Emergent behaviors of the continuum thermodynamic Kuramoto model in a large coupling regime

Abstract

We study the emergent dynamics of the continuum thermodynamic Kuramoto model which arises from the continuum limit of the lattice thermodynamic Kuramoto (TK) model [17]. The continuum TK model governs the time-evolution of the Kuramoto phase field in a temperature field, and the solution to the lattice TK model corresponds to the simple function-valued solution to the continuum TK model on a compact spatial region. Asymptotic emergent estimates for the continuum TK model consist of two sequential processes (temperature homogenization and phase-locking). First, we show that the temperature field relaxes to a positive constant temperature exponentially fast pointwise depending on the nature of the communication weight function. In contrast, the emergent dynamics of phase field exhibits a more rich phenomena. For the phase field in a constant natural frequency field, the phase field concentrates to either one-point cluster or bi-polar cluster, whereas in a nonconstant natural frequency field, the phase field exhibits a phased-locked state asymptotically. We also provide several numerical simulations and compare them with analytical results.

Introduction

Collective behaviors of oscillatory systems are often observed in nature [1], [2], [5], [19], [24], [25], [27], [33], [34]. Recently, some of the present authors in [17] proposed a lattice TK model via a rigorous reduction from the thermomechanical Cucker-Smale (TCS) model [15], [18] under the assumptions that the spatial domain is two-dimensional and the ratio between the modulus of velocity and temperature (thermal velocity) takes a uniform constant $\eta /T_{⁎}$ independent of lattice points. In this paper, we further continue the studies which were begun in [17] on the emergent behaviors of the Kuramoto ensemble in an external temperature field. To prepare our discussion, we begin with a brief description of the lattice TK model governing the phase dynamics under the effect of temperature field [23], [26], [35].

Let Ω be a compact region in ${\mathbb{R}}^d$ with positive Lebesgue measure, and $\mathrm{\Gamma }\subset \mathrm{\Omega }$ be a lattice with index set $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$. Since Ω is compact and Γ is discrete, the index set Λ is a finite set. Let ${\theta }_{\alpha }={\theta }_{\alpha }(t),T_{\alpha }=T_{\alpha }(t)$ and ${\nu }_{\alpha }$ be phase, temperature and natural frequency of the internal (Kuramoto) oscillator at the lattice point $x_{\alpha }\in \mathrm{\Gamma }$ and at time t. Then, the temporal evolution of phase-temperature configuration $({\theta }_{\alpha },T_{\alpha })$ is governed by the following lattice TK model:

$$\{\begin{array}{c}{\dot{\theta }}_{\alpha }={\nu }_{\alpha }+\frac{{\kappa }_1}{|\mathrm{\Lambda }|}\sum _{\beta \in \mathrm{\Lambda }}\frac{{\psi }_{\alpha \beta }}{T_{\alpha }}\sin ({\theta }_{\beta }-{\theta }_{\alpha }),t>0,\hfill \\ {\dot{T}}_{\alpha }=\frac{{\kappa }_2}{|\mathrm{\Lambda }|}\sum _{\beta \in \mathrm{\Lambda }}\frac{{\zeta }_{\alpha \beta }{T}_{⁎}^2}{(T_{⁎}^2+{\eta }^2{T}_{\alpha })}(\frac{1}{T_{\alpha }}-\frac{1}{T_{\beta }}),\alpha \in \mathrm{\Lambda },\hfill \end{array}$$

where ${\kappa }_i$ is nonnegative coupling strength, respectively, $|\mathrm{\Lambda }|$ is the cardinality of the index set Λ, and the communication weight functions ${\psi }_{\alpha \beta }$ and ${\zeta }_{\alpha \beta }$ are assumed to be symmetric and positive:

$${\psi }_{\alpha \beta }={\psi }_{\beta \alpha }>0,{\zeta }_{\alpha \beta }={\zeta }_{\beta \alpha }>0,\forall \alpha ,\beta \in \mathrm{\Lambda }.$$

Moreover, η is a constant parameter with speed dimension, and $T_{⁎}$ is a constant reference temperature that in the following we put both equal to 1.

For a mechanical case with common constant temperatures, with

$$\mathrm{\Lambda }=\{1,\cdots ,N\},T_{\alpha }\equiv T^{\infty }\forall \alpha \in \mathrm{\Lambda },{\psi }_{\alpha \beta }=1,{\tilde{\kappa }}_1=\frac{{\kappa }_1}{T^{\infty }},$$

the lattice TK model (1.1) reduces to the Kuramoto model:

$${\dot{\theta }}_{\alpha }={\nu }_{\alpha }+\frac{{\tilde{\kappa }}_1}{N}\sum _{\beta =1}^N\sin ({\theta }_{\beta }-{\theta }_{\alpha }),\alpha =1,\cdots ,N.$$

From a mathematical point of view, the Kuramoto model can be obtained as a limiting case of TK model in which the relaxation time $1/{\kappa }_2$ tends to zero.

