Achieving Synchronization in Arrays of Coupled Differential Systems with Time-Varying Couplings

Abstract

We study complete synchronization of the complex dynamical networks described by linearly coupled ordinary differential equation systems (LCODEs). Here, the coupling is timevarying in both network structure and reaction dynamics. Inspired by our previous paper (Lu et al. (2007-2008)), the extended Hajnal diameter is introduced and used to measure the synchronization in a general differential system. Then we find that the Hajnal diameter of the linear system induced by the time-varying coupling matrix and the largest Lyapunov exponent of the synchronized system play the key roles in synchronization analysis of LCODEs with identity inner coupling matrix. As an application, we obtain a general sufficient condition guaranteeing directed time-varying graph to reach consensus. Example with numerical simulation is provided to show the effectiveness of the theoretical results.

1. Introduction

Complex networks have widely been used in theoretical analysis of complex systems, such as Internet, World Wide Web, communication networks, and social networks. A complex dynamical network is a large set of interconnected nodes, where each node possesses a (nonlinear) dynamical system and the interaction between nodes is described as diffusion. Among them, linearly coupled ordinary differential equation systems (LCODEs) are a large class of dynamical systems with continuous time and state.

The LCODEs are usually formulated as follows: 𝑥̇^𝑖(𝑡)=𝑓𝑥^𝑖(𝑡)+𝜎^𝑚_𝑗=1𝑙_𝑖𝑗𝐵𝑥^𝑗(𝑡),𝑖=1,2,…,𝑚, (1) where 𝑡∈ℝ⁺=[0,+∞) stands for the continuous time and 𝑥^𝑖(𝑡)∈ℝ^𝑛 denotes the variable state vector of the 𝑖th node, 𝑓∶ℝ^𝑛→ℝ^𝑛 represents the node dynamic of the uncoupled system, 𝜎∈ℝ⁺=(0,+∞) denotes coupling strength, 𝑙_𝑖𝑗≥0 with 𝑖≠𝑗 denotes the interaction between the two nodes, and 𝑙_𝑖𝑖=−∑^𝑚_𝑗≠𝑖𝑙_𝑖𝑗, 𝐵∈ℝ^𝑛,𝑛 denotes the inner coupling matrix. The LCODEs model is widely used to describe the model in nature and engineering. For example, the authors study spike-burst neural activity and the transitions to a synchronized state using a model of linearly coupled bursting neurons in [1]; the dynamics of linearly coupled Chua circuits are studied with application to image processing and many other cases in [2].

For decades, a large number of papers have focused on the dynamical behaviors of coupled systems [3–5], especially the synchronizing characteristics. The word “synchronization” comes from Greek; in this paper the concept of local complete synchronization (synchronization for simplicity) is considered (see Definition 3). For more details, we refer the readers to [6] and the references therein.

Synchronization of coupled systems have attracted a great deal of attention [7–9]. For instances, in [7], the authors considered the synchronization of a network of linearly coupled and not necessarily identical oscillators; in [8], the authors studied globally exponential synchronization for linearly coupled neural networks with time-varying delay and impulsive disturbances. Synchronization of networks with time-varying topologies was studied in [10–16]. For example, in [10], the authors proposed the global stability of total synchronization in networks with different topologies; in [16], the authors gave a result that the network will synchronize with the time-varying topology if the time-average is achieved sufficiently fast.

Synchronization of LCODEs has also been addressed in [17–19]. In [17], mathematical analysis was presented on the synchronization phenomena of LCODEs with a single coupling delay; in [18], based on geometrical analysis of the synchronization manifold, the authors proposed a novel approach to investigate the stability of the synchronization manifold of coupled oscillators; in [19], the authors proposed new conditions on synchronization of networks of linearly coupled dynamical systems with non-Lipschitz right-handsides. The great majority of research activities mentioned above all focused on static networks whose connectivity and coupling strengths are static. In many applications, the interaction between individuals may change dynamically. For example, communication links between agents may be unreliable due to disturbances and/or subject to communication range limitations.

In this paper, we consider synchronization of LCODEs with time-varying coupling. Similar to [17–19], time-varying coupling will be used to represent the interaction between individuals. In [6, 13], they showed that the Lyapunov exponents of the synchronized system and the Hajnal diameter of the variational equation play key roles in the analysis of the synchronization in the discrete-time dynamical networks. In this paper, we extend these results to the continuous-time dynamical network systems. Different from [11, 16], where synchronization of fast-switching systems was discussed, we focus on the framework of synchronization analysis with general temporal variation of network topologies. Additional contributions of this paper are that we explicitly show that (a) the largest projection Lyapunov exponent of a system is equal to the logarithm of the Hajnal diameter, and (b) the largest Lyapunov exponent of the transverse space is equal to the largest projection Lyapunov exponent under some proper conditions.

The paper is organized as follows: in Section 2, some necessary definitions, lemmas, and hypotheses are given; in Section 3, synchronization of generalized coupled differential systems is discussed; in Section 4, criteria for the synchronization of LCODEs are obtained; in Section 5, we obtain a sufficient condition ensuring directed time-varying graph reaching consensus; in Section 6, example with numerical simulation is provided to show the effectiveness of the theoretical results; the paper is concluded in Section 7.

Notions. 𝑒^𝑛_𝑘=[0,0,…,0,1,0,…,0]^𝖳∈ℝ^𝑛 denotes the 𝑛-dimensional vector with all components zero except the 𝑘th component 1, 𝟏_𝑛 denotes the 𝑛-dimensional column vector with each component 1; for a set in some Euclidean space 𝑈, 𝑈 denotes the closure of 𝑈, 𝑈^𝑐 denotes the complementary set of 𝑈, and 𝐴⧵𝐵=𝐴∩𝐵^𝑐; for 𝑢=[𝑢₁,…,𝑢_𝑛]^𝖳∈ℝ^𝑛, ‖𝑢‖ denotes some vector norm, and for any matrix 𝐴=(𝑎_𝑖𝑗)∈ℝ^𝑛,𝑚, ‖𝐴‖ denotes some matrix norm induced by vector norm, for example, ‖𝑢‖₁=∑^𝑛_𝑖=1|𝑢_𝑖| and ‖𝐴‖₁=max_𝑗∑^𝑛_𝑖=1|𝑎_𝑖𝑗|; for a matrix 𝐴=(𝑎_𝑖𝑗)∈ℝ^𝑛,𝑚, |𝐴| denotes a matrix with |𝐴|=(|𝑎_𝑖𝑗|); for a real matrix 𝐴, 𝐴^𝖳 denotes its transpose and for a complex matrix 𝐵, 𝐵^∗ denotes its conjugate transpose; for a set in some Euclidean space 𝑊, 𝒪(𝑊,𝛿)={𝑥∶dist(𝑥,𝑊)<𝛿}, where dist(𝑥,𝑊)=inf_𝑦∈𝑊‖𝑥−𝑦‖; #𝐽 denotes the cardinality of set 𝐽; ⌊𝑧⌋ denotes the floor function, that is, the largest integer not more than the real number 𝑧; ⊗ denotes the Kronecker product; for a set in some Euclidean space 𝑊, 𝑊^𝑚 denote the Cartesian product 𝑊×⋯×𝑊 (𝑚 times).

2. Preliminaries

In this section we will give some necessary definitions, lemmas, and hypotheses. Consider the following general coupled differential system: 𝑥̇^𝑖(𝑡)=𝑓^𝑖𝑥¹(𝑡),𝑥²(𝑡),…,𝑥^𝑚(𝑡),𝑡,𝑖=1,2,…,𝑚, (2) with initial state 𝑥(𝑡₀)=[𝑥¹(𝑡₀)^𝖳,…,𝑥^𝑚(𝑡₀)^𝖳]^𝖳∈ℝ^𝑛𝑚, where 𝑡₀∈ℝ⁺ denotes the initial time, 𝑡∈ℝ⁺ denotes the continuous time, and 𝑥^𝑖(𝑡)=[𝑥^𝑖₁(𝑡),…,𝑥^𝑖_𝑛(𝑡)]∈ℝ^𝑛 denotes the variable state of the 𝑖th node, 𝑖=1,2,…,𝑚.

For the functions 𝑓^𝑖∶ℝ^𝑛𝑚×ℝ⁺→ℝ^𝑛, 𝑖=1,2,…,𝑚, we make the following assumption.

Assumption 1. (a) There exists a function 𝑓∶ℝ^𝑛→ℝ^𝑛 such that 𝑓^𝑖(𝑠,𝑠,…,𝑠,𝑡)=𝑓(𝑠) for all 𝑖=1,2,…,𝑚, 𝑠∈ℝ^𝑛, and 𝑡≥0; (b) for any 𝑡≥0, 𝑓^𝑖(⋅,𝑡) is 𝐶¹-smooth for all 𝑥=[𝑥^1𝖳,…,𝑥^𝑚𝖳]^𝖳∈ℝ^𝑛𝑚, and by 𝐷𝐹^𝑡(𝑥)=((𝜕𝑓^𝑖/𝜕𝑥^𝑗)(𝑥,𝑡))^𝑚_𝑖,𝑗=1∈ℝ^{𝑛𝑚,𝑛𝑚} denotes the Jacobian matrix of 𝐹(𝑥,𝑡)=[𝑓^𝑖(𝑥,𝑡)^𝖳,…,𝑓^𝑚(𝑥,𝑡)^𝖳]^𝖳 with respect to 𝑥∈ℝ^𝑛𝑚; (c) there exists a locally bounded function 𝜙(𝑥) such that ‖𝐷𝐹^𝑡(𝑥)‖≤𝜙(𝑥) for all (𝑥,𝑡)∈ℝ^𝑛𝑚×ℝ⁺; (d) 𝐷𝐹^𝑡(𝑥) is uniformly locally Lipschitz continuous: there exists a locally bounded function 𝐾(𝑥,𝑦) such that 𝐷𝐹^𝑡(𝑥)−𝐷𝐹^𝑡𝑦≤𝐾𝑥,𝑦𝑥−𝑦 (3) for all 𝑡≥0 and 𝑥,𝑦∈ℝ^𝑛𝑚; (e) 𝑓^𝑖(𝑥,𝑡) and 𝐷𝐹^𝑡(𝑥) are both measurable for 𝑡≥0.

We say a function 𝑔(𝑦)∶ℝ^𝑞→ℝ^𝑝 is locally bounded if for any compact set 𝐾⊂ℝ^𝑞, there exists 𝑀>0 such that ‖𝑔(𝑦)‖≤𝑀 holds for all 𝑦∈𝐾.

The first item of Assumption 1 ensures that the diagonal synchronization manifold 𝒮=𝑥^1𝖳,𝑥^2𝖳,…,𝑥^𝑚𝖳𝖳∈ℝ^𝑛𝑚∶𝑥^𝑖𝖳=𝑥^𝑗𝖳,𝑖,𝑗=1,2,…,𝑚 (4) is an invariant manifold for (2).

If 𝑥¹(𝑡)=𝑥²(𝑡)=⋯=𝑥^𝑚(𝑡)=𝑠(𝑡)∈ℝ^𝑛 is the synchronized state, then the synchronized state 𝑠(𝑡) satisfies 𝑠̇(𝑡)=𝑓(𝑠(𝑡)). (5)

Since 𝑓(⋅) is 𝐶¹-smooth, then 𝑠(𝑡) can be denoted by the corresponding continuous semiflow 𝑠(𝑡)=𝜗^(𝑡)𝑠₀ of the intrinsic system (5). For 𝜗^(𝑡), we make following assumption.

Assumption 2. The system (5) has an asymptotically stable attractor: there exists a compact set 𝐴⊂𝑅^𝑛 such that (a) 𝐴 is invariant through the system (5), that is, 𝜗^(𝑡)𝐴⊂𝐴 for all 𝑡≥0; (b) there exists an open bounded neighborhood 𝑈 of 𝐴 such that ⋂_𝑡≥0𝜗^(𝑡)𝑈=𝐴; (c) 𝐴 is topologically transitive; that is, there exists 𝑠₀∈𝐴 such that 𝜔(𝑠₀), the 𝜔 limit set of the trajectory 𝜗^(𝑡)𝑠₀, is equal to 𝐴 [3].

