Partial Exponential Stability Analysis of Slow-Fast Systems Via Periodic Averaging Yuzhen Qin, Student Member, IEEE, Yu Kawano, Member, IEEE, and Brian D

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Partial Exponential Stability Analysis of Slow-fast Systems via Periodic Averaging Yuzhen Qin, Student Member, IEEE, Yu Kawano, Member, IEEE, and Brian D. O. Anderson, Life Fellow, IEEE, and Ming Cao, Senior Member, IEEE

Abstract—This paper presents some new criteria for partial biological, neuroscientific, and ecological systems [10, Chap. exponential stability of a slow-fast nonlinear system with a fast 20], and one often needs to study the partial stability of them. scalar variable using periodic averaging methods. Unlike classical Therefore, there is a great need to further develop new criteria averaging techniques, we construct an averaged system by averaging over this fast scalar variable instead of the time variable. for partial stability analysis, in particular in the setting of slow- We then show that partial exponential stability of the averaged fast systems. In this paper, we aim at developing new criteria system implies partial exponential stability of the original one. for exponential partial stability of a particular type of slow-fast As some intermediate results, we also obtain a new converse systems, wherein the fast variable is a scalar. Various practical Lyapunov theorem and some perturbation theorems for partially systems can be modeled by this type of slow-fast systems, such exponentially stable systems. We then apply our established criteria to study remote synchronization of Kuramoto-Sakaguchi as semiconductor lasers [11], and mixed-mode oscillations in oscillators coupled by a star network with two peripheral nodes. chemical systems [12], where the fast scalar variables are the We analytically show that detuning the natural frequency of the photon density and a chemical concentration, respectively. In central mediating oscillator can increase the robustness of the particular, fast time-varying systems can always be modeled remote synchronization against phase shifts. in this way since the time variable t can be taken as the fast Index Terms—Partial exponential stability, averaging, remote scalar [13]. synchronization, Kuramoto-Sakaguchi oscillators. In full-state stability analysis of fast time-varying systems, averaging methods are widely used to establish criteria for I.INTRODUCTION full-state exponential stability [14, Chap. 10] [15] and also ARTIAL stability describes a property of a dynamical asymptotic stability [16]. Inspired by these works, we utilize P system that only a given part of its states, instead of periodic averaging techniques and establish some criteria for all, are stable. The earliest study on partial stability dates partial exponential stability of the considered type of slow-fast back to a century ago in Lyapunov’s seminal work in 1892, systems. Unlike what is usually done in standard averaging, and some comprehensive and well-known results of which we construct an averaged system by averaging the original one were documented by Vorotnikov in his book [1]. Different over the fast scalar variable, which is in general different from from standard full-state stability theory which usually deals the time variable. We show that partial exponential stability with stability of point-wise equilibria, partial stability is more of the averaged system implies partial exponential stability of associated with stability of motions lying in a subspace [2]. It the original one. Compared to the existing criteria for full- provides a unified and powerful framework to study a range of state exponential stability [13], [14], [16], the analysis in our application-motivated theoretical problems, such as spacecraft case is much more challenging since some state variables stabilization by rotating masses [1], inertial navigation systems are unstable. To construct the proof, we also develop a new [3], transient stability of power systems [4], output regulation converse Lyapunov theorem and some perturbation theorems [5], and synchronization in complex networks [6], [7]. for partially exponentially stable systems. Unlike the converse Lyapunov theorem in [2, Theorem 4.4], we present two arXiv:1910.07465v1 [eess.SY] 16 Oct 2019 Some Lyapunov criteria have been established to study partial stability of nonlinear systems [1], [2, Chap. 4], [8], bounds for the partial derivatives of the Lyapunov function [9]. However, when it comes to the analysis of multi-time with respect to the stable and unstable states, respectively. scale systems, the existing results are usually difficult to Moreover, our obtained perturbation theorems are the first- apply. In fact, time scale separation is pervasive in physical, known ones for partial exponential stability analysis, although their counterparts [2, Chap. 9 and 10] for full-state stability Y. Qin and M. Cao are with Engineering and Technology Institute have been widely used to analyze perturbed systems. (ENTEG), University of Groningen, the Netherlands (emails: {y.z.qin, We then apply our obtained results to study the partial m.cao}@rug.nl). Y. Kawano is with the Graduate School of Engineering, Hi- roshima University, Higashi-Hiroshima, Japan (email: ykawano@hiroshima- exponential stability of a concrete slow-fast system that arises u.ac.jp). B.D.O. Anderson is with School of Automation, Hangzhou Di- from the remote synchronization problem of coupled phase anzi University, Hangzhou, 310018, China, and Data61-CSIRO and Re- oscillators. Remote synchronization describes the phenomenon search School of Electrical, Energy and Materials Engineering, Aus- tralian National University, Canberra, ACT 2601, Australia (email: that oscillators coupled indirectly get synchronized, but the [email protected]). The work of M. Cao is supported in part by ones connecting them are not synchronized with them [6]. the European Research Council (ERC-CoG-771687) and the Netherlands In fact, remote synchronization is ubiquitous in nature. For Organization for Scientific Research (NWO-vidi-14134). The work of B.D.O. Anderson is supported by the Australian Research Council (ARC) under grants example, distant cortical regions without apparent direct neural DP-190100887 and DP-160104500, and by Data61-CSIRO. links in the human brain are detected to behave coherently 2

[17]. Unlike the emergence of the classical synchronization, II.PROBLEM FORMULATION for which strong connections are required [18]–[20], remote A wide range of systems exhibit multi-timescale dynamics, synchronization is more associated with morphological sym- and among them many have a fast changing variable that is metry. For example, nodes located remotely in a network scalar. This motivates us to study a class of slow-fast systems might be able to swap their positions without changing the in this paper, whose dynamics are described by functioning of the overall system [21]. In this paper, we consider the perhaps simplest network that is morphologically x˙ = f1(x, y, z), (1a) symmetric (i.e., a star network with two peripheral nodes), y˙ = f2(x, y, z), (1b) and study remote synchronization of oscillators coupled by εz˙ = f (x, y, z), (1c) this network. Despite its simple structure, this fundamentally 3 important network has been shown to give rise to zero-lag where x ∈ Rn, y ∈ Rm, z ∈ R, and ε > 0 is a small constant. synchronization of remotely separated neuronal populations, That is, x, y are the states of slow dynamics, and z is the n+m+1 n even in the presence of considerable synaptic conduction state of fast dynamics. All the maps, f1 : R → R , n+m+1 m n+m+1 delays [22]. Experiments have also demonstrated that this net- f2 : R → R , f3 : R → R, are continuously work can render isochronous synchronization of delay-coupled differentiable, and T -periodic in z, i.e., fi(x, y, z + T ) = semiconductor lasers [23], and chaotic electronic circuits [24]. fi(x, y, z) for all i = 1, 2, 3. We further assume that the fi’s The central element in this network, which is dynamically are such that the solution to the system (1) exists for all t ≥ 0. relaying or mediating the dynamics of the peripheral ones, is Moreover, x = 0 is a partial equilibrium point of the system n believed to play an essential role. Some recent works show (1), i.e., f1(0, y, z) = 0 for any y ∈ R and z ∈ R. Also, that the remote synchronization can be enhanced if some f2(0, y, z) = 0 for any y and z. However, f3 is not required parameter mismatch or heterogeneity is introduced to the to satisfy f3(0, y, z) = 0. central element [25], [26]. We seek to analytically study this We are interested in studying partial exponential stability of interesting experimental finding, which is quite challenging. the system (1). Let us first define uniform partial exponential Towards this end, we employ the Kuramoto-Sakaguch stability. model to describe the dynamics of the oscillators [27]. Differ- Definition 1 ([2, Chap. 4]): A partial equilibrium point x = ent from the classical Kuramoto model, there is an additional 0 of the system (1) is exponentially x-stable uniformly in y and phase shift term in the Kuramoto-Sakaguchi model. This phase z if there exist c1, c2, δ > 0 such that kx(0)k < δ implies that shift is usually used to model small time delays [28], e.g., −c2t m kx(t)k ≤ c1kx(0)ke for any t ≥ 0 and (y0, z0) ∈ R ×R. synaptic connection delays [29]. We then detune the natural Note that when we refer to this definition, we also say that frequency of the central oscillator to investigate whether the x = 0 of the system (1) is partially exponentially stable or introduction of parameter heterogeneity can strengthen remote the system (1) is partially exponentially stable with respect to synchronization. When there is no natural frequency detuning, x, without causing any confusion since only uniform partial we show that a large phase shift can destabilize remote exponential stability is studied in this paper. Some noticeable synchronization. In sharp contrast, remote synchronization efforts have been made to study partial stability of nonlinear surprisingly becomes more robust against phase shifts when systems, [1], [2, Chap. 4], [9]. Although these results do not there is sufficiently large natural frequency detuning. Actually, explicitly utilize the slow-fast structure, it is possible to apply it remains exponentially stable for any phase shift. In other them to slow-fast systems. Some Lyapunov criteria have been words, we provide a rigorous proof for the first time explain- established in [2, Chap. 4]. However, it is not always easy to ing the phenomenon experimentally demonstrated by [25], verify partial stability by using such criteria. As a motivating [26]. Compared to the classical synchronization of coupled example, we consider the following academic but suggestive Kuramoto oscillators where frequency synchronization usually model. occurs, the analysis is much more challenging since the central Example 1: Consider a nonlinear system whose dynamics oscillator has a different frequency than the peripheral ones are described by when remote synchronization occurs. Modeling the problem into a slow-fast system, we construct the proof using our x˙ = −x − 0.2x sin y − 2x cos z, obtained criteria for partial stability. y˙ = 2x cos y + x sin z, The rest of the paper is structured as follows. Section II εz˙ = 3 − sin x + cos y. introduces the model of a slow-fast system and formulates the problem. The main results and corresponding analysis are As it will be shown later, for sufficiently small ε > 0, it is provided in Section III. As an application, remote synchroniza- possible to prove that the partial equilibrium point x = 0 tion of Kuramoto-Sakaguchi oscillators is studied in Section is exponentially stable uniformly in y and z. However, it is IV. Brief concluding remarks appear in Section V. difficult to construct an appropriate Lyapunov function using Notations: Let R denote the set of real numbers. For any the criteria in [2, Chap. 4]. For example, we might reasonably n 2 δ > 0, let Bδ := {x ∈ R : kxk < δ}. Given two vectors x ∈ choose V = x as a Lyapunov function candidate. Its time Rn and y ∈ Rm, denote col(x, y) = (x>, y>)>. Denote the derivative is V˙ = −2(1 + 0.2 sin y + 2 cos z)x2, which can be unit circle by S1, and a point on it is called a phase since the positive for some y and z, while it is required by [2, Theorem point can be used to indicate the phase angle of an oscillator. 1, Chap. 4] to be negative for any x 6= 0, y, and z in order to Let Sn = S1 × · · · × S1. show the partial exponential stability. 4 3

