Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration

1.   Introduction

Collective behaviors of oscillatory complex systems are ubiquitous in our nature, i.e., flashing of fireflies, beating of cardiac pacemaker cells, and arrays of Josephson junctions [1,10,27,30,32] etc. Recently, collective behaviors have received lots of attention from distinct scientific disciplines such as control theory, physics, neuroscience due to its applications in robot system, sensor network, and unmanned aerial vehicle. Among them, we are interested in the synchronization representing adjustment of rhythms of oscillators. The rigorous and systematic study for synchronization goes back to two pioneers Winfree and Kuramoto in a half century ago. In this paper, our focus lies in the Kuramoto model with frustration [29] (sometimes called the Kuramoto-Sakaguchi model). In order to fix the idea, let $\theta_i = \theta_i(t) \in \mathbb{T}$ and $\alpha$ be the phase of the $i$-th Kuramoto oscillator and the uniform frustration (phase shift) between oscillators. Then the phase dynamics of Kuramoto oscillators is governed by the following first-order system [20,21,28,29]:

where $\omega_i$, $\kappa$ and $N$ are the natural frequency of the $i$-th Kuramoto oscillator, coupling strength, and the number of oscillators, respectively.

Note that the K-S model (1.1) can be rewritten as

The terms in the R.H.S. of (1.2) correspond to the natural frequency, synchronization enforcing force and integrable forcing term, respectively. When the system size $N$ is sufficiently large, state of system (1.1) can be approximated by the continuity equation with nonlocal velocity field. More precisely, let $F = F(t, \theta, \omega)$ be a one-particle distribution function, and $f = f(t, \theta, \omega)$ be the conditional probability density function defined by the following relation:

where $g(\omega)$ is the probability density for natural frequency. Then, for $\omega \in \mathbb{R}$, we define a measurable map $\omega \longmapsto f_{\omega} : = f(\cdot, \cdot, \omega)$ from $\mathbb{R}$ to ${\mathcal P}(\mathbb{R}_+ \times \mathbb{T})$ (set of all probability measures on $\mathbb{T}$). Then the dynamics of conditional probability density $f_{\omega}$ satisfies

where differential operator $\partial_{\theta}$ is the derivation on $\mathbb{T}$ interpreted in the sense of distribution: for any $\mathcal{C}^k$ function $\phi$ on the circle, the action of $\partial_{\theta}(\mathcal{V}[f_\omega] f_{\omega})$ is given by

In the absence of frustration $\alpha = 0$, there have been lots of literatures on the Kuramoto model (1.1), to name a few [4,5,8,9,14,15,17,19,31] etc. As discussed in [6,7,22,25,26,33], frustration is needed for the realistic modeling of physical and biological oscillators. The mere change (1.1) of the Kuramoto model by adding frustration causes several analytical difficulties in the study of emergent dynamics. For example, the total phases:

are not conserved quantitites, and gradient flow structure for (1.1) is also destroyed. Thus, we cannot use the useful machineries for the Kuramoto model in the study of emergent dynamics for (1.1) and (1.3). So far, there are only few works on the Kuramoto model with frustration [2,12,23]. Recently, the work [16] investigated the emergent property for a finite-N particle model with frustration $|\alpha| < \frac{\pi}{2}$, and Ha and his collaborators studied the stability and instability of incoherent solution in [11,18], if initial data are sufficient regular. However, As far as the authors know, there are no results concerning all measurable stationary solutions and their stability, if the initial datum is just a Radon measure for system (1.3). Thus, in this paper, we address the following questions:

● (Q1): Are there measurable stationary solutions for the K-S equation (1.3)?

● (Q2): If there exists a stationary solution, are they nonlinearly stable?

The purpose of this paper is to answer the above questions. First, we discuss the first question (Q1). Mirollo and Strogatz [24] presented some special stationary solutions in the absence of frustration. In contrast, when frustration effect is present, many tricks employed in the Kuramoto model do not work mainly due to the non-conservation of the total phase (see Lemma 2.1 for details). Hence, we can not expect genuine stationary solutions. For this, we define a new variable $\widetilde{\theta}(\theta, t)$:

By studying a new equation formulated in terms of ${\tilde \theta}$, we present all measurable stationary solutions for the reformulated equation. Second, we study a nonlinear stability of stationary solutions in more general case. Let $\mathcal{M}(\mathbb{T}\times{\mathbb{R}})$ be the set of non-negative Radon measures on $\mathbb{T}\times{\mathbb{R}}$. Then, for a Radon measure $\mu \in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$, we use the standard duality relation:

where $\mathcal{C}_0^{\infty}$ denotes the set of smooth functions vanishing at infinity.

Consider the stability of stationary solutions arising from the Cauchy problem:

where $\mu_t(d\theta d\omega) : = g(\omega) df_{\omega} d\omega$.

