodal point decays to zero exponentially fast.


1.
Introduction. The purpose of this paper is to continue the study begun in [22,35] on the phase concentration of the kinetic K-S type equations via the dynamics of order parameters. Since Christiaan Huygens's novel observation on the asynchronous behavior of two pendulum clocks, collective behaviors of a large population of coupled oscillators is often observed in biology, control theory, physics, and social sciences, etc, see [38,42]. However, its rigorous studies have been started a half century ago by two pioneers, Arthur Winfree and Yoshiki Kuramoto in [30,41]. Recently, collective behaviors of multi-agent systems and networks have received a lot of attentions due to increasing needs from the control of sensor networks, power networks, multi-agent UAVs and internet networks. In this paper, we are interested in the collective behaviors of the Kuramoto model with frustration, which was discussed in the seminal work of Sakaguchi and Kuramoto [39] as a prototype example of synchronization of phase-coupled oscillators.
Let θ i = θ i (t) ∈ T and α ∈ (− π 2 , π 2 ) be the phase of the i-th Kuramoto oscillator and the uniform frustration between oscillators, respectively. Then the dynamics of Kuramoto oscillators is governed by the following system [1,29,30,39,40]: sin (θ j − θ i + α) , t > 0, where ν i , κ > 0, and N ∈ N + denote the natural frequency, the coupling strength, and the number of oscillators, respectively. Since the right-hand side of (1.1) is Lipschitz continuous and bounded, a global well-posedness of equation (1.1) can be followed from the standard Cauchy-Lipschitz theory. Thus, one of remaining interesting issues is the large-time collective dynamics of (1.1) such as the existence of attractors, identification of basin of attractors and the relaxation process. The need of frustration is suggested from empirical facts in physics and biology [1,5,10,11,32,36,37,43]. The presence of frustration can cause numerous mathematical difficulties in the analysis of synchronization behaviors underlying oscillatory systems. To name a few, total phase N j=1 θ j is not conserved, and system (1.1) is not a gradient flow any more. Thus, analytical tools employed in the study of the Kuramoto model with zero frustration cannot be used as it is, e.g., energy method breaks down for system (1.1) with α = 0. Thus, analyzing large-time behavior of physical systems without conservation laws is one of challenging problems in nonlinear dynamics. For the Kuramoto model, detailed analysis for the emergence of synchronization has been extensively studied in [7,9,13,14,15,19,28,34] and references therein, in contrast, existing nonzero frustration problems are mostly based on numerical approaches due to the aforementioned difficulties. As far as we know, there are only few analytical results studying on the nonzero frustration case. In [20,21,33], the authors showed synchronization of oscillators for (1.1) with initial configurations confined in a half circle. These results were further extended to a region a slightly larger than a half circle for identical case in [18], Recently, the work [23] showed the emergence of phase-locking for system (1.1) with general initial data using the analysis of Kuramoto order parameters. At present, for an initial configuration which does not lie in a half circle, whether does system (1.1) with a distributed natural frequencies exhibit phase-locking in a large coupling regime is not known yet.
In this paper, we address synchronization phenomenon (or phase concentration) for the corresponding kinetic K-S equation which can be obtained from the particle model (1.1) in a mean-field limit. More precisely, we look for a sufficient framework leading to the phase concentration of the kinetic K-S equation. As the number of oscillators tends to infinity, i.e., N → ∞, the dynamics of a large particle system (1.1) can be effectively approximated by the corresponding mean-field kinetic equation. Below, we briefly introduce the mean-field kinetic equation which can be derived formally from the particle model (1.1) via the BBGKY hierarchy. Let f = f (t, ν, θ) be an one-oscillator probability density function at phase θ with natural frequency ν at time t. Then, the dynamics of f is governed by the following Cauchy problem to the K-S equation with frustration: A global existence of measure-valued solution for (1.2) was studied in [25], and the stability and instability analysis of the incoherent solution are studied in [17,24]. For zero frustration case, we refer to [2,3,4,6,8,12,16,26,27,31] and references therein for a global existence and asymptotic behaviors. In [22], the authors proved the amplitude order parameter has a positive lower bound near unity under the assumption that the coupling strength is sufficiently large, and they showed the total mass concentrates on the phase order parameter. Thus, following the same line of their study, we investigate the emergence of phase concentration for the K-S equation with frustration. More precisely, we study how the frustration affects the phase concentration dynamics for the K-S equation. In the presence of frustration, does the amplitude order parameter still has a lower bound near unity? If then under what condition? Does the mass still concentrate? If then, how does the concentration look like? Via a detailed analysis on the Kuramoto order parameters R(t), φ(t) , we show that the amplitude order parameter R(t) does have a positive lower bound R ∞ = R ∞ (κ, α): in a regime that the coupling strength is sufficiently large and the size of frustration is sufficiently small. We also present a positive lower bound R ∞ = R ∞ (κ) for the zero frustration case, which is somewhat different from the one given in [22], but it has the same property, i.e., it tends to unity as the coupling strength tends to infinity. Furthermore, we show the mass outside a time-dependent region D ∞ (t) which is away from the translated average phase φ(t) + α decays exponentially fast: For a detailed main result, we refer to Section 3.
The rest of this paper is organized as follows. In Section 2, we provide several estimates on the order parameters R(t), φ(t) , masses around the average phase φ(t) + α and φ(t) + α + π, and their relations. These estimates will be used crucially in later analysis. In Section 3, we provide phase concentration estimate for the K-S equation with frustration. In Section 4, we provide a proof of Proposition 3.1 which is used in section 3 without proof. In Appendix A, we provide the Lipschitz continuities ofṘ(t) and M + γ (t).