Motivated by the above example, in the next section we time axis z(t), the slow-fast system becomes aim at further developing Lyapunov theory for partial stability dx(t) dx(t) dt f (x(t), y(t), z(t)) analysis of slow-fast systems. = = ε 1 , dz(t) dt dz(t) f3(x(t), y(t), z(t)) dy(t) dy(t) dt f (x(t), y(t), z(t)) = = ε 2 , III.PARTIAL STABILITY OF SLOW-FAST DYNAMICS dz(t) dt dz(t) f3(x(t), y(t), z(t)) dz(t) In this section, our goal is to provide a new Lyapunov = 1, dz(t) criterion for partial stability of slow-fast systems. Our analysis consists of several steps. We first construct reduced slow The first two subsystems can be viewed as time-varying sys- dynamics. Under some practically reasonable assumptions, the tems with the new time variable z(t). Note that, since t 7→ z(t) partial stability of the constructed slow dynamics and the is a global diffeomorphism, the partial stability with respect original slow-fast system are shown to be equivalent. That to x of the first two time-varying subsystems in the new time is, analysis reduces to the partial stability analysis of the axis is equivalent to the partial stability with respect to x of constructed slow dynamics. Moreover, since the original slow- the system (1) in the original time axis. Therefore, hereafter fast system is periodic, the constructed one is also periodic. we focus on the first two time-varying subsystems in the new Next, in order to study the partial stability of the constructed time axis. For the sake of simplicity of description, the first slow dynamics, we use an averaging method. For periodic two time-varying subsystems are rewritten into systems, averaging methods have been widely used to establish dx = εh (x, y, z), (3a) criteria for the standard full-state exponential stability [14, dz 1 Chap. 10] [15] and also asymptotic stability [16]. Inspired dy = εh (x, y, z), (3b) by these works, we will develop a new criterion for partial dz 2 stability of the fast periodic dynamics via averaging. Accord- ing to our new criterion, if the averaged system is partially where h1 = f1/f3, and h2 = f2/f3. From the properties of f1, f2, and f3, it follows that both h1(0, y, z) = 0 and exponentially stable, then the slow dynamics and consequently m the original periodic slow-fast system is partially exponentially h2(0, y, z) = 0 for any y ∈ R and z ∈ R, and the solution to stable for sufficiently small ε > 0. It is worth emphasizing that the system (3) exists for all z ≥ 0. Moreover, the constructed compared with the case of full-state stability, partial stability slow dynamics (3) is again T -periodic in z. analysis is much more challenging since some states are not stable. B. Partial Stability Conditions via Averaging In order to study partial stability with respect to x of the constructed periodic slow dynamics (3), we use an averaging A. Slow Dynamics technique for periodic systems. The averaged system obtained As the first step, we construct a reduced slow dynamics from the slow dynamics (3) can be used for the partial stability studied in the following subsections. One important fact is analysis of the slow dynamics (3) before averaging. From the that the partial stability of the constructed slow dynamics is discussion in the previous subsection, the partial stability of equivalent to that of the original slow-fast system (1) under the slow dynamics (3) is equivalent to that of the original the assumption below. slow-fast system (1). Assume that for the fast subsystem: Since the slow dynamics (3) is T -periodic in z, it is possible to apply an averaging method. We associate it with a partially n m f3(x, y, z) ≥ ϑ, ∀x ∈ R , y ∈ R , z ∈ R, (2) averaged system given by dw or f (x, y, z) ≤ −ϑ, where ϑ > 0. Note that we only consider = εh (w, v), (4a) 3 dz av the first inequality in this paper since these two inequalities are dv essentially the same. This assumption (2) is naturally satisfied = εh (w, v, z), (4b) dz 2 for some practical problems such as vibration suppression of where the function h is defined by rotating machinery where f3 is the angular velocity [30], and av spin stabilization of spacecrafts where f3 describes the spin 1 Z T rate [31]. hav(w, v) = h1(w, v, τ)dτ, (5) T The assumption (2) implies that t 7→ z(t) can be interpreted 0 m as a change of time (recall that z is a scalar). In fact, (2) where hav(0, v) = 0 for any v ∈ R from h1(0, v, z) = 0. implies that for any given initial state (x(0), y(0), z(0)) of Note that only the dynamics of w is averaged with respect the slow-fast system (1), a part of the solution z(t) is a to z. strictly increasing function of t. That is, t 7→ z(t) is a In fact, if the averaged system (4) is partially exponentially global diffeomorphism1 from [0, ∞) to [0, ∞). In the new stable with respect to w, then the periodic slow system (3) is partially exponentially stable with respect to x for sufficiently 1This follows from the earlier assumption that the solution to the system small ε > 0. This implies that partial exponential stability of (1) exists for all t ≥ 0. the original slow-fast system (1) can be verified by using the 4

ˆ averaged system (4). This fact is stated formally as follows, where the function hav is deﬁned by which is one of the main results in this paper. Z T ˆ 1 Theorem 1: Suppose that w = 0 of the averaged system (4) hav(w) = h1(w, τ)dτ, (11) is partially exponentially stable uniformly in v, i.e., there exists T 0 δ > 0 such that for any z0 ∈ R and w(0) ∈ Bδ, As expected, if the averaged system (10) is exponentially