We first study the stability of the complete phase-locked state (stationary solution with $R = 1$, see Definition 2.2 for details), and we show that the complete phase-locked state is nonlinearly stable in space of Radon measures in which the size of initial phase diameter is smaller than $\pi - 2|\alpha|$ (see Theorem 2.3 for more details). On the other hand, we discuss the stability of partial phase-locked state. Let $\mu_t, \nu_t \in \mathcal{C}_{w} \big( [0, T]; \mathcal{M}(\mathbb{T}\times{\mathbb{R}}) \big)$ be measurable solutions to system (1.4) (see Definition 2.4 for details). One of difficulty that we confront with is the lack of exponential stability (see [3]):

where $W_p(\mu_t, \nu_t)$ is the $p$-Wassertein distance between two measures $\mu_t$ and $\nu_t$. In fact, let $\phi_i, i = 1,2$ be the pseudo-inverse functions associated to $\mu_t$ and $\nu_t$, denote $R_i$ by the order parameters of $\phi_i$ respectively, then by Lemma 2.1 and proof of Theorem $5.1$ in [3], $\frac{d}{dt}W_p(\mu_t, \nu_t)$ is bounded by $|\kappa R_1^2(t) \sin\alpha - \kappa R_2^2(t) \sin\alpha|$, so we have no knowledge whether $W_p(\mu_t, \nu_t)$ will converge to zero. For this, we introduce the characteristic function for the K-S system:

and consider the dynamics of the following quantity:

where $B_{\theta}(t)$ is the orthogonal $\theta$-projection of $\mbox{supp}(\mu_t)$, and then show

which yields that $\lim_{t \rightarrow \infty}(\Theta_1 - \Theta_2)$ exists and finite, that is, phase-locked state emerge asymptotically. Moreover, we obtain the semi-stability of the partially phase-locked state in space of Radon measures under suitable conditions (see Theorem 2.6 for details). Finally, we study the stability of incoherent state to system (1.4) for identical oscillators, i.e., the natural frequency is a constant, and without loss of generality, we set $\omega = 0$ for simplicity. In a large frustration regime, the order parameter $R$ tends to zero, as time tends to infinity, i.e., the incoherent solution is stable, whereas in a small frustration regime, the order parameter will tends to zero if there exists a constant $M$ such that $\mu_t(d\theta) \leq M \mu_e(d\theta)$ with $\mu_e$ a renormalized measure satisfying $\mu_e(\mathbb{T}) = 1$, otherwise, the probability distribution concentrates and forms a Dirac mass.

The rest of this paper is organized as follows. In Section 2, we briefly present main results of this paper. In Section 3, we discuss measurable stationary solutions. We first verify that the incoherent state is unique, then we find out all phase-locked states. In Sections 4 and 5, we study the stability of phase-locked states and incoherent state by providing suitable frameworks respectively.

2.   Presentation of main results

In this section, we present sufficient frameworks and main results on the existence of stationary solutions and their stabilities for the kinetic K-S equation.

2.1.   Existence of stationary solutions

In this subsection, we introduce the order parameters and present stationary solutions for the K-S equation (1.3). We first define the order parameters $(R, \psi)$ as follows:

We divide both sides of relation (2.1) by $e^{i \psi}$, and separate the real and imaginary part to obtain

Moreover, we further divide both sides of relation (2.1) by $e^{i (\theta- \alpha)}$, and compare the real and imaginary parts of the resulting relation to get

Lemma 2.1. Suppose the natural frequency distribution $g = g(\omega)$ is integrable. Then, one has

Proof. We use (2.3) to rewrite the nonlocal velocity ${\mathcal V}$ as

Then we use (2.5) and (2.2) to get

By Lemma 2.1, we can not get equilibria for system (1.3) unless $R \equiv 0$. Thus, we can get a relative equilibrium which is an equilibrium in a rotating frame with a constant velocity. For this, we define

and consider $f_{\omega} (t, \tilde{\theta}, \omega)$ instead of $f_{\omega}(t, \theta, \omega)$. For simplicity, we still write $\theta$ for $\tilde{\theta}$. Then our original K-S equation (1.3) can be rewritten as

Next, we recall several distinguished states in the following definition.

Definition 2.2. Let $f$ be a measurable stationary solution to (2.4), and $R = R_f$ is a corresponding order parameter associated with $f$ in (2.1).

1. If $R \equiv 0$, then $f$ is called an incoherent state.

2. If $0 < R< 1$, then $f$ is called a partial phase-locked state.

3. If $R \equiv 1$, then $f$ is called the complete phase-locked state.

Now we are ready to present our first main theorem as follows.

Theorem 2.3. Suppose that the natural frequency distribution $g = g(\omega) \in L^1(\mathbb{R})$ and

Then, the following assertions hold.

1. The incoherent state of (2.4) and (2.5) is unique, and $f_{\omega} = \mu_e$ is the normalized Lebesgue measure on $\mathbb{T}$ with $\mu_e(\mathbb{T}) = 1$.