We refer to [1], [4], [6], [7], [8], [9], [10], [13], [15], [16], [20], [25], [30], [31], [32] for the emergent dynamics of the Kuramoto model (1.3). Then, by letting the lattice size tends to zero, we can formally obtain a system of integro-differential equations for phase and temperature fields $(\theta ,T)$ on ${\mathbb{R}}_{+}\times \mathrm{\Omega }$:

$$\{\begin{array}{c}{\partial }_t\theta (t,x)=\nu (x)+\frac{{\kappa }_1}{|\mathrm{\Omega }|}\underset{\mathrm{\Omega }}{\int }\frac{\psi (x,y)}{T(t,x)}\sin (\theta (t,y)-\theta (t,x))dy,(t,x)\in {\mathbb{R}}_{+}\times \mathrm{\Omega },\hfill \\ {\partial }_{t}T(t,x)=\frac{{\kappa }_2}{|\mathrm{\Omega }|}\underset{\mathrm{\Omega }}{\int }\frac{\zeta (x,y)}{1+T(t,x)}(\frac{1}{T(t,x)}-\frac{1}{T(t,y)})dy,\hfill \end{array}$$

where $|\mathrm{\Omega }|$ is the Lebesgue measure of the set Ω in ${\mathbb{R}}^d$. When Ω is given by a rectangular domain, the limiting process from (1.2) to (1.4) can be justified rigorously by using the Lebesgue differentiation theorem as in [11], [22], [28], [29]. However, we here provide an improved argument which allows any compact Ω having positive Lebesgue measure with negligible boundary (see Section 3). The lattice model (1.2) can be viewed as the evaluation of the continuum model (1.4) at lattice points. Especially, for the following special setting:

$$\mathrm{\Omega }=[0,1],T\equiv 1,\psi \equiv 1,$$

system (1.4) reduces to the continuum Kuramoto model in [11], [21], [22], [28], [29]:

$${\partial }_t\theta (t,x)=\nu (x)+{\kappa }_1\underset{0}{\overset{1}{\int }}\sin (\theta (t,y)-\theta (t,x))dy,(t,x)\in {\mathbb{R}}_{+}\times [0,1].$$

Note that the temperature equation ${\text{(1.4)}}_2$ is completely decoupled from the phase equation, and the appearance of temperature variable in the denominator of R.H.S. of ${\text{(1.4)}}_2$ causes several analytical difficulties, when we try to obtain a local well-posedness, different from [11], [22], [28], [29]. To guarantee the positivity of temperature T, we adopt a coldness observable defined as a reciprocal of the temperature and design an implicit successive approximation scheme to guarantee the preservation of positivity of temperature in each iteration step, and then we show that the iterations are convergent in a suitable norm and the limit provides a local classical solution to ${\text{(1.4)}}_2$. Moreover, we can also extend this local solution to a global one by inductive argument. Once we have a global solution for temperature field equation, we can also construct a global solution to the phase equation using the standard contraction mapping principle as in [12], [22]. The detailed arguments were presented in Appendix A.

In this paper, we mainly focus on the emergent dynamics of the continuum TK model (1.4). More precisely, we address the following question:

“What are the conditions on system functions, parameters and initial data leading to the emergent dynamics of (1.4)?”

Since the dynamics for temperature field is completely decoupled from that of phase field, after we analyze the temperature homogenization first, and then using this explicit relaxation estimate toward the constant temperature field, we can study emergence of phase-locking under suitable conditions. More precisely, our main results are three-fold.

First, we deal with the relaxation of the temperature field toward the common positive temperature which is completely determined by the initial data (see Remark 2.2). If coupling strength, communication weight and initial temperature satisfy

$${\kappa }_2>0,\underset{x,y}{\min }\zeta (x,y)=:{\zeta }_m>0,\zeta \in C({\mathrm{\Omega }}^2),\underset{x}{\min }{T}^{in}(x)>0,T^{in}\in C(\mathrm{\Omega }),$$

there exists a positive constant ${\lambda }_0:=\lambda ({\zeta }_m,T^{in})$ such that

$$|\underset{x\in \mathrm{\Omega }}{\max }T(t,x)-\underset{x\in \mathrm{\Omega }}{\min }T(t,x)|\lesssim e^{-{\kappa }_2{\lambda }_{0}t}\text{as }t\to \infty .$$

We refer to Theorem 4.1 for more details.