−λ(z−z0) stable, then the partial stability of (8) is ensured as long as kw(z)k ≤ kkw(0)ke , k, λ > 0, ∀z ≥ z0. (6) ε > 0 is sufficiently small, which is formally stated in the n Assume that there are L1,L2 > 0 such that for any x ∈ following corollary. If f3(x, z) = 1 for all x ∈ and z ∈ , m R R Bδ, y ∈ R , z ∈ R, the functions h1 and h2 in (3) satisfy this corollary reduces to the criteria in [14, Chap. 10] and [15] z ∂h1 ∂h2 for exponential stability of fast time-varying systems since (x, y, z) ≤ L1, (x, y, z) ≤ L2. (7) ∂x ∂x and t are the same. Corollary 1: Suppose that w = 0 is exponentially stable for Then, there exists ε > 0 such that, for any ε < ε , 1 1 the averaged system (10). Assume that there is L > 0 such the partial equilibrium point x = 0 of the system (3) is that for any x ∈ B , z ∈ the function h in (9) satisfies exponentially stable uniformly in y. As a consequence, for δ R 1 any ε < ε1, the partial equilibrium point x = 0 of the system ∂h1 (x, z) ≤ L. (12) (1) is exponentially stable uniformly in y and z. 4 ∂x The following subsections are dedicated to proving this Then, there exists ε > 0 such that, for any ε < ε , the partial theorem. Before providing the proof, we illustrate its utility. 1 1 equilibrium x = 0 of the system (8) is partially exponentially First, let us look back at Example 1, and see how the obtained stable uniformly in z. 4 results can be applied. In next section, we show how our results on partial ex- Continuation of Example 1: As 3 − sin x + cos y ≥ 1 for ponential stability can be applied to remote synchronization any x, y, the property (2) holds. Then, one can construct the in a simple network of Kuramoto oscillators. Before that, we averaged system (4) of the system in Example 1 as construct the proof of Theorem 1 in the following subsections. dw −w − 0.2w sin v = ε , dz 3 − sin w + cos v dv 2w cos v + w sin z C. Preparatory Results for Proving Theorem 1 = ε . dz 3 − sin w + cos v In the following subsections, our objective is to prove Choose a Lyapunov function candidate V (w, v, z) = w2. Theorem 1, that is to show the partial exponential stability Then, it holds that of the averaged system (4) implies that of the periodic slow system (3). Recall that partial stability of the slow system (3) is dV 1 + 0.2 sin v 8 = −2ε · w2 ≤ − εw2. equivalent to that of the original slow-fast system (1) under (2). dz 3 − sin w + cos v 15 For conventional full-state exponential stability analysis, According to [2, Theorem 1], w = 0 of the averaged system the original system is regarded as a perturbed system of the is partially exponentially stable. From Theorem 1, one can averaged one. As long as the perturbation characterized by ε conclude that x = 0 of the original system in Example 1 is is sufficiently small, the exponential stability of the original partially exponentially stable if ε > 0 is sufficiently small. 4 system is ensured [15], [14, Chap. 10]. Similar ideas are used in this paper for partial exponential stability analysis. Instead By using averaging techniques, Theorem 1 provides a new of full-state stability, we only require the averaged system (4) way to study partially stability of slow-fast systems for which to be partially exponentially stable. the existing criteria are difficult to apply. As another applica- In order to show the partial exponential stability of the tion, we can recover the conventional criteria [14, Chap. 10] periodic slow system (3), we use Lyapunov theory. First, we and [15] for exponential stability of fast time-varying systems, construct a Lyapunov function for a partially exponentially sta- as follows. Consider the following system with respect to x, ble averaged system. Then, by using this Lyapunov function,

x˙ = f1(x, z), (8a) we show the partial exponential stability of the periodic slow system if the perturbation is sufﬁciently small. This subsection εz˙ = f (x, z), (8b) 3 is dedicated to constructing a Lyapunov function. In effect, we where f1 and f3 satisfy all the assumption made for sys- provide a new converse Lyapunov theorem for partial stability. tem (1). The difference from (1) is that there is no variable y. As a generalized form of (4), we consider the following To study partial stability with respect to x, we apply the change time-varying systems in this section of time-axis, t → z. Then, we have dw dx = ϕ1(w, v, z), (13a) = εh1(x, z), (9) dz dz dv = ϕ2(w, v, z), (13b) where h1 := f1/f3. Next, compute the averaged system of the dz fast subsystem where w ∈ Rn, v ∈ Rm, z ∈ R, and the functions, n+m+1 n n+m+1 m dw ˆ ϕ1 : R → R , ϕ2 : R → R are continuously = εhav(w), (10) dz differentiable. Moreover, it holds that ϕ1(0, v, z) = 0 and 5

m ϕ2(0, v, z) = 0 for any v ∈ R . We further assume that for Particularly, we assume that the perturbation terms satisfy the any z0 the solution to the system (13) exists for all z ≥ z0. bounds Now, we provide a converse theorem for exponential partial kg (w , v , z)k ≤ γ (z)kw k + ψ (z), (21) stability of the system (13), which is directly applicable to the 1 p p 1 p 1 averaged system (4). kg2(wp, vp, z)k ≤ γ2(z)kwpk + ψ2(z), (22) Theorem 2: Suppose that w = 0 is partially exponentially where γ1, γ2 : R → R are nonnegative and continuous for all stable uniformly in v for the system (13), i.e., there exists z ∈ R, and ψ1, ψ2 : R → R are nonnegative, continuous and δ > 0 such that for any z ∈ and w(0) ∈ B , 0 R δ bounded for all z ∈ R. Notice that the bounds on the right are independent of v . kw(z)k ≤ kkw(0)ke−λ(z−z0), k, λ > 0, ∀z ≥ z . (14) p 0 The following theorem presents some results on the asymptotic behavior of the perturbed system (20) when the nominal Also, assume that there are L1,L2 > 0 such that system (13) has a partially exponentially stable equilibrium

∂ϕ1 ∂ϕ2 wp = 0. The proof is based on the constructed Lyapunov (w, v, z) ≤ L1, (w, v, z) ≤ L2, (15) ∂w ∂w function in Theorem 2; for the proof, see Appendix VI-B. m Theorem 3: Suppose that the nominal system (13) satisﬁes for any w ∈ Bδ, v ∈ R , z ∈ R. Then, there exists a function m all the assumptions in Theorem 2. Also, assume that the per- V : Bδ ×R ×R → R that satisﬁes the following inequalities: turbation terms g1(wp, vp, z) and g2(wp, vp, z) are respectively 2 2 bounded as in (21) and (22) for γ , γ and ψ , ψ satisfying c1kwk ≤ V (w, v, z) ≤ c2kwk , (16) 1 2 1 2 the following inequalities ∂V ∂V ∂V 2 + ϕ1(w, v, z) + ϕ2(w, v, z) ≤ −c3kwk , (17) Z z Z z ∂z ∂w ∂v c γ (τ)dτ + c γ (τ)dτ ≤ κ(z − z ) + η, (23) ∂V 4 1 5 2 0 z0 z0 ≤ c4kwk, (18) ∂w where

∂V c1c3 ≤ c5kwk, (19) 0 ≤ κ < , η ≥ 0; (24) ∂v c2 for some positive constants c1, c2, c3, c4 and c5. 4 and The same uniform boundedness assumptions on the partial 2c1k1δ c4ψ1(z) + c5ψ2(z) < , ∀z ≥ z0, (25) derivatives of the functions ϕ1 and ϕ2 are actually made k2 in Theorem 2 and Theorem 4.4 of [2]. Unlike the theorem where in [2], we work on time-varying systems. Moreover, we c3 κ η obtain additional two bounds for the partial derivative of V , k1 = − , k2 = exp . (26) 2c2 2c1 2c1 namely (18) and (19) by assuming ϕ2(0, v, z) = 0 for any v and z. Thus, our proof is more involved; it can be found in Then, the solution to the perturbed (20) satisﬁes Appendix VI-A. rc 2 −k1(z−z0) kwp(z)k ≤k2 kwp(z0)ke c1 Z z D. Analysis of Perturbed Systems k2 + e−k1(z−τ)ψ(τ)dτ, ∀z ≥ z . (27) 2c 0 In the previous subsection, we have constructed a converse 1 z0 n Lyapunov theorem. By applying this to a partially expo- for any initial time z0 ∈ R and any initial state wp(z0) ∈ R m nentially stable averaged system (4), one can construct a and vp(z0) ∈ R such that Lyapunov function satisfying all conditions in Theorem 2. By r δ c1 using this Lyapunov function, we consider to study the partial kwp(z0)k < . (28) exponential stability of the periodic slow dynamics (3). As k2 c2 typically done in averaging methods, we consider periodic 4 slow dynamics (3) as a perturbed system of its averaged A similar result is found in [14, Lemma 9.4], where the system (4). Then, we conclude the partial exponential stability nominal system is assumed to be exponentially stable. With of the periodic slow dynamics (3) by using the Lyapunov some perturbation, the asymptotic behavior of the full state is function for its averaged system (4). reported there. In contrast, the nominal system studied here is To this end, we study the following perturbed system of the only assumed to be partially exponentially stable in Theorem system (13) in the previous subsection: 3, and we show that the asymptotic behavior of a part of the states wp for the perturbed system follows some speciﬁc dwp = ϕ1(wp, vp, z) + g1(wp, vp, z), (20a) rule, without being concerned about how the other part of dz v dv the states, p, is evolving. We next consider a particular case p = ϕ (w , v , z) + g (w , v , z), (20b) where g and g in (20) are vanishing perturbations, i.e., ψ (z) dz 2 p p 2 p p 1 2 1 and ψ2(z) in (21) and (22) satisfy ψ1(z) = ψ2(z) = 0, and n+m+1 n n+m+1 m where g1 : R → R and g2 : R → R are obtain the next corollary. In fact, this corollary is used to prove piecewise continuous in z and locally Lipschitz in (wp, vp). Theorem 1. 6