2. If the phase-locked state of (2.4) and (2.5) exists, then

and the following relations must hold:

where $0 \leq \eta(\omega) \leq 1$ is a constant, $C_{s}$ is given by

with signs determined by $\int_{\mathbb{T}}\frac{C_s}{s- \kappa R \sin(\theta - \alpha)} = 1$, phases $\theta_{\omega} - \alpha \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ and $\theta_{\omega}^* - \alpha = \pi - (\theta_{\omega} -\alpha)$ are the roots of equation:

3. If $g(\omega)$ is non-increasing on $[0, L]$, then

Remark 1. From the proof of Theorem 2.3, we will see that the same results hold for identical case except the uniqueness of the incoherent state.

2.2.   Stability of phase-locked states

In this subsection, we present main results on the stability of complete phase-locked state and partial phase-locked state. Denote by ${\mathcal{C}}_w \big([0, T); \mathcal{M}(\mathbb{T}\times{\mathbb{R}})\big)$ the space of weakly continuous time-dependent measures. First, we present a definition of measure-valued solution for (1.4) as follows.

Definition 2.4 (Measure-valued solution). For $T\in (0, \infty]$, we say $\mu_t \in {\mathcal{C}}_w \big([0, T); \\ \mathcal{M}(\mathbb{T}\times{\mathbb{R}})\big)$ be a measure-valued solution to (1.4) with an initial Radon measure $\mu_0 \in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$ if $\mu_t$ satisfies the following conditions:

(ⅰ) $\langle \mu_t, h \rangle$ is continuous as a function of time $t$, for any $h \in \mathcal{C}_0^{\infty}(\mathbb{T} \times \mathbb{R})$.

(ⅱ) For any $h \in\mathcal{C}_0^{\infty}([0, T] \times \mathbb{T}\times{\mathbb{R}})$,

where $\mathcal{V}[\mu](s,\theta, \omega)$ is defined by

Remark 2. Below, we provide a brief comment on measure-valued solutions.

1. Recall that $\mbox{supp}(\mu)$ (the support of a measure $\mu$) is the closure of the set consisting of all points $(x,v) \in {\mathbb R}^{2d}$ such that $\mu(B_r((x,v))) > 0,\; \; \forall r > 0$. For a finite measure with a compact support, we can use $h \in C^1({\mathbb R}^{2d})$ as a test function in (2.6). Thus, we choose $h = 1, \omega$ in (2.6) to get

2. Let $(\theta_i(t), \omega_i(t))$ be a solution of following ODE system:

Then, the empirical measure

is a measure-valued solution to system (1.4).

Now let $\mu_t \in \mathcal{C}_{w} \big( [0, T]; \mathcal{M}(\mathbb{T}\times{\mathbb{R}}) \big)$ be a measurable solution to system (1.4). We define $B_{\theta}(t)$ and $B_{\omega}(t)$ be the orthogonal $\theta$ and $\omega$-projections of $supp \,\mu_t$, respectively, i.e.,

where $M$ and $\mbox{diam} A$ are given as follows.

We use Remark 2 to see

For identical oscillators with $g(\omega) = \delta_{\omega_c}$, we set

Our second main result deals with the stability of complete phase-locked state (i.e. for identical oscillators, $g(\omega) = \delta_{\omega_c(0)})$ as follows.

Theorem 2.5. Suppose that the initial datum $\mu_0 \in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$ satisfies

Then the measure-valued solution $\mu_t$ to system (1.4) satisfies

where $\Lambda_0 : = \frac{2}{\pi} \cos\big(\frac{1}{2}D_{\theta}(\mu_0) + |\alpha| \big)$ is a positive constant, and $\mu_{\infty}(d\theta d\omega) = \delta_{\theta_c(t)} \times \delta_{\omega_c(0)}$. In particular, we obtain the stability of the complete phase-locked state in space of Radon measure-valued solution whose initial data satisfy (2.7).

For nonidentical oscillators, a difficulty we need to overcome is that the exponential decay (see [3]):

where $W_p(\mu_t, \nu_t)$ is the $p$-Wasserstein distance between two measures $\mu_t$ and $\nu_t$. As discussed in the introduction, $\frac{d}{dt}W_p(\mu_t, \nu_t)$ is bounded by $|\kappa R_1^2(t) \sin\alpha - \kappa R_2^2(t) \sin\alpha|$ with $R_i, i = 1,2$ the order parameter of the pseudo-inverse functions associated to $\mu_t$ and $\nu_t$, so we have no knowledge if $W_p(\mu_t, \nu_t)$ will converge to zero. For this, we study the characteristic function for system (1.4):

and define

Our third main result deals with the exponential stability of $D_\theta^1$.

Theorem 2.6. Suppose the initial measure $\mu_0 \in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$ satisfies

and let $\mu_t$ be a measure-valued solution to (1.4). Then, there exists a positive time $t_0$ such that

where $D^{\infty} \in (0, \frac{\pi}{2} - |\alpha|)$ is the solution of

In particular, we obtain the semi-stability of the partial phase-locked state in space of Radon measure-valued solution whose initial data satisfy (2.8).