Definition 3. Local complete synchronization (synchronization for simplicity) is defined in the sense that the set 𝒮𝐴^𝑚=𝑥^𝖳,𝑥^𝖳,…,𝑥^𝖳𝖳∈ℝ^𝑛𝑚∶𝑥^𝖳∈𝐴 (6) is an asymptotically stable attractor in ℝ^𝑛𝑚. That is, for the coupled dynamical system (2), differences between components converge to zero if the initial states are picked sufficiently near 𝒮⋂𝐴^𝑚, that is, if the components are all close to the attractor 𝐴 and if their differences are sufficiently small.

Next we give some lemmas which will be used later, and the proofs can be seen in the appendix.

Lemma 4. Under Assumption 1, one has ^𝑚_𝑗=1𝜕𝑓^𝑖𝜕𝑥^𝑗(𝑠,̂𝑡)=𝜕𝑓(𝑠),𝜕𝑠 (7) for all 𝑠∈𝑅^𝑛 and 𝑡≥0, where 𝑠̂=[𝑠^𝖳,𝑠^𝖳,…,𝑠^𝖳]^𝖳.

Lemma 5. Under Assumptions 1 and 2, there exists a compact neighborhood 𝑊 of 𝐴 such that 𝜗^(𝑡)𝑊⊂𝜗^(𝑡′)𝑊 for all 𝑡≥𝑡^′≥0 and ⋂_𝑡≥0𝜗^(𝑡)𝑊=𝐴.

Let 𝛿𝑥(𝑡)=[𝛿𝑥¹(𝑡)^𝖳,…,𝛿𝑥^𝑚(𝑡)^𝖳]^𝖳∈ℝ^𝑛𝑚, where 𝛿𝑥^𝑖(𝑡)=𝑥^𝑖(𝑡)−𝑠(𝑡)∈ℝ^𝑛. We have the following variational equation near the synchronized state 𝑠(𝑡): 𝛿𝑥̇^𝑖(𝑡)=^𝑚_𝑗=1𝜕𝑓^𝑖𝜕𝑥^𝑗(𝑠̂(𝑡),𝑡)𝛿𝑥^𝑗(𝑡),𝑖=1,2,…,𝑚, (8) or in matrix form: 𝛿𝑥̇(𝑡)=𝐷𝐹^𝑡(𝑠(𝑡))𝛿𝑥(𝑡), (9) where 𝐷𝐹^𝑡(𝑠(𝑡)) denotes the Jacobin matrix 𝐷𝐹^𝑡(𝑠(̂𝑡)) for simplicity.

From [20], we can give the results on the existence, uniqueness, and continuous dependence of (2) and (9).

Lemma 6. Under Assumption 2, each of the differential equations (2) and (9) has a unique solution which is continuously dependent on the initial condition.

Thus, the solution of the linear system (9) can be written in matrix form.

Definition 7. Solution matrix 𝑈(𝑡,𝑡₀,𝑠₀) of the system (9) is defined as follows. Let 𝑈(𝑡,𝑡₀,𝑠₀)=[𝑢¹(𝑡,𝑡₀,𝑠₀),…,𝑢^𝑛𝑚(𝑡,𝑡₀,𝑠₀)], where 𝑢^𝑘(𝑡,𝑡₀,𝑠₀) denotes the 𝑘th column and is the solution of the following Cauchy problem: 𝛿𝑥̇(𝑡)=𝐷𝐹^𝑡(𝑠(𝑡(𝑡),𝑠𝑡))𝛿𝑥₀=𝑠₀,𝛿𝑥𝑡₀=𝑒_𝑘^𝑛𝑚. (10)

Immediately, according to Lemma 6, we can conclude that the solution of the following Cauchy problem 𝛿𝑥̇(𝑡)=𝐷𝐹^𝑡(𝑠(𝑡(𝑡),𝑠𝑡))𝛿𝑥₀=𝑠₀,𝛿𝑥𝑡₀=𝛿𝑥₀ (11) can be written as 𝛿𝑥(𝑡)=𝑈(𝑡,𝑡₀,𝑠₀)𝛿𝑥₀.

We define the time-varying Jacobin matrix 𝐷𝐹^𝑡 by the following way: 𝐷ℱ∶ℝ⁺×𝑅^𝑛⟶2^{ℝ𝑛𝑚,𝑛𝑚},𝑡₀,𝑠₀⟼𝐷𝐹^𝑡(𝑠(𝑡))_𝑡≥𝑡0 (12) with 𝑠(𝑡₀)=𝑠₀, where 2^{ℝ𝑛𝑚,𝑛𝑚} is the collection of all the subsets of ℝ^{𝑛𝑚,𝑛𝑚}.

Definition 8. For a time varying system denoted by 𝐷ℱ, we can define its Hajnal diameter of the variational system (9) as follows: diam𝐷ℱ,𝑠₀=lim_𝑡→∞sup_𝑡0≥0diam𝑈𝑡,𝑡₀,𝑠₀^1/𝑡, (13) where for a ℝ^{𝑛𝑚,𝑛𝑚} matrix in block matrix form: 𝑈=(𝑈_𝑖𝑗)^𝑚_𝑖,𝑗=1 with 𝑈_𝑖𝑗∈𝑅^𝑛,𝑛, its Hajnal diameter is defined as follows: diam(𝑈)=max_𝑖,𝑗𝑈_𝑖−𝑈_𝑗, (14) where 𝑈_𝑖=[𝑈_𝑖1,𝑈_𝑖2,…,𝑈_𝑖𝑚].

Lemma 9 (Grounwell-Beesack's inequality). If function 𝑣(𝑡) satisfies the following condition: 𝑣(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑣(𝜏)𝑑𝜏, (15) where 𝑏(𝑡)≥0 and 𝑎(𝑡) are some measurable functions, then one has 𝑣(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑎(𝜏)𝑒^{∫𝑡𝜏𝑏(𝜃)𝑑𝜃}𝑑𝜏,𝑡≥0. (16)

Based on Assumption 1, for the solution matrix 𝑈, we have the following lemma.

Lemma 10. Under Assumption 1, one has the following:(1)∑^𝑚_𝑗=1𝑈_𝑖𝑗(𝑡,𝑡₀,𝑠₀)=𝑈̆(𝑡,𝑡₀,𝑠₀), where 𝑈̆(𝑡,𝑡₀,𝑠₀) denotes the solution matrix of the following Cauchy problem: 𝑢̇=𝜕𝑓(𝑠(𝑡𝑠𝑡𝜕𝑠))𝑢,₀=𝑠₀; (17)(2)for any given 𝑡≥0 and the compact set 𝑊 given in Lemma 5, 𝑈(𝑡+𝑡₀,𝑡₀,𝑠₀) is bounded for all 𝑡₀≥0 and 𝑠₀∈𝑊 and equicontinuous with respect to 𝑠₀∈𝑊.

Let 𝑃=(𝑃_𝑖𝑗)^𝑚_𝑖,𝑗=1 be a ℝ^{𝑛𝑚,𝑛𝑚} matrix with 𝑃_𝑖𝑗∈ℝ^𝑛,𝑛 satisfying (a) 𝑃_𝑖1=(1/√𝑚)𝑃₀ for some orthogonal matrix 𝑃₀∈ℝ^𝑛,𝑛 and all 𝑖=1,2,…,𝑚; (b) 𝑃 is also an orthogonal matrix in ℝ^{𝑛𝑚,𝑛𝑚}. We also write 𝑃 and its inverse 𝑃⁻¹=𝑃^𝖳 in the form 𝑃=𝑃₁,𝑃₂,𝑃^𝖳=𝑃^𝖳₁𝑃^𝖳₂, (18) where 𝑃₁=(1/√𝑚)𝟏_𝑚⊗𝑃₀ and 𝑃₂∈ℝ^{𝑛𝑚,𝑛(𝑚−1)}. According to Lemma 10, we have 𝑈𝑡,𝑡₀,𝑠₀𝑃₁=1√𝑚𝟏_𝑚⊗𝑈̆𝑡,𝑡₀,𝑠₀𝑃₀. (19) Since 𝑃^𝖳₂𝑃₁=0 which implies that each row of 𝑃^𝖳₂ is located in the subspace orthogonal to the subspace {𝟏_𝑚⊗𝜉,𝜉∈ℝ^𝑛}, we can conclude that 𝑃^𝖳₂𝑈(𝑡,𝑡₀,𝑠₀)𝑃₁=0. Then, we have 𝑃⁻¹𝑈𝑡,𝑡₀,𝑠₀𝑃𝑃=^𝖳0𝑈̆𝑡,𝑡₀,𝑠₀𝑃₀𝛼𝑡,𝑡₀,𝑠₀0𝑈𝑡,𝑡₀,𝑠₀, (20) where 𝑈̆(𝑡,𝑡₀,𝑠₀) denotes the common row sum of 𝑈(𝑡,𝑡₀,𝑠₀)=(𝑈_𝑖𝑗)^𝑚_𝑖,𝑗=1 as defined in Lemma 10, 𝑈(𝑡,𝑡₀,𝑠₀)=𝑃^𝖳₂𝑈(𝑡,𝑡₀,𝑠₀)𝑃₂∈ℝ^{𝑛(𝑚−1),𝑛(𝑚−1)}, 𝛼(𝑡,𝑡₀,𝑠₀)∈ℝ^{𝑛,𝑛(𝑚−1)} denotes a matrix, and we omit its accurate expression. One can see that 𝑈(𝑡,𝑡₀,𝑠₀) is the solution matrix of the following linear differential system.

Definition 11. We define the following linear differential system by the projection variational system of (9) along the directions 𝑃₂: 𝜙̇=𝐷_𝑃𝐹^𝑡(𝑠(𝑡𝑠𝑡))𝜙,₀=𝑠₀, (21) where 𝐷_𝑃𝐹^𝑡(𝑠(𝑡))=𝑃^𝖳₂𝐷𝐹^𝑡(𝑠(𝑡))𝑃₂.

Definition 12. For any time varying variational system 𝐷ℱ∶ℝ⁺×ℝ^𝑛→2^{ℝ𝑛𝑚,𝑛𝑚}, we define the Lyapunov exponent of the variational system (9) as follows: 𝜆𝐷ℱ,𝑢,𝑠₀=lim_𝑡→∞sup_𝑡0≥01𝑡log𝑈𝑡,𝑡₀,𝑠₀𝑢, (22) where 𝑢∈ℝ^𝑛𝑚 and 𝑠(𝑡₀)=𝑠₀.

Similarly, we can define the projection Lyapunov exponents by the following projection time-varying variation: 𝐷_𝑃ℱ∶ℝ⁺×ℝ^𝑛⟶2^{ℝ𝑛(𝑚−1),𝑛(𝑚−1)},𝑡₀,𝑠₀⟼𝐷_𝑃𝐹^𝑡(𝑠(𝑡))_𝑡≥𝑡0, (23) that is, 𝜆𝐷_𝑃ℱ,𝑢,𝑠₀=lim_𝑡→∞sup_𝑡0≥01𝑡log𝑈𝑡,𝑡₀,𝑠₀𝑢, (24) where 𝑢∈ℝ^{𝑛(𝑚−1)} and 𝑠(𝑡₀)=𝑠₀. Let 𝜆_𝑃𝐷ℱ,𝑠₀𝑢=max_{∈ℝ𝑛(𝑚−1)}𝜆𝐷_𝑃ℱ,𝑢,𝑠₀. (25)

Then, we have the following lemma.

Lemma 13. 𝜆_𝑃(𝐷ℱ,𝑠₀)=logdiam(𝐷ℱ,𝑠₀).

Remark 14. From Lemma 13, we can see that the largest projection Lyapunov exponent is independent of the choice of matrix 𝑃.