Preliminaries.
In this section, we study preliminary materials to be used in later sections. We first introduce real-valued Kuramoto order parameters and study their dynamics, and then we define two time-dependent intervals L + γ (t) and L − γ (t), with constant γ less than π 2 to be determined later, and study the evolution of masses on these intervals. Finally, we present several relations between R(t) and masses on L + γ (t) and L + γ (t).
First, we state a basic priori estimates on the positivity of f and conservation law. , Proof. Since a proof of can be found in [17,24], here we omit details here.
2.1. The order parameters. First, we introduce the real-valued order parameters R(t) ∈ [0, 1] and φ(t) ∈ T as follows: We divide the both sides of (2.1) by e iφ(t) , and compare the real and imaginary parts of the resulting relation to see If we divide the both sides of (2.1) by e i(θ−α) , and compare the imaginary part of the resulting relation to get Thus, equation (1.2) can be rewritten using order parameters: Let f = f (t, ν, θ) be a classical solution to (1.2), and let R(t) and φ(t) be the order parameters defined in (2.2). Then we have Proof. We first differentiate (2.1) with respect to t to geṫ Then we divide both sides of (2.5) by e iφ to obtaiṅ Now we use equation (2.3) and the periodicity of θ to havė We compare the real and imaginary parts of equation (2.6) to get estimates (2.4).
Note that Lemma 2.2 yieldṡ For the study ofṘ(t), we define Remark 2.1. By definition of R(t) and Lemma 2.1, we have Proof. We use relation (2.7) to obtain d dt Recall that equation (2.3) gives We substitute (2.9) into (2.8) and integrate the resulting relation by parts to have By using the positivity of f and relation (2.7), we get (2.11) and Thus, the remaining thing is to estimate 2ṘS and 2κR 2 T×R ν sin(θ − φ) cos(θ − φ)f dθdν.

2.2.
Evolution of masses on L ± γ (t). In this subsection, we study temporal evolution of masses on time-dependent intervals L + γ (t) and L − γ (t) around φ(t) + α and φ(t) + α + π: with γ less than π 2 to be determined later. We define 1 and 2 functionals on L + γ (t) and L − γ (t) as follows: where B + and B − are functions as follows: where B + and B − are functions as follows: Proof. By straightforward calculation, one has Now we come to estimate the second R.H.S. term of (2.24). Again we use relation Note that Now we substitute (2.25) into (2.24) to get