2.3.   Stability of the incoherent state

In this subsection, we provide stability and instability estimates for the incoherent state. First, we discuss the small frustration case:

where $|\alpha| < \frac{\pi}{2}$. Our main results are as follows.

Proposition 2.7. (Small frustration) Suppose frustration and initial datum satisfy

and let $\mu_t$ be a measure-valued solution for system (2.9). Then, the following assertions hold.

1. If there exist a positive constant $M$ such that $\mu_t(d\theta) \leq M \mu_e(d\theta)$, then we have

2. If there does not exists a positive constant $M$ such that $\mu_t(d\theta) \leq M \mu_e(d\theta)$, then we have

Next, we consider a large frustration $|\alpha| \geq \frac{\pi}{2}$ case. For this, we set $\hat{\alpha} : = \alpha - \frac{\pi}{2}.$ Then, the original identical system can be rewritten as

Then, our last result is as follows.

Proposition 2.8. (Large frustration) Suppose frustration and initial datum satisfy

and let $\mu$ be a measure-valued solution for system (2.10). Then we have

3.   Existence of stationary solutions

In this section, we look for all measurable stationary solutions for the kinetic K-S equation (2.4). Without loss of generality, we may assume that phase order parameter $\psi = 0$.

Note that the stationary solution of (2.4) satisfies the equation:

Recall that the distribution $\xi$ on $\mathbb{T}$ satisfying $\partial_{\theta} \xi = 0$ is equal to a constant $\tilde{C}_{\omega}$ multiples of normalized Lebesgue measure $\mu_e$ on $\mathbb{T}$. In other words, $d\mu_e = \frac{1}{2\pi} d\theta$. Hence, the stationary solution for (2.4) should satisfy

In the following proposition, we show that a unique incoherent state is given by $\mu_e$.

Proposition 3.1. The incoherent state for (2.4) and (2.5) is unique, and

where $\mu_e$ is the normalized Lebesgue measure on $\mathbb{T}$.

Proof. Note that the incoherent solutions for (2.4) satisfies

Since $R = 0$, we have

Thus, we have

On the other hand, note that

Thus,

Next, we classify all the phase-locked states.

Proposition 3.2. The phase-locked state for (2.4) and (2.5) is

where $\eta(\omega)$ is a positive constant, and $C^{\pm}_{\omega - \kappa R^2 \sin\alpha}$ is given by

with signs determined by

phases $\theta_{\omega} - \alpha \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, $\theta_{\omega}^* - \alpha : = \pi - (\theta_{\omega} -\alpha)$ are the roots of equation:

Proof. For a proof, we divide the domain of the natural frequency $\omega$ into two cases:

$\bullet$ Case A. Suppose that

Then we use relation (2.4) and assumption $\psi = 0$ to get

Hence, we use relation (3.1) to obtain

We use formula $\int_{\mathbb{T}} df_{\omega} = 1$ to get

By direct calculation, one has

We set

with signs of $C_{\omega - \kappa R^2 \sin\alpha}$ determined by

Hence, we have

$\bullet$ Case B. Suppose that

In this case, we claim:

Proof of claim (3.4). Suppose not, then for $\theta \notin \big\{\omega - \kappa R^2 \sin\alpha - \kappa R \sin(\theta - \alpha) = 0 \big\}$

Then, there exists $\theta \notin \big\{\omega - \kappa R^2 \sin\alpha - \kappa R \sin(\theta - \alpha) = 0 \big\}$ such that $d f_{\omega} < 0$. This gives a contradiction since $df_{\omega} \geq 0$, then our claim holds. Hence, we have

We use $\int_{\mathbb{T}} d f_{\omega} = 1$ to get

i.e.,

Thus, we can obtain

where $\theta_{\omega} - \alpha \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ is the solution of equation:

and $\theta_{\omega}^* = \pi - \theta_{\omega} + 2\alpha$. Now we summarize the value of $f_{\omega}$ as follows:

In Proposition 3.2, we have determined the ansatz for $f_{\omega}$ in terms of $R$. It is clear that $f_{\omega}$ should satisfy the following relations with respect to order parameters:

Lemma 3.3. The phase-locked state in (3.2) satisfies the following relations:

where $C_{s} = \pm \frac{1}{2\pi} \sqrt{s^2 - (\kappa R)^2}$ for any variable $s$, $\eta(\omega)$ is a constant satisfying $0 \leq \eta(\omega) \leq 1$.

