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Synchronization of weakly coupled oscillators is a collective behavior often found in complex biological systems such as groups of fireflies, neurons, and cardiac pacemaker cells [1,3,6,36,39,40]. It was first reported in scientific literature by physicist Christiaan Huygens around the middle of the seventh century; however, its rigorous mathematical treatment was conducted by Winfree [39] and Kuramoto [24] only several decades ago. Since then, several extensions of the Kuramoto model have been extensively investigated in several scientific fields such as applied mathematics, control theory, neuroscience, physics and engineering [1,4,10,12,13,14,26,29,30,32]. In this paper, we are primarily interested in the {generalized Kuramoto model with intrinsic dynamics [20,21]. Here, the intrinsic dynamics refer to smooth functions which describe the individual dynamics when the couplings vanish. Note that the classic Kuramoto oscillator has simple intrinsic dynamics governed by its natural frequency so that the uncoupled Kuramoto oscillator's phase has Kronecker dynamics on the unit circle
The main purpose of this work is to study the emergence of local exponential synchronization of Kuramoto oscillators in networks with heterogeneous intrinsic dynamics. To better understand our goal, consider a mixture of two homogeneous ensembles denoted by
$ \label{dec1} {\dot \zeta}_i = F_1(\zeta_i, t), \;\;{\dot \eta}_j = F_2 (\eta_j, t), \;\;1 \leq i \leq N_1, \;\;1 \leq j \leq N_2. $ | (1) |
The functions
$
˙ζi=F1(ζi,t)+KsN1N1∑k=1sin(ζk−ζi)+KdN2N2∑k=1sin(ηk−ζi),i=1,…,N1,˙ηi=F2(ηi,t)+KsN2N2∑k=1sin(ηk−ηi)+KdN1N1∑k=1sin(ζk−ηi),i=1,…,N2.
$
|
(2) |
Note that (1) gives the intrinsic dynamic for each oscillator in (2). The system (2) can be viewed as an interaction of two generalized Kuramoto models with external forcing terms [2,31,33,34,35]; moreover, it can be used to model the sleep-wake cycle. The sleep-wake cycle and circadian rhythms are phase-locked to each other in a 24-hour time period. Many biological experiments have shown that in isolation from a 24-hour periodic environment such as the light-dark cycle, various circadian rhythms, e.g., feeding, body temperature, and neuroendocrine variables, as well as the pattern of sleep and wakefulness, are maintained. However, a certain internal desynchronization phenomenon occurs, i.e., separate rhythmic variables oscillate with different periods. Many mathematical models have been developed to explain these phenomena, notably the generalized Kuramoto model with an external periodic force
One of the most notable results of this study is the emergence of exponential local synchronization in each homogeneous group under suitable frameworks. Here let us introduce what the local synchronization means in this paper.
Definition 1.1. Let
$
limt→∞|ζi(t)−ζj(t)|=0,i≠j,i,j=1,…,N1,limt→∞|ηi(t)−ηj(t)|=0,i≠j,i,j=1,…,N2.
$
|
In Theorems 3.3, 4.3, 5.3, and 5.4, we show that the state-diameter of each homogeneous group converges to zero exponentially fast, although the total ensemble diameter does not converge to zero as
The rest of this paper is organized as follows. In Section 2, we discuss the generalized Kuramoto model with intrinsic dynamics and review earlier results on its practical synchronization. To motivate our analysis, in Sections 3 and 4 we consider the interaction of two homogeneous groups of oscillators with
Notation: Throughout this paper, we will use the following notation. For state vectors
$D(ζ):=maxi,j|ζj−ζi|,D(η):=maxi,j|ηj−ηi|,D(F):=maxi,j‖Fi−Fj‖L∞,D∗(ζ,η):=max{D(ζ),D(η),max1≤i1≤N1,1≤i2≤N2|ζi1−ηi2|}. $
|
In this section, we first consider the interactions of two homogeneous systems with
Consider the four-oscillator system with
$
˙ζ1=F1(ζ1,t)+Ks2sin(ζ2−ζ1)+Kd2(sin(η1−ζ1)+sin(η2−ζ1)),˙ζ2=F1(ζ2,t)+Ks2sin(ζ1−ζ2)+Kd2(sin(η1−ζ2)+sin(η2−ζ2)),˙η1=F2(η1,t)+Ks2sin(η2−η1)+Kd2(sin(ζ1−η1)+sin(ζ2−η1)),˙η2=F2(η2,t)+Ks2sin(η1−η2)+Kd2(sin(ζ1−η2)+sin(ζ2−η2)).
$
|
(3) |
To illustrate the dynamics of (3), we perform two numerical simulations. Figure 1 shows the temporal evolution of the phases of the four oscillators with the following network structures:
$ \text{Bipartite coupling}: K_d=0.5, \, K_s=0, \text{and} \; \text{All-to-all coupling}: K_d=0.5, \, K_s=5.$ |
Different colors are used to represent the different oscillators. Note that the emergence of local synchronization is observed in each case; however, system (3) does not show asymptotic complete synchronization, even in in Fig. 1(b) if the intra coupling strength
Let
$
˙ζi=Fi(ζi,t)+KNN∑k=1sin(ζk−ζi),t>0,i=1,…,N,ζi(0)=ζi0.
