科技部補助專題研究計畫成果報告
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耦合系統的全局動態行為之研究及其在生物模型上的應用
計 畫 類 別 ： 個別型計畫 計 畫 編 號 ： MOST 102-2115-M-004-004- 執 行 期 間 ： 102 年 08 月 01 日至 103 年 10 月 30 日 執 行 單 位 ： 國立政治大學應用數學學系 計 畫 主 持 人 ： 曾睿彬 計畫參與人員： 碩士班研究生-兼任助理人員：王柏堯 大專生-兼任助理人員：鄭喨元 報 告 附 件 ： 出席國際會議研究心得報告及發表論文 處 理 方 式 ： 1.公開資訊：本計畫涉及專利或其他智慧財產權，2 年後可公開查詢 2.「本研究」是否已有嚴重損及公共利益之發現：否 3.「本報告」是否建議提供政府單位施政參考：否中 華 民 國 104 年 01 月 29 日
中 文 摘 要 ： 在這近幾十年來，耦合系統的同步化行為與多重穩定性已經 成為相當重要的研究課題。在現有文獻中，用來處理線性耦 合系統同步化問題的方法往往依賴於特定的耦合形式；也因 此它們的應用往往受到了限制。現有處理同步問題的方法大 多要求耦合矩陣是與時間無關的、或對稱的，或者要求耦合 矩陣之行的總和須為零、或其非對角線元素必須為非負、或 其所有非零特徵值需具有負實部、或滿足節點的平衡等等。 在這個研究中，我們發展出一套可以處理具更一般耦合矩陣 形式之耦合系統的同步化方法。另外在這個研究中，我們也 發展一個可處理具多重穩定平衡點之神經網路的全局收斂性 的方法；此方法可適用於具平滑的 S 形(sigmoidal)耦合函數 或分段線性耦合函數。經由此方法，我們可推導了具各種不 同平衡點個數的條件，並研究系統的收斂性。 中文關鍵詞： 耦合系統、耦合矩陣、同步化、多重穩定性、收斂性、平衡 點
英 文 摘 要 ： Synchronization and multistability of coupled systems have been important research topics in recent
decades. In the literature, much of the existing methods for the synchronization of coupled systems strong rely on specific forms of the coupling structure； their applications are consequently limited. Most of the existing approaches to the synchronization problems require the connectivity matrix to be time-independent, symmetric, with zero row-sums, with nonnegative off-diagonal entries, with all nonzero eigenvalues having negative real part, or with node balance, etc. In this project, we develope an approach to the synchronization of a network of coupled oscillators under which the connection matrix could be quite general. Moreover, we also develope a new approach to conclude the global convergence to multiple equilibrium points of the neural networks. This approach accommodates both smooth sigmoidal and piecewise linear activation functions. Based on this approach, we derive several criteria which lead to disparate numbers of equilibria, and investigate the convergence of the systems
英文關鍵詞： Coupled systems, Connection matrix, Synchronization, miltistability, convergence, equilibrium point
前言
在近幾十年來，一群耦合動態系統所組成複雜網路已被廣泛地利用來模擬許多在科 學領域、工程領域與社會領域等領域的複雜系統。因此關於耦合動態系統的研究已 受到了相當多的關注, 其中一些重要的問題包含耦合系統的同步化、耦合系統之多 平衡點的存在性與穩定性。在這個研究計劃中，我們進行了耦合系統的全局動態分 析。其中的工作包括了一個具時間延遲之Hopfield型式神經網路的多平衡點之存在 性及其穩定性分析；我們對此系統平衡點之存在性做了多樣性的分類並研究其平衡 點的穩定性，同時我們也研究系統的全局收斂性。另外，我們也研究了一個線性耦 合系統的全局同步化行為，在我們的同步化理論下，此耦合系統的耦合結構可以是 非常的一般化。 研究目的與文獻探討Complex networks of coupled dynamical systems have been widely exploited to model many complex systems in sciences, engineering and society, and have attracted much attention in recent decades. Synchronization is a crucial and common phenomenon in various biological and physical systems. Therefore, synchronization in coupled dynamical systems has been a topic of continuous interest [1, 2, 3, 4, 5, 6, 7, 8]; in particular, synchronization of chaotic systems has been an important research topic in mathematics, physics, and engineering [3, 9, 10, 11, 12]. Among the existing synchronization researches in coupled systems, some conclude local synchronization which is concerned with the stability of synchronization manifold or solution behavior in a neighborhood of certain synchronous solution, while others obtain global synchronization by showing that all solutions converge to the synchronization manifold or some synchronous solution. The master stability function, developed by Pecora and Carroll [3, 4], was a well-known approach to studying local synchronization of coupled chaotic systems. This method is based on computing the eigenvalues of the connectivity matrix and the Lyapunov exponent of the associated variational equation to determine the stability of the synchronization manifold for the coupled systems. Methodologies for concluding global synchronization largely involve the notion of Lyapunov functions. For example, Belykh, Belykh, and Hasler developed the “connection graph stability method” combined with the Lyapunov function approach to studying global synchronization in symmetrically [12] and asymmetrically [13, 14] coupled networks of chaotic systems. From the viewpoint of feedback control, Nijmeijer and collaborators introduced the notion of passivity and semipassivity and constructed a Lyapunov-Razumikhin function to study global synchronization in coupled systems [15, 16, 17]. Other works employing the Lyapunov function/functional technique include [18, 19, 20, 21, 22, 23]. Actually, much of the existing methods for the study of synchronization of coupled systems strong rely on specific form of the coupling structure; their applications are consequently limited. For instance, the methods which are based on the eigenvalues of the connection matrix and the Lyapunov exponents may fail to work if the connection matrix is time-dependent since the
stability theory may be not valid for the corresponding linearized systems. Some other restrictions include requiring the connectivity matrix need to be symmetric [12], with zero row-sums [12, 13, 14, 18, 24], with nonnegative off-diagonal entries [12, 13, 14, 18], with all nonzero eigenvalues having negative real part [18, 24], or with node balance (with both zero row and column sums) [13], etc. Therefore, in this project, we want to develop a approach to the synchronization of a network of chaotic oscillators under which the connection matrix could be general (time-dependent, asymmetric, without zero row-sums, with negative off-diagonal entries, etc).