Proof. We split the proof into two cases:

$\bullet$ Step A. We first estimate the integral:

We use the relation (3.3) and

to get

where

In (3.6), we use the estimate (3.7) to see

where we have used the relations:

Similar to (3.8), we get

$\bullet$ Step B. Next, we consider the integral:

By direct calculation, one has

Note that

In (3.10), we combine estimates (3.11) and (3.12) to obtain

Similarly, we get

Now we substitute estimates (3.8), (3.9), (3.13) and (3.14) into (3.5) to obtain

and

We multiply relation (3.15) by $\sin\alpha$, and multiply relation (3.16) by $\cos\alpha$, and then take the difference of the two resulting relations to derive

Next, we substitute relation (3.17) into (3.15) to get

Since $\alpha \in (-\frac{\pi}{2}, \frac{\pi}{2})$, we know $\cos\alpha \neq 0$. Thus, we have

Lemma 3.4. If the phase-locked state for (2.4) and (2.5) exists, and $g(\omega)$ is non-increasing on $[0, L]$, then the upper bound $L$ of natural frequency should satisfy

Proof. Recall that Lemma 3.3 yields

By direct calculation, we have

Thus, we have

Next, we will prove only $\alpha \geq 0$ case. The other case $\alpha < 0$ can be treated similarly. Suppose that

Then, we will derive a contradiction by ruling out the following two cases:

$\bullet$ Case A $(\ell \geq \kappa R + \kappa R^2 \sin \alpha)$. In this case, we analyze R.H.S term of relation (3.18) by using the non-increasing property of $g$ on $[0, +\infty)$ to get

This contradicts to relation (3.18).

$\bullet$ Case B $(\kappa R -\kappa R^2 \sin \alpha < \ell < \kappa R + \kappa R^2 \sin \alpha)$. Note that

This also contradicts to relation (3.18). Hence, we derived the desired estimate.

Proof of Theorem 2.3. First, Proposition 3.1 gives $(1)$ of Theorem 2.3 directly. Next, Proposition 3.2 shows if the phase-locked state exists, then

where $0 \leq \eta(\omega) \leq$ is a constant, $C^{\pm}_{\omega - \kappa R^2 \sin\alpha}$ is given by

with signs determined by

and phases $\theta_{\omega} - \alpha \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, $\theta_{\omega}^* - \alpha : = \pi - (\theta_{\omega} -\alpha)$ are the roots of equation:

Furthermore, Lemmas 3.3 and 3.4 show that the following relations must hold:

4.   Stability of phase-locked states

In this section, we study stability estimates of the phase-locked states for equation (1.4), i.e., stability of the complete phase-locked state and partial phase-locked state, respectively.

4.1.   Existence of measure-valued solutions

We first derive a global existence of measure-valued solution for the corresponding mean-field model.

Recall that the order parameters $R$ and $\psi$ can be redefined as follows.

Then as in Section 2, one has

and

Thus, equation (1.4) can be rewritten as

4.1.1.   Emergence of phase-locked states

In this subsection, we list the emergent estimates for the Kuramoto-Sakaguchi model for later use. Since the methodology for proofs is similar to the arguments given in [12,13,23], we leave detailed proofs in Appendix A. Consider the following N-particle Kuramoto model with frustration:

and for a given phase vector $\Theta = (\theta_1, \cdots, \theta_N)$, we define

Below, we state two asymptotic phase-locking for identical and non-identical ensembles.

Proposition 4.1. The following assertions hold.

1. Suppos natural frequencies, coupling strength and initial data satisfy

and let $\theta_i$ be a solution to (A.1). Then, we have exponential synchronization:

2. Suppose natural frequencies, coupling strength and initial data satisfy

Then, we have

where $D^{\infty} \in (0, \frac{\pi}{2} - |\alpha|)$ is the root of the following trigonometric equation:

Proof. We leave its proof in Appendix A.

4.1.2.   Measure-valued solutions

Next, we briefly study a global existence of measure-valued solution for system (1.4) and its property.

Theorem 4.2. For any $\mu_0\in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$, let $\mu_t$ be a unique measure-valued solution to system (1.4) with the initial data $\mu_0$. Then $\mu_t$ can be approximated by a sequence of empirical measures:

Furthermore, one has

Proof. Since the proof is nearly the same as in [3] using the N-particle theory in Section 4.1.1, we omit its proof.

Lemma 4.3. Suppose the initial measure satisfies

and let $\mu_t$ be a measure-valued solution of system (1.4). Then, for $t\geq{0}$, one has

Proof. We take $h = \theta$ in relation (2.6) and use system (4.3) to get

Note that assumption on initial datum and Remark 2 $(1)$ yield

Thus, we use relation (4.2) to obtain

4.2.   Complete phase-locked state

In this subsection, we study the stability estimate of the complete phase-locked state. Let $\mu_t \in \mathcal{C} \big( [0, T]; \mathcal{M}(\mathbb{T}\times{\mathbb{R}}) \big)$ be a measurable weak solution to system (1.4).

Recall that

where

Since Remark 2 $(1)$ gives that $\langle\mu_t, \omega\rangle = \langle\mu_0, \omega \rangle$, one has

Note that for identical oscillators, without loss of generality, we may assume $\omega_c(0) = 0$. Thus,

Now we use Theorem 4.2 and Proposition 4.1 to get an exponential decay of $D_\theta(\mu_t)$.