$
|
(4) |
For the zero coupling strength
$ \label{dec} {\dot \zeta}_i = F_i(\zeta_i, t), \;\;i =1, \dots, N. $ |
When the intrinsic dynamics
Definition 2.1.[21] Let
1. The solution
$ \lim\limits_{t \to \infty} |\zeta_i(t)-\zeta_j(t)| = 0,\;\; \forall i\neq j. $ |
2. The solution
$ \lim\limits_{K \to \infty} \limsup\limits_{t \to \infty} D(\zeta(t)) = 0. $ |
The synchronization property of (4) was studied in [21] under the relaxed concept of practical synchronization. When the forcing function
$ \label{KM-o} {\dot \zeta}_i = \Omega_i + \frac{K}{N} \sum\limits_{k=1}^{N}\sin(\zeta_k - \zeta_i). $ | (5) |
One nice feature of the Kuramoto model in (5) is that it can be rewritten as a gradient flow system (see [38]), i.e., for
$ \dot\zeta(t)=-\nabla V(\zeta), $ |
where the potential
$ V(\zeta) :=-\sum\limits_{i=1}^N\Omega_i \zeta_i+\frac{K}{2N}\sum\limits_{i, j=1}^N \big(1-\cos(\zeta_j-\zeta_i)\big). $ |
Since the potential function
A natural question that arises is whether this gradient flow approach can be extended to a general model of oscillators with intrinsic dynamics. For (4), a possible candidate for the potential function is
$ {\tilde V}(\zeta, t) :=-\sum\limits_{i=1}^N\int_{0}^{\zeta_i}F_i(\xi, t)d\xi+\frac{K}{2N}\sum\limits_{i, j=1}^N \big(1-\cos(\zeta_j-\zeta_i)\big).$ |
Note that for the non-autonomous system in (4), a natural way of converting the non-autonomous system in (4) into an autonomous system on
$ {\tilde \zeta} :=(\zeta_1, \dots, \zeta_N, t). $ |
To reformulate the gradient flow, it is required that
$ \dot t=-\frac{\partial \tilde V}{\partial t},\;\; \mathrm{i.e., } \;\; \frac{\partial \tilde V}{\partial t}\equiv -1.$ |
This means
$\sum\limits_{i=1}^N\int_{0}^{\zeta_i}\frac{\partial}{\partial t}{F_i(\xi, t)}d\xi=1, $ |
which is not true for a family of non-constant analytic functions
However, a rudimentary approach based on the Lyapunov functional approach can be taken for the state-diameter. In [21] the practical synchronization of (4) was studied by the method of energy estimate. The main result for system (4) is summarized as follows: under suitable conditions on the heterogeneous dynamics
$ \limsup\limits_{t \to \infty} D(\zeta(t)) \leq \mathcal{O}(1)K^{-1}.$ |
For more details we refer to ([21,Theorem 4.1]). In [20], the authors considered the Kuramoto model with inertia and heterogeneous intrinsic dynamics:
$ \label{GKM} \displaystyle m {\ddot \zeta}_i + {\dot \zeta}_i = F_i(\zeta_i, t) + \frac{K}{N} \sum\limits_{j=1}^{N} \sin(\zeta_j - \zeta_i), \;\;1\leq i \leq N. $ | (6) |
The following practical synchronization for (6) has been obtained under suitable conditions on the heterogeneous dynamics
$ \mathop {\lim \sup }\limits_{t \to \infty } D(\zeta (t)) \le {\cal O}(1){K^{ - \frac{1}{2}}}\;\;\;\;{\rm{as}}\;K \to \infty $ |
Note that the practical synchronization result in [21] focuses on the "global (or complete) synchronization" for the whole ensemble, i.e., asymptotic dynamics of phase diameter for the whole ensemble. However, as can be seen in Figure 1(b), the phase diameter for the whole ensemble does not converge to a constant value as time goes on, in fact, it is only bounded, but fluctuates in time. In contrast, our results in this paper focus on the dynamics of phase diameter for the local (or partial) ensemble consisting of oscillators with the same intrinsic dynamics, thus our paper deal with "local synchronization" for the ensemble. This is the difference between the results in [20,21] and our main results described in Theorem 3.1, Theorem 4.1 and Theorem 5.1. Moreover, our results shows that under suitable conditions on the intrinsic dynamics, coupling strength and initial configuration, the local synchronization occurs exponentially fast. In contrast, for the second-order flocking models in [8], it is known that local flockings occur algebraically slowly even for all-to-all interactions.
Throughout the paper, we frequently use the time derivative of group phase diameter
Lemma 2.2. Let
$\label{elemineq} \frac{\sin x}{ x} > \frac{\sin a}{ a}. $ |
In the following two sections, we consider the following cases for arbitrary finite
$ \mbox{Either}\;K_d > 0, K_s = 0, \;\;\mbox{or} \;K_d > 0, K_s > 0. $ |
The first case is called bipartite interactions and the latter case is called all-to-all interactions, between two homogeneous groups.