Corollary 2: Suppose that the nominal system (13) sat- of the obtained dynamics from that of the averaged system. isfies all the assumptions in Theorem 3. Also, assume that Our goal is to show that the partial stability of the obtained the perturbation terms g1(wp, vp, z) and g2(wp, vp, z) are dynamics (32) implies that of the original slow system (3). respectively bounded by kg1(wp, vp, z)k ≤ γ1(z)kwpk and Let us represent the obtained dynamics (32) by a perturba- i kg2(wp, vp, z)k ≤ γ2(z)kwpk for γ1 and γ2 satisfying (23) tion of the averaged system (4). For k = 1, 2, let hk be the and (24), i.e. ψ1(·) = 0 and ψ2(·) = 0. Then, wp = 0 ith component of hk. From the mean value theorem, for each i i of the system (20) is partially exponentially stable uniformly k = 1, 2, there exists λk = λk(wp, vp, z, ε) > 0 such that in v . Moreover, the solution to (20) satisfies the following p i i specialisation of (27): hk(wp + εu, vp, z) − hk (wp, vp, z) ∂hi rc k i 2 −k1(z−z0) = (wp + ελku, vp, z) · εu. kwp(z)k ≤ k2 kwp(z0)ke , ∀z ≥ z0, ∂wp c1 Let us denote for any initial time z0 ∈ R and any initial condition wp(z0) ∈ n m and vp(z0) ∈ satisfying (28). R R H1(wp, vp, z, εu) In the next subsection, we show how the results obtained in 1 n > the previous and this subsection enable us to use averaging ∂h1 1 ∂h1 n = (wp + ελ1u, vp, z),..., (wp + ελ1 u, vp, z) techniques to study the partial stability of a periodic slow ∂wp ∂wp dynamics (3) from its averaged system (4). H2(wp, vp, z, εu) 1 n > ∂h2 1 ∂h2 n E. Proof of Theorem 1 = (wp + ελ2u, vp, z),..., (wp + ελ2 u, vp, z) . ∂wp ∂wp Now, we are ready to provide the proof of Theorem 1. Proof of Theorem 1: First, in order to describe the Then we have original slow system (3) as a perturbation of the averaged h1(wp + εu, vp, z) − h1 (wp, vp, z) system (4), we introduce the following change of variables = H (w , v , z, εu) · εu, (33) into the original slow system (3): 1 p p h2(wp + εu, vp, z) − h2 (wp, vp, z) x = wp + εu(wp, vp, z), (29a) = H2(wp, vp, z, εu) · εu, (34) y = vp, (29b) where both H1(wp, vp, z, εu) and H1(wp, vp, z, εu) are where bounded since from (7) each ∂hi /∂w is. Due to the bounded- Z z k ness of k∂u/∂zk, k∂u/∂wpk, and k∂u/∂vpk from Proposition u(wp, vp, z) = ∆(wp, vp, τ)dτ, (30) 0 3 in Appendix VI-C, it is clear that the matrix P (ε) is nonsingular for sufficiently small ε > 0, and its inverse can be with ∆(wp, vp, z) = h1(wp, vp, z) − hav(wp, vp). From the described as P −1(ε) = I + O(ε) with some O(ε). Applying definition of hav in (5), it holds that this fact together with the equalities (33) and (34) to (32), Z T 0 one can show that there are bounded H1(wp, vp, z, εu) and ∆(wp, vp, τ)dτ = 0. (31) 0 0 H2(wp, vp, z, εu) such that After substituting (29) into (3), we obtain the following dwp 2 0 = εhav (wp, vp) + ε H1(wp, vp, z, εu)u, (35a) dx dwp ∂u ∂u dwp ∂u dvp dz = + ε + ε + ε , dv dz dz ∂z ∂wp dz ∂vp dz p 2 0 = εh2(wp, vp, z) + ε H2(wp, vp, z, εu)u. (35b) dy dv dz = p . dz dz This is a perturbation of the averaged system (4). Substituting (3) and (29a) into the above equations yields Next, we apply Corollary 2 to show that the partial exponential stability of the averaged system (4) implies that of " dwp # dz its perturbation (35) for sufficiently small ε > 0. From the P (ε) dvp = dz definition of hav, we have εh1(wp + εu, vp, z) − εh1 (wp, vp, z) + εhav (wp, vp) Z T , ∂hav 1 ∂h1 εh2(wp + εu, vp, z) (wp, vp) = (wp, vp, τ)dτ ≤ L1 (36) ∂w T ∂w (32) p 0 p m where for any wp ∈ Bδ, v ∈ R . By the assumption (7), ∂h2/∂wp ≤ I + ε ∂u ε ∂u L2. Therefore, both inequalities in (15) are satisfied. Since P (ε) = ∂wp ∂y . the system (4) is assumed to be partially exponentially stable, 0 I all the assumptions in Theorem 3 are satisfied. To apply We then show that the obtained dynamics (32) can be viewed Corollary 2, it remains to show that the perturbation terms are as a perturbation of the averaged system (4). Therefore, bounded linearly in kwpk. Let b1 > 0 and b2 > 0 be constants 0 0 Corollary 2 can be used in order to show the partial stability such that kH1(w, v, z, εu)k ≤ b1 and kH2(wp, vp, z, εu)k ≤ 7

b2. From (66) in Appendix VI-C, it holds that ku(wp, vp, s)k ≤ 0 2 2TL1kwk, and then the perturbation terms satisfy 2 0 2 kε H1(wp, vp, z, εu)uk ≤ 2ε b1TL1kwpk, 2 0 2 kε H2(wp, vp, z, εu)uk ≤ 2ε b2TL1kwpk. 1

Moreover, for sufficiently small ε1 > 0, any ε < ε1 makes sure both inequalities (23) and (24) are satisfied. Therefore, Fig. 1. A simple network motif: central node 0 and peripherals 1 and 2. Corollary 2 implies that wp = 0 is partially exponentially stable for the perturbation (35), or equivalently, (32). In other where θ ∈ 1 is the phase of the ith oscillator; ω > 0 0 0 0 i S words, there are δ > 0 and k , λ > 0 such that wp(z0) ∈ Bδ0 0 is the uniform natural frequency of each oscillator; Ai > 0 0 −λ (z−z0) implies kwp(z)k ≤ k kwp(z0)ke , for all z ≥ z0. is the coupling strength between the central node 0 and the Finally, we show that the partial exponential stability of the peripheral node i; α ∈ (0, π/2) is the phase shift; and u ≥ 0 system (32) implies that of the slow dynamics (3). From (29a) is the natural frequency detuning (For the case u < 0, one and (66) in the Appendix, one obtains can obtain virtually identical results to those obtained below). |1 − 2εT L1| · kwp(z)k ≤ kx(z)k ≤ |1 + 2εT L1| · kwp(z)k, Note that, the phase shift α is often used to model delays arising in synaptic connections in neural networks [29]. for all z ≥ z0. Then, it follows that > 3 Let θ = (θ0, θ1, θ2) ∈ S . To study the remote syn- |1 + 2εT L | 0 0 1 −λ (z−z0) chronization in our considered network, we define a remote kx(z)k ≤ k kx(z0)ke , ∀z ≥ z0, |1 − 2εT L1| synchronization manifold as follows. proving the partial exponential stability of x = 0 for the Definition 2 (Remote Synchronization Manifold): The re- system (3) for sufficiently small ε > 0. Finally, one can mote synchronization manifold is defined by conclude that x = 0 is also partially exponentially stable for M := θ ∈ 3 : θ = θ . the original slow-fast system (1) uniformly in y and z under S 1 2 asssumption (2). A solution θ(t) to (37) is said to be remotely synchronized if it holds that θ(t) ∈ M for all t ≥ 0. It is shown in [21] that IV. REMOTE SYNCHRONIZATION IN A NETWORKOF network symmetries are critical to the occurrence of remote KURAMOTO OSCILLATORS synchronization. In our considered network in Fig. 1, we say In this section, we apply the results on partial stability oscillators 1 and 2 are symmetric if A1 = A2. It can be analysis to studying remote synchronization of oscillators. observed that the requirement A1 = A2 is necessary for the Remote synchronization is a phenomenon arising in coupled system (37) to have a remote synchronized solution since the networks of oscillators when two oscillators without a direct equation connection become synchronized without requiring the inter- ˙ ˙ mediate ones on a path linking the two oscillators to also θ1 − θ2 = A1 sin(θ0 − θ1 − α) − A2 sin(θ0 − θ2 − α) = 0 be synchronized with them [6]. We restrict our attention to has a solution θ = θ only if A = A . Therefore, we assume remote synchronization in a network motif shown in Fig. 1. 1 2 1 2 that the coupling strengths satisfy2 This network is simple, but has been shown to surprisingly account for the emergence of zero-lag synchronization in A1 = A2 = A. (38) remote cortical regions of the brain, even in the presence of large synaptic conduction delays [22]. Experiments have also Evidently, the network in Fig. 1 is the simplest symmetric net- evidenced that the same network can give rise to isochronous work. In what follows, we study the exponential stability of the synchronization of delay-coupled semiconductor lasers [23]. remote synchronization manifold M under assumption (38). 3 The central element 0 in this network plays a critical role in Define a δ-neighborhood of M by Uδ = {θ ∈ S : mediating or relaying the dynamics of the peripheral 1 and dist(θ, M) < δ}, where dist(θ, M) is the minimum distance 2. A recent work reveals that detuning the parameters of the from θ to a point on M, that is, dist(θ, M) = infy∈M kθ−yk. central element from those of the peripheral ones can actually Let us define the exponential stability of the remote synchro- enhance remote synchronization [25]. To study this interesting nization manifold M. finding analytically, we employ Kuramoto-Sakaguchi model Definition 3: For the system (37), the remote synchroniza- [27] and detune the natural frequency of the central oscillator. tion manifold M is said to be exponentially stable along the system (37) if there is δ > 0 such that for any initial phase 3 A. Problem Statements θ(0) ∈ S satisfying θ(0) ∈ Uδ it holds that Using Kuramoto-Sakaguchi model and detuning the natural dist(θ(t), M) = k · dist(θ(0), M) · e−λt, ∀t ≥ 0, frequency of the central oscillator, the dynamics of the oscil- 2 lators are described by This assumption requires the two coupling strengths, A1 and A2, to ˙ be strictly identical. This is somewhat demanding since it is not easy to θi = ω + Ai sin(θ0 − θi − α), i = 1, 2; (37a) be fulfilled in practical situations. Simulation results show that the phase 2 difference between the peripheral oscillators remains bounded if A1 and A2 X θ˙ = ω + A sin(θ − θ − α) + u, (37b) are only approximately the same, although exact phase synchronization cannot 0 j j 0 take place. It is quite interesting to study this problem in the future, though j=1 we only consider A1 = A2 in this paper. 8