Lemma 4.4. Suppose the initial datum $\mu_0 \in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$ satisfies

and let $\mu_t$ be a measure-valued solution to system (1.4). Then, there exists a positive constant $\Lambda_0 : = \frac{2}{\pi} \cos\big(\frac{1}{2}D_{\theta}(\mu_0) + |\alpha| \big)$ such that

Proof. We define $\mu_0^N$ as in reference [3]. Note that Proposition 4.1 gives that the approximate measure valued solution $\mu_t^N \in \mathcal{M}(\mathbb{T} \times \mathbb{R})$ satisfies

Now we use Theorem 4.2:

to see

Hence, our desired stability estimate is obtained.

Now, we set

Theorem 4.5. Suppose the initial datum $\mu_0 \in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$ satisfies

and let $\mu_t$ be a measure-valued solution to system (1.4). Then, one has

Proof. Let $h \in \mathcal{C}(\mathbb{T})$ be an arbitrary test function satisfying

Then we have

where

We use Theorem 4.2 to conclude that

As a corollary of Theorem 4.5, we obtain the exponential stability of the complete phase-locked state in the space of Radon measure-valued solutions.

Corollary 4.6. Suppose that the initial datum $\mu_0 \in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$ satisfies

and let $\mu_t$ be a measure-valued solution to system (1.4). Then, $\mu_t$ is asymptotically phase-locked. In particular, we obtain the stability of the complete phase-locked state in the space of Radon measure solution with initial datum satisfying (4.6).

Proof. Let $\Theta_1^0$ and $\Theta_2^0$ be initial measures satisfying

Then, it follows from Theorem 4.5 that

where $\dot{\Theta}(t, \theta, \omega) = \kappa \int_{\mathbb{T} \times \mathbb{R}} \sin(\Theta_* - \Theta + \alpha) \mu(d\theta_* d\omega_*)$ is the characteristic function.

4.3.   Partial phase-locked state

In this subsection, we study the nonlinear stability of partial phase-locked state to the K-S equation (1.4).

Consider the characteristic function defined by the following system:

In this case, we set

Then, the new variable $\Phi(t)$ satisfies

We also define

Note that $0 \leq D_{\theta}^1(\mu_t) < \infty$. In fact, for any $\Theta(t) \in B_{\theta}(t)$, we use the compactness of $g$ in Lemma 3.4 to have

In the sequel, we will prove that the quantity $D_{\theta}^1(\mu_t)$ tends to zero exponentially fast as $t \to \infty$.

Lemma 4.7. Let $\mu_0 \in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$ be a given initial measure such that

and let $\mu_t$ be a measure-valued solution to equation (1.4) with an initial datum $\mu_0$. Then, there exists a time $t_0$ such that

where $D^{\infty} \in (0, \frac{\pi}{2} - |\alpha|)$ is the solution of

Proof. For given $N> 0$, we consider the approximation $\mu_0^N$ for $\mu_0$:

We solve the Cauchy problem for the following N-particle system:

with the initial data $(\theta_i^0, \omega_i^0)$. We use Theorem 4.2 to obtain

Furthermore, Proposition 4.1 shows that there exists time $t_0$:

with $D^{\infty, N} + |\alpha| = \arcsin(D_{\theta}(\mu_0^N) + |\alpha|) \in (0, \frac{\pi}{2})$ such that

for $N$ large enough. Now let $N$ tends to infinity to obtain the desired results.

Theorem 4.8. Let $\mu_0 \in \mathcal{M}(\mathbb{T}\times{\mathbb{R}})$ be a given initial measure such that

and let $\mu_t$ be a measure valued solution to equation (1.4). Then there exists a positive time $t_0$ such that

Proof. For $t > t_0$ and any $\varepsilon \in (0, \frac{1}{8})$, we take any $\Theta_{M, \varepsilon} \in B_{M, \varepsilon}$ and $\Theta_{m, \varepsilon} \in B_{m, \varepsilon}$ such that

Then, we have

Below, we estimate $\mathcal{J}_{1i},\; i = 1,2,3$ one by one.

$\bullet$ Case A (Estimate on $\mathcal{J}_{11}$). It follows from relation (4.7) and Lemma 4.7 that

$\bullet$ Case B (Estimate on $\mathcal{J}_{12}$). Similar to Case A, we get

$\bullet$ Case C (Estimate on $\mathcal{J}_{13}$). We use Lemma 4.7 to obtain

Now, we combine all estimates to derive

Thus, we can derive

Finally, we use Gronwall's lemma to get

Corollary 4.9. Let $\mu_0 \in \mathcal{M}(\mathbb{T} \times{\mathbb{R}})$ be an initial measure satisfying

Then, the measure-valued solution $\mu_t$ to equation (1.4) is asymptotically phase-locked. In particular, we obtain the semi-stability of the partially phase-locked state in space of Radon measure solution whose initial data satisfy (2.8).