Multistability is a notion to describe the coexistence of multiple stable equilibria or cycles. Such dynamics is essential in several applications of neural networks, including pattern recognition and associative memory storage. Hopfield-type neural networks, with or without consideration of transmission delays, have been the primary models that attract a great deal of research interests on multistability. Multistability for Hopfield-type neural networks has been studied in several papers. The existence of 3n equilibria for n-coupled
Hopfield-type neural networks was established in [27, 28]. Therein, it was also shown that the 2 equilibria out of those n 3n equilibria are stable, and each of them is located in a
positively invariant region. Later, different criteria for stability was obtained in [29] by using the Lyapunov functional and matrix inequality techniques. There are several works that are strongly restricted to the class of piecewise linear activation functions, cf. [30,31,32]. Convergence of dynamics (also called complete stability) which means that every solution of the system converges to one of the equilibrium points, is a key ingredient in multistability. Quasi-convergence of dynamics for n-coupled Hopfield-type neural networks was established in [28]. It indicates that almost every orbit converges to one of the 3n equilibria, and 2n among these 3n equilibria are stable, under a
delay-dependent criterion. Chua & Roska (1990) have demonstrated that if the interconnection matrix is irreducible, and the neuron activations are modeled by sigmoidal functions, then the solution flow generated by a cooperative cellular neural network without delay is eventually strongly monotone. According to the standard theory of cooperative dynamical systems, the flow enjoys the so-called limit set dichotomy, and generically, the solution converges to the set of equilibria.Di Marco et al. extended the limit set dichotomy to Hopfield-type neural networks with piecewise-linear activation functions, nonsymmetric cooperative interconnection matrix, and without delays [33], and with delay [34]. There were other studies on complete stability, which consider only piecewise-linear activation functions, such as [35,36]. Actually, almost all the existing results on the number of multiple equilibria of neural networks are in terms of n-power of the number of saturated (or near-saturated) regions, in a n-neuron system. In our research, we plan to derive the criteria under which the numbers of equilibria are not in power of n. These new multistability scenarios demonstrates the strength of the our methodology, as they are inaccessible by other treatments.
研究方法、結果、討論與計畫成果自評
究中，我們改良[26]的方法並將其運用在藕合混沌系統上，進而得到在一般形式的 藕合矩陣(無需 circular coupling 條件)之下藕合混沌系統的全局同步化理論。在 我們的同步化理論之下，耦合矩陣是可與時間有關的具無須對稱，耦合矩陣之行的 總和無須為零，耦合矩陣之非對角線元素可為負值等等。另一方面，我們利用並改 良了[27,28,37,38]的方法來研究具時間延遲之 Hopfield 型式神經網路的多平衡點 之存在性及其穩定性；我們對此系統平衡點之存在性做了多樣性的分類並研究其平 衡點的穩定性，同時我們也研究系統的全局收斂性。 在這個研究工作中，我們所完成的兩個主要工作如下：
a. Multistability in Delayed Neural Networks via Sequential Contracting. (joint with Chang-Yuan Cheng, Kuang-Hui Lin, Chih-Wen Shih)
b. Synchronizations for Networks of Coupled Oscillators. (joint with Chih-Wen Shih)
其中 Multistability in Delayed Neural Networks via Sequential Contracting 這份成果已
完成論文撰寫並投至IEEE TNNLS，目前正在審查中。
In the former project, we develop a novel approach to establishing the global synchronization of a network of linearly coupled systems. Under this framework, the coupling configuration of the coupled systems can be quite general. The connection matrices are free from the commonly imposed conditions and can be time-dependent, asymmetric, with nonzero row-sums or non-positive off-diagonal entries. We apply the present approach to study the global synchronization of coupled Lorenz equations. We first establish the dissipative property of coupled Lorenz equations. From this property, we derive a criterion for the global synchronization of Lorenz equations under general coupling scheme. The criterion can be expressed transparently and examined by straightforward computations. For non-diffusively coupled Lorenz equations,
we show that the chaotic behavior can emerge, or conversely, be suppressed, as the coupled equations synchronize under the synchronization criterion.
In the latter project, we explore a variety of new multistability scenarios in the general delayed neural network system. Geometric structure imbedded in the equations is exploited and incorporated into the analysis to elucidate the underlying dynamics. Criteria derived from different geometric configurations lead to disparate numbers of equilibria. A new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system. The formulation accommodates both smooth sigmoidal and piecewise linear activation functions. Several numerical examples illustrate the present analytic theory.
Reference:
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[17] E. Steur, H. Nijmeijer, IEEE Trans. Circ. Syst. I, 58(6) (2011) 1358-1371. [18] Wenlian Lu, Tianping Chen, Phys. D 213 (2006) 214-230 .
[19] T. Chen, Z. Zhu, Int. J. Bifurc. Chaos 17 (2007) 999-1005.