Consider the time-varying driven by some metric dynamical system MDS(Ω,ℬ,ℙ,𝜚^(𝑡)), where Ω is the compact state space, ℬ is the 𝜎-algebra, ℙ is the probability measure, and 𝜚^(𝑡) is a continuous semiflow. Then, the variational equation (9) is independent of the initial time 𝑡₀ and can be rewritten as follows: 𝜙̇=𝐷𝐹𝑠(𝑡),𝜚^(𝑡)𝜔₀𝑠(0)𝜙,=𝑠₀. (26) In this case, we denote the solution matrix, the projection solution matrix, and the solution matrix on the synchronization space by 𝑈(𝑡,𝑠₀,𝜔₀), 𝑈(𝑡,𝑠₀,𝜔₀), and 𝑈̆(𝑡,𝑠₀,𝜔₀), respectively. For simplicity, we write them as 𝑈(𝑡), 𝑈(𝑡), and 𝑈̆(𝑡), respectively. Also, we write the Lyapunov exponents and the projection Lyapunov exponent as follows: 𝜆𝐷ℱ,𝑢,𝑠₀,𝜔₀=lim_𝑡→∞1𝑡log𝑈𝑡,𝑠₀,𝜔₀𝑢,𝜆𝐷ℱ,𝑠₀,𝜔₀=max_{𝑢∈ℝ𝑛𝑚}𝜆𝐷ℱ,𝑢,𝑠₀,𝜔₀,𝜆_𝑃𝐷ℱ,𝑢,𝑠₀,𝜔₀=lim_𝑡→∞1𝑡log𝑈𝑡,𝑠₀,𝜔₀𝑢,𝜆_𝑃𝐷ℱ,𝑠₀,𝜔₀=max_{𝑢∈ℝ𝑛(𝑚−1)}𝜆𝐷_𝑃ℱ,𝑢,𝑠₀,𝜔₀. (27) We add the following assumption.

Assumption 15. (a) 𝜚^(𝑡) is a continuous semiflow; (b) 𝐷𝐹(𝑠,𝜔) is a continuous map for all (𝑠,𝜔)∈ℝ^𝑛×Ω.

The following are involving linear differential systems. For more details, we refer the readers to [21]. For a continuous scalar function 𝑢(𝑡), we denote its Lyapunov exponent by 𝜒[𝑢(𝑡)]=lim_𝑡→∞1𝑡log|𝑢(𝑡)|. (28) The following properties will be used later: (1)𝜒[∏^𝑛_𝑘=1𝑐_𝑘𝑢_𝑘(𝑡)]≤∑^𝑛_𝑘=1𝜒[𝑢_𝑘(𝑡)], where 𝑐_𝑘, 𝑘=1,2,…,𝑛, are constants;(2)if lim_𝑡→∞(1/𝑡)log|𝑢(𝑡)|=𝛼, which is finite, then 𝜒[1/(𝑢(𝑡))]=−𝛼;(3)𝜒[𝑢(𝑡)+𝑣(𝑡)]≤max{𝜒[𝑢(𝑡)],𝜒[𝑣(𝑡)]};(4)for a vector-value or matrix-value function 𝑈(𝑡), we define 𝜒[𝑈(𝑡)]=𝜒[‖𝑈(𝑡)‖].

For the following linear differential system: 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡), (29) where 𝑥(𝑡)∈ℝ^𝑛, a transformation 𝑥(𝑡)=𝐿(𝑡)𝑦(𝑡) is said to be a Lyapunov transformation if 𝐿(𝑡) satisfies(1)𝐿(𝑡)∈𝐶¹[0,+∞);(2)𝐿(𝑡), 𝐿̇(𝑡), 𝐿⁻¹(𝑡) are bounded for all 𝑡≥0.It can be seen that the class of Lyapunov transformations forms a group and the linear system for 𝑦(𝑡) should be 𝑦̇(𝑡)=𝐵(𝑡)𝑦(𝑡), (30) where 𝐵(𝑡)=𝐿⁻¹(𝑡)𝐴(𝑡)𝐿(𝑡)−𝐿⁻¹(𝑡)𝐿̇(𝑡). Then, we say system (30) is a reducible system of system (29). We define the adjoint system of (29) by 𝑥̇(𝑡)=−𝐴^∗(𝑡)𝑥(𝑡). (31) If letting 𝑉(𝑡) be the fundamental matrix of (29), then [𝑉⁻¹(𝑡)]^∗ is the fundamental matrix of (31). Thus, we say the system (29) is a regular system if the adjoint systems (29) and (31) have convergent Lyapunov exponent series: {𝛼₁,…,𝛼_𝑛} and {𝛽₁,…,𝛽_𝑛}, respectively, which satisfy 𝛼_𝑖+𝛽_𝑖=0 for 𝑖=1,2,…,𝑛, or its reducible system (30) is also regular.

Lemma 16. Suppose that Assumptions 1, 2, and 15 are satisfied. Let {𝜎₁,𝜎₂,…,𝜎_𝑛,𝜎_𝑛+1,…,𝜎_𝑛𝑚} be the Lyapunov exponents of the variational system (26), where {𝜎₁,…,𝜎_𝑛} correspond to the synchronization space and the remaining correspond to the transverse space. Let 𝜆_𝑇(𝐷ℱ,𝑠₀,𝜔₀)=max_{𝑖≥𝑛+1}𝜎_𝑖 and 𝜆_𝑆(𝐷ℱ,𝑠₀,𝜔₀)=max_{1≤𝑖≤𝑛}𝜎_𝑖. If (a) the linear system (17) is a regular system, (b) ‖𝐷𝐹(𝑠(𝑡),𝜚^(𝑡)𝜔₀)‖≤𝑀 for all 𝑡≥0, (c) 𝜆_𝑃(𝐷ℱ,𝑠₀,𝜔₀)≠𝜆_𝑆(𝐷ℱ,𝑠₀,𝜔₀), then 𝜆_𝑇(𝐷ℱ,𝑠₀,𝜔₀)=𝜆_𝑃(𝐷ℱ,𝑠₀,𝜔₀).

3. General Synchronization Analysis

In this section we provide a methodology based on the previous theoretical analysis to judge whether a general differential system can be synchronized or not.

Theorem 17. Suppose that 𝑊∈ℝ^𝑛 is the compact subset given in Lemma 5, and Assumptions 1 and 2 are satisfied. If sup_𝑠0∈𝑊diam𝐷ℱ,𝑠₀<1, (32) then the coupled system (2) is synchronized.

Proof. The main techniques of the proof come from [3, 6] with some modifications. Let 𝜗^(𝑡) be the semiflow of the uncoupled system (5). By the condition (32), there exist 𝑑 satisfying sup_𝑠0∈𝑊diam(𝐷ℱ,𝑠₀)<𝑑<1 and 𝑇₁≥0 such that 𝑑^𝑇1<1/3, and 𝑟₀=inf{𝑟>0,𝒪(𝜗^(𝑇1)𝑊,𝑟)⊂𝑊}>0. For each 𝑠₀∈𝑊, there must exist 𝑡(𝑠₀)≥𝑇₁ such that diam(𝑈(𝑡₀+𝑡(𝑠₀),𝑡₀,𝑠₀))<𝑑^𝑡(𝑠0) for all 𝑡₀≥0. According to the equicontinuity of 𝑈(𝑡₀+𝑡(𝑠₀),𝑡₀,𝑠₀), there exists 𝛿>0 such that for any 𝑠^′₀∈𝒪(𝑠₀,𝛿), diam(𝑈(𝑡₀+𝑡(𝑠₀),𝑡₀,𝑠^′₀))<𝑑^𝑡(𝑠0) for all 𝑡₀≥0. According to the compactness of 𝑊, there exists a finite positive number set 𝒯={𝑡₁,𝑡₂,…,𝑡_𝑣} with 𝑡_𝑗≥𝑇₁ for all 𝑗=1,2,…,𝑣 such that for any 𝑠₀∈𝑊, there exists 𝑡_𝑗∈𝒯 such that diam(𝑈(𝑡₀+𝑡_𝑗,𝑡₀,𝑠₀))<1/3 for all 𝑡₀≥0. Let 𝑥(𝑡) be the collective states {𝑥¹(𝑡),…,𝑥^𝑚(𝑡)} which is the solution of the coupled system (2) with initial condition 𝑥^𝑖(𝑡₀)=𝑥^𝑖₀, 𝑖=1,2,…,𝑚. And let 𝑠(𝑡) be the solution of the synchronization state equation (5) with initial condition 𝑠(𝑡₀)=𝑥₀=(1/𝑚)∑^𝑚_𝑗=1𝑥^𝑗₀∈𝑊. Then, letting Δ𝑥^𝑖(𝑡)=𝑥^𝑖(𝑡)−𝑠(𝑡), we have Δ𝑥̇^𝑖_𝑘(𝑡)=𝑓^𝑖_𝑘𝑥¹(𝑡),…,𝑥^𝑚(𝑡),𝑡−𝑓_𝑘(𝑠(𝑡=))_{𝑚𝑛𝑗=1𝑙=1}𝜕𝑓^𝑖_𝑘𝜕𝑥^𝑗_𝑙𝜉^𝑖𝑗_𝑘𝑙(𝑡),𝑡Δ𝑥^𝑗_𝑙(𝑡), (33) where 𝜉^𝑖𝑗_𝑘𝑙(𝑡)∈ℝ^𝑚𝑛, 𝑖,𝑗=1,2,…,𝑚, 𝑘,𝑙=1,2…,𝑛, are obtained by the mean value principle of the differential functions. Letting 𝐷𝐹^𝑡(𝜉(𝑡))=((𝜕𝑓^𝑖_𝑘/𝜕𝑥^𝑖_𝑙)(𝜉^𝑖𝑗_𝑘𝑙(𝑡),𝑡)), we can write the equations above in matrix form: Δ𝑥̇(𝑡)=𝐷𝐹^𝑡(𝜉(𝑡))Δ𝑥(𝑡), (34) and denote its solution matrix by 𝑈(𝑡+𝑡₀,𝑡₀,𝑥₀)=(𝑈_𝑖𝑗(𝑡+𝑡₀,𝑡₀,𝑥₀))^𝑚_𝑖,𝑗=1. Then, for any 𝑡>0 there exists 𝐾₂>0 such that ‖𝐷𝐹^𝑡+𝑡0(𝜉(𝑡+𝑡₀))‖≤𝐾₂ for all 𝑡∈𝒯 and 𝑡₀≥0 according to the 3th item of Assumption 1. Then, we have Δ𝑥^𝑖_𝑘𝑡+𝑡₀=𝑥^𝑖₀−𝑥_0𝑘+^𝑡+𝑡0_𝑡0^𝑚𝑛_𝑗=1𝑙=1𝜕𝑓^𝑖_𝑘𝜕𝑥^𝑗_𝑙𝜉^𝑖𝑗_𝑘𝑙(𝜏),𝜏Δ𝑥^𝑗_𝑙(𝜏)𝑑𝜏,_{𝑚𝑛𝑗=1𝑘=1}Δ𝑥^𝑖_𝑘𝑡+𝑡₀≤^𝑚𝑛_𝑗=1𝑘=1𝑥^𝑖_0𝑘−𝑥_0𝑘+𝐾₂^𝑡+𝑡0_𝑡0^𝑚𝑛_𝑗=1𝑙=1Δ𝑥^𝑗_𝑙(𝜏)𝑑𝜏. (35) By Lemma 9, we have ^𝑚𝑛_𝑗=1𝑙=1Δ𝑥^𝑗_𝑙𝑡+𝑡₀≤𝑒^{𝐾2𝑡𝑚𝑛}_𝑗=1𝑙=1𝑥^𝑗_0𝑙−𝑥_0𝑙. (36) Let 𝑊_𝛼=𝑥=𝑥^1𝖳,…,𝑥^𝑚𝖳𝖳∶𝑥∈𝑊,^𝑚_𝑗=1𝑥^𝑗−𝑥≤𝛼. (37) Picking 𝛼 sufficiently small such that for each 𝑥₀∈𝑊_𝛼, there exists 𝑡∈𝒯 such that ∑^𝑚_𝑗=1‖Δ𝑥^𝑗(𝑡+𝑡₀)‖<𝑟₀/2 and diam(𝑈(𝑡+𝑡₀,𝑡₀,𝑥₀))<1/2 for all 𝑡₀≥0.
Thus, we are to prove synchronization step by step.
For any 𝑥₀∈𝑊_𝛼, there exists 𝑡^′=𝑡(𝑥₀)∈𝒯 such that 𝑥^𝑖𝑡^′+𝑡₀−𝑥^𝑗𝑡^′+𝑡₀=Δ𝑥^𝑖𝑡^′+𝑡₀−Δ𝑥^𝑗𝑡^′+𝑡₀≤^𝑚_𝑘=1𝑈_𝑖𝑘𝑡^′+𝑡₀,𝑡₀,𝑥₀−𝑈_𝑗𝑘𝑡^′+𝑡₀,𝑡₀,𝑥₀Δ𝑥^𝑘₀≤diam𝑈𝑡^′+𝑡₀,𝑡₀,𝑥₀max_𝑖,𝑗𝑥^𝑖₀−𝑥^𝑗₀≤12max_𝑖,𝑗𝑥^𝑖₀−𝑥^𝑗₀. (38) Therefore, we have max_𝑖,𝑗‖𝑥^𝑖(𝑡^′+𝑡₀)−𝑥^𝑗(𝑡^′+𝑡₀)‖≤(1/2)max_𝑖,𝑗‖𝑥^𝑖₀−𝑥^𝑗₀‖, which implies that 𝑥(𝑡^′+𝑡₀)∈𝑊 and 𝑥(𝑡^′+𝑡₀)∈𝑊_𝛼/2.
Then, reinitiated with time 𝑡^′+𝑡₀ and condition 𝑥(𝑡^′+𝑡₀), continuing with the phase above, we can obtain that lim_𝑡→∞max_𝑖,𝑗‖𝑥^𝑖(𝑡)−𝑥^𝑗(𝑡)‖=0. Namely, the coupled system (2) is synchronized. Furthermore, from the proof, we can conclude that the convergence is exponential with rate 𝑂(𝛿^𝑡) where 𝛿=sup_𝑠0∈𝑊diam(𝐷ℱ^𝑡,𝑠₀), and uniform with respect to 𝑡₀≥0 and 𝑥₀∈𝑊_𝛼. This completes the proof.