for some k > 0 and λ > 0. role in determining the stability of M1. The proof using the In fact, remote synchronization behavior for the net- linearization method can be found in Appendix VI-D. work (37) can be categorized into two different types, depend- Theorem 4: Assume that (38) is satisfied. For any nonzero ing on whether phase locking occurs or not. Phase locking A, the following statements hold: √ is a phenomenon where every pairwise phase difference is a 1) if α < arctan 3, there exists a unique exponentially constant, θi −θj = ci,j, ∀i, j (the phenomenon when ci,j = 0, stable remote synchronization manifold in M, that is ∀i, j, is especially called phase synchronization). Phase lock- M1; √ ing is also called frequency synchronization because this is 2) if α > arctan 3, there does not exist an exponen- equivalent to the frequencies of all oscillators being syn- tially stable remote synchronization manifold in M . chronized, θ˙ = ··· = θ˙ . If the network is phase locked, 1 n Consistent with the findings in [22] and [23], remote syn- it is remotely synchronized in the sense that the peripheral chronization emerges thanks to the central mediating oscillator, oscillator frequencies θ˙ and θ˙ are the same. However, the 1 2 and it is exponentially stable for a wide range of phase converse is not always true. In remote synchronization, the √ shift, i.e., α ∈ (0, arctan( 3)). Nevertheless, an even larger frequency of the central oscillator, θ˙ , is allowed to be different 0 phase shift α out of this range can destabilize the remote from the peripheral ones, θ˙ , θ˙ . 1 2 synchronization. In the next subsection, we detune the natural We will study these two categories of remote synchroniza- frequency of the central oscillator by letting u 6= 0, which tion in the next two subsections, where we assume u = 0 and is similar to the introduction of parameter impurity in [25], u 6= 0, respectively, to reveal the role that the natural frequency and show how a sufficiently large natural frequency detuning detuning u plays. The phase-locked case is relatively easy to can lead to robust remote synchronization that is exponential analyze as demonstrated in the next subsection. In contrast, stable for any phase shift α ∈ (0, π/2). the analysis of the other case is technically involved, but is possible thanks to our results on partial stability established in the previous section. Although in the phase-locked case, C. Natural frequency detuning u 6= 0 the results on partial stability are not utilized, we analyze this In this subsection, we consider the case when the natural property for the sake of ensuring a complete account of remote frequency detuning u > 0, and show how it can give rise to synchronization. robust remote synchronization. Note that if u > 3A, there does not exists a phase- B. Natural frequency detuning u = 0 locked solution to (37). This is because the equations x˙ i = P2 In this subsection, we assume that the natural frequency u + j=1 A sin(−xj − α) − A sin(xi − α) = 0, i = 1, 2, do detuning u = 0. We will show that if remote synchronization not have a solution. Nevertheless, the remote synchronization occurs, it necessarily entails phase locking. A linearization can still be exponentially stable for a sufficiently large control method is the tool we will use to show the stability of the input u. In other words, the natural frequency detuning can ac- remote synchronization manifold M. First, we observe that for tually stabilize the remote synchronization, although it makes any α ∈ (0, π/2), there always exists a remotely synchronized phase locking impossible. In fact, the remote synchronization manifold M as a whole becomes exponentially stable for any solution to (37) that is phase locked. To show this, let xi := α with this control input. The following is the main result of θ0 − θi for i = 1, 2. The time derivative of xi is this section. 2 X Theorem 5: There is a positive constant ρ > 3A such that for x˙ = A sin(−x − α) − A sin(x − α). (39) i j i any u satisfying u > ρ, the remote synchronization manifold j=1 M is exponentially stable for the system (37) for any phase Any remotely synchronized solution satisfies x1 = x2. Solving shift α ∈ (0, π/2). the equation x˙ i = 0 with x1 = x2 we obtain two isolated The proof is technically involved and is based on the equilibrium points of the system (39) in the interval [0, 2π]: results on partial stability established in the previous section. ∗ ∗ ∗ ∗ 0 1) x1 = x2 = c(α); 2) x1 = x2 = c (α), where Before providing the proof, we first define some variables, sin α and associate the remote synchronization manifold M with c(α) = − arctan , c0(α) = π + c(α). (40) 3 cos α an equivalent set defined on the new variables. We then prove that this set is exponentially stable. Note that other equilibrium points outside of [0, 2π] are equiva- Let us define z1 and z2 by lent to these two, and it is thus sufficient to only consider them. 0 2 Any solution satisfies θ1(t) = θ2(t) and θ0(t) − θ1(t) = c (α) 1 X z1 := cos(θ0 − θj), (41a) (or θ0(t) − θ1(t) = c(α)) is a phase-locked and remotely 2 synchronized solution. To capture this type of remote syn- j=1 2 chronization, we define M1 := {θ ∈ M : θ0 − θ1 = c(α)}, 1 X 0 0 z2 := sin(θ0 − θj). (41b) M1 := {θ ∈ M : θ0 − θ1 = c (α)}, and refer to them as 2 the phase-locked remote synchronization manifolds. It is not j=1 hard to see that they are two positively invariant manifolds of Then, it is clear to see z1, z2 ∈ R satisfy |z1| ≤ 1 and |z2| ≤ 0 3 the system (37). We show in the following theorem that M1 1. Note that for any initial condition θ(0) ∈ S , the unique is always unstable, and the phase shift α plays an essential solution θ(t) to (37) exists for all t ≥ 0. As a consequence, 9