[20] M. Porfiri and R. Pigliacampo, SIAM Appl. Dynam. Systems, 7 (3) (2008), 825-842. [21] M. Porfiri and R. Pigliacampo, Chaos Solitons Fractals, 41(1) (2009), 245-262. [22] K. Xiao and S. Guo, Math. Methods Appl. Sci., 33(7) (2010),892-903.
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[26] Chih-Wen Shih, Jui-Pin Tseng, SIAM Appl. Dynam. Systems, 12(3) (2013) 1354-1393.
[27]C.-Y. Cheng, K.-H. Lin, and C.-W. Shih (2006). Multistability in recurrent neural networks. SIAM J. Appl. Math., 66(4), 1301-1320.
[28] C.-Y. Cheng, K.-H. Lin, and C.-W. Shih (2007). Multistability and convergence in delayed neural networks. Phys. D: Nonlin. Phenomena, 225(1), 61-74.
[29]G. Huang, and J. Cao (2010). Delay-dependent multistability in recurrent neural networks. Neural Netw., 23(2), 201-209.
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Synchronizations for Networks of
Coupled Oscillators
Chih-Wen Shih
∗Jui-Pin Tseng
†January 29, 2015
Abstract
This investigation presents a novel approach to establishing the global synchronization of a network of linearly coupled systems. Under this frame-work, the coupling configuration of the coupled systems can be quite general. The connection matrices are free from the commonly imposed conditions and can be time-dependent, asymmetric, with nonzero row-sums or non-positive off-diagonal entries. We apply the present approach to study the global syn-chronization of coupled Lorenz equations. We first establish the dissipative property of coupled Lorenz equations. From this property, we derive a cri-terion for the global synchronization of Lorenz equations under general cou-pling scheme. The criterion can be expressed transparently and examined by straightforward computations.
Keywords: Coupled system; Synchronization; Lorenz equation
1
Introduction
Synchronization is a common phenomenon and crucial mechanism in various bi-ological and physical systems. Therefore, synchronization in coupled dynamical systems has been a topic of wide-ranging and continuous interests, see the books by [29, 41, 42]. In particular, synchronization of chaotic systems has attracted a great deal of research interests which has led to fruitful applications in physics and engineering [1, 7, 8, 9, 51, 13, 14, 16, 17, 18, 19, 22, 24, 27, 28, 31, 32, 36, 38, 45, 52]. Various synchronization notions and scenarios have been studied in the past decades,
∗Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 30010 ([email protected])
†Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan 11605 (Corresponding author, [email protected]. Tel.: +886-2-2938-7046; fax: +886-2-2938-7905.)
including complete or identical synchronization [12, 26], phase synchronization [33], lag synchronization [34], generalized synchronization [25, 35], and almost synchro-nization [11]. The complete identical synchrosynchro-nization was the first discovered and is the simplest form of synchronization in chaotic systems. This phenomena was first shown to take place when two identical chaotic systems are coupled unidirectionally, provided that the conditional Lyapunov exponents for the subsystems are all neg-ative [26]. For more details about chaotic synchronization, the readers are referred to the report [4].
Among synchronization studies in coupled systems, some concluded local syn-chronization which is concerned with the stability of synchronous manifold or so-lution behavior in a neighborhood of certain synchronous soso-lution, while others obtained global synchronization by showing that all solutions converge to the syn-chronous manifold or some synsyn-chronous solution. The master stability function, developed by Pecora and Carroll [26, 28], is a well-known approach to studying lo-cal synchronization of coupled chaotic systems. This method is based on computing the eigenvalues of the connectivity matrix and the Lyapunov exponents of the asso-ciated variational equation to determine the stability of the synchronous manifold for the coupled systems. These Lyapunov exponents are usually called conditional Lyapunov exponents if considering drive-response systems; alternatively, they are called transverse Lyapunov exponents. However, the calculation of Lyapunov ex-ponents cannot be made analytical and thus requires numerical operation [15, 44]. In addition, it is well known that the negativity of the largest transverse Lyapunov exponent is a necessary condition for local synchronization.
Methodologies for concluding global synchronization largely involve the notion of Lyapunov functions. For example, Belykh et al. developed the connection-graph-stability method combined with the Lyapunov function approach to study global synchronization in symmetrically coupled networks [1] and asymmetrically coupled networks [2, 3] of dynamical systems. From the viewpoint of feedback control, Nijmeijer and collaborators introduced the notion of passivity and semipassivity and constructed a Lyapunov-Razumikhin function to study global synchronization in coupled systems [30, 39, 40]. Lyapunov’s direct method has also been applied to study the synchronization in networks [46, 47, 48, 49]. The other works employing the Lyapunov function/functional technique include [5, 23, 31, 32, 50, 53].
In this investigation, we shall employ a new approach disparate from Lya-punov method to investigate synchronization in general linearly coupled systems.