Remark 18. According to Assumption 2 that attractor 𝐴 is asymptotically stable and the properties of the compact neighbor 𝑊 given in Lemma 5, we can conclude that the quantity sup_𝑠0∈𝑊diam𝐷ℱ,𝑠₀ (39) is independent on the choice of 𝑊.

If the timevariation is driven by some MDS(Ω,ℬ,ℙ𝑃,𝜚^(𝑡)) and there exists a metric dynamical system {𝑊×Ω,𝐅,𝐏,𝜋^(𝑡)}, where 𝐅 is the product 𝜎-algebra on 𝑊×Ω, 𝐏 is the probability measure, and 𝜋^(𝑡)(𝑠₀,𝜔)=(𝜃^(𝑡)𝑠₀,𝜚^(𝑡)𝜔). From Theorem 17, we have the following.

Corollary 19. Suppose that the conditions in Lemma 16 are satisfied, 𝑊×Ω is compact in the topology defined in this MDS, the semiflow 𝜋^(𝑡) is continuous, and on 𝑊×Ω the Jacobian matrix 𝐷𝐹(𝜃^(𝑡)𝑠₀,𝜚^(𝑡)𝜔) is continuous. Let {𝜎_𝑖}^𝑛𝑚_𝑖=1 be the Lyapunov exponents of this MDS with multiplicity and {𝜎_𝑖}^𝑛_𝑖=1 correspond to the synchronization space. If sup_{𝐏∈Erg𝜋(𝑊×Ω)}sup_{𝑖≥𝑛+1}𝜎_𝑖<0, (40) where Erg_𝜋(𝑊×Ω) denotes the ergodic probability measure set supported in the MDS {𝑊×Ω,𝐅,𝐏,𝜋^(𝑡)}, then the coupled system (2) is synchronized.

4. Synchronization of LCODEs with Identity Inner Coupling Matrix and Time-Varying Couplings

In this section we study synchronization in linearly coupled ordinary differential equation systems (LCODEs) with time-varying couplings. Considering the following LCODEs with identity inner coupling matrix: 𝑥̇^𝑖(𝑡)=𝑓𝑥^𝑖(𝑡)+𝜎^𝑚_𝑗=1𝑙_𝑖𝑗(𝑡)𝑥^𝑗(𝑡),𝑖=1,2,…,𝑚, (41) where 𝑥^𝑖(𝑡)∈ℝ^𝑛 denotes the state variable of the 𝑖th node, 𝑓(⋅)∶ℝ^𝑛→ℝ^𝑛 is a differential map, 𝜎∈ℝ⁺ denotes coupling strength, and 𝑙_𝑖𝑗(𝑡) denotes the coupling coefficient from node 𝑗 to 𝑖 at time 𝑡, for all 𝑖≠𝑗, which are supposed to satisfy the following assumption. Here, we highlight that the inner coupling matrix is the identity matrix.

Assumption 20. (a) 𝑙_𝑖𝑗(𝑡)≥0, 𝑖≠𝑗 are measurable and 𝑙_𝑖𝑖(𝑡)=−∑^𝑚_{𝑗=1,𝑗≠𝑖}𝑙_𝑖𝑗(𝑡); (b) there exists 𝑀₁>0 such that |𝑙_𝑖𝑗(𝑡)|≤𝑀₁ for all 𝑖,𝑗=1,2,…,𝑚.
Similarly, we can define the Hajnal diameter of the following linear system: 𝑢̇(𝑡)=𝜎𝐿(𝑡)𝑢(𝑡). (42) Let 𝑉(𝑡)=(𝑣_𝑖𝑗(𝑡))^𝑚_𝑖,𝑗=1 be the fundamental solution matrix of the system (42). Then, its solution matrix can be written as 𝑉(𝑡,𝑡₀)=𝑉(𝑡)𝑉(𝑡₀)⁻¹. Thus, the Hajnal diameter of the system (42) can be defined as follows: diam(ℒ)=lim_𝑡→∞sup_𝑡0≥0diam𝑉𝑡,𝑡₀^1/𝑡. (43)

By Theorem 17, we have the following theorem.

Theorem 21. Suppose Assumptions 1, 2, and 20 are satisfied. Let 𝜇 be the largest Lyapunov exponent of the synchronized system 𝑠(̇𝑡)=𝑓(𝑠(𝑡)), that is, 𝜇=sup_𝑠0∈𝑊max_{𝑢∈ℝ𝑛}𝜆𝐷𝑓,𝑢,𝑠₀. (44) If log(diam(ℒ))+𝜇<0, then the LCODEs (41) is synchronized.

Proof. Considering the variational equation of (41): 𝛿𝑥̇(𝑡)=𝐼_𝑚⊗𝐷𝑓(𝑠(𝑡))+𝜎𝐿(𝑡)⊗𝐼_𝑛𝛿𝑥(𝑡). (45) Let 𝑈̆(𝑡,𝑡₀,𝑠₀) be the solution matrix of the synchronized state system (17) and 𝑉(𝑡,𝑡₀)=(𝑣_𝑖𝑗(𝑡,𝑡₀))^𝑚_𝑖,𝑗=1 be the solution matrix of the linear system (42). We can see that 𝑉(𝑡,𝑡₀)⊗𝑈̆(𝑡,𝑡₀,𝑠₀) is the solution matrix of the variational system (45). Then, diam𝑉𝑡+𝑡₀,𝑡₀⊗𝑈̆𝑡+𝑡₀,𝑡₀,𝑠₀=max^𝑚_{𝑖,𝑗=1,…,𝑚𝑘=1}𝑣_𝑖𝑘𝑡+𝑡₀,𝑡₀−𝑣_𝑗𝑘𝑡+𝑡₀,𝑡₀×𝑈̆𝑡+𝑡₀,𝑡₀,𝑠₀=diam𝑉𝑡+𝑡₀,𝑡₀𝑈̆𝑡+𝑡₀,𝑡₀,𝑠₀. (46) This implies that the Hajnal diameter of the variational system (45) is less than 𝑒^𝜇diam(ℒ). This completes the proof according to Theorem 17.

For the linear system (42), we firstly have the following lemma.

Lemma 22 (see [22]). 𝑉(𝑡,𝑡₀) is a stochastic matrix.

From Lemmas 13 and 16, we have the following corollary.

Corollary 23. logdiam(ℒ)=𝜆_𝑃(ℒ), where 𝜆_𝑃(ℒ) denotes the largest one of all the projection Lyapunov exponents of system (41). Moreover, if the conditions in Lemma 16 are satisfied, then logdiam(ℒ)=𝜆_𝑇(ℒ), where 𝜆_𝑇(ℒ) denotes the largest one of all the Lyapunov exponents corresponding to the transverse space, that is, the space orthogonal to the synchronization space.

If 𝐿(𝑡) is periodic, we have the following.

Corollary 24. Suppose that 𝐿(𝑡) is periodic. Let 𝜍_𝑖, 𝑖=1,2,…,𝑚, are the Floquet multipliers of the linear system (42). Then, there exists one multiplier denoted by 𝜍₁=1 and diam(ℒ)=max_𝑖≥2𝜍_𝑖.

If 𝐿(𝑡)=𝐿(𝜚^(𝑡)𝜔) is driven by some MDS(Ω,ℬ,𝑃,𝜚^(𝑡)), from Corollaries 19 and 23, we have the following corollary.

Corollary 25. Suppose 𝐿(𝜔) is continuous on Ω and conditions in Lemma 16 are satisfied. Let 𝜇=sup_𝑠0∈𝑊max_𝑢∈ℝ^𝑛𝜆(𝐷𝑓,𝑢,𝑠₀), 𝜍_𝑖, 𝑖=1,2,…,𝑚, be the Lyapunov exponents of the linear system (42) with 𝜍₁=0, and 𝜍=sup_{𝑃∈𝐸𝑟𝑓𝜃(Ω)}max_𝑖≥2𝜍_𝑖. If 𝜇+𝜍<0, then the coupled system (41) is synchronized.

Let ℐ be the set consisting of all compact time intervals in [0,+∞) and 𝒢 be the the set consisting of all graph with vertex set 𝒩={1,2,…,𝑚}.

Define 𝐺∶ℐ×𝑅⁺𝑡⟶𝒢,𝐼=₁,𝑡₂,𝛿⟼𝐺(𝐼,𝛿), (47) where 𝐺(𝐼,𝛿)={𝒩,ℰ} is a graph with vertex set 𝒩 and its edge set ℰ is defined as follows: there exists an edge from vertex 𝑗 to vertex 𝑖 if and only if ∫^𝑡2_𝑡1𝑙_𝑖𝑗(𝜏)𝑑𝜏>𝛿. Namely, we say that there is a 𝛿-edge from vertex 𝑗 to 𝑖 across 𝐼=[𝑡₁,𝑡₂].

Definition 26. We say that the LCODEs (41) has a 𝛿-spanning tree across the time interval 𝐼 if the corresponding graph 𝐺(𝐼,𝛿) has a spanning tree.

For a stochastic matrix 𝑉=(𝑣_𝑖𝑗)^𝑚_𝑖,𝑗=1, let 𝜂(𝑉)=min_𝑖,𝑗𝑣_𝑖∧𝑣_𝑗1, (48) where 𝑣_𝑖=[𝑣_𝑖1,…,𝑣_𝑖𝑚], 𝑖=1,2,…,𝑚, and 𝑣_𝑖∧𝑣_𝑗=[min{𝑣_𝑖1,𝑣_𝑗1},…,min{𝑣_𝑖𝑚,𝑣_𝑗𝑚}]^𝖳. Then, we can also define that 𝑉 is 𝛿-scrambling if 𝜂(𝑉)>𝛿.

Theorem 27. Suppose Assumption 20 is satisfied. diam(ℒ)<1 if and only if there exist 𝛿>0 and 𝑇>0 such that the LCODEs (41) has a 𝛿-spanning tree across any 𝑇-length time interval.

Remark 28. Different from [16], we do not need to assume that 𝐿(𝑡) has zero column sums and the timeaverage is achieved sufficiently fast.