In this section, we study bipartite interactions so that oscillators do not interact in the same groups directly; instead, they interact indirectly through oscillators in other groups:
$ K_s=0 \;\;\mbox{and} \;\;K_d > 0. $ |
In this setting, system (2) can be simplified as follows:
$
˙ζi=F1(ζi,t)+KdN2N2∑k=1sin(ηk−ζi),i=1,…,N1,˙ηi=F2(ηi,t)+KdN1N1∑k=1sin(ζk−ηi),i=1,…,N2.
$
|
(7) |
Lemma 3.1.(Cohesiveness) Suppose that the size of the local groups and initial data, intrinsic dynamics, and inter coupling strength satisfy the following conditions:
$
(ⅰ)N1≥2,N2≥2,D∗(ζ0,η0)<D∞forsomeconstantD∞∈(0,π).(ⅱ)D(F)<∞,supζ,i,t∂Fi∂ζ<∞.(ⅲ)Kd>N1N2N1+N2(D(F)sinD∞+D∞sinD∞supζ,i,t∂Fi∂ζ).
$
|
Then for any solution
1. The functional
$ \sup\limits_{0\leq t < \infty} D^*(t) \leq D_\infty. $ |
2. The functional
$ \dot{D}^*(t)\leq D(\mathcal F)+ {D_\infty} \sup\limits_{ {\zeta, i, t}}\frac{\partial F_i}{\partial \zeta} - K_d\left(\frac{1}{N_1}+\frac{1}{N_2}\right)\sin D^*(t), for\, \, t > 0. $ |
Proof.(1) First, we apply the continuity argument. Set
$\mathcal T:=\{a\in \mathbb R^+: D^*(t) < D_\infty \, \, \text{for all}\; t\in [0, a)\},\;\; \text{and} \;T_*:=\sup \;\mathcal T.$ |
Since
$\label{contradiction1} {D^*(T_*)=D_\infty.}$ |
We now derive an estimate for the time-derivative of
$dD∗(t)dt|t=T∗ =F1(ζM,T∗)−F1(ζm,T∗)+KdN2N2∑k=1[sin(ηk−ζM)−sin(ηk−ζm)]=(∂F1∂ζ(ζ∗,T∗))(ζM−ζm)+KdN2N2∑k=1[sin(ηk−ζM)−sin(ηk−ζm)]≤(supζ,t∂F1∂ζ)(ζM−ζm)+KdsinD∞N2D∞N2∑k=1[(ηk−ζM)−(ηk−ζm)]=(D∞sinD∞supζ,t∂F1∂ζ−Kd)sinD∞<0. $
|
Here, we use the elementary inequality in Lemma 2.2. Similarly, if the extremal phases are attained in
$dD∗(t)dt|t=T∗≤(D∞sinD∞supζ,t∂F2∂ζ−Kd)sinD∞<0. $
|
$dD∗(t)dt|t=T∗=F1(ζM,T∗)−F2(ηm,T∗)+KdN2N2∑k=1sin(ηk−ζM)−KdN1N1∑k=1sin(ζk−ηm)≤F1(ζM,T∗)−F2(ζM,T∗)+F2(ζM,T∗)−F2(ηm,T∗) +KdN2sin(−D∞)−KdN1sinD∞≤D(F)+(∂F2∂ζ(ζ∗,T∗))(ζM−ηm)−Kd(1N1+1N2)sinD∞≤D(F)+D∞supζ,t∂F2∂ζ−Kd(1N1+1N2)sinD∞<0, $
|
where the condition (ⅲ) and the elementary inequality in Lemma 2.2. If the extremal phases are
(2) Since the uniform bound
Lemma 3.2. Under the same assumptions as Lemma 3.1, there exists
$D^*(t)\leq {\bar D}_\infty,\ \ \forall \, t\geq t_0.$ |
Here,
$\sin {\bar D}_\infty=\sin D_\infty, \ \ \bar D_\infty \in \Big(0, \frac{\pi}{2}\Big).$ |
In other words,
proof. The proof is similar to the proof in Appendix A.1 in [9], thus, it is omitted here.
We are now ready to prove local exponential synchronization in each group.
Theorem 3.3. Suppose that the size of the local groups and initial data, intrinsic dynamics, and inter coupling strength satisfy the following conditions:
$
(ⅰ)N1≥2, N2≥2, D∗(0)<D∞ for some D∞∈(0,π).(ⅱ)D(F)<∞, supζ,i,t∂Fi∂ζ<∞.(ⅲ)Kd >max{N1N2N1+N2(D(F)sinD∞+D∞sinD∞supζ,i,t∂Fi∂ζ),ˉD∞supζ,i,t∂Fi∂ζ2cosˉD∞sin(ˉD∞2)}.