2.3.
Relations among R and M ± γ . In this subsection, we study relations among R(t) and masses on L ± γ (t). Lemma 2.7. Let f = f (t, ν, θ) be a classical solution to equation (1.2). Then we have Proof. We use relation (2.2) to get 3. Emergence of phase concentration. In this section, we show the emergence of phase concentration for the K-S equation in a regime that the coupling strength is sufficiently large, the size of frustration is sufficiently small, and the frequency Note that θ is the only dynamic variable for f , whereas ν variable acts like a parameter in the dynamics of f , so ν-support for f is propagated in time without changing. Using this observation, we combine the equation (1.2), Lemma 2.1, and relation (3.1) to see Now, we set where is a sufficiently small constant and ξ ∞ is defined as Then, our main result on phase concentration can be stated as follows.
with initial data f in continuously differentiable and R 0 := R(0) > 0. Suppose the coupling strength κ is sufficiently large, the size of frustration |α| is sufficiently small, and supp g(ν) ⊂ [− , ], then the following assertions hold: Remark 3.1. By analogue argument on nonidentical equation, our result holds for identical equation, i.e., g(ν) = δ(ν) case, and we do not need to assume the coupling strength κ is sufficiently large in this case. The exponential decay of density function f on T\D ∞ (t) means that f will concentrate to interval D ∞ (t)(centered at φ(t) + α and with width arccos( 1 − ξ 2 ∞ )+ ). This coincide with the result in [23], where the authors show that if the frustration is small and the order parameter has some lower bound, the particles will be phase-locked to some point in [φ(t) + α − σ, φ(t) + α + σ] except at most one particle which will be phase-locked to some point in [φ(t) + α + π −σ, φ(t)+α+π +σ] with σ a constant. Hence, we can expect under the assumption that frustration |α| is sufficiently small (the coupling strength κ is sufficiently large and the frustration |α| is sufficiently small for nonidentical equation, respectively), there exists some interval D ∞ (t) centered at φ(t) + α such that the density function f will concentrate to D ∞ (t) and decay to zero outside the concentration interval (on T\D ∞ (t)).
In the sequel, we present a proof of Theorem 3.1 by verifying each estimate in Theorem 3.1 one by one.
3.1. Derivation of the first estimate in (3.2). In this part, we show that there exists R ∞ which tends to 1 as α tends to zero and |κ| tends to infinity such that Since the derivation of (3.2) 1 is rather lengthy, we split the verification of the first estimate into several steps as follows. bullet Step A.1: We first derive a Riccati type differential inequality (see Lemma 3.1):Ṙ where C is a positive constant depending on γ, α. bullet Step A.2: Let κ be large enough and |α| be small enough, we show that there exists a positive constant µ such that R(t) > R, and lim sup where R ∈ (0, R 0 ) is a constant, and t 0 is any instant satisfying R(t 0 ) ≥ R 0 > 0. (see Proposition 3.1).
bullet Step A.3: We derive an exponential decay of M − γ (see Lemma 3.2): (3.4) bullet Step A.4: Finally, we use (3.3), (3.4) and comparison principle for ODE to derive a desired positive lower bound for R.
Next, we will present several useful lemmas as follows.
Let t 0 ≥ 0 be an instant such that R(t 0 ) ≥ R 0 = R(0) > 0. Then the following assertion holds: Proof. Since the proof is very lengthy, we postpone its proof in Section 4.

3.2.
Derivation of the second estimate in (3.2). In this part, we show the mass on T\D tends to zero exponentially. For R ∈ (0, 1], we set Then for κ 1 and |α| 1, Proposition 3.2 yields that R ∞ ≈ 1. Hence, we can get ξ ∞ := ξ(R ∞ ) < 1. In particular, there exists > 0 such that for any We consider two equations as follows: and (3.19) where θ ∈ T and p ∈ (− 1 − ξ 2 * , 1 − ξ 2 * ). For the second estimate in Theorem 3.1, we take the following two steps: bullet Step B.1: For any ν ∈ R, we show that the mass on D(t) decays to zero exponentially fast (see Proposition 3.3): (see Proposition 3.4).
Next, we perform the above outlined analysis one by one.
• (Exponential decay of mass on D(t)): In the sequel, we perform analysis for the mass on D(t). For this, we begin with the estimate on cos θ − φ(t * ) − α . Lemma 3.3. Suppose the assumption of Theorem 3.1 holds, and let Θ and P be solutions to (3.18) and (3.19), respectively. Then, the following assertions hold.
Note thatQ Now we use Lemma 2.4 (ii) and R(t) ≥ R * for any t ≥ T * to get Hence, we can geṫ Hence, together with the assumption cos θ − φ(t * ) − α ≤ p, we get for all t ∈ [T * , t * ).
(ii) It follows from the standard ODE that the solution P (t) of (3.19) satisfies Now set p = −δ, we use (3.20) to obtain Recall that assertion (i) yields for all t ∈ [T * , t * ), (3.22) Thus, we combine relations (3.21) and (3.22) to get for all t ∈ [T * , t * ). Now, we show that the mass on the region D(t) decays exponentially fast. Proposition 3.3. Suppose the assumption of Theorem 3.1 holds, and let Θ be a solution to (3.18). Then for any ν ∈ R, one has Proof. Take any t > T * and θ ∈ D(t), we use Lemma 3.2 (ii) to obtain Hence, we can obtain Now we use relation (3.23) to get • (Exponential decay of mass on T\D(t)): Below, we show that the mass on T\D tends to zero exponentially fast.
Lemma 3.4. Suppose the assumption of Theorem 3.1 holds, and let Θ be a solution to equation (3.18). If we take any θ ∈ T, δ ∈ (0, 1 − ξ 2 * ) and Proof. We first claim that Proof of claim (3.24). Recall that the solution P (t) of (3.19) satisfies Hence, for any δ ∈ (0, 1 − ξ 2 * ), there exists d such that Furthermore, we use equation (3.19) to get Thus, the constant d has a upper bound as follows which yields claim (3.25). Now let p = δ, then by assumption cos θ − φ(t) − α < δ = q and (i) of Lemma 3.3, we get Furthermore, we use relation (3.25) to obtain Hence, we have Proposition 3.4. Suppose the assumption of Theorem 3.1 holds, and let Θ be a solution to equation (3.18). Then for any ν ∈ R, we have Proof. Take any t > T * + D * , and θ ∈ T\D(t), we use Lemma 3.4 to get Analogue analysis as in Proposition 3.3 yields Thus, we can get Now we combine (3.27) and (3.28) to obtain That is