Secondly, we present asymptotic phase-locking for a constant natural frequency field:

$$\nu (x)\equiv \text{constant}.$$

In this case, we introduce the order parameters $(R,\varphi )$ defined by the following implicit relation:

$$R(t)e^{\mathrm{i}\varphi (t)}=\frac{1}{|\mathrm{\Omega }|}\underset{\mathrm{\Omega }}{\int }{e}^{\mathrm{i}\theta (t,x)}dx.$$

Then, our second result documented in Theorem 4.2 says that if coupling strengths and the initial data satisfy

$${\kappa }_1>0,{\kappa }_2>0,\underset{x\in \mathrm{\Omega }}{\min }{T}^{in}(x)>0,R^{in}:=\left|\frac{1}{|\mathrm{\Omega }|}\underset{\mathrm{\Omega }}{\int }{e}^{\mathrm{i}{\theta }^{in}(x)}dx\right|>0,\nu \equiv 0,\psi \equiv 1,$$

then there exists a time-dependent average phase function $\varphi (t)$ such that

$$\underset{t\to \infty }{\lim }\underset{\mathrm{\Omega }}{\int }|\sin (\varphi (t)-\theta (t,x)){|}^{2}dx=0,\underset{t\to \infty }{\lim }|\dot{\varphi }(t)|=0.$$

Hence, the first estimate implies a formation of one-point cluster or bi-polar configuration asymptotically (see Theorem 4.2 for more details).

Thirdly, our last result deals with a situation in which the natural frequency field $\nu =\nu (x)$ is not constant. In this case, we show the emergence of complete phase-locking in which the frequency field ${\partial }_t\theta$ tends to the average value of natural frequency field asymptotically. More precisely, if coupling strengths and initial data satisfy

$${\kappa }_1\gg \text{diam}\{\nu (x):x\in \mathrm{\Omega }\},{\kappa }_2>0,\mathcal{D}({\mathrm{\Theta }}^{in})\ll \frac{\pi }{2},$$

phase-locking emerges asymptotically (see Theorem 4.3):

$$\underset{t\to \infty }{\lim }{\partial }_t\theta (t,x)=\frac{1}{|\mathrm{\Omega }|}\underset{\mathrm{\Omega }}{\int }\nu (y)dy,\text{a.e.}x\in \mathrm{\Omega }.$$

The rest of this paper is organized as follows. In Section 2, we briefly review the basic properties of the lattice model and corresponding continuum model. In Section 3, we show that the solution to the lattice model can be derived from the continuum model on the compact region with positive Lebesgue measure and a nice boundary. In Section 4, we study the emergent dynamics of the continuum model such as temperature homogenization and emergence of phase-locking. In Section 5, we provide several numerical examples and compare them with analytical results. Finally, Section 6 is devoted to a brief summary of our main results and some remaining issues to be investigated in a future work. In Appendix A, we introduce a concept of classical solutions and provide a global existence of classical solutions to the continuum model. In Appendices B, C and D, we provide proofs of Lemma 3.1, Theorem 3.1 and Lemma 4.5, respectively.

Notation: The constant C denotes a generic positive constant which may differ from line to line, and the relation $A(t)\lesssim B(t)$ represents an inequality $A(t)\le CB(t)$ for a generic positive constant C and all $t\ge 0$. For notational simplicity, we also use handy notation from time to time:

$$\underset{x}{\max }:=\underset{x\in \mathrm{\Omega }}{\max },\underset{x}{\min }:=\underset{x\in \mathrm{\Omega }}{\min },\underset{x}{\sup }:=\underset{x\in \mathrm{\Omega }}{\sup },\underset{x}{\inf }:=\underset{x\in \mathrm{\Omega }}{\inf },\sum _{\alpha }:=\sum _{\alpha \in \mathrm{\Lambda }}.$$

Also, for any real vector $X=(x_1,\cdots ,x_N)$, we denote by $\mathcal{D}(X)$ a maximal difference between $x_{\alpha }$'s:

$$\mathcal{D}(X):=\underset{1\le \alpha ,\beta \le N}{\max }|x_{\alpha }-x_{\beta }|.$$

Finally, for a given measurable function $F:\mathrm{\Omega }\times \mathrm{\Omega }\to \mathbb{R}$, we set a mixed norm and $L^p$-norm as follows:

$${\Vert F\Vert }_{L_x^{\infty }(L_y^1)}:=\underset{x\in \mathrm{\Omega }}{\sup }\underset{\mathrm{\Omega }}{\int }|F(x,y)|dy,{\Vert F\Vert }_{L_y^{\infty }(L_x^1)}:=\underset{y\in \mathrm{\Omega }}{\sup }\underset{\mathrm{\Omega }}{\int }|F(x,y)|dx,{\Vert F\Vert }_{L^p}={\Vert F\Vert }_{L_{x,y}^p}:={(\underset{{\mathrm{\Omega }}^2}{\iint }|F(x,y){|}^{p}dydx)}^{\frac{1}{p}},1\le p<\infty ,{\Vert F\Vert }_{L^{\infty }}={\Vert F\Vert }_{L_{x,y}^{\infty }}:={\text{ess.sup}}_{(x,y)\in {\mathrm{\Omega }}^2}|F(x,y)|.$$