Proof. Note that Theorem 4.8 yields

Then we have

This means that for any $\Theta_1$ and $\Theta_2$,

Now we use Theorem 4.8 and definition of order parameters to get

We set

Then, we can get

Hence, for any $\Theta(0)$ satisfy assumption (2.8), $\Theta(t)$ approaches to an equilibrium described in $(2)$ of Theorem 2.3. In particular, the partially phase-locked state we obtained in Section 3 is semi-stable.

5.   Stability of the incoherent state

In this section, we study the stability of incoherent solution to the K-S equation with a frustration for identical oscillators. For this, we define a new quantity:

Lemma 5.1. Let $\mu_t : = \frac{\mu_e}{2\pi}$ be a uniform distribution on ${\mathbb S}^1$ such that

Then we can have

Proof. By direct estimate, we have

By Lemma 5.1, we can see that $\mathcal{I}(t)$ works well (at least for the uniform distribution on $S^1$). As a corollary of Lemma 5.1, we have upper and lower bound estimates for ${\mathcal I}$.

Lemma 5.2. Suppose there exists positive constants $m$ and $M$ such that

Then, the quantity $\mathcal{I}(t)$ satisfies

Proof. We use the assumptions (5.1) to get that for all $t \geq 0$,

Note that

Thus, we have

5.1.   Small frustrations

In this subsection, we consider small frustration case with $\alpha \in (-\frac{\pi}{2}, \frac{\pi}{2})$. Similar to system (4.3), we use order parameters to rewrite equation (2.9) as

We differentiate both sides of (4.1) with respect to $t$ and use equation (5.2) to get

Now we divide both sides of relation (5.3) by $e^{{\mathrm i}\psi}$, and compare the real and imaginary parts to obtain

Lemma 5.3. Let $\mu_t \in \mathcal{M}(\mathbb{T} \times{\mathbb{R}})$ be a measure-valued solution for equation (2.9). Then, for all $t \geq 0$,

Proof. $(i)$ We use equation (5.2) to obtain

Integrations by parts yield

By direct calculation, one has

Now we use the above estimates and relation:

to obtain

We use relation (4.2) to derive

Thus, we have

This yields

$(ii)$ Now we differentiate relation (5.6) with respect to time $t$ to get

Hence, we use relation (5.4) to obtain

Proposition 5.4. Suppose the frustration and initial datum satisfy

and let $\mu$ be a measure-valued solution for equation (2.9).

1. If there exist constant $M$ such that $\mu(d\theta) \leq M \mu_e(d\theta)$, then we have

2. If not, we have

Proof. (1) We use Lemma 5.3 to see that $\frac{d}{dt} \mathcal{I}(t)$ is uniformly continuous and $\mathcal{I}(t)$ is decreasing. Now if there exist a constant $M$ such that $\mu(d\theta) \leq M \mu_e(d\theta)$, we can use Lemma 5.2 to get

Hence, we deduce

Together with initial assumption

we obtain

Thus, we can use Barbalat's Lemma to conclude

Now we use

to see

(2) If we can not find a positive constant $M$ such that $\mu(d\theta) \leq M \mu_e(d\theta)$, then we obtain

Furthermore, we can see

Thus, we can deduce

5.2.   Large frustrations

In this subsection, we consider a large frustration case ($|\alpha| \geq \frac{\pi}{2}$). For this, we define $\hat{\alpha} = \alpha - \frac{\pi}{2}$. Then, the original K-S equation becomes

Note that the assumption on $\alpha$ and definition of $\hat{\alpha}$, we have $\hat{\alpha} \in [0, \pi]$, and since $\alpha \in [\frac{\pi}{2}, \pi]$, it is easy to see that $\hat{\alpha} \in [0, \frac{\pi}{2}]$. As $\alpha \in (- \pi, -\frac{\pi}{2}]$, we have

We divide both sides of relation (4.1) by $e^{{\mathrm i}(\theta - \hat{\alpha})}$ and take the real part of the resulting relation to have

We use relation (5.8) to rewrite the equation (5.7) to rewrite as

We differentiate both sides of (4.1) with respect to $t$ and use system (5.9) to get

Now, we divide both sides of above relation by $e^{i\psi}$ and separate the real and imaginary parts to obtain

Now, we define the first phase moment $m_1(t)$ as follows.

Lemma 5.5. Let $\mu_t \in \mathcal{M}(\mathbb{T} \times{\mathbb{R}})$ be a measure-valued solution for equation (2.10). Then one has

Proof. $(i)$ We use relation (5.9) to obtain that

We use integration by parts to get

By direct calculation, one has

Now, we use (5.11), (5.12) and relation

to obtain

Similar to the derivation of (5.5) in Lemma 4.7, we have

Hence, we conclude

We use relation (5.9) to obtain

$(ii)$ Now we differentiate relation (5.13) with respect to time $t$ to get

We use relation (5.10) to deduce

Corollary 5.6. Let $\mu_t \in \mathcal{M}(\mathbb{T} \times{\mathbb{R}})$ be a measure-valued solution for equation (2.10). Then, the quantity $\cos \hat{\alpha} \mathcal{I}(t) - \sin\hat{\alpha} m_1(t)$ is invariant with respect to time $t$:

Proof. We combine estimate $(i)$ of Lemma 5.5 $(i)$ to get

This yields our desired estimate.