z1(t) and z2(t) exist for all t ≥ 0. We then define the following where with (41), unit circle by using z and z : v 1 2 u 2 2 1u X 2 X 2 r := t cos(θ0 − θj) + sin(θ0 − θj) , (47) n 2 2 2 o 2 L := z ∈ R : z1 + z2 = 1 , (42) j=1 j=1 P2 ! j=1 sin(θ0 − θj) where z = (z , z )>. In fact, this set L has a strong relation ζ := arctan . (48) 1 2 P2 cos(θ − θ ) with remote synchronization as follows. j=1 0 j 2 2 2 Proposition 1: Let M and L be defined in Definition It follows from (45) and (46) that z1 (t) + z2 (t) = r . Thus, 2 and (42), respectively. The following two statements are the distance from z(t) to the circle L, denoted by µ(t) is equivalent: µ(t) := dist(z(t), L) = 1 − r(t). (49) 1) θ belongs to the remote synchronization manifold M. > In fact, the dynamics of µ(t) and ζ(t) can be described by 2) z = (z1, z2) belongs to the set L. dµ(t) Proof: From (41), there holds that = −A(1 − (1 − µ)2) cos(ζ − α), (50a) dt dζ(t) 2 2 = u − A(1 − µ)2 sin(ζ + α) + sin(ζ − α), (50b) z1 +z2 = dt 2 2 1 X 2 1 X 2 where µ(t) ∈ [0, 1] and ζ(t) ∈ . The proof is given by Propo- cos(θ − θ ) + sin(θ − θ ) . (43) R 4 0 j 4 0 j sition 4 in Appendix VI-E. Therefore, remote synchronization j=1 j=1 analysis reduces to partial stability analysis of µ(t). We are now ready to provide the proof of Theorem 5 based on the The right-hand side of the equality (43) can be simplified to results of partial stability obtained in the previous section. 2 2 Proof of Theorem 5: As we have shown, in order to 1 X 2 1 X 2 cos(θ − θ ) + sin(θ − θ ) prove the exponential stability of M, it is sufficient to prove 4 0 j 4 0 j j=1 j=1 the set L is partially exponentially stable along the system 2 (37). In other words, we show that µ = 0 of the system (50) 1 X = cos2(θ − θ ) + sin2(θ − θ ) is exponentially stable uniformly in ζ based on Corollary 1. 4 0 j 0 j j=1 To this end, we show that the system satisfies the conditions 2 2 in Corollary 1. + cos(θ − θ ) cos(θ − θ ) + sin(θ − θ ) sin(θ − θ ) 4 0 1 0 2 4 0 1 0 2 First, we confirm the requirements for the system (8) taking 1 1 µ = x1 and ζ = z. These are simply that µ = 0 is a partial = + cos(θ − θ ), (44) 2 2 1 2 equilibrium of (50a), and the system is 2π-periodic in ζ, which one can check immediately. Next, from the assumption u > where the last equality has used the trigonometric identity ρ > 3A, there exists ϑ > 0 such that dζ(t)/dt ≥ ϑ for any µ cos a cos b + sin a sin b = cos(a − b). We first prove that and ζ, which means that assumption (2) is satisfied. 1) implies 2). If θ ∈ M, one obtains θ1 = θ2 from Next, we compute the equivalent of (9) by applying the the definition of the remote synchronization manifold M. It change of time-axis, t 7→ ζ, and then obtain the equivalent of follows from (44) that the right-hand side of (43) equals 1. We its averaged system (10). The derivative of µ with respect to now prove that 2) implies 1). If z ∈ L, from (44) we obtain ζ can be computed as 1/2+1/2 cos(θ1−θ2) = 1, which means that cos(θ1−θ2) = 1. dµ dr dζ This, in turn, proves that θ ∈ M. The proof is complete. = / = εf(µ, ζ) (51) dζ dt dt Proposition 1 provides us an alternative way to study remote where, corresponding to h1, synchronization. Any pair of (z1, z2) belongs to L if and only if the corresponding θ ∈ R3 is included in the remote −A(2 − µ)µ cos(ζ − α) f(µ, ζ) = , synchronization manifold M. Further, if θ1(0) = θ2(0), it A 1 − u (1 − µ) 2 sin(ζ + α) − sin(ζ − α) can be seen from (37) that θ1(t) = θ2(t) for all t ≥ 0. In other words, z(0) ∈ L implies that z(t) ∈ L for all t ≥ 0, and ε = 1/u. Note that for any given u, there is L > 0 such which means that the set L is a positively invariant set of the that the equivalent of (12) holds. Then, the associated averaged system (37). To show the exponential stability of the remote system is synchronization manifold, it suffices to show the positively Z 2π ˙ invariant set L is exponentially stable along the system (37) µˆ = fav(ˆµ) := f(ˆµ, τ)dτ, (52) 0 using the distance dist(z, L) = infy∈L kz − yk. where To proceed with the analysis, we represent z1 and z2 in the 2π polar coordinates Z 4π(2 − µˆ)ˆµ fav(ˆµ) = f(ˆµ, τ)dτ = · g(ˆµ), 0 (1 − µˆ)(5 + 4 cos 2α) ! z1 = r cos ζ, (45) 1 1 g(ˆµ) = − . z2 = r sin ζ, (46) u pu2 − 5A2(1 − µˆ)2 − 4A2(1 − µˆ)2 cos 2α 10

According to Corollary 1, it remains to check the expo- V. CONCLUDING REMARKS nential stability of the averaged system. By the assumption Using periodic averaging methods, we have obtained some u > 3A, it follows that g(ˆµ) < 0 and thus fav(ˆµ) < 0 for any criteria for partial exponential stability of a type of slow-fast 0 < µˆ < 1. Moreover, for any µˆ satisfying 0 < µˆ < ξ < 1, it nonlinear systems in this paper. We have associated the orig- holds that inal system with an averaged one by averaging over the fast varying variable. We have shown that the partial exponential fav(ˆµ) < −cµ,ˆ (53) stability of the averaged system implies that of the original where the constant c is given by one. As some intermediate results, we have also obtained: 1) ! a converse Lyapunov theorem; and 2) some perturbation the- 4π 1 1 orems that are the first known ones for partially exponentially c = p − . 9 u2 − 9A2(1 − ξ)2 u stable systems. We have applied our results to stabilization of remote synchronization in a network of Kuramoto oscillators Choose V (ˆµ) =µ ˆ2 as a Lyapunov candidate, and it is easy to with a phase shift, showing how our results can be used to see V˙ ≤−cµˆ2, which implies that µˆ = 0 is exponentially stable prove stability of limit cycles. In the future, we are interested along the averaged system (52) for any u > 3A. According in developing new criteria for partial asymptotic stability of to Corollary 1, there exists ε∗ > 0 such that if ε < ε∗, the nonlinear systems using averaging techniques. Moreover, it is system (50) is partially exponentially stable with respect to interesting to study stability of remote synchronization in more µ. As ε = 1/u, it is equivalent to saying that there exists complex networks. ρ > 3A such that if the input u > ρ, the system (50) is partially exponentially stable respect to µ. Thus, the remote synchronization manifold M is exponential stable for any VI.APPENDIX phase shift α. A. Proof of Theorem 2 Consistent with the experimental findings in [25], [26], Before proving Theorem 2, we first present an intermediate Theorem 5 rigorously shows that by detuning the natural result. frequency one is able to stabilize the remote synchronization Proposition 2: Consider a continuously differentiable func- manifold even when the phase shift is quite large. Interestingly, tion h : n1 × n2 → n1 , which satisfies h(0, v) = 0 for any the central oscillator has a different frequency θ˙ from the R R R 0 v ∈ n2 . Suppose that there exists a connected set D ⊂ n1 peripheral ones θ˙ and θ˙ when remote synchronization occurs R R 1 2 containing the origin x = 0 such that under the assumption u > ρ > 3A. ∂h In fact, what we have proven in Theorem 5 is that L is an n2 (w, v) ≤ l1, ∀w ∈ D, v ∈ R , exponentially stable limit cycle. We now show that it is also ∂w periodic. It can be observed that the set L defined in (42) is for a positive constant l1. Then, there exists l2 > 0 such that compact. From (45) and (46), the time derivatives of z1 and ˙ ˙ ∂h z2 are z˙1 =r ˙ cos ζ − r sin ζ · ζ and z˙2 =r ˙ sin ζ + r cos ζ · ζ, n2 (w, v) ≤ l2kwk, ∀w ∈ D, v ∈ R . respectively. Since r˙ = −µ˙ and r = 1 − µ, it follows that ∂v ˙ z˙1 = −µ˙ cos ζ − (1 − µ) sin ζ · ζ, (54a) Proof: Due to the continuous differentiability of h, it ˙ follows from the mean value theorem that for any w ∈ D, z˙2 = −µ˙ sin ζ + (1 − µ) cos ζ · ζ. (54b) and v, δ ∈ Rn2 , there exists 0 ≤ λ ≤ 1 such that From (49), one knows that µ = 0 and µ˙ = 0 if col(z1, z2) ∈ L, ∂hi in which case the dynamics (54) become z˙ = −ζ˙ sin ζ, z˙ = hi(w, v + δ) − hi(w, v) = (w, v + λδ) · δ. 1 2 ∂v ζ˙ cos ζ. One can observe that this system has no equilibrium in L since it holds that ζ˙ > 0 from (50b). It is implied Since k∂h/∂wk ≤ l1 and h(0, v) = 0, ∀v, we have by Poincare-Bendixson´ Theorem [32, Theorem 9.3] that L is khi(w, v + δ) − hi(w, v)k actually a periodic orbit. In other words, L is an exponentially stable periodic orbit. ≤ khi(w, v + δ) − hi(0, v + δ)k + khi(w, v) − hi(0, v)k ˙ ˙ ˙ Let v1 = θ1 + θ2 and v2 = θ0, and then we can rewrite the ≤ 2l1kwk, (56) set (42) into where the last inequality has used the fact that k∂hi/∂wk ≤ n > 2 k∂h/∂wk ≤ l . We then observe that the inequality C : = (v1, v2) ∈ R : 1 2 2 ∂h (v1 + v2 − 3ω − u) (v1 − v2 − ω + u) o i + = 1 , (55) (w, v + λδ) · δ ≤ 2l1kwk 16A2 sin2 α 16A2 cos2 α ∂v

n2 which is also a limit cycle for the variables v1 and v2. Note holds for any w ∈ D, and v, δ ∈ R , which implies that for 0 0 that v1 is the sum of the peripheral oscillators’ frequencies. any i there exists l1 > 0 such that k∂hi/∂vk ≤ l1kwk. As a One can say the remote synchronization is reached if and only consequence, there is l2 > 0 so that k∂h/∂vk ≤ l2kwk. if the frequencies v1 and v2 reach the limit cycle C. Theorem We are now ready to prove Theorem 2. 5 also implies the exponential stability of the limit cycle C, Proof of Theorem 2: Let φ1(τ; w, v, z) and φ2(τ; w, v, z) and furthermore, C is also periodic. denote the solution to the system (13a) and (13b) that starts 11