$
|
Then for any solution
$ D(\zeta (t)),{\mkern 1mu} \;{\mkern 1mu} D(\eta (t)) \le {D^*}(0){e^{ - {\Lambda _1}t}}\;\;for\;t\;sufficiently\;l\arg e{\rm{.}} $ |
Proof. It follows from Lemma 3.2 that there exists a finite time
$ {D^*}\left( t \right) \le {\bar D_\infty } < \frac{\pi }{2}\;\;\;\;{\rm{for\ all}}\;t > {t_0}. $ |
Now, consider the intra group phase diameter
$˙ζi−˙ζj=F1(ζi,t)−F1(ζj,t)+KdN2N2∑k=1(sin(ηk−ζi)−sin(ηk−ζj))=∂F1∂ζ(ζ∗t,t)(ζi−ζj)+2KdN2N2∑k=1cos(ηk−ζi2+ηk−ζj2)sinζj−ζi2. $
|
On the other hand, note that
$\cos\left(\frac{\eta_k-\zeta_M}{2}+\frac{\eta_k-\zeta_m}{2}\right) \geq \cos {\bar D}_\infty, \ \ D(\zeta(t))\leq D^*(t)\leq {\bar D}_\infty, \ \ \forall t > t_0.$ |
Thus, we have
$
˙D(ζ)≤(supζ,t∂F1∂ζ)D(ζ)+2KdN2(N2cosˉD∞)sin−D(ζ)2≤(supζ,t∂F1∂ζ)D(ζ)−2KdcosˉD∞sinˉD∞2ˉD∞2D(ζ)2=(supζ,t∂F1∂ζ−2KdcosˉD∞sin(ˉD∞2)ˉD∞)D(ζ),∀t>t0,
$
|
(8) |
where the relation
$ \label{C-2} \dot D (\eta) \leq \left(\sup\limits_{\eta, t} \frac{\partial F_2}{\partial \eta}- \frac{2K_d\cos {\bar D}_\infty \sin \left(\frac{{\bar D}_\infty}{2}\right)}{{\bar D}_\infty}\right) D(\eta), \ \ \forall t > t_0. $ | (9) |
Note that assumption (ⅲ) implies
$ \sup\limits_{\zeta, t} \frac{\partial F_1}{\partial \zeta}- \frac{2K_d\cos {\bar D}_\infty \sin \left(\frac{{\bar D}_\infty}{2}\right)}{{\bar D}_\infty},\ \ \sup\limits_{\eta, t} \frac{\partial F_2}{\partial \eta}- \frac{2K_d\cos {\bar D}_\infty \sin \left(\frac{{\bar D}_\infty}{2}\right)}{{\bar D}_\infty} < 0. $ |
By combining (8) and (9), the desired estimate is obtained.
In this section, we consider the case of all-to-all couplings with
$
˙ζi=F1(ζi,t)+KsN1N1∑k=1sin(ζk−ζi)+KdN2N2∑k=1sin(ηk−ζi), i=1,…,N1,˙ηi=F2(ηi,t)+KsN2N2∑k=1sin(ηk−ηi)+KdN1N1∑k=1sin(ζk−ηi), i=1,…,N2.
$
|
(10) |
For the case
$ {\displaystyle N_1=N_2,\ \ K_s=K_d=\frac{K}{2}, }$ |
the coupled system (10) reduces to the generalized Kuramoto model [21]. In the following, we present a framework for exponential local synchronization. We begin with a lemma on the cohesiveness of the total ensemble.
Lemma 4.1. (Cohesiveness) Suppose that
$
(ⅰ)N1≥2,N2≥2,D∗(0)<D∞forsomeD∞∈(0,π).(ⅱ)D(F)<∞,supζ,i,t∂Fi∂ζ<∞.(ⅲ)2min{Ks,Kd}>D(F)+D∞supζ,i,t∂Fi∂ζsinD∞.
$
|
then for any solution
1. The functional
$ \sup\limits_{0\leq t < \infty} D^*(t) \leq D_{\infty}. $ |
2. The functional
$ {\dot D}^*(t)\leq D(\mathcal F)+ D_\infty\sup\limits_{\zeta, t}\frac{\partial F_i}{\partial \zeta} - 2\min\{K_s, K_d\} \sin D^*(t), \ \ {i=1, 2, } \ \ for \ t > 0. $ |
Proof.(1) Let
$\mathcal T:=\{a\in \mathbb R^+: D^*(t) < D_\infty \, \, \text{for all} \ t\in [0, a)\}, \text{and}\ T_*:=\sup \mathcal T.$ |
Since
$\label{contradiction} {D^*(T_*)=D_\infty.} $ |
We now derive an estimate for the time derivative of
$dD∗(t)dt|t=T∗=F1(ζM,T∗)−F1(ζm,T∗)+KsN1N1∑k=1[sin(ζk−ζM)−sin(ζk−ζm)]+KdN2N2∑k=1[sin(ηk−ζM)−sin(ηk−ζm)]≤D∞supζ,t∂F1∂ζ−(Ks+Kd)sinD∞<0. $
|
Here, we use the relation
$dD∗(t)dt|t=T∗≤D∞supζ,t∂F2∂ζ−(Ks+Kd)sinD∞<0. $
|
$dD∗(t)dt|t=T∗=F1(ζM,T∗)−F2(ηm,T∗)+KsN1N1∑k=1sin(ζk−ζM)−KsN2N2∑k=1sin(ηk−ηm)+KdN2N2∑k=1sin(ηk−ζM)−KdN1N1∑k=1sin(ζk−ηm)≤F1(ζM,T∗)−F2(ζM,T∗)+F2(ζM,T∗)−F2(ηm,T∗)+min{Ks,Kd}N1N1∑k=1[sin(ζk−ζM)−sin(ζk−ηm)]+min{Ks,Kd}N2N2∑k=1[sin(ηk−ζM)−sin(ηk−ηm)]≤D(F)+D∞∂F2∂ζ(ζ∗,T∗)−2min{Ks,Kd}sinD∞<0. $
|
The analysis in Case 1 and Case 2 contradicts the definition of
(2) Since the uniform bound
Lemma 4.2. Similar to Lemma 3.2, there exists
$D^*(t) \leq \bar D_\infty,\ \ \forall \, t\geq t_0. $ |
where
Next, we consider the exponential local synchronization. We will derive this by following the arguments in Theorem 3.3.