Remark 3. For all $t \geq 0$, for $\hat{\alpha} = 0$, Lemma 5.5 gives

and for $\hat{\alpha} = \frac{\pi}{2}$, Lemma 5.5 gives $m_1(t) = m_1(0)$.

Proposition 5.7. Suppose frustration and initial datum satisfy

and let $\mu_t \in \mathcal{M}(\mathbb{T} \times{\mathbb{R}})$ be a measure-valued solution for equation (2.10). Then we have

Proof. It follows from Lemma 5.5 that

This yields that $\frac{d}{dt} \mathcal{I}(t)$ is uniformly continuous. Furthermore, we use the assumption on $\mu_0$ to get

Thus, we apply Barbalat's Lemma to conclude

Now we use the relation $\frac{d}{dt} \mathcal{I}(t) = \kappa R^2 \sin\hat{\alpha}$, $\hat{\alpha} \in (0, \pi)$ and (5.14) to obtain

Appendix A.   Proof of Proposition 4.1

In this appendix, we provide proofs for Proposition 4.1 on the emergent dynamics of the Kuramoto-Sakaguchi equation.

A.1.   Proof of the first part

Consider an ensemble of identical oscillators. Without loss of generality, we may assume

In this case, the Kuramoto model becomes

Lemma A.1. (Phase coherence) Suppose the coupling strength and initial data satisfy

and let $\{ \theta_i \}$ be a solution to (A.1). Then, the following relations hold:

Proof. (ⅰ) The first relation can be obtained from Lemma $3.1$ in [12].

(ⅱ) Note that the phase diameter $D(\Theta)$ satisfies

where we have used the following relations:

Now, we are ready to provide the first estimate in Proposition 4.1.

Let $\Theta$ be a solution to system (A.1). By definition of $D(\Theta)$, one has

Since

one has

Then, we use (A.3) to get

This yields

Now we substitute (A.4) into (A.2) to derive a differential inequality:

Since $\frac{\theta_M - \theta_m}{2} \in (0, \pi)$, one has

Now we combine (A.5) and (A.6) to obtain

We integrate the differential inequality (A.7) with respect to time $t$ to get

A.2.   Proof of the second part

In this subsection, we consider the non-identical Kuramoto-Sakaguchi equation (4.4). We find the exact time $t_0$ such that all the particles trapped in half circle after time $t_0$, which is important for asymptotic phase-locking of non-identical oscillators in Section 4.3.

Lemma A.2. Suppose initial data, natural frequencies and coupling strength satisfy

Then we have

Proof. We will use the continuity argument. For this, we define

We claim:

Suppose not, i.e., $T_* < +\infty$. Then there exists $t_0$ such that

Then we use continuity of $D(\Theta)(t)$ to see

On the other hand, note that the system (4.4) can be rewritten as

Hence, we use definition of $D(\Theta)$ in (4.5) to obtain

We use (A.9) to see that for $t \in (0, t_0)$

and

Hence, for $t \in (0, t_0)$, one has

Now we substitute (A.12) and (A.13) into (A.11) to obtain

Thus, we use (A.14) to get

This contradicts (A.10). Thus, we deduce that $T_* = +\infty$, and our assertion holds.

Now, we are ready to provide a proof of the second part.

Suppose initial data, natural frequencies and coupling strength satisfy

Then we claim: for all $t > t_0 = \frac{D(\Theta^{in}) - D^{\infty}}{(1 -\frac{\kappa}{\kappa_e}) D(\Omega)}$,

where $D^{\infty} \in (0, \frac{\pi}{2} - |\alpha|)$ is a root of the equation:

We split the proof of claim (A.15) into two steps.

$\bullet$ Step A. We need to find time $t_0$ such that

For this, we consider two cases.

$\diamond$ Case $I$. Suppose

Then, one has

Thus, it follows from Lemma A.2 that we have the desired estimate (A.15).

$\diamond$ Case $II$. Suppose

Then, it follows from Lemma A.2 that

Now we claim:

Proof of Claim (A.16). Since $(D^{\infty} + |\alpha|, D(\theta_0) + |\alpha|) \subseteq (|\alpha|, \pi - |\alpha|)$, we have

We use relation (A.11), relation (A.17) and assumption of $\kappa$ to get

In fact, we can get

Hence, we have

Thus, in order to have $D(\Theta(t))< D^{\infty}$, we need

$\bullet$ Step B. We need to verify

Suppose that there exists a time $t_1 > t_0$ such that

This yields

However, we use relation (A.11) again to obtain that

This contradicts to the relation (A.18). Thus, our assertion holds.