We consider the following N coupled oscillators with time-dependent coupling: ˙xi(t) = F(xi(t), t) + c N X j=1 aij(t)Dxj(t), for i ∈ N , t ≥ t0, (1.1)
where N := {1, 2, . . . , N }, xi(t) = (xi,1(t), xi,2(t) . . . , xi,K(t)) ∈ RK, i ∈ N ,
rep-resents the state of the ith oscillator at time t, F = (F1, F2, . . . , FK) is a smooth
function which depicts the intrinsic dynamics of each oscillator, and c ∈ R is the coupling strength. Each function aij(t) is bounded and denotes the coupling
coeffi-cient from oscillator j to oscillator i; moreover, a signal sent from oscillator j reaches oscillator i at time t if aij(t) 6= 0, whereas no signal is received at oscillator i from
oscillator j at time t if aij(t) = 0. The time-dependent matrix A(t) := [aij(t)]N ×N
refers to the outer-coupling (or connection) matrix and the K × K matrix D rep-resents the inner-coupling matrix. The connection matrix is time-independent if A(t) = A = [aij] is a constant matrix. System (1.1) is a system of linearly coupled
ordinary differential equations and has been widely adopted to model various phe-nomena and processes in nature and engineering [10, 20]. In particular, it has been largely adopted to study the synchronization of chaotic oscillators in the literature [1, 2, 3, 4, 6, 15, 16, 19, 22, 23, 24, 30, 39, 46, 47, 48, 52]. System (1.1) is said to attain global (identical) synchronization if
xi,k(t) − xj,k(t) → 0, as t → ∞, for all i, j ∈ N , k ∈ K
for every solution (x1(t), x2(t), . . . , xN(t)) of (1.1), where xi(t) = (xi,1(t), xi,2(t), . . . , xi,K(t)),
K := {1, 2, . . . , K}.
As mentioned above, Lyapunov-function-based methods have been employed extensively to study global synchronization of coupled systems in the literature. However, applications of these methods to system (1.1) bear certain constraints. First, those methods commonly rely strongly on specific forms of the coupling struc-ture (connection matrix). For instance, those methods which rely on manipulation of eigenvalues of the connection matrix and the Lyapunov exponents may fail to re-main effective if the connection matrix is time-dependent, as the stability theory may be invalid for the corresponding linearized systems. However, in many real-world networks, such as biological and social networks, the network topology can evolve over time. In addition, some of those methods require the connection matrix to be symmetric (aij = aji for all i, j ∈ N ) [1, 39], or have zero row-sums (Pj∈Naij = 0
for all i, j ∈ N and i 6= j) [1, 2, 3, 23, 39, 49], or all eigenvalues nonzero with negative real part [15, 16, 23], or node balance (with both zero row and column sums) (i.e. P
j∈Naij =
P
j∈N aji = 0 for all i ∈ N ) [2], or be irreducible [52]. Regarding the
applications of those methods in concluding synchronization, one other restriction is about the complication in verifying the synchronization criteria. To the best of our knowledge, some of those criteria are actually difficult to examine in applications. In some work, numerical examples are not provided to demonstrate the effectiveness of their synchronization theories. In some other works, examples are given without verifying the criteria to support their synchronization theories. The difficulty in ver-ifying those synchronization criteria arises naturally as accompanying the operation of mathematical approach. It may also arise due to the lack of information on the dissipative property of the coupled systems.
The dissipative property is concerned with the existence of a compact region (called attracting region) to which every solution of the coupled system converges. A rigorous and explicitly elucidated criterion of global synchronization for a coupled system such as (1.1) should depend not only on the contents of intrinsic dynamics and coupling terms but also on the attracting region for the system [30, 39]. Indeed, the dissipative property of coupled systems is a nontrivial, even challenging task. In some studies on global synchronization problems, the dissipative property of coupled systems was neglected in spite of its importance in assuring the global existence of solutions of the systems. Moreover, lacking information on the attracting region results in insufficiently transparent description of synchronization criterion in terms of system parameters and obstructs practical application of the synchronization theory.
Recently, a novel approach, named sequential contracting, to investigating the identical synchronization of coupled systems was developed in [37]. Therein, a gen-eral framework for establishing synchronization of N -cell systems under circulant coupling was presented. Applications of this framework to neuronal models, neural networks, and gene regulatory networks were reported therein. In this paper, we shall employ and improve the approach developed in [37] to study the synchroniza-tion of networks of coupled oscillators and pay special attensynchroniza-tion to chaotic oscillators (1.1) under general linear coupling. In particular, the circulant coupling structure is no longer necessary in the present work.
Now, let us introduce the assumption on the connection matrix in system (1.1):
Condition (S): There exists α(t) such that P
j∈Naij(t) = α(t) for every i ∈ N and
t ≥ t0.
This assumption on the connection matrix A(t) is quite weak, even the weakest in the literature, to the best of our knowledge. Under such condition, the connection matrix A(t) can be time-dependent, without node balance, reducible, or have non-zero row-sums, negative off-diagonal entries, some nonnon-zero eigenvalues with positive real parts. Moreover, it can be asymmetric, and thus the coupling between any two systems may be either absent, unidirectional, or bidirectional with not necessarily equal coupling coefficients for both directions.
Notably, under condition (S), every solution of system (1.1) evolved from the synchronous set {(x1, x2, . . . , xN) : x1 = x2 = · · · = xN ∈ RK} will remain on the
set, and thus synchronous, for all t ≥ t0. Notably, the “diffusive condition”
X
j∈N
aij(t) = 0, for every i ∈ N , and t ≥ t0 (1.2)
which is a commonly adopted assumption in the literature, is a special case satisfying condition (S). Another example satisfying condition (S) is the so-called circulant coupling [37], that is,
A(t) = [aij(t)]N ×N = circ(a1(t), a2(t), . . . , aN(t)), (1.3)
a circulant matrix.
We shall present the synchronization theory for system (1.1) in Section 2. As an illustration and application, we discuss synchronization of the coupled Lorenz equations in Section 3. We choose such an application, as coupled Lorenz equations is a benchmark for chaotic synchronization and has received extensive studies. We first establish the dissipative property for the coupled Lorenz equations, which leads to an estimation on the attracting region for the coupled Lorenz equations. From this estimation, we derive a criterion for global synchronization of the coupled Lorenz equations.