−(λ−L)δ at (w, v, z), respectively; note that φ1(z; w, v, z) = x and with c4 = 2k(1 − e )/(λ − L), which proves the φ2(z; w, v, z) = y. Let inequality (18). Note that the case when λ = L can always be Z z+δ ruled out since one can replace L with L + σ for any σ > 0 2 V (w, v, z) = kφ1(τ; w, v, z)k2dτ. (57) if that happens. z Following a similar line, one can show there exists c5 > 0 Following similar steps as those in Theorem 4.14 of [14] and such that (19) are satisfied, which completes the proof. Theorem 4.4 of [2], one can show that 1 k2 −2L1δ 2 −2λδ 2 (1 − e )kwk2 ≤ V (w, v, z) ≤ (1 − e )kwk2, B. Proof of Theorem 3 2L1 2λ Proof: From the assumption for the nominal system (13), and V˙ (w, v, z) = −(1 − k2e−2λδ)kwk2, which proves the in- 2 there is a Lyapunov function V satisfying all conditions in equalities (16) and (17). Note that V˙ denotes a total derivative Theorem 2. Here, we use this V to estimate the convergence along trajectories with the independent variable z rather than speed of the perturbed system (20). t. m We next prove the inequalities (18) and (19). Let φ = For any wp ∈ Bδ and vp ∈ R , z ∈ R, the total derivative of the Lyapunov function V (wp, vp, z) along the trajectories col(φ1, φ2), ϕ = col(ϕ1, ϕ2), and for k = 1, 2, denote of the perturbed system (20) satisfies ∂ φ (τ; w, v, z) = φ (τ; w, v, z). k,w ∂w k dV ∂V ∂V = + (ϕ1(wp, vp, z) + g1(wp, vp, z)) 0 dz ∂z ∂w and φw = ∂φ/∂w. Note that φ1(τ; w, v, z) and φ2(τ; w, v, z) p satisfy ∂V + (ϕ (w , v , z) + g (w , v , z)) τ 2 p p 2 p p Z ∂vp φ1(τ; w, v, z) = w + ϕ1(φ1(s; w, v, z), φ2(s; w, v, z))ds, 2 2 z ≤ − c3kwpk + (c4γ1(z) + c5γ2(z)) kwpk Z τ + (c4ψ1(z) + c5ψ2(z)) kwpk, φ2(τ; w, v, z) = v + ϕ2(φ2(s; w, v, z), φ2(s; w, v, z))ds. z where the last inequality follows from the inequalities (17), Then, the partial derivative φ0 is w (18), and (19) together with the bounds (21) and (22). For Z τ 0 I 0 ∂ϕ 0 simplicity, denote γ(z) = c4γ1(z) + c5γ2(z) and ψ(z) = φw(τ; w, v, z) = + φx(s; w, v, z)ds. (58) 0 0 z ∂φ c4ψ1(z) + c5ψ2(z). By the inequality (16) one can obtain an ˙ Recall that ϕ1 and ϕ2 are continuously differentiable, and upper bound for V given by satisfy ϕ1(0, v, z) = 0 and ϕ2(0, v, z) = 0 for any v and z, dV c3 1 1 √ and k∂ϕ1/∂wk2 ≤ L1 and k∂ϕ2/∂wk2 ≤ L2 for any w ∈ Bδ ≤ − − γ(z) V + √ ψ(z) V. dz c2 c1 c1 and v ∈ Rm, t ∈ R. It then follows from Proposition 2 that 0 0 p k∂ϕ1/∂vk2 ≤ L1 and k∂ϕ2/∂vk2 ≤ L2 for some positive Let W (wp, vp, z) = V (wp, vp, z), and when V 6= 0 the total 0 0 constants L1 and L2, since kwk is upper bounded by some derivative of W satisfies constant in a compact set B . Equivalently, there exists L > 0 δ dW dV/dz 1 c 1 1 such that = √ ≤ − 3 − γ(z) W + √ ψ(z). dz 2 V 2 c2 c1 2 c1 ∂ϕ m ≤ L, ∀X ∈ Bδ × R , (59) (61) ∂X 2 When V = 0, following the same idea as the proof of Lemma where X = col(w, v). Consequently, k∂ϕ/∂φk2 ≤ L, and it then follows from (58) that 9.4 in [14] one can show that the Dini derivative of W satisfies Z τ 1 0 0 + kφw(τ; w, v, z)k2 ≤ 1 + L kφw(s; , w, v, z)k2ds, D W ≤ √ ψ(z), z 2 c1 0 L(τ−z) + which implies that kφw(τ; w, v, z)k2 ≤ e by Gronwall’s¨ which implies that D W satisfies (61) for all values of V . lemma [2, Lemma 2.2]. Since kφ1,wk2 ≤ kφ1,w, φ2,wk2 = Using the comparison lemma [14, Lemma 3.4], one can show 0 kφw(τ; w, v, z)k2, it holds that that W (z) satisfies the following inequality L(τ−z) kφ1,w(τ; w, v, z)k2 ≤ e . (60) 1 Z z W (z) ≤ Φ(z, z )W (z ) + Φ(z, τ)ψ(τ)dτ, −λ(τ−z) 0 0 √ (62) From (14), it holds that kφ1(τ; w, v, z)k ≤ Kkwk2e . 2 c1 z0 Then, the partial derivative ∂V/∂w satisfies where the transition function is

∂V Z z+δ Z z = 2φ>(τ; w, v, z)φ (τ; w, v, z)dτ c3 1 1 1,w Φ(z, z0) = exp − (z − z0) + γ(τ)dτ . (63) ∂w 2 z 2 2c2 2c1 z0 Z z+δ Substituting (23) and (24) into (63), we have ≤ 2 kφ1(τ; w, v, z)k · kφ1,w(τ; w, v, z)k2dτ z −k1(z−z0) Z z+δ Φ(z, z0) ≤ k2e , (64) −λ(τ−z) L(τ−z) ≤ 2kkwk2e e dτ = c4kwk2, z with k1 and k2 given by (26). 12

√ From the inequality (16), we obtain c kw k ≤ W ≤ In turn, this implies that for any w ∈ B , v ∈ m, and √ 1 p p δ p R c2kwpk. Then, for any z ≥ z0 such that kwp(z)k ∈ Bδ, z ∈ R, it follows from the inequalities (62) and (64) that Z z00 ∂u ∂∆ r (wp, vp, z) ≤ (wp, vp, τ) dτ c2 ∂w ∂w kw (z)k ≤ k kw (z )ke−k1(z−z0) p 0 p p 2 p 0 00 c1 ≤ 2z L1 ≤ 2TL1. k Z z 2 −k1(z−τ) + e ψ(τ)dτ, (65) Finally, we prove k∂u/∂vpk is bounded. Since 2c1 z0 h1(0, vp, z) = 0 for any vp and z, from Proposition 2, 0 for any z ≥ z0 such that kwp(z)k ∈ Bδ. Under the assumption there exists L1 > 0 such that for any wp ∈ Bδ, and m (25), if the initial condition satisﬁes (28), it then holds that vp ∈ R , z ∈ R,

−k1(z−z0) −k1(z−z0) ∂h1 0 kwp(z)k < δe + δ(1 − e ) = δ, ∀z ≥ z0, (wp, vp, z) ≤ L1kwpk. (67) ∂vp which ensures that the inequality (65) holds for any z ≥ z . 0 Then, the partial derivative of ∆ with respect to vp satisﬁes The proof is complete.