Before proving this theorem, we need the following lemma.

Lemma 29. If the LCODEs (41) has a 𝛿-spanning tree across any 𝑇-length time interval, then there exist 𝛿₁>0 and 𝑇₁>0 such that 𝑉(𝑡,𝑡₀) is 𝛿₁-scrambling for any 𝑇₁-length time interval.

Proof of Theorem 27. Sufficiency. From Lemma 29, we can conclude that there exist 𝛿₁>0, 𝛿^′>0, and 𝑇₁>0 such that 𝑉(𝑡,𝑡₀) is 𝛿₁-scrambling across any 𝑇₁-length time interval and inf_𝑡0≥0𝜂(𝑉(𝑇₁+𝑡₀,𝑡₀))>𝛿^′. For any 𝑡≥𝑡₀, let 𝑡−𝑡₀=𝑝𝑇₁+𝑇^′, where 𝑝 is an integer and 0≤𝑇^′<𝑇₁ and 𝑡_𝑙=𝑡₀+𝑙𝑇₁, 0≤𝑙≤𝑝. Then, we have diam𝑉𝑡,𝑡₀=diam𝑉𝑡,𝑡_𝑝^𝑝_𝑙=1𝑉𝑡_𝑙,𝑡_𝑙−1≤diam^𝑝_𝑙=1𝑉𝑡_𝑙,𝑡_𝑙−1≤2^𝑝_𝑙=11−𝜂𝑉𝑡_𝑙,𝑡_𝑙−1≤21−𝛿^{′⌊(𝑡−𝑡0)/𝑇1⌋}. (49) For the first inequality, we use the results in [23, 24]. This implies diam(ℒ)≤(1−𝛿^′)^1/𝑇1<1.
Necessity. Suppose that for any 𝑇≥0 and 𝛿>0, there exists 𝑡₀=𝑡₀(𝑇,𝛿), ∫^𝑇+𝑡0_𝑡0𝐿(𝜏)𝑑𝜏 does not have a 𝛿-spanning tree. According to the condition, there exist 1>𝑑>diam(ℒ), 𝜖>0, and 𝑇^′>0 such that diam(𝑉(𝑡+𝑡₀))<𝑑^𝑡 for all 𝑡₀≥0 and 𝑡≥𝑇^′ and 𝑑^𝑇′<1−𝜖. Thus, picking 𝑇>𝑇^′, 𝛿=𝑚⁻³𝑒^{−𝑀1𝑚𝑇}𝜖/2, 𝑡₁=𝑡₀(𝑇,𝛿), and 𝐿^′=(𝑙^′_𝑖𝑗)^𝑚_𝑖,𝑗=1=(∫^𝑇+𝑡1_𝑇𝑙_𝑖𝑗(𝜏)𝑑𝜏)^𝑚_𝑖,𝑗=1, there exist two vertex set 𝐽₁ and 𝐽₂ such that 𝑙^′_𝑖𝑗≤𝛿 if 𝑖∈𝐽₁ and 𝑗∉𝐽₁, or 𝑖∈𝐽₂ and 𝑗∉𝐽₂. For each 𝑖∈𝐽₁ and 𝑗∉𝐽₁, we have 𝑣̇_𝑖𝑗(𝑡)=𝑙_𝑖𝑖(𝑡)𝑣_𝑖𝑗(𝑡)+^𝑘≠𝑖_𝑘∈𝐽1𝑙_𝑖𝑘(𝑡)𝑣_𝑘𝑗(𝑡)+_𝑘∉𝐽1𝑙_𝑖𝑘(𝑡)𝑣_𝑘𝑗(𝑡)≤𝑀₁^𝑘≠𝑖_𝑘∈𝐽1𝑣_𝑘𝑗(𝑡)+_𝑘∉𝐽1𝑙_𝑖𝑘(𝑡). (50) Then, 𝑣̇_{𝑖∈𝐽1,𝑗∉𝐽1𝑖𝑗}(𝑡)≤𝑀₁^{𝑘≠𝑖,𝑗∉𝐽1}_{𝑖∈𝐽1,𝑘∈𝐽1}𝑣_𝑘𝑗(𝑡)+^𝑗∉𝐽1_{𝑖∈𝐽1,𝑘∉𝐽1}𝑙_𝑖𝑘(𝑡)=𝑀₁#𝐽₁−1^𝑗∉𝐽1_𝑘∈𝐽1𝑣_𝑘𝑗(𝑡)+𝑚−#𝐽₁^𝑘∉𝐽1_𝑖∈𝐽1𝑙_𝑖𝑘(𝑡). (51) Let 𝑣(𝑡)=∑_{𝑖∈𝐽1,𝑗∉𝐽1}𝑣_𝑖𝑗(𝑡). According to Lemma 9, we have 𝑣𝑇+𝑡₁≤𝑒^{𝑀1(#𝐽1−1)𝑇}𝑚−#𝐽₁^𝑇+𝑡1_𝑡1^𝑗∉𝐽1_𝑖∈𝐽1𝑙_𝑖𝑗(𝜏)≤𝑑𝜏𝑚−#𝐽₁𝑒^{𝑀1(#𝐽1−1)𝑇}#𝐽₁𝑚−#𝐽₁𝛿≤𝑚³𝑒^{𝑚𝑀1𝑇}𝜖𝛿≤2. (52) Similarly, we can conclude that ∑_{𝑖∈𝐽𝑙,𝑗∉𝐽𝑙}𝑣_𝑖𝑗(𝑇+𝑡₁)≤𝜖/2 for all 𝑙=1,2. Without loss of generality, we suppose 𝐽₁={1,2,…,𝑝} and 𝐽₂={𝑝+1,𝑝+2,…,𝑝+𝑞}, where 𝑝 and 𝑞 are integers with 𝑝+𝑞≤𝑚. Then, we can write 𝑉(𝑇+𝑡₁,𝑡₁) in the following matrix form: 𝑉𝑇+𝑡₁,𝑡₁=𝑋₁₁𝑋₁₂𝑋₁₃𝑋₂₁𝑋₂₂𝑋₂₃𝑋₃₁𝑋₃₂𝑋₃₃, (53) where 𝑋₁₁∈ℝ^𝑝,𝑝 and 𝑋₂₂∈𝑅^𝑞,𝑞 correspond to the vertex subset 𝐽₁ and 𝐽₂, respectively. Immediately, we have ‖𝑋₁₂‖_∞+‖𝑋₁₃‖_∞+‖𝑋₂₁‖_∞+‖𝑋₂₃‖_∞≤𝜖. Let 𝑣=^𝟏𝑝₀₀. We let 𝑉𝑡₁+𝑇,𝑡₁𝑋𝑣=¹¹𝟏_𝑝𝑋₂₁𝟏_𝑝𝑋₃₁𝟏_𝑝. (54) Let 𝑢=^𝑢1_𝑢²_𝑢3=[𝑢₁,…,𝑢_𝑚]^𝖳 with 𝑢^𝑖=[𝑢^𝑖₁,…,𝑢^𝑖_𝑝𝑖]^𝖳=𝑋_𝑖1𝟏_𝑝 and 𝑝₁=𝑝, 𝑝₂=𝑞, 𝑝₃=𝑚−𝑝−𝑞. Then, max_𝑖,𝑗𝑢_𝑖−𝑢_𝑗≥max_𝑘,𝑙𝑢¹_𝑘−𝑢²_𝑗≥1−𝑋_12∞−𝑋_13∞−𝑋_21∞−𝑋_23∞≥1−𝜖. (55) Also, max_𝑖,𝑗𝑢_𝑖−𝑢_𝑗≤diam𝑉𝑡₁+𝑇,𝑡₁≤𝑑^𝑇. (56) This implies 𝑑^𝑇≥1−𝜖 which leads contradiction with 𝑑^𝑇<1−𝜖. Therefore, we can conclude the necessity.

5. Consensus Analysis of Multiagent System with Directed Time-Varying Graphs

If we let 𝑛=1, 𝑓≡0, and 𝜎=1 in system (41), then we have 𝑥̇^𝑖(𝑡)=^𝑚_𝑗=1𝑙_𝑖𝑗(𝑡)𝑥^𝑗(𝑡),𝑖=1,2,…,𝑚. (57) In this case, if Assumption 20 is satisfied, then the synchronization analysis of system (57) becomes another important research field named consensus problems.

Definition 30. We say the differential system (57) reaches consensus if for any 𝑥(𝑡₀)∈ℝ^𝑚, ‖𝑥^𝑖(𝑡)−𝑥^𝑗(𝑡)‖→0 as 𝑡→∞ for all 𝑖,𝑗∈𝒩.

In graph view, the coefficients matrix of (57) 𝐿(𝑡)=(𝑙_𝑖𝑗(𝑡))∈ℝ^𝑚,𝑚 is equal to the negative graph Laplacian associated with the digraph 𝐺(𝑡) at time 𝑡, where 𝐺(𝑡)=(𝒱,ℰ(𝑡),𝒜(𝑡)) is a weighted digraph (or directed graph) with 𝑚 vertices, the set of nodes 𝒱={𝑣₁,…,𝑣_𝑚}, set of edges ℰ(𝑡)⊆𝒱×𝒱, and the weighted adjacency matrix 𝒜(𝑡)=(𝑎_𝑖𝑗(𝑡)) with nonnegative adjacency elements 𝑎_𝑖𝑗(𝑡). An edge of 𝐺(𝑡) is denoted by 𝑒_𝑖𝑗(𝑡)=(𝑣_𝑖,𝑣_𝑗)∈ℰ(𝑡) if there is a directed edge from vertex 𝑖 to vertex 𝑗 at time 𝑡. The adjacency elements associated with the edges of the graph are positive, that is, 𝑒_𝑖𝑗(𝑡)∈ℰ(𝑡)⇔𝑎_𝑖𝑗(𝑡)>0, for all 𝑖,𝑗∈𝒩. It is assumed that 𝑎_𝑖𝑖(𝑡)=0 for all 𝑖∈𝒩. The indegree and outdegree of node 𝑣_𝑖 at time 𝑡 are, respectively, defined as follows: deg_in𝑣_𝑖(𝑡)=^𝑁_𝑗=1𝑎_{𝑗𝑖(𝑡)},deg_out𝑣_𝑖(𝑡)=^𝑁_𝑗=1𝑎_{𝑖𝑗(𝑡)}. (58) The degree matrix of digraph 𝐺(𝑡) is defined as 𝐷(𝑡)=diag(deg_out(𝑣₁(𝑡)),…,deg_out(𝑣_𝑚(𝑡))) at time 𝑡. The graph Laplacian associated with the digraph 𝐺(𝑡) at time 𝑡 is defined as −𝐿(𝑡)=ℒ(𝐺(𝑡))=𝐷(𝑡)−𝒜(𝑡). (59)

Let 𝐺(𝐼,𝛿) defined as before. We say that the digraph 𝐺(𝑡) has a 𝛿-spanning tree across the time interval 𝐼 if 𝐺(𝐼,𝛿) has a spanning.

Theorem 31. Suppose Assumption 20 is satisfied. The system (57) reaches consensus if and only if there exist 𝛿>0 and 𝑇>0 such that the corresponding digraph 𝐺(𝑡) has a 𝛿-spanning tree across any 𝑇-length time interval.

Proof. Since 𝑓≡0, we have 𝜇=0 in Theorem 21. This completes the proof according to Theorems 27 and 21.

Remark 32. This theorem is a part of Theorem 17 in [25].

6. Numerical Examples

In this section, a numerical example is given to demonstrate the effectiveness of the presented results on synchronization of LCODEs with time-varying couplings. The Lyapunov exponents are computed numerically. By this way, we can verify the the synchronization criterion and analyze synchronization numerically. We use the Rössler system [16, 26] as the node dynamics 𝑥̇₁(𝑡)=−𝑥₂(𝑡)−𝑥₃(𝑡),𝑥̇₂(𝑡)=𝑥₁(𝑡)+𝑎𝑥₂(𝑡),𝑥̇₃(𝑡)=𝑏+𝑥₃(𝑡)𝑥₁(𝑡)−𝑐, (60) where 𝑎=0.165, 𝑏=0.2, and 𝑐=10. Figure 1 shows the dynamical behaviors of the Rössler system (60) with random initial value in [0,1] that includes a chaotic attractor [16, 26].