Theorem 4.3. Suppose that the sizes of the subgroups, intrinsic dynamics, and coupling strengths satisfy the following conditions:
$
(ⅰ)N1≥2, N2≥2, D∗(0)<D∞for someD∞∈(0,π).(ⅱ)D(F)<∞, supζ,i,t∂Fi∂ζ<∞.(ⅲ)2min{Ks,Kd}>D(F)+D∞supζ,i,t∂Fi∂ζsinD∞.(ⅳ)supζ,i,t∂Fi∂ζ−sinˉD∞ˉD∞(Ks+KdcosˉD∞)<0.
$
|
then there exists a positive constant
$ D(\zeta (t)),{\mkern 1mu} {\mkern 1mu} \;D(\eta (t)) \le {e^{ - \Lambda t}}\;\;for\;t\;sufficiently\;l\arg e.$ |
Proof. Consider the intra group phase diameter
$˙ζi−˙ζj=∂F1∂ζ(ζ∗t,t)(ζi−ζj)+2KsN1N1∑k=1cos(ζk−ζi2+ζk−ζj2)sinζj−ζi2+2KdN2N2∑k=1cos(ηk−ζi2+ηk−ζj2)sinζj−ζi2. $
|
Note that
$cos(ζk−ζM2+ζk−ζm2)≥cosD(ζ)2, D(ζ(t))≤D∗(t)≤ˉD∞,cos(ηk−ζM2+ηk−ζm2)≥cosˉD∞, ∀t>t0. $
|
This yields
$
˙D(ζ)≤(supζ,t∂F1∂ζ)D(ζ)−2KsN1N1cosD(ζ)2sinD(ζ)2−2KdN2N2cosˉD∞sinD(ζ)2=(supζ,t∂F1∂ζ)D(ζ)−KssinD(ζ)−2KdcosˉD∞sinD(ζ)2≤[supζ,t∂F1∂ζ−sinˉD∞ˉD∞(Ks+KdcosˉD∞)]D(ζ), ∀t>t0.
$
|
(11) |
Here, we used the relation
$
˙D(η)≤[sup∂F2∂η−sinˉD∞ˉD∞(Ks+KdcosˉD∞)]D(η), ∀t>t0.
$
|
(12) |
Therefore, the desired results follow from (11) and (12).
Remark 1. 1. For the special case when
$ F_1 \equiv \Omega_1,\ \ F_2 \equiv \Omega_2, \ \ { N_1=N_2,\ \ K_s=K_d=\frac{K}{2}}, $ |
complete synchronization of (10) is addressed in [15].
2. For the Kuramoto system with two distinct constant natural frequencies, i.e.,
In this section, we extend the results in Sections 3 and 4 to the case of synchronous dynamics of Kuramoto oscillators with
$
˙ζ(1)i=F1(ζ(1)i,t)+KN1N1∑k=1sin(ζ(1)k−ζ(1)i)+∑l≠1K1lNlNl∑k=1sin(ζ(l)k−ζ(1)i), i=1,…,N1,⋮˙ζ(n)i=Fn(ζ(n)i,t)+KNnNn∑k=1sin(ζ(n)k−ζ(n)i)+∑l≠nKnlNlNl∑k=1sin(ζ(l)k−ζ(n)i), i=1,…,Nn.
$
|
(13) |
Here, the system consists of
$D(s)(ζ):=max1≤i,j≤Ns|ζ(s)i−ζ(s)j|,D∗(ζ):=max1≤s1,s2≤n,1≤i≤Ns1,1≤j≤Ns2|ζ(s1)i−ζ(s2)j|,D(s1s2)(ζ):=max{D(s1),D(s2),max1≤i≤Ns1,1≤j≤Ns2|ζ(s1)i−ζ(s2)j|}. $
|
In this subsection, we present a scheme for local synchronization in subgroups.
Lemma 5.1. (Cohesiveness) Suppose that the sizes of the subensembles, intrinsic dynamics, and coupling strengths satisfy the following conditions:
$
(ⅰ)N1≥2, N2≥2, D∗(ζ0)<D∞for someD∞∈(0,π).(ⅱ)D(F)<∞, supζ,s,t∂Fs∂ζ<∞.(ⅲ)2min{K,Kij}+∑l≠i,jmin{Kil,Kjl}>D(F)+D∞supζ,s,t∂Fs∂ζsinD∞,∀i≠j.