Lower bounds of R(t).
Note that, ifṘ(t) > 0 for any t ≥ 0, we can get the lower bound of R(t) by only assuming R 0 > 0. However, the conditionṘ(t) > 0 is not always true. Thus, to prove existence of a positive lower bound of R(t), we need to consider either the caseṘ(t 0 ) ≤ 0, or the caseṘ(t 0 ) ≤ µ with small enough µ.
Then by relation of d 0 , we know d 0 = 1 3κ cos α (ln a − ln b). Now we integrate (4.9) over [T 0 , T 0 + d 0 ) to obtain Hence, we have Now we are ready to prove our first estimate on the positive lower bound as follows.
Thus, to prove Proposition 4.1, we only need to show T is not empty and T * = ∞. • Step A: We check T is not empty. Now we discuss the following two cases: • Case A.1: IfṘ(t 0 ) > 0, then we use the continuity ofṘ(t) in Lemma A.1 to see that there exists a small constant η > 0 such thaṫ Thus, [t 0 − η, t 0 + η] ⊆ T .
• Case A.2: IfṘ(t 0 ) ≤ 0, then we use Lemma 4.1 to conclude Then by the the continuity of M + γ (t) proved in Lemma A.2, we see that there exists a small constant η > 0 such that That is, [t 0 − η, t 0 + η] ⊆ T . • Step B (T * = ∞): Suppose not, i.e., we assume If claim (4.12) holds, by the continuity of M + γ (t), these exists small η such that which contradicts definition of T * . Thus, T * = ∞.
(Proof of (4.12)): To prove (4.12), we first give two other claims: Firstly, it is clear to see By the continuity of R(t), we see that there exists a small constant η > 0 such that Secondly, we want to prove S * = T * .

Now we can get
which is a contradiction. Thus, the desired assertion holds.
Appendix A. Lipschitz continuities ofṘ(t) and M + γ (t). In this appendix, we show the Lipschitz continuities ofṘ(t) and M + γ (t) by establishing the uniform boundedness of their first derivatives.
Lemma A.1. (Uniform boundedness and Lipschitz continuity ofṘ) Let f and R be a classical solution to system (1.2), and the order parameter defined in (2.2), respectively. Then, the following assertions hold: (i) The order parameter R is Lipschitz continuous in [0, ∞), and its Lipschitz constant is given by |Ṙ(t)| < + κ, t ∈ [0, ∞).
(ii) Suppose that the following a priori condition holds: inf 0≤t≤T R(t) ≥ R for some T ∈ (0, ∞) and R > 0.
Then there exists η > 0 such thatṘ(t) is Lipschitz continuous in [0, T + η), and the Lipschitz constant is given by Proof. In the sequel, we estimate the terms J 2i one by one.
Then we complete our proof.
Lemma A.2. Let f = f (t, ν, θ) be a classical solution to (1.2) with the initial data f in satisfying f in L ∞ (T×R) < ∞ and a priori condition: inf 0≤t≤T R(t) > R.
Then the following assertions hold.
(ii) We first claim (Proof of claim (A.5)): Firstly, we define a forward characteristics Θ(t) issued from θ 0 ∈ T at time t = 0 as a solution to the Cauchy problem as follows.
Hence, we use assertion (i) and Claim (A.5) to obtain