2
Main result
For simplicity of presentation, we shall consider the inner coupling matrix to be diagonal, i.e., D = diag(d1, d2, · · · , dK), in this investigation. Accordingly, system
(2.1) can be written in the following component form: ˙xi,k(t) = Fk(xi(t), t) + cdk N X j=1 aij(t)xj,k(t), for i ∈ N , k ∈ K, t ≥ t0. (2.1)
Our approach is still valid for the general matrix D. To investigate the synchro-nization of system (2.1), we further make two assumptions on (2.1). The first one is related to the dissipative property of the whole coupled systems:
Condition (D): All solutions of system (2.1) converge to some compact set QN := Q × Q × · · · × Q, where Q := [ˇq
1, ˆq1] × [ˇq2, ˆq2] × · · · × [ˇqK, ˆqK] ⊂ RK.
Herein, we say that a function y(t) converges to compact set Ω if d(y(t), Ω) := inf{ky(t) − ζk : ζ ∈ Ω} → 0, as t → ∞. The second assumption is concerned with the intrinsic dynamics determined by F = (F1, F2, . . . , FK). We first decompose
functions Fk(Ξ1, t) − Fk(Ξ2, t), k ∈ K, as follows:
Fk(Ξ1, t) − Fk(Ξ2, t) = ¯hk(ξ1,k, ξ2,k, t) + ¯wk(Ξ1, Ξ2, t), (2.2)
for some functions ¯hk and ¯wk, where Ξi = (ξi,1, ξi,2, . . . , ξi,K) ∈ RK, i = 1, 2. Such
a decomposition is always achievable since a trivial choice is ¯hk ≡ 0. We hope that
such decomposition admits a structure depicted in the following condition:
Condition (H): For each k ∈ K, there exist ˇ`k, ˆ`k ∈ R and ¯`kl ≥ 0 for every
l ∈ K − {k}, such that the following two properties hold for all (Ξ1, Ξ2) ∈ Q × Q,
and all t ≥ t0: (H-i): ˇ `k ≤ ¯hk(ξ1,k, ξ2,k, t)/[ξ1,k − ξ2,k] ≤ ˆ`k if ξ1,k − ξ2,k 6= 0, ¯ hk(ξ1,k, ξ2,k, t) = 0 if ξ1,k − ξ2,k = 0, (H-ii): | ¯wk(Ξ1, Ξ2, t)| ≤ P l∈K−{k}`¯kl|ξ1,l− ξ2,l|.
where Ξi = (ξi,1, ξi,2, · · · , ξi,K) ∈ RK, i = 1, 2. Herein, Q is defined in condition (D).
The spirit of making a decomposing (2.2) satisfying condition (H) is to collect the key terms in Fk(Ξ1, t) − Fk(Ξ2, t), and lump the remaining terms. Conditions (H-i)
and (H-ii) can then be anticipated to hold when (Ξ1, Ξ2) is restricted to Q × Q, after
making certain estimates.
To introduce the main synchronization result, we consider the linear system:
Mv = 0, (2.3)
where M := DM − LM − UM = [M(k˜k)]1≤k,˜k≤K, and DM, −LM and −UM
repre-sent the diagonal, strictly lower-triangular and strictly upper-triangular parts of M, respectively; M(k˜k), k, ˜k ∈ K, are blocks of (N − 1) × (N − 1) matrices defined by
M(k˜k) := (
−¯`k˜kI if k, ˜k ∈ K and k 6= ˜k, [m(kk)ij ]1≤i,j≤N −1 if k = ˜k ∈ K,
where I is the (N − 1) × (N − 1) identity matrix, and m(kk)ij are defined by
m(kk)ij := −ˆµ`(i,k)˜ if i = j ∈ N − {N },
−cdkα¯ij if i, j ∈ N − {N } and i 6= j.
Theorem 2.1. Assume that ˆµ`(i,k)˜ < 0 for every (i, k) ∈ A, and conditions (S), (D),
and (H) hold. Then system (2.1) achieves global synchronization if the Gauss-Seidel iteration for linear system (2.3) converges to zero, the unique solution of (2.3), or equivalently,
λsyn:= max
1≤σ≤K×(N −1){|λσ| : λσ : eigenvalue of (DM− LM) −1
UM} < 1. (2.4)
3
Application to coupled Lorenz equations
Lorenz equations, proposed by Edward Lorenz in 1963, depict a simplified convection rolls in the atmosphere [21]. The existence of strange attractor was justified in 1999 [43]. It is well known that Lorenz equations
˙x1 = σ(x2− x1) ˙x2 = x1(ρ − x3) − x2 ˙x3 = x1x2− βx3 (3.1)
exhibits chaotic behavior when
σ = 10, β = 8/3, ρ = 28. (3.2)
The synchronization of coupled Lorenz equations has been investigated extensively, see [1, 2, 6, 7, 13, 15, 16, 18, 19, 22, 30, 52] and the references therein. In this section, based on Theorem 2.1, we shall investigate synchronization of the following coupled Lorenz equations:
˙xi,1(t) = σ(xi,2(t) − xi,1(t)) + cd1
PN
j=1aij(t)xj,1(t)
˙xi,2(t) = xi,1(t)(ρ − xi,3(t)) − xi,2(t) + cd2PNj=1aij(t)xj,2(t)
˙xi,3(t) = xi,1(t)xi,2(t) − βxi,3(t) + cd3
PN
j=1aij(t)xj,3(t),
where i ∈ N , σ, ρ, β, d1, d2, d3 ≥ 0, and aij(t) satisfy condition (S). Note that the
case with dl = 0 for some l = 1, 2, 3 refers to partial-component coupling in (3.3).