$
|
Then for any solution
1. The functional
$ \sup\limits_{0\leq t < \infty} D^*(\zeta(t)) \leq D_{\infty}. $ |
2. The functional
$dD∗(ζ(t))dt≤D(F)+D∞supζ,s,t∂Fs∂ζ−(2min{K,Kij}+∑l≠i,jmin{Kil,Kjl})sinD∗(ζ(t)). $
|
Proof. (1) Set
$\mathcal T:=\{a\in \mathbb R^+: D^*(\zeta(t)) < D_\infty \, \, \text{for all}\ t\in [0, a)\}, \text{and}\ T_*:=\sup \mathcal T.$ |
$ {D^*(\zeta(T_*))=D_\infty.} $ |
We now estimate the time {derivative} of
$dD∗(ζ(t))dt|t=T∗=F1(ζ(1)M,T∗)−F1(ζ(1)m,T∗)+KN1N1∑k=1[sin(ζ(1)k−ζ(1)M)−sin(ζ(1)k−ζ(1)m)]+∑l≠1K1lNlNl∑k=1[sin(ζ(l)k−ζ(1)M)−sin(ζ(l)k−ζ(1)m)]=F1(ζ(1)M,T∗)−F1(ζ(1)m,T∗)+2KN1N1∑k=1cos(ζ(1)k−ζ(1)M2+ζ(1)k−ζ(1)m2)sinζ(1)m−ζ(1)M2+∑l≠12K1lNlNl∑k=1cos(ζ(l)k−ζ(1)M2+ζ(l)k−ζ(1)m2)sinζ(1)m−ζ(1)M2≤∂F1∂ζ(ζ(1)∗,T∗)(ζ(1)M−ζ(1)m)−KsinD∞−∑l≠1K1lsinD∞≤D∞supζ,s,t∂Fs∂ζ−(K+∑l≠1K1l)sinD∞<0. $
|
$dD∗(ζ(t))dt|t=T∗=F1(ζ(1)M,T∗)−F2(ζ(2)m,T∗)+KN1N1∑k=1sin(ζ(1)k−ζ(1)M)−KN2N2∑k=1sin(ζ(2)k−ζ(2)m)+∑l≠1K1lNlNl∑k=1sin(ζ(l)k−ζ(1)M)−∑l≠2K2lNlNl∑k=1sin(ζ(l)k−ζ(2)m)=F1(ζ(1)M,T∗)−F2(ζ(1)M,T∗)+F2(ζ(1)M,T∗)−F2(ζ(2)m,T∗)+KN1N1∑k=1sin(ζ(1)k−ζ(1)M)−K21N1N1∑k=1sin(ζ(1)k−ζ(2)m)−KN2N2∑k=1sin(ζ(2)k−ζ(2)m)+K12N2Nl∑k=1sin(ζ(2)k−ζ(1)M)+∑l≠1,2[K1lNlNl∑k=1sin(ζ(l)k−ζ(1)M)−K2lNlNl∑k=1sin(ζ(l)k−ζ(2)m)]≤D(F)+∂F2∂ζ(ζ∗,T∗)(ζ(1)M−ζ(2)m)−min{K,K21}sinD∞−min{K,K12}sinD∞−∑l≠1,2min{K1l,K2l}sinD∞≤D(F)+D∞supζ,s,t∂Fs∂ζ−(2min{K,K12}+∑l≠1,2min{K1l,K2l})sinD∞<0. $
|
The analysis in Case 1 and Case 2 contradicts the definition of
(2) Since the uniform bound
Lemma 5.2. By the same argument as Lemma 3.2, there exists
$D^*(\zeta(t)) \leq \bar D_\infty,\ \ \forall \, t\geq t_0. $ |
where
Now, we consider the complete phase synchronization in subgroups.
Theorem 5.3. (Local synchronization) Suppose that the sizes of the subgroups, intrinsic dynamics, and coupling strengths satisfy the following conditions:
$
(ⅰ)Ns≥2, for s=1,…,n.(ⅱ)D∗(ζ0)<D∞for someD∞∈(0,π), D(F)<∞, supζ,s,t∂Fs∂ζ<∞.(ⅲ)2min{K,Kij}+∑l≠i,jmin{Kil,Kjl}>D(F)+D∞supζ,s,t∂Fs∂ζsinD∞,∀i≠j.(ⅳ)K+cosˉD∞∑l≠sKsl>ˉD∞sinˉD∞supζ,t∂Fs∂ζ, for s∈S⊂{1,2,…,n}.
$
|
Then for each
${D^{(s)}}(\zeta (t)) \le {D^*}({\zeta _0}){e^{ - {\Lambda _s}t}},\;\;\;for\;t\;sufficiently\;l\arg e{\rm{.}}$ |
Proof. Consider the local phase diameter
$˙ζ(s)i−˙ζ(s)j=∂Fs∂ζ(ζ(s)∗,t)(ζ(s)i−ζ(s)j)+2KNsNs∑k=1cos(ζ(s)k−ζ(s)i2+ζ(s)k−ζ(s)j2)sinζ(s)j−ζ(s)i2+∑l≠s2KslNlNl∑k=1cos(ζ(l)k−ζ(s)i2+ζ(l)k−ζ(s)j2)sinζ(s)j−ζ(s)i2. $
|
Note that
$cos(ζ(s)k−ζ(s)i2+ζ(s)k−ζ(s)j2)≥cosD(s)(ζ)2, D(s)(ζ(t))≤D∗(ζ(t))≤ˉD∞,cos(ζ(l)k−ζ(s)i2+ζ(l)k−ζ(s)j2)≥cosˉD∞, ∀t>t0. $
|
This yields
$
dD(s)(ζ(t))dt≤(supζ,t∂Fs∂ζ)D(s)(ζ)−2KcosD(s)(ζ)2sinD(s)(ζ)2−cosˉD∞sinD(s)(ζ)2∑l≠s2Ksl≤(supζ,t∂Fs∂ζ)D(s)(ζ)−KsinD(s)(ζ)−cosˉD∞sinˉD∞ˉD∞D(s)(ζ)∑l≠sKsl≤[supζ,t∂Fs∂ζ−sinˉD∞ˉD∞(K+cosˉD∞∑l≠sKsl)]D(s)(ζ), ∀t>t0.