Figure 1 

The dynamical behavior of the Rössler system (60) with 𝑎=0.165, 𝑏=0.2, and 𝑐=10.

The network with time-varying topology we used here is NW small-world network with a time-varying coupling, which was introduced as the blinking model in [11, 27]. The time-varying network model algorithm is presented as follows: we divide the time axis into intervals of length 𝜏, in each interval: (a) begin with the nearest neighbor coupled network consisting of 𝑚 nodes arranged in a ring, where each node 𝑖 is adjacent to its 2𝑘-nearest neighbor nodes; (b) add a connection between each pair of nodes with probability 𝑝, which usually is a random number between [0,0.1]; for more details, we refer the readers to [11]. Figure 2 shows the time-varying structure of shortcut connections in the blinking model with 𝑚=50 and 𝑘=3.

(a)

(b)

Figure 2 

The blinking model of shortcuts connections. Probability of switchings 𝑝=0.04, the switching time step 𝜏=1.

In this example, the parameters are taken values as 𝑚=50, 𝑘=3, 𝜏=1, and 𝑝=0.04. Then blinking small-world network can be generated with the coupling graph Laplacian ℒ(𝐺(𝑡))=−𝐿(𝑡). The dynamical network system can be described as follows: 𝑥̇^𝑖₁(𝑡)=−𝑥^𝑖₂(𝑡)−𝑥^𝑖₃(𝑡)+𝜎^𝑚_𝑗=1𝑙_𝑖𝑗(𝑡)𝑥^𝑗₁(𝑡),𝑥̇^𝑖₂(𝑡)=𝑥^𝑖₁(𝑡)+𝑎𝑥^𝑖₂(𝑡)+𝜎^𝑚_𝑗=1𝑙_𝑖𝑗(𝑡)𝑥^𝑗₂(𝑡),𝑥̇^𝑖₃(𝑡)=𝑏+𝑥^𝑖₃(𝑡)𝑥^𝑖₁(𝑡)−𝑐+𝜎^𝑚_𝑗=1𝑙_𝑖𝑗(𝑡)𝑥^𝑗₃(𝑡),𝑖=1,2,…,𝑚. (61)

Let 𝑒(𝑡)=max_{1≤𝑖<𝑗≤50}‖𝑥^𝑖(𝑡)−𝑥^𝑗(𝑡)‖ denotes the maximum distance between nodes at time 𝑡. Let 𝐸=∫_𝑇^𝑇+𝑅𝑒(𝑡)𝑑𝑡, for some sufficiently large 𝑇>0 and 𝑅>0. Let 𝐻=𝜇+𝜍 defined in Corollary 25. As described in Corollary 25, two steps are needed for verification: (a) calculating the largest Lyapunov exponent of the uncoupled synchronized system (60), 𝜇 and (b) calculating the second largest Lyapunov exponent of the linear system (42). In detail, we use Wolf's method [28] to compute 𝜇 and the Jacobian method [29] to compute Lyapunov spectra of (42). More details can be found in [28–30]. Figure 3 shows convergence of the maximum distance between nodes during the topology evolution with a different coupling strength 𝜎. It can be seen from Figure 3 that the dynamical network system (61) can be synchronized with 𝜎=0.4 and 𝜎=0.5.

Figure 3 

Convergence of the maximum distance between nodes with a different coupling strength 𝜎.

We pick the time length 200. Let 𝑇=190 and 𝑅=10. And choose initial state randomly from the interval [0,1]. Figure 4 shows the variation of 𝐸 and 𝐻 with respect to the coupling strength 𝜎. It can be seen that the parameter (coupling strength 𝜎) region where 𝐻 is negative coincides with that of synchronization, that is, where 𝐸 is near zero. This verified the theoretical result (Corollary 25). In addition, we find that 𝜎≈0.38 is the threshold for synchronizing the coupled systems in this case.

Figure 4 

Variation of 𝑒 and 𝐻 with respect to 𝜎 for the blinking topology.

7. Conclusions

In this paper, we present a theoretical framework for synchronization analysis of general coupled differential dynamical systems. The extended Hajnal diameter is introduced to measure the synchronization. The coupling between nodes is timevarying in both network structure and reaction dynamics. Inspired by the approaches in [6, 13], we show that the Hajnal diameter of the linear system induced by the time-varying coupling matrix and the largest Lyapunov exponent of the synchronized system play the key roles in synchronization analysis of LCODEs. These results extend synchronization analysis of discrete-time network in [6] to continuous-time case. As an application, we obtain a very general sufficient condition ensuring directed time-varying graph reaching consensus, and the way we get this result is different from [25]. An example of numerical simulation is provided to show the effectiveness the theoretical results. Additional contributions of this paper are that we explicitly show that the largest projection Lyapunov exponent, the Hajnal diameter, and the largest Lyapunov exponent of the transverse space are equal to each other in coupled differential systems (see Lemmas 13 and 16), which was proved in [6] for couple discrete-time systems.

Appendix

Proof of Lemma 5. Let 𝑈 be a bounded open neighborhood of 𝐴 satisfying ⋂_𝑡≥0𝜗^(𝑡)𝑈=𝐴 and 𝑈_𝑡={𝑥∈𝑅^𝑛∶𝜗^(𝜏)𝑥∈𝑈,0≤𝜏≤𝑡}. This implies 𝑈_𝑡⊃𝑈_𝑡′ if 𝑡^′≥𝑡≥0, 𝑈_𝑡 is an open set due to the continuity of the semiflow 𝜗^(𝑡), and 𝜗^(𝛿)𝑈_𝑡⊂𝑈_𝑡−𝛿 for all 𝑡≥𝛿≥0. Let 𝑉=⋂_𝑡≥0𝑈_𝑡. We claim that there exists 𝑡₀≥0 such that 𝑉=𝑈_𝑡 for all 𝑡≥𝑡₀.
For any 𝛿>0, let 𝑡_𝑛=𝑛𝛿 and 𝑈_𝑛=𝑈_𝑡𝑛. We can conclude that 𝑉=⋂^∞_𝑛=1𝑈_𝑛. We will prove in the following that there exists 𝑛₀ such that 𝑉=𝑈_𝑛0. Otherwise, there always exists 𝑥_𝑛∈𝑈_𝑛⧵𝑈_𝑛+1 for 𝑛≥0. Let 𝑦_𝑛=𝜗^(𝑡𝑛+1)𝑥_𝑛. We have (i) 𝑦_𝑛∈⋂^𝑛_𝑘=0𝜗^(𝑡𝑘)𝑈 and (ii) 𝑦_𝑛∉𝑈. For any limit point 𝑦 of 𝑦_𝑛, 𝑦 can be either finite or infinite. For both cases, 𝑦∉𝑈 which implies 𝑦∉𝐴. However, the claim (i) implies that 𝑦∈𝐴, which contradicts with the claim (ii). This completes the proof by letting 𝑊=𝑉.

Proof of Lemma 10. (a) For any initial condition with the form 𝛿𝑥₀=𝟏_𝑚⊗𝑢₀, the solution of (11) can be 𝑈(𝑡,𝑡₀,𝑠₀)(𝟏_𝑚⊗𝑢₀)=𝟏_𝑚⊗𝑈̆(𝑡,𝑡₀,𝑠₀)𝑢₀ according to Lemma 4. This implies the first claim in this lemma.
(b) According to Lemma 5, there exists 𝐾₁>0 such that 𝑠(𝑡), the solution of (5), satisfies ‖𝑠(𝑡)‖≤𝐾₁ for all 𝑠₀∈𝑊 and 𝑡≥0. So, there exists 𝐾>0 such that ‖𝐷𝐹^𝑡(𝑠(𝑡))‖≤𝐾 according to the 3th item of Assumption 1. Write the solution of (11) 𝛿𝑥(𝑡)=𝑈(𝑡,𝑡₀,𝑠₀)𝛿𝑥₀ as 𝛿𝑥𝑡+𝑡₀=𝛿𝑥₀+^𝑡+𝑡0_𝑡0𝐷𝐹^𝜏(𝑠(𝜏))𝛿𝑥(𝜏)𝑑𝜏. (A.1) Then, 𝛿𝑥𝑡+𝑡₀≤𝛿𝑥₀+^𝑡+𝑡0_𝑡0𝐷𝐹^𝜏(𝑠(𝜏(𝜏≤))‖𝛿𝑥)‖𝑑𝜏𝛿𝑥₀+𝐾^𝑡₀𝛿𝑥𝜏+𝑡₀𝑑𝜏. (A.2) According to Lemma 9, we have ‖𝛿𝑥(𝑡+𝑡₀)‖≤‖𝛿𝑥₀‖+𝐾∫^𝑡₀‖𝛿𝑥₀‖𝑒^{(𝑡−𝜏)𝐾}𝑑𝜏=𝑒^𝐾𝑡‖𝛿𝑥₀‖. This implies that ‖𝑈(𝑡+𝑡₀,𝑡₀,𝑠₀)‖≤𝑒^𝐾𝑡 for all 𝑠₀∈𝑊 and 𝑡₀≥0.
For any 𝑠₀,𝑠^′₀∈𝑊, let 𝑠(𝑡) and 𝑠^′(𝑡) be the solution of the synchronized state equation (5) with initial condition 𝑠(𝑡₀)=𝑠₀ and 𝑠^′(𝑡₀)=𝑠^′₀, respectively. We have 𝑠𝑡+𝑡₀−𝑠^′𝑡+𝑡₀=^𝑡+𝑡0_𝑡0𝑓(𝑠(𝜏))−𝑓𝑠^′(𝜏)𝑑𝜏+𝑠𝑡₀−𝑠^′𝑡₀,𝑠𝑡+𝑡₀−𝑠^′𝑡+𝑡₀≤𝑠𝑡₀−𝑠^′𝑡₀+𝐾^𝑡+𝑡0_𝑡0𝑠(𝜏)−𝑠^′(𝜏)𝑑𝜏. (A.3) By Lemma 9, we have ‖𝑠(𝑡+𝑡₀)−𝑠^′(𝑡+𝑡₀)‖≤𝑒^𝐾𝑡‖𝑠₀−𝑠^′₀‖ for all 𝑡₀,𝑡≥0 and 𝑠₀,𝑠^′₀∈𝑊. Also, according to the 4th item of Assumption 1, there must exist 𝐾₂>0 such that ‖𝐷𝐹^𝑡(𝑠(𝑡))−𝐷𝐹^𝑡(𝑠^′(𝑡))‖≤𝐾₂‖𝑠(𝑡)−𝑠^′(𝑡)‖ for all 𝑡≥0 and 𝑠₀,𝑠^′₀∈𝑊. Then, let 𝛿𝑥(𝑡)=𝑈(𝑡,𝑡₀,𝑠₀)𝛿𝑥₀, 𝛿𝑦(𝑡)=𝑈(𝑡,𝑡₀,𝑠^′₀)𝛿𝑥₀, and 𝑣(𝑡)=𝛿𝑥(𝑡)−𝛿𝑦(𝑡). We have 𝑣𝑡+𝑡₀=^𝑡+𝑡0_𝑡0𝐷𝐹^𝜏(𝑠(𝜏))𝛿𝑥(𝜏)−𝐷𝐹^𝜏𝑠^′(𝜏)(𝜏=𝛿𝑦)𝑑𝜏_{𝑡+𝑡0𝑡0}𝐷𝐹^𝜏(𝑠(𝜏))−𝐷𝐹^𝜏𝑠^′(𝜏(𝜏)+)𝛿𝑥𝑑𝜏_{𝑡+𝑡0𝑡0}𝐷𝐹^𝜏𝑠^′(𝜏)𝑣(𝜏)𝑣𝑑𝜏,𝑡+𝑡₀≤^𝑡+𝑡0_𝑡0𝐷𝐹^𝜏(𝑠(𝜏))−𝐷𝐹^𝜏𝑠^′(𝜏)(𝜏+‖𝛿𝑥)‖𝐷𝐹^𝜏𝑠^′(𝜏)‖𝑣(𝜏)‖𝑑𝜏≤𝐾₂^𝑡₀𝑒^2𝐾𝜏𝑑𝜏𝛿𝑥₀𝑠₀−𝑠^′₀+𝐾^𝑡0_𝑡^+𝑡₀‖𝑣(𝜏)‖𝑑𝜏. (A.4) According to Lemma 9, 𝑣𝑡+𝑡₀≤𝐾₂𝑒^2𝐾𝑡−𝑒^𝐾𝑡𝐾𝛿𝑥₀𝑠₀−𝑠^′₀. (A.5) This implies 𝑈𝑡+𝑡₀,𝑡₀,𝑠₀−𝑈𝑡+𝑡₀,𝑡₀,𝑠^′₀≤𝐾₂𝑒^2𝐾𝑡−𝑒^𝐾𝑡𝐾𝑠₀−𝑠^′₀ (A.6) for all 𝑠₀,𝑠^′₀∈𝑊. This completes the proof.