System (3.3) is in the form of (2.1) with F = (F1, F2, F3) : R3 → R3 given by
F1(Ξ) = σ(ξ2− ξ1), (3.4)
F2(Ξ) = ξ1(ρ − ξ3) − ξ2, (3.5)
F3(Ξ) = ξ1ξ2− βξ3, (3.6)
where Ξ = (ξ1, ξ2, ξ3). In addition, we assume that there exists a ¯λmax such that
¯
λmax≥ sup{λmax(t) : t ≥ t0} (3.7)
where
λmax(t) := max
1≤i≤N{λi(t) : λi(t) is an eigenvalue of [A(t) + A T
(t)]/2}. Proposition 3.1. Consider system (3.3) and assume that
σ − cd1λ¯max> 0, 1 − cd2λ¯max> 0, and β − cd3λ¯max> 0, (3.8)
and there exist γ1 and γ2 such that
γ1 ≤ β − cd3 N X i=1 aij(t) ≤ γ2, for all j ∈ N , t ≥ t0, (3.9) and 2(β − cd3λ¯max) ≤ γ2+ √ N ¯γ, (3.10) where ¯γ := max{|γ1|, |γ2|}. Then every solution of system (3.3) converges to
˜ QN := ˜Q × ˜Q × · · · × ˜Q (3.11) where ˜Q := [ ˇR1, ˆR1] × [ ˇR2, ˆR2] × [ ˇR3, ˆR3] ⊂ R3 with ˆ Ri = − ˇRi := p N [(¯σ1)2+ (¯σ2)2+ (max{|ˆσ3− (σ + ρ)|, |ˇσ3 − (σ + ρ)|})2], i = 1, 2, ˆ R3 := p N [(¯σ1)2+ (¯σ2)2+ (max{|ˆσ3− (σ + ρ)|, |ˇσ3− (σ + ρ)|})2] + σ + ρ, ˇ R3 := − p N [(¯σ1)2+ (¯σ2)2+ (max{|ˆσ3− (σ + ρ)|, |ˇσ3 − (σ + ρ)|})2] + σ + ρ, and ¯ σ1 := (σ + ρ)¯γ q N/[(β − cd3λ¯max)(σ − cd1λ¯max)]/2, ¯ σ2 := (σ + ρ)¯γ q N/[(β − cd3λ¯max)(1 − cd2λ¯max)]/2, ˆ σ3 := (γ2+ √ N ¯γ)(σ + ρ)/[2(β − cd3λ¯max)], ˇ σ3 := (γ1− √ N ¯γ)(σ + ρ)/[2(β − cd3¯λmax)]. 8
The attracting region derived in Proposition 3.1 is dependent on the coupling strength c, and the scale N of coupled system. The following situation leads to a c-independent estimate of the attracting region. Consider the following conditions: every eigenvalue of [A(t) + AT(t)]/2 is nonpositive for all t ≥ t0, (3.12)
and
X
i∈N
aij(t) = 0 for every j ∈ N and t ≥ t0. (3.13)
Subsequently, we can choose ¯λmax = 0 and γ1 = γ2 = β. This then leads to the
following c-independent result.
Corollary 3.2. Consider system (3.3) and assume that (3.12) and (3.13) hold. Then every solution of system (3.3) converges to
˜ QN := ˜Q × ˜Q × · · · × ˜Q (3.14) where ˜Q := [ ˇR1, ˆR1] × [ ˇR2, ˆR2] × [ ˇR3, ˆR3] ⊂ R3 with ˆ Ri = − ˇRi := (σ + ρ) q N2β(1/σ + 1) + N (√N + 1)2/2, i = 1, 2, ˆ R3 := (σ + ρ)[2 + q N2β(1/σ + 1) + N (√N + 1)2]/2, ˇ R3 := (σ + ρ)[2 − q N2β(1/σ + 1) + N (√N + 1)2]/2.
Proposition 3.3. Under the conditions in Proposition 3.1 (resp., Corollary 3.2), system (3.3) satisfies condition (H) with
ˇ `j = ˆ`j = −σ, j = 1, −1, j = 2, −β, j = 3, and ¯`j,k = σ, (j, k) = (1, 2), 0, (j, k) = (1, 3), ¯ ρ, (j, k) = (2, 1), ˆ R1, (j, k) = (2, 3) or (3, 2), ˆ R2, (j, k) = (3, 1),
where ¯ρ := max{|ρ − ξ| : ξ ∈ [ ˇR3, ˆR3]} and ˆR1, ˆR2, ˇR3, and ˆR3 are defined in
Proposition 3.1 (resp., Corollary 3.2).
Theorem 3.4. Assume that condition (S) and the conditions in Proposition 3.1 hold, and ˆµ`(i,k)˜ < 0 for every (i, k) ∈ A. Then the coupled Lorenz equations (3.3)
attains global synchronization if the Gauss-Seidel iteration for linear system (2.3), with ˇ`j, ˆ`j and ¯`j,k defined in Proposition 3.3, converges to zero, the unique solution
4
Discussions
In this section, we summarize the present investigations on synchronization of cou-pled systems and make a comparison with previous studies.