$
|
(14) |
Here, we used the relation
In this subsection, we study local synchronization without cohesiveness of phases. In this case, we do not confine the initial total diameter
Theorem 5.4. Suppose that for
$
\begin{array}{l} (\mathit{i})N_s \geq 2, D(\mathcal F) < \infty, \ \ \sup\limits_{\zeta, t}\frac{\partial F_s}{\partial \zeta} < \infty. \cr (\mathit{ii}) D^{(s)}(\zeta_0) < D^{(s)}_\infty for\ some D^{(s)}_\infty \in (0, \pi). \cr (\mathit{iii})\begin{array}{l} K > \frac{{D_\infty ^{(s)}\mathop {\sup }\limits_{\zeta ,t} \frac{{\partial {F_s}}}{{\partial \zeta }} + \max \{ 2,D_\infty ^{(s)}\} \sum {_{l \ne s}{K_{sl}}} }}{{\sin D_\infty ^{(s)}}}.\\ \end{array} \cr
\end{array}
$
|
Then there exists a positive constant
${D^{(s)}}(\zeta (t)) \le {D^{(s)}}({\zeta _0}){e^{ - {\Lambda _s}t}},\;\;\;\;for\;t\;sufficiently\ {\rm{ }}l\arg e.$ |
Proof. First, we claim that the local phase diameter
$ \sup\limits_{0\leq t < \infty} D^{(s)}(\zeta(t)) \leq D^{(s)}_{\infty}. $ | (15) |
Proof of claim (15): Set
$\mathcal T^{(s)}=\{a\in \mathbb R^+: D^{(s)}(\zeta(t)) < D^{(s)}_\infty \, \, \text{for all}\ t\in [0, a)\}, \text{and}\ T^{(s)}_*=\sup \mathcal T^{(s)}.$ |
Since
$ D^{(s)}(\zeta(T^{(s)}_*))=D^{(s)}_\infty. $ |
However,
$dD(s)(ζ(t))dt|t=T(s)∗=Fs(ζ(s)M,T(s)∗)−Fs(ζ(s)m,T(s)∗)+KNsNs∑k=1[sin(ζ(s)k−ζ(s)M)−sin(ζ(s)k−ζ(s)m)] +∑l≠sKslNlNl∑k=1[sin(ζ(l)k−ζ(s)M)−sin(ζ(l)k−ζ(s)m)]=Fs(ζ(s)M,T(s)∗)−Fs(ζ(s)m,T(s)∗) +2KNsNs∑k=1cos(ζ(s)k−ζ(s)M2+ζ(s)k−ζ(s)m2)sinζ(s)m−ζ(s)M2 +∑l≠s2KslNlNl∑k=1cos(ζ(l)k−ζ(s)M2+ζ(l)k−ζ(s)m2)sinζ(s)m−ζ(s)M2≤D(s)∞supζ,t∂Fs∂ζ−KsinD(s)∞+2∑l≠sKsl<0. $
|
This contradicts the definition of
$˙D(s)(ζ)≤(supζ,t∂Fs∂ζ−KsinD(s)∞D(s)∞)D(s)(ζ)+2∑l≠sKslsinD(s)(ζ)2≤(supζ,t∂Fs∂ζ−KsinD(s)∞D(s)∞+∑l≠sKsl)D(s)(ζ). $
|
Here, we use the relation
In this section, we provide several numerical simulations for the frameworks discussed in the previous sections. For all numerical simulations, we use the fourth-order Runge-Kutta method.
In this subsection, we consider the mixture of two homogeneous Kuramoto ensembles with different intrinsic dynamics under two network structures: a bipartite network and an all-to-all network.
In this subsection, we present numerical simulations for a bipartite network. The conditions for the simulations in Fig. 2 are as follows:
$
N1=N2=50, Ks=0, Kd=1,F1(ζ,t)=0.3sin(t−ζ), F2(η,t)=0.2cos(4t−η),
$
|
and the initial phases are randomly chosen from the interval
In Fig. 2(a), snapshots of the particle distribution at
In Fig. 2(b), the local ensemble diameters
In Fig. 2(c), the decay rate of the local diameters is shown in log scale. Notice that the Lyapunov exponent is close to
In Fig. 2(d), we present a three-dimensional plot of the local diameter
In this subsection, we present simulations for an all-to-all network under the same initial configuration as in Subsection 6.1.1. We set
In Fig. 3(a), we present snapshots of the particle distribution at
In Figs. 3(b) and (c), the local diameters
In Fig. 3(d), we present a three-dimensional plot of the local diameter
In this subsection, we perform simulations under the conditions of Theorem 5.4. The conditions for the simulations are as follows:
$
n=3, N1=N2=N3=50, D(s)∞=2 for s=1,2,3,K=2.5,K12=K23=K31=0.01,F1(ζ,t)=sin(2t−ζ), F2(η,t)=2sin(4t−η), F3(ψ,t)=cos(2t−ψ).