Proof of Lemma 13. We define the projection joint spectral radius as follows: 𝜌_𝑃𝐷ℱ,𝑠₀=lim_𝑡→∞sup_𝑡0≥0𝑈𝑡,𝑡₀,𝑠₀^1/𝑡. (A.7) First, we will prove that diam(𝐷ℱ,𝑠₀)=𝜌_𝑃(𝐷ℱ,𝑠₀). For any 𝑑>𝜌_𝑃(𝐷ℱ,𝑠₀), there exists 𝑇≥0 such that ‖𝑈(𝑡+𝑡₀,𝑡₀,𝑠₀)‖≤𝑑^𝑡 for all 𝑡₀≥0 and 𝑡≥𝑇. This implies that 𝑃⁻¹𝑈𝑡+𝑡₀,𝑡₀,𝑠₀𝐼𝑃−^𝑛0⋮0×𝑃^𝖳₀𝑈̆𝑡+𝑡₀,𝑡₀,𝑠₀𝑃₀,𝛼𝑡+𝑡₀,𝑡₀,𝑠₀=0𝑈00𝑡+𝑡₀,𝑡₀,𝑠₀≤𝐶₁𝑑^𝑡 (A.8) for some 𝐶₁>0, all 𝑡₀≥0 and all 𝑡≥𝑇. Thus, there exist some 𝐶₂>0 and some matrix function 𝑞(𝑡)∈ℝ^{𝑛,𝑛𝑚} such that 𝑈𝑡+𝑡₀,𝑡₀,𝑠₀−𝟏_𝑚(𝑡=𝑈⊗𝑞)𝑡+𝑡₀,𝑡₀,𝑠₀𝐼−𝑃^𝑛0⋮0×𝑃^𝖳₀𝑈̆𝑡+𝑡₀,𝑡₀,𝑠₀𝑃₀,𝛼𝑡+𝑡₀,𝑡₀,𝑠₀𝑃⁻¹𝑣≤𝐶₂𝑑^𝑡 (A.9) for all 𝑡₀≥0 and 𝑡≥𝑇, where 𝑞(𝑡)∈ℝ^{𝑛,𝑛𝑚} denotes a matrix, and we omit its accurate expression. So, we can conclude that diam(𝑈(𝑡+𝑡₀,𝑡₀,𝑠₀))≤𝐶₃𝑑^𝑡 for some 𝐶₃>0, all 𝑡₀≥0, and 𝑡≥𝑇. This implies that diam(𝐷ℱ,𝑠₀)≤𝑑, that is, diam(𝐷ℱ,𝑠₀)≤𝜌_𝑃(𝐷ℱ,𝑠₀) due to the arbitrariness of 𝑑≥𝜌_𝑃(𝐷ℱ,𝑠₀). Conversely, for any 𝑑>diam(𝐷ℱ,𝑠₀), there exists 𝑇>0 such that 𝑈𝑡+𝑡₀,𝑡₀,𝑠₀−𝟏_𝑚⊗𝑈₁≤𝐶₄𝑑^𝑡 (A.10) for some 𝐶₄>0, all 𝑡₀≥0, and 𝑡≥𝑇, where 𝑈₁=[𝑈₁₁,𝑈₁₂,…,𝑈_1𝑚] the first 𝑛 rows of 𝑈(𝑡+𝑡₀,𝑡₀,𝑠₀). Then, 𝑃⁻¹𝑈𝑡+𝑡₀,𝑡₀,𝑠₀𝑃−𝑃⁻¹𝟏_𝑚⊗𝑈₁𝑃=𝑃⁻¹𝑈𝑡+𝑡₀,𝑡₀,𝑠₀𝛾(𝑡)𝛽(𝑡)=(𝑡)0𝑈𝑃−000𝛽𝑡+𝑡₀,𝑡₀,𝑠₀≤𝐶₅𝑑^𝑡 (A.11) for some 𝐶₅>0, all 𝑡₀≥0, and 𝑡≥𝑇, where 𝛾(𝑡)=𝑃^𝖳₀𝑈̆(𝑡,𝑡₀,𝑠₀)𝑃₀∈ℝ^𝑛,𝑛 and 𝛽(𝑡)∈ℝ^{𝑛,𝑛(𝑚−1)} denotes a matrix, and we omit its accurate expression. This implies that ‖𝑈(𝑡+𝑡₀,𝑡₀,𝑠₀)‖≤𝐶₆𝑑^𝑡 holds for some 𝐶₆>0, all 𝑡₀≥0, and 𝑡≥𝑇. Therefore, we can conclude that 𝜌_𝑃(𝐷ℱ,𝑠₀)≤𝑑. So, 𝜌_𝑃(𝐷ℱ,𝑠₀)=diam(𝐷ℱ,𝑠₀).
Second, it is clear that log𝜌_𝑃(𝐷ℱ,𝑠₀)≥𝜆_𝑃(𝐷ℱ,𝑠₀). We will prove that log𝜌_𝑃(𝐷ℱ,𝑠₀)=𝜆_𝑃(𝐷ℱ,𝑠₀). Otherwise, there exists some 𝑟,𝑟₀>0 satisfying 𝜌_𝑃(𝐷ℱ,𝑠₀)>𝑟>𝑟₀>𝑒^{𝜆𝑃(𝐷ℱ,𝑠0)}. If so, there exists a sequence 𝑡_𝑘↑∞ as 𝑘→∞, 𝑡^𝑘₀≥0, and 𝑣_𝑘∈ℝ^{𝑛(𝑚−1)} with ‖𝑣_𝑘‖=1 such that ‖𝑈(𝑡_𝑘+𝑡^𝑘₀,𝑡^𝑘₀,𝑠₀)𝑣_𝑘‖>𝑟^𝑡𝑘 for all 𝑘∈𝒩. Then, there exists a subsequence 𝑣_𝑘𝑙 with lim_𝑙→∞𝑣_𝑘𝑙=𝑣^∗. Let {𝑒₁,𝑒₂,…,𝑒_{𝑛(𝑚−1)}} be a normalized orthogonal basis of ℝ^{𝑛(𝑚−1)}. And, let 𝑣_𝑘𝑙−𝑣^∗=∑^{𝑛(𝑚−1)}_𝑗=1𝜉^𝑘𝑙_𝑗𝑒_𝑗. We have lim_𝑙→∞𝜉^𝑘𝑙_𝑗=0 for all 𝑗=1,…,𝑛(𝑚−1). Thus, there exists 𝐿>0 such that 𝑈𝑡_𝑘𝑙+𝑡^𝑘𝑙₀,𝑡^𝑘𝑙₀,𝑠₀𝑣^∗≥𝑈𝑡_𝑘𝑙+𝑡^𝑘𝑙₀,𝑡^𝑘𝑙₀,𝑠₀𝑣_𝑘𝑙−𝑈𝑡_𝑘𝑙+𝑡^𝑘𝑙₀,𝑡^𝑘𝑙₀,𝑠₀𝑣_𝑘𝑙−𝑣^∗≥𝑟^𝑡𝑘𝑙−^{𝑛(𝑚−1)}_𝑗=1𝜉^𝑘𝑙_𝑗𝑈𝑡_𝑘𝑙+𝑡^𝑘𝑙₀,𝑡^𝑘𝑙₀,𝑠₀𝑒_𝑗≥𝑟^𝑡𝑘𝑙−𝑟^𝑡𝑘𝑙₀>𝑟^𝑡𝑘𝑙₀ (A.12) for all 𝑙≥𝐿. This implies 𝑒^{𝜆(𝐷𝑃ℱ,𝑣∗,𝑠0)}≥𝑟₀ which contradicts with 𝑒^{𝜆𝑃(𝐷ℱ,𝑠0)}<𝑟₀. This implies 𝜌_𝑃(𝐷ℱ,𝑠₀)=𝑒^{𝜆𝑃(𝐷ℱ,𝑠0)}. Therefore, we can conclude logdiam(𝐷ℱ,𝑠₀)=𝜆_𝑃(ℱ,𝑠₀). The proof is completed.