∂∆ (wp, vp, z) ∂vp C. Proposition 3 ∂h 1 Z T ∂h 1 1 Proposition 3: Consider the function u(w , v , z) deﬁned = (wp, vp, z) − (wp, vp, τ)dτ p p ∂vp T ∂vp m 0 in (30). For any wp ∈ Bδ, vp ∈ R , z ∈ R, ku(wp, vp, z)k, 0 ≤ 2L kwpk, k∂u/∂wpk, and k∂u/∂vpk are all bounded. 1

Proof: First, we prove that ku(wp, vp, z)k is bounded. ∂u 0 0 which implies that ∂v (wp, vp, z) ≤ 2TL1kwpk ≤ 2TL1δ One can observe that u(wp, vp, z) is T -periodic in z since p for any wp ∈ Bδ. ∆(wp, vp, z) is. For any z ≥ 0, there exists a nonnegative 0 0 integer N1 and z satisfying 0 ≤ z < T . such that z = 0 N1T + z . Then, using (31) we have D. Proof of Theorem 4 Z z Proof: We prove this theorem by two steps. We first 0 ∆(wp, vp, τ)dτ demonstrate that M1 and M1 are the only two positively 0 0 invariant manifolds in M for any α by proving that starting Z N1T Z z from any point in M/M0 , the solution to (37), θ(t), converges = ∆(wp, vp, τ)dτ + ∆(wp, vp, τ)dτ, 1 0 0 to M1 asymptotically. Then, we investigate the stability of z0 0 Z M1 and M1 under different assumptions of α. = ∆(wp, vp, τ)dτ. We start with the first step. When θ ∈ M, there holds that 0 x1 = x2. Then, the dynamics of x1 and x2 are described by Next, the partial derivative of ∆ with respect to w satisfies x˙ = −2A sin(x + α) − A sin(x − α). (68) i i i ∂∆ (wp, vp, z) Note that x2 has the same dynamics of x1, and it is thus ∂wp sufficient to only investigate the asymptotic behavior of x1. For ∂h 1 Z T ∂h 0 1 1 any initial condition θ(0) ∈ M/M1, there hold that θ1(0) = = (wp, vp, z) − (wp, vp, τ)dτ ≤ 2L1, ∂wp T 0 ∂wp θ2(0) and θ0(0) − θ1(0) ∈ (−π, π + c(α)) ∪ (π + c(α), π), which means x1(0) = x2(0) and x1(0) ∈ (−π, π + c(α)) ∪ where the triangle inequality and (7) have been used. Using (π +c(α), π). When x1(0) ∈ (−π, π +c(α)), we choose V1 = this inequality and ∆(0, vp, z) = 0 in (30) yields 1 2 2 (x1 − c(α)) as a Lyapunov candidate. Its time derivative ˙ Z z0 is V1 = −A(x1 − c(α)) 2 sin(xi + α) + sin(xi − α) , which ˙ ˙ ku(wp, vp, z)k ≤ k∆(wp, vp, τ) − ∆(0, vp, τ)kdτ satisfies V < 0 for any x1 ∈ (−π, π+c(α)) and V = 0 if x1 = 0 c(α). Thus, starting from (−π, π + c(α)), x1(t) converges to 0 ≤ 2z L1kwpk ≤ 2TL1kwpk. (66) c(α) asymptotically. When x1(0) ∈ (π + c(α), π), we choose V = 1 (x − 2π − c(α))2 as a Lyapunov candidate. Likewise, w ∈ B v ∈ m, z ∈ 1 2 1 For any p δ and p R R, it is now clear that one can show starting from (π + c(α), π), x (t) converges to ku(w , v , z)k ≤ 2TL δ 1 p p 1 . 2π + c(α) asymptotically. Since c(α) and 2π + c(α) represent Second, we prove k∂u/∂wpk is bounded. Since ∂u/∂wp is 1 the same point on , the two equilibrium points of (68), x1 = R T ∂u S T -periodic in z and satisfies (wp, vp, τ)dτ = 0, for any 0 ∂wp x2 = c(α) and x1 = x2 = 2π + c(α), correspond to the same z ≥ 0 N z00 3 , there exists a nonnegative integer 2 and satisfying manifold M1 of θ ∈ S . Then, there is no positively invariant 0 ≤ z00 ≤ T such that z = N T + z00, implying then that 0 2 manifolds in M other than M1 and M1, since starting from 0 z00 any point in M/M1, θ(t) converges to M1. ∂u Z ∂∆ 0 (wp, vp, z) = (wp, vp, τ)dτ. Second, it remains to study the stability of M1 and M1 for ∂wp 0 ∂wp different values of α since they are the only positively invariant 13

2 θ1−θ2 2 manifolds in M. The Jacobian matrix of (39) evaluated at the where sin ( 2 ) = 1 − r and the trigonometric identity > x = (x1, x2) is cos β1 cos β2 = (cos(β1 − β2) + cos(β1 + β2))/2 have been used to obtain the second equality. We further observe that hcos (x1 + α) + cos (x1 − α) cos (x2 + α) i J(x) = −A cos (x + α) cos (x + α) + cos (x − α) . 2 1 2 2 X √ cos(θ0 − θj − α) If α < arctan( 3), all the eigenvalues of J(c(α)) are j=1 M 2 2 negative, which proves that 1 is exponentially stable; on X X the other hand, J(π + c(α)) has a positive eigenvalue, which = cos α cos(θ0 − θj) + sin α sin(θ0 − θj). 0 j=1 j=1 means M1 is unstable. Then, there is a unique exponentially stable remote synchronization manifold, that is M . Following 1 P2 0 From (41), there hold that cos(θ0 − θj) = 2z1 and similar lines,√ one can show both M1 and M1 are unstable if j=1 P2 sin(θ − θ ) = 2z . It then follows from (45) and (46) α > arctan( 3), which proves 2). This implies the remote√ j=1 0 j 2 synchronization manifold M is unstable if α > arctan( 3). that 2 X cos(θ0 − θj − α) j=1 E. Proposition 4 = 2r cos α cos ζ + 2r sin α sin ζ = 2r cos(ζ − α). (72)

Proposition 4: The dynamics of µ(t) in (49) and ζ(t) in (48) Substituting (72) into (71) one obtains are given by (50a) and (50b), respectively. 2 dµ(t) Proof: We observe that (cos(θ0 − θ1) + cos(θ0 − θ2)) + = −A(1 − r2) cos(ζ − α), 2 (sin(θ0−θ1)+sin(θ0−θ2)) = 2+2 cos(θ0−θ1) cos(θ0−θ2)+ dt 2 sin(θ0 − θ1) sin(θ0 − θ2). Using the trigonometric identity which is nothing but (50a) since r = 1 − µ. cos β1 cos β2 + sin β1 sin β2 = cos(β1 − β2) one obtains that 2 2 We next derive the time derivative of ζ(t) given in (50b). (cos(θ0 −θ1)+cos(θ0 −θ2)) +(sin(θ0 −θ1)+sin(θ0 −θ2)) = It holds that ζ = arctan(z2/z1), and then the time derivative 2+2 cos(θ1 −θ2). Substituting this equality into (47) we have of ζ satisﬁes r dζ(t) 1 1p 2 θ1 − θ2 r = 2 + 2 cos(θ1 − θ2) = cos , (69) = 2 2 (z1z˙2 − z2z˙1). (73) 2 2 dt z1 + z2 where the second equality has used the trigonometric identity The time derivatives of z1 and z2 can be calculated using (41), cos 2β = 2 cos2 β−1. The time derivative of µ(t) then satisﬁes and it then follows that

dµ(t) dr(t) z1z˙2 − z2z˙1 = − dt dt 2 2 1 X X ˙ ˙ ˙ ˙ = cos(θ0 − θj) cos(θ0 − θj) · (θ0 − θj) 1 θ1 − θ2 θ1 − θ2 θ1 − θ2 4 = q 2 cos( ) sin( ) j=1 j=1 2 θ1−θ2 2 2 2 2 cos 2 2 2 1 X X ˙ ˙ 1 θ − θ θ − θ + sin(θ0 − θj) sin(θ0 − θj) · (θ0 − θj) = sin( 1 2 ) cos( 1 2 )(θ˙ − θ˙ ). (70) 4 2r 2 2 1 2 j=1 j=1 2 2 1 X 1 X It follows from (37a) that = (θ˙ − θ˙ ) + cos(θ − θ ) cos(θ − θ ) 4 0 j 4 0 j 0 −j j=1 j=1 ˙ ˙ θ1 − θ2 = A sin(θ0 − θ1 − α) − A sin(θ0 − θ2 − α) ˙ ˙ + sin(θ0 − θj) sin(θ0 − θ−j) · (θ0 − θ−j), θ − θ θ + θ = −2A sin( 1 2 ) cos(θ − 1 2 − α), 2 0 2 where −j is deﬁned in a way so that −j = 2 if j = 1, and −j = 1 otherwise. By using the trigonometric identity where the second equality has used the trigonometric identity cos β cos β + sin β sin β = cos(β − β ), we have β1−β2 β1+β2 1 2 1 2 1 2 sin β1 − sin β2 = 2 sin( 2 ) cos( 2 ). Substituting this ˙ ˙ 2 2 expression of θ1 − θ2 into (70) yields 1 X 1 X z z˙ − z z˙ = (θ˙ − θ˙ ) + cos(θ − θ ) (θ˙ − θ˙ ) 1 2 2 1 4 0 j 4 1 2 0 j dµ(t) A θ − θ j=1 j=1 = − sin2( 1 2 ) dt r 2 2 1 θ1 − θ2 X θ − θ θ + θ = cos2 (θ˙ − θ˙ ) × cos( 1 2 ) cos(θ − 1 2 − α) 2 2 0 j 2 0 2 j=1 2 2 A X 1 X = − (1 − r2) cos(θ − θ − α), (71) = r2 (θ˙ − θ˙ ), 2r 0 j 2 0 j j=1 j=1 14

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