$
|
In Fig. 4(a), particle distributions are displayed at
In Figs. 4(b) and (c), we observe the local exponential synchronization.
In Fig. 4(d), we present a three-dimensional plot of the local diameter
In this paper, we presented several sufficient conditions for the emergence of local exponential synchronization in a mixture of homogeneous Kuramoto ensembles. Our main tool for studying local synchronization was the Lyapunov functional supplemented by the continuity argument. We also extended the result for binary mixture to multi-mixture case. Restricting the initial configuration was crucial for estimating the diameter of the oscillators, although numerical simulations demonstrated that local exponential synchronization is plausible for generic initial configurations such as the classical Kuramoto model. In terms of local exponential synchronization, we were able to observe practical synchronization in a more general framework than in [21]. We found that it is not only the intra coupling strength, but also the inter coupling strength that make a contribution to the emergence of local exponential synchronization in each group. This is due to the fact that the inter coupling strength plays the role of the indirect coupling strength among the oscillators in a group. As evidenced by the numerical simulations, our local exponential synchronization seems to be true for generic initial configurations. However, as discussed in Section 2.2, when homogeneous ensembles have nontrivial intrinsic dynamics, the resulting coupled system may not be a gradient flow. Even if this is true, uniform boundedness of the ensemble is not generally guaranteed. Thus, the gradient flow arguments cannot be applied to our mixture case, unlike the Kuramoto model. Therefore, the emergence of local exponential synchronization for a generic initial configurations is left for a future work.
[1] | Amer. Natur., 139 (1992), 663-668. |
[2] | Biotechnol. Bioeng., 19 (1977), 1375-1386. |
[3] | Math. Biosci., 118 (1993), 127-180. |
[4] | J. Math. Biol., 28 (1990), 99-111. |
[5] | Biochem. J., 85 (1962), 440-447. |
[6] | Proc. Amer. Math. Soc., 96 (1986), 425-430. |
[7] | J. Diff. Equ., 63 (1986), 255-263. |
[8] | J. Math. Biol., 24 (1986), 167-191. |
[9] | Math. Biosci., 83 (1987), 1-48. |
[10] | J. Biol. Syst., 16 (2008), 547-564. |
[11] | Eco. Let., 5 (2002), 302-315. |
[12] | Heath, Boston, 1965. |
[13] | Heidelberg, Springr-Verlag, 1977. |
[14] | J. Plankton Res., 23 (2001), 389-413. |
[15] | Dyna. Stabi. Syst., 11 (1996), 347-370. |
[16] | Bull. Math. Biol., 61 (1999), 303-339. |
[17] | J. Theor. Biol., 191 (1998), 353-376. |
[18] | in Ocean. Sound Scat. Predic. (eds. N. R. Anderson and B. G. Zahurance), Plenum, New York, 1977, 749-765. |
[19] | in Proceedings of the First International Conference on Mathematical Modeling (eds. J. R. Avula), Vol. IV, University of Missouri Press, Rolla, 1977, 2081-2088. |
[20] | J. Math. Biol., 5 (1978), 261-280. |
[21] | Sci., 207 (1980), 1491-1493. |
[22] | Amer. Natur., 144 (1994), 741-771. |
[23] | SIAM J. Appl. Math., 34 (1978), 760-763. |
[24] | Yale University Press, New Haven, 1961. |
[25] | Comput. Math. Appl., 49 (2005), 375-378. |
[26] | J. Bacteriol., 113 (1976), 834-840. |
[27] | Ph.D. dissertation, New Mexico State University, Las Cruces, New Mexico, U.S.A., 2012. |
[28] | J. Theor. Biol., 50 (1975), 185-201. |
[29] | J. Math. Anal. and Appl., 242 (2000), 75-92. |
[30] | W. A. Benjamin, N.Y., 1971. |
[31] | Hermann et Cie, Paris, 1942. |
[32] | Ecol. Model., 198 (2006), 163-173. |
[33] | Third edition, Springer, 2001. |
[34] | Amer. Natur., 105 (1971), 575-587. |
[35] | Theor. Popul. Biol., 75 (2009), 68-75. |
[36] | J. Theor. Biol., 208 (2001), 15-26. |
[37] | J. Math. Biol., 31 (1993), 633-654. |
[38] | J. Theor. Biol., 244 (2007), 218-227. |
[39] | J. Plankton Res., 14 (1992), 157-172. |
[40] | Math. Biosci. Eng., 10 (2013), 913-923. |
[41] | J. Math. Biol., 30 (1992), 755-763. |
[42] | J. Appl. Math., 52 (1992), 222-233. |
[43] | Biotechnol. Bioeng., 17 (1975), 1211-1235. |
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