Proof of Lemma 16. Let 𝜙=𝑃⁻¹𝜙. We have ̇𝜙=𝑃⁻¹𝐷𝐹𝑠(𝑡),𝜚^(𝑡)𝜔₀𝑃𝜙=𝑃^𝖳₀𝜕𝑓(𝑠(𝑡𝑃𝜕𝑠))₀𝛼(𝑡)0𝐷𝐹𝑠(𝑡),𝜚^(𝑡)𝜔₀𝜙. (A.13)
Write 𝜙=^𝑦(𝑡)_𝑧(𝑡), where 𝑦(𝑡)∈ℝ^𝑛. Then, we have 𝑧̇(𝑡)=𝐷𝐹𝑠(𝑡),𝜚^(𝑡)𝜔₀𝑧(𝑡),𝑦̇(𝑡)=𝑃^𝖳₀𝜕𝑓(𝑠(𝑡𝑃𝜕𝑠))₀𝑦(𝑡)+𝛼(𝑡)𝑧(𝑡). (A.14) Thus, we can write its solution by 𝑧(𝑡)=𝑈(𝑡)𝑧₀,𝑦(𝑡)=𝑃^𝖳₀𝑈̆(𝑡)𝑃₀𝑦₀+^𝑡₀𝑃^𝖳₀𝑈̆(𝑡)𝑈̆⁻¹(𝜏)𝑃₀𝛼(𝜏)𝑈(𝜏)𝑧₀𝑑𝜏. (A.15)
We write 𝜆_𝑃(𝐷ℱ,𝑠₀,𝜔₀), 𝜆_𝑆(𝐷ℱ,𝑠₀,𝜔₀), and 𝜆_𝑇(𝐷ℱ,𝑠₀,𝜔₀) by 𝜆_𝑃, 𝜆_𝑆, and 𝜆_𝑇, respectively for simplicity.
Case 1 (𝜆_𝑃>𝜆_𝑆). We can conclude that 𝜒[𝑧(𝑡)]≤𝜆_𝑃 and 𝜒𝑦(𝑡)≤max𝜒𝑃^𝖳₀𝑈̆(𝑡)𝑃₀𝑦₀,𝜒^𝑡₀𝑃^𝖳₀𝑈̆(𝑡)𝑈̆⁻¹(𝜏)𝑃₀𝛼(𝜏)×𝑈(𝜏)𝑧(0)𝑑𝜏. (A.16) From Cauchy-Buniakowski-Schwarz inequality, we have 𝜒^𝑡₀𝑃^𝖳₀𝑈̆(𝑡)𝑈̆⁻¹(𝜏)𝑃₀𝛼(𝜏)𝑈(𝜏)𝑑𝜏≤𝜒^𝑡₀𝑈̆(𝑡)𝑈̆⁻¹(𝜏)²𝑑𝜏^1/2+𝜒^𝑡₀𝛼(𝜏)𝑈(𝜏)𝑑𝜏^1/2. (A.17)
Claim 1 (𝜒(∫^𝑡₀‖𝑈̆(𝑡)𝑈̆⁻¹(𝜏)‖²𝑑𝜏)≤0). Considering the linear system 𝑢̇(𝑡)=𝜕𝑓(𝑠(𝑡𝑢(𝑡),𝜕𝑠)) (A.18) due to its regularity and the boundedness of its coefficients, there exists a Lyapunov transform 𝐿(𝑡) such that letting 𝑢(𝑡)=𝐿(𝑡)𝑣(𝑡), consider the transformed linear system 𝑣̇(𝑡)=𝐿⁻¹(𝑡)𝜕𝑓(𝑠(𝑡𝐿(𝑡)𝜕𝑠))−𝐿⁻¹(𝑡)𝐿̇(𝑡𝑣(𝑡)=𝐴̆(𝑡)𝑣(𝑡).) (A.19) Let solution matrix 𝑉̆(𝑡)=(𝑣̆_𝑖𝑗(𝑡))^𝑛_𝑖,𝑗=1, 𝐴̆(𝑡)=(𝑎̆_𝑖𝑗(𝑡))^𝑛_𝑖,𝑗=1 which satisfies that 𝐴̆(𝑡) and 𝑉̆(𝑡) are lowertriangular. And its Lyapunov exponents can be written as follows: 𝜎_𝑖=lim_𝑡→∞1𝑡^𝑡₀𝑎̆_𝑖𝑖(𝜏)𝑑𝜏, (A.20) which are just the Lyapunov exponents of the regular linear system (A.18), 𝑖=1,2,…,𝑛. We have 𝜒[𝑣̆_𝑖𝑖(𝑡)]=𝜎_𝑖 and 𝑣̆_𝑘+1,𝑘(𝑡)=𝑒^{∫𝑡0̆𝑎𝑘+1,𝑘+1(𝜏)𝑑𝜏𝑡}₀𝑒^{−∫𝜏0̆𝑎𝑘+1,𝑘+1(𝜗)𝑑𝜗}𝑎̆_𝑘+1,𝑘(𝜏)𝑣̆_𝑘,𝑘(𝜏)𝑑𝜏. (A.21) This implies 𝜒𝑣̆_𝑘+1,𝑘(𝑡)≤𝜎_𝑘+1−𝜎_𝑘+1+0+𝜎_𝑘=𝜎_𝑘. (A.22) By induction, we can conclude that 𝜒[𝑣̆_𝑗𝑘(𝑡)]≤𝜎_𝑘 for all 𝑗>𝑘. For 𝑗<𝑘, 𝜒[𝑣̆_𝑗𝑘(𝑡)]=−∞ due to the lower-triangularity of the matrix 𝑉̆(𝑡).
Considering the lower-triangular matrix 𝑉̆⁻¹(𝑡)=(𝑤̆_𝑖𝑗)^𝑛_𝑖,𝑗=1, its transpose (𝑉̆⁻¹(𝑡))^𝖳 can be regarded as the solution matrix of the adjoint system of (A.18): 𝑤̇(𝑡)=−𝐴̆^𝖳(𝑡)𝑤(𝑡), (A.23) which is also regular. By the same arguments, we can conclude that 𝜒[𝑤̆_𝑘𝑘]=−𝜎_𝑘 for all 𝑘=1,2,…,𝑛, 𝜒[𝑤̆_𝑗𝑘]≤−𝜎_𝑘 for all 𝑘>𝑗, and 𝜒[𝑤̆_𝑗𝑘]=−∞ for all 𝑘<𝑗. Therefore, for each 𝑖>𝑗, max_𝑖,𝑗𝜒^𝑡₀𝑈̆(𝑡)𝑈̆⁻¹(𝜏)_𝑖𝑗𝑑𝜏≤max_𝑖,𝑗𝜒^𝑡₀𝑉̆(𝑡)𝑉̆⁻¹(𝜏)_𝑖𝑗𝑑𝜏≤max_𝑖,𝑗𝜒^𝑡_{0𝑗≤𝑘≤𝑖}𝑣̆_𝑖𝑘(𝑡)𝑤̆_𝑘𝑗(𝜏)𝑑𝜏≤max_𝑖,𝑗max_{𝑗≤𝑘≤𝑖}𝜒^𝑡₀𝑣̆_𝑖𝑘(𝑡)𝑤̆_𝑘𝑗(𝜏)𝑑𝜏≤max_𝑖,𝑗max_{𝑗≤𝑘≤𝑖}𝜎_𝑘−𝜎_𝑘=0. (A.24) This implies that 𝜒[∫^𝑡₀‖𝑈̆(𝑡)𝑈̆⁻¹(𝜏)𝑑𝜏‖²𝑑𝜏]≤0.
Noting that 𝜒^𝑡₀𝛼(𝜏)𝑈(𝜏)²₂𝑑𝜏≤𝜒𝛼(𝑡)𝑈(𝑡)²≤2𝜆_𝑃. (A.25) So, 𝜒[𝑦(𝑡)]≤max{𝜆_𝑆,𝜆_𝑃}=𝜆_𝑃. This leads to 𝜒[𝜙(𝑡)]≤𝜆_𝑃. This implies that 𝜆_𝑃=max{𝜆_𝑆,𝜆_𝑇}. Thus, 𝜆_𝑃=𝜆_𝑇 can be concluded due to 𝜆_𝑃>𝜆_𝑆.
Case 2 (𝜆_𝑃<𝜆_𝑆). For any 𝜖 with 0<𝜖<(𝜆_𝑆−𝜆_𝑃)/3, there exists 𝑇>0 such that 𝑈̆⁻¹(𝜏)≤𝑒^{(−𝜆𝑆+𝜖)𝜏},‖𝛼(𝜏)‖≤𝑒^𝜖𝜏,𝑈(𝜏)≤𝑒^{(𝜆𝑃+𝜀)𝜏} (A.26)
for all 𝑡≥𝑇. Define the subspace of ℝ^𝑛𝑚: 𝑦𝑧𝑉=∶𝑦=−^∞₀𝑃^𝖳₀𝑈̆⁻¹(𝜏)𝑃₀𝛼(𝜏)𝑈(𝜏)𝑑𝜏𝑧, (A.27) which is well defined due to ‖𝑃^𝖳₀𝑈̆⁻¹(𝜏)𝑃₀𝛼(𝜏)𝑈(𝜏)‖≤𝑒^{(3𝜖−𝜆𝑆+𝜆𝑃)𝜏}∈𝐿([𝑇,+∞)). For each 𝜙(𝑡) with initial condition ^𝑦_𝑧∈𝑉, we have 𝜒[𝑧(𝑡)]≤𝜆_𝑃 and 𝜒𝑦(𝑡)=𝜒−𝑃₀⁻¹𝑈̆(𝑡)𝑃₀^∞₀𝑃^𝖳₀𝑈̆⁻¹(𝜏)𝑃₀𝛼(𝜏)𝑈(𝜏)𝑑𝜏+𝑃^𝖳₀𝑈̆(𝑡)𝑃₀^𝑡₀𝑃^𝖳₀𝑈̆⁻¹(𝜏)𝑃₀𝛼(𝜏)𝑈(𝜏)𝑑𝜏𝑧=𝜒−𝑃^𝖳₀𝑈̆(𝑡)^∞_𝑡𝑈̆⁻¹(𝜏)𝑃₀𝛼(𝜏)𝑈(𝜏)𝑑𝜏𝑧≤𝜆_𝑃 (A.28) according to the arguments above. Thus, we have max_𝑢∈𝑉𝜆(𝐷ℱ,𝑢,𝑠₀,𝜔₀)=𝜆_𝑃. Since dim(𝑉)=𝑛(𝑚−1), 𝑉 define the transverse space and 𝜆_𝑇=𝜆_𝑃. This completes the proof.

Proof of Lemma 22. Since 𝐿(𝑡) satisfies Assumption 20, if the initial condition is 𝑢(𝑡₀)=𝟏_𝑚, then the solution must be 𝑢(𝑡)=𝟏_𝑚, which implies that each row sum of 𝑉(𝑡,𝑡₀) is one. Then, we will prove all elements in 𝑉(𝑡,𝑡₀) are nonnegative. Consider the 𝑖th column of 𝑉(𝑡,𝑡₀) denoted by 𝑉^𝑖(𝑡,𝑡₀) which can be regarded as the solution of the following equation: ̇(𝑡)𝑢𝑡𝑢=𝜎𝐿𝑢,₀=𝑒^𝑚_𝑖. (A.29) For any 𝑡≥𝑡₀, if 𝑖₀=𝑖₀(𝑡) is the index with 𝑢_𝑖0(𝑡)=min_{𝑖=1,2,…,𝑚}𝑢_𝑖(𝑡), we have 𝑢̇_𝑖0(𝑡)=∑^𝑚_𝑗=1𝜎𝑙_𝑖0𝑗(𝑢_𝑗(𝑡)−𝑢_𝑖0(𝑡))≥0. This implies that min_{𝑖=1,2,…,𝑚}𝑢_𝑖(𝑡) is always nondecreasing for all 𝑡≥𝑡₀. Therefore, 𝑢_𝑖(𝑡)≥0 holds for all 𝑖=1,2,…,𝑚 and 𝑡≥𝑡₀. We can conclude that 𝑉(𝑡,𝑡₀) is a stochastic matrix. The proof is completed.

Proof of Lemma 29. Consider the following Cauchy problem: 𝑢̇_𝑖(𝑡)=^𝑚_𝑗=1𝜎𝑙_𝑖𝑗(𝑡)𝑢_𝑗(𝑡),𝑢_𝑖𝑡₀=1,𝑖=𝑘,0,otherwise,𝑖=1,2,…,𝑚. (A.30) Noting that 𝑢̇_𝑘(𝑡)≥𝜎𝑙_𝑘𝑘𝑢_𝑘, we have 𝑢_𝑘(𝑡)≥𝑒^{−𝑀1(𝑡−𝑡0)}. For each 𝑖≠𝑘, since 𝑢_𝑖(𝑡)≥0 for all 𝑖=1,2,…,𝑚 and 𝑡≥𝑡₀, we have 𝑢_𝑖(𝑡)=_𝑗≠𝑖^𝑡_𝑡0𝑒^{∫𝑡𝜏𝜎𝑙𝑖𝑖(𝜗)𝑑𝜗}𝜎𝑙_𝑖𝑗(𝜏)𝑢_𝑗(𝜏)≥𝑑𝜏_𝑡𝑡0𝑒^{∫𝑡𝜏𝜎𝑙𝑖𝑖(𝜗)𝑑𝜗}𝜎𝑙_𝑖𝑘(𝜏)𝑢_𝑘(𝜏)≥𝑑𝜏_𝑡𝑖0𝑒^{−𝑀1(𝑡−𝜏)}𝑒^{−𝑀1(𝜏−𝑡0)}𝜎𝑙_𝑖𝑘(𝜏)𝑑𝜏=𝑒^{−𝑀1(𝑡−𝑡0)𝑡}_𝑡0𝜎𝑙_𝑖𝑘(𝜏)𝑑𝜏. (A.31) So, if there exists a 𝛿-edge from vertex 𝑗 to 𝑖 across [𝑡₀,𝑡₀+𝑇], then we have 𝑣_𝑖𝑗(𝑡₀+𝑇,𝑡₀)≥𝑒^−𝑀1𝑇𝛿. Let 𝛿₂=min{𝑒^−𝑀1𝑇,𝑒^−𝑀1𝑇𝛿}. We can see that 𝑉(𝑡,𝑡₀) has a 𝛿₂ spanning tree across any 𝑇-length time interval. Therefore, according to [31, 32], there exist 𝛿₁>0 and 𝑇₁=(𝑚−1)𝑇 such that 𝑉(𝑡,𝑡₀) is 𝛿₁ scrambling across any 𝑇₁-length time interval. The Lemma is proved.

Acknowledgments

This work is jointly supported by the National Key Basic Research and Development Program (no. 2010CB731403), the National Natural Sciences Foundation of China under Grant (nos. 61273211 and 61273309), the Shanghai Rising-Star Program (no. 11QA1400400), the Marie Curie International Incoming Fellowship from the European Commission under Grant FP7-PEOPLE-2011-IIF-302421, and the Laboratory of Mathematics for Nonlinear Science, Fudan University.