The master-stability-function method is a well-known approach to investigating local synchronization of coupled systems [26, 28]. Basically, such a method strongly relies on time-invariance and the configuration of the connection matrices. In a recent work [15], the following network of N coupled oscillators is considered:
˙xi = F(xi) + c N
X
j=1
aijH(xj), i ∈ N , (4.15)
where xi ∈ Rn, i ∈ N , connection matrix A = [aij]N ×N is time-independent,
diago-nalizable and satisfies the diffusion condition (1.2), and H is the coupling function which can be linear or nonlinear. The largest Lyapunov exponent determined from the variational equation about a synchronous solution defines the master stability function. A necessary condition for local synchronization of the coupled systems (4.15) is that the master stability function takes negative values. If time-dependent connection matrix A(t) is considered, block-diagonalizing the linearized equation and the computation for the Lyapunov exponents will become rather complicated.
The connection-graph-stability method has provided some synchronization con-ditions for systems similar to (1.1). This method relies on an assumption that a quadratic function is a Lyapunov function for an associated auxiliary system. The assumption implicates that the auxiliary system can be globally stabilized. It took certain effort to justify this assumption for the Lorenz systems coupled in the first component (d1 = 1, d2 = d3 = 0) in [1], as an application of the theory therein. An
estimate of the attracting region for the coupled system is needed to determine when the quadratic function is a Lyapunov function for the associated auxiliary system in the connection-graph-stability method. The matrix measure approach was adopted to combine with the connection-graph-stability method in [6]. Therein, that the auxiliary system associated with the Lorenz equations coupled in all components can be globally stabilized was obtained by numerical simulations. In addition to the difficulty of justifying this assumption via mathematical arguments, computa-tion of the associated integrals of matrix measure of more general time-dependent connection matrices adds another challenge in such an approach. It will be even more complicated to treat the systems with time-dependent intrinsic terms by the
connection-graph-stability-based method.
As mentioned in the Introduction, the dissipative property or ultimate bound-edness of solutions for coupled system is often needed in mathematical arguments for concluding global synchronization. In fact, concluding the dissipative property or ultimate boundedness is a nontrivial and even challenging task in dynamical sys-tems. There has not been a general methodology to justifying this property and so such assertions are case-dependent in general. Locating the range of attracting re-gion for a coupled system with various parameters or time-dependent parameters is even more demanding. For example, finding an estimate of the attracting region for Lorenz equations under general coupling is a challenging task. In [52], the attracting region was estimated for (3.3) under y-component coupling (d1 = d3 = 0, d2 = 1)
with the connection matrix being irreducible, with nonnegative off-diagonal entries, and time-independent. Similar result was obtained in [19], with y-component cou-pling and irreducible time-independent connection matrix with zero-row-sum and nonnegative off-diagonal entries. Our Proposition 3.2 has advanced such estimates, as it accommodates time-dependent connection matrix and both partial-component and all-component couplings.
In [30, 39], a diffusively coupled system
˙zj = q(zj, yj)
˙
yj = a(zj, yj) + CBuj
(4.16)
was studied, where j = 1, 2, · · · , k, zj(t) ∈ Rn−m, yj(t) ∈ Rm, q, a are two functions,
C, B are constant matrices, and the coupling terms are CBuj with
uj = −γj1(yj − y1) − γj2(yj− y2) − · · · − γjk(yj − yk),
and γij = γji ≥ 0 . Ultimate boundedness of |zi(t)| and |yi(t)| is needed to
accom-pany the use of Lyapunov function arguments in establishing the global synchro-nization.
Our research goal in the investigation of synchronization is to develop a frame-work which carries less restriction on the equations, and hence accommodates wider class of coupled systems. The established framework in this paper allows the in-dividual oscillators to be non-autonomous and the connection matrix to be time-dependent and non-diffusive.
Under the present theory, the problem of establishing the global synchroniza-tion of coupled systems reduces to solving a corresponding linear algebraic equasynchroniza-tion.
Our criterion for synchronization is indeed a sufficient condition for global synchro-nization and does not rely on the existence of certain Lyapunov functions which are required in some previous studies. The present synchronization criterion can be examined through numerical linear algebraic computation, and thus is advantageous in applications. Finally, we remark that the sequential contracting can also conclude local synchronization if a positively invariant region around synchronous set for a coupled system is known a priori.
Acknowledgements. This work is partially supported by the National Science Council of Taiwan.
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科技部補助專題研究計畫出席國際學術會議心得報告
日期：103 年 9 月 11 日
一、 參加會議經過
此 次 參 加 韓 國 首 爾 所 舉 辦 的International Congress of Mathematicians(ICM2014) （ 期 間 ： 2014,8/15-8/19）。國際數學家大會（ICM）是數學界最大的會議；在國際數學聯盟（IMU）的主 持之下，它每四年舉行一次。特別地，在會議第一天的開幕式，大會會頒發Fields Medals、Nevanlinna Prize、Gauss Prize與Chern Medal得獎者。此會議所包含的議題相當的廣泛；共分成19個sections, 分別為Logic and Foundations，Algebra，Number Theory，Algebraic and Complex Geometry，Geometry， Topology，Lie Theory and Generalizations，Analysis and its Applications，Dynamical Systems and Ordinary Differential Equations, Partial Differential Equations, Mathematical Physics, Probability and Statistics, Combinatorics, Mathematical Aspects of Computer Science, Numerical Analysis and Scientific Computing, Control Theory and Optimization, Mathematics in Science and Technology, Mathematics Education and Popularization of Mathematics, History of Mathematics。大會議程主要分為 Plenary lectures，Invited Sessions lectures，Invited Panels，Short communications，Poster sessions。 在這個會議，我所參加的部分是Dynamical Systems and Ordinary Differential Equations這個section 的Poster sessions，而時間被安排於會議的第四天(8/14)。這是我第一次在國際會議中以海報的型 式報告自已的工作，當有人來提問時心中難免緊張。除了在會議的海報時間之外，依時間安排我 也聽了數個Dynamical Systems and Ordinary Differential Equations這個section的演講；同時我也去聽 了Fields Medals得主的演講，感受一下大師在安排、鋪陳演講內容的風範, 同時也想藉此了解一下 目前數學界中最受重視的研究問題是什麼？ 計畫編號 NSC 102-2115-M-153 -002 - 計畫名稱 耦合系統的全局動態行為之研究及其在生物模型上的應用 出國人員姓 名 曾睿彬 服務機構 及職稱 國立政治大學應用數學系， 助理教授 會議時間 2014 年 8 月 15 日 至 2014 年 8 月 19 日 會議地點 韓國，首爾
(Coex, Seoul, Korea)
會議名稱 (中文)
(英文) SEOUL ICM 2014 International Congress of Mathematicians 發表題目 (中文)
計畫主持人參加會議之照片 二、 與會心得 這個會議相當的大型，但會議中的各項安排，如場地的規劃與設備、各項指示設備、會 場人員的協助與服務等等都相當地完善。除了學術活動之外，大會還在一些時段特別安 排了一些韓國當地的民俗遊戲讓與會學者參與，感受韓國當地的風俗，讓人可以感受到 主辨單位的用心。整體來言，我覺得這個會議辨地相當成功。也因為這個會議相當地大 型，所討論的議題相當的廣泛，所以即使是我較熟悉的Dynamical Systems and Ordinary Differential Equations這個section的講題也大多為我較不熟悉的研究題目。雖然我在學術研 究上較難有所立即的收獲，但從中我仍可藉此接觸到一些不同於自已熟悉研究課題，增 廣自已的見聞，對我長期的研究發展上也是相當有幫助的。這次參加會議與我同行的還 有台灣師大數學系陳賢修教授。他所報告的的主題是Dynamics of a continuum
Hindmarsh-Rose type equation with recurrent neural feedback，我也用藉這次參加會議的機 會與陳教授討論了彼此的研究工作，尋求未來可能的研究合作機會。
三、 發表論文全文或摘要
The investigation presents a novel approach to establish the global synchronization of networks of linearly coupled systems. Under our framework, the coupling configuration of the coupled systems could be quite general, with the coupling matrix not assumed to be time-independent, symmetric, with zero row-sums, or with positive off-diagonal entries. We apply the present approach to study the global synchronization of coupled Lorenz equations. We first establish the dissipative property of coupled Lorenz equations. Based on the dissipative property, we derive the criterion of global synchronization for chaotic Lorenz equations under general coupling matrices; moreover, the criterion can be verified easily. The synchronization criterion could depend on the scale of the coupled equations. We shall show that certain coupled Lorenz equations can satisfy the synchronization criterion and hence achieves synchronization as the scale is small; however, the synchrony will be lost as the scale gets larger. For non-diffusively coupled Lorenz equations, we can show that chaotic behavior can emerge; conversely, that
chaotic behavior can be suppressed, as the coupled equations are synchronized under our synchronization criterion. 四、 建議 此次非常感謝國科會給我這個機會，讓我能出國見識這個國際問最大型、最受重視的數學會議。 藉這次參加會議，我認識了一些學者也對他們的研究有初步的了解，接觸到一些自已較不熟悉的 研究課題，增廣自已的見聞，覺得自己又學到了許多東西，並且瞭解自己的不足。在這麼大型的 會議中我可以更能感受到國際學術之間的交流；這之間包含了學者與學者之間的連結、知識之間 的傳遞。建議貴部未來可以多補助國內的相關學術研究單位可多多爭取舉辦大型的國際研討會， 相信這對台灣的學術發展會有很大的幫助。 五、 攜回資料名稱與內容： 與會名牌、註冊收據
科技部補助計畫衍生研發成果推廣資料表
日期:2015/01/29科技部補助計畫
計畫名稱: 耦合系統的全局動態行為之研究及其在生物模型上的應用 計畫主持人: 曾睿彬 計畫編號: 102-2115-M-004-004- 學門領域: 常微分方程無研發成果推廣資料
102 年度專題研究計畫研究成果彙整表
計畫主持人：曾睿彬 計畫編號： 102-2115-M-004-004-計畫名稱：耦合系統的全局動態行為之研究及其在生物模型上的應用 量化 成果項目 實際已達成 數（被接受 或已發表） 預期總達成 數(含實際已 達成數) 本計畫實 際貢獻百 分比 單位 備 註 （ 質 化 說 明：如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ... 等） 期刊論文 0 0 0% 研究報告/技術報告 0 0 0% 研討會論文 0 0 0% 篇 論文著作 專書 0 0 0% 申請中件數 0 0 0% 專利 已獲得件數 0 0 0% 件 件數 0 0 0% 件 技術移轉 權利金 0 0 0% 千元 碩士生 0 0 0% 博士生 0 0 0% 博士後研究員 0 0 0% 國內 參與計畫人力 （本國籍） 專任助理 0 0 0% 人次 期刊論文 0 2 100% 研究報告/技術報告 0 0 0% 研討會論文 0 0 0% 篇 論文著作 專書 0 0 0% 章/本 申請中件數 0 0 0% 專利 已獲得件數 0 0 0% 件 件數 0 0 0% 件 技術移轉 權利金 0 0 0% 千元 碩士生 0 0 0% 博士生 0 0 0% 博士後研究員 0 0 0% 國外 參與計畫人力 （外國籍） 專任助理 0 0 0% 人次其他成果
