Pinning Synchronization of Nonlinearly Coupled Complex Dynamical Networks on Time Scales

In this paper, we study the synchronization problem for nonlinearly coupled complex dynamical networks on time scales. To achieve synchronization for nonlinearly coupled complex dynamical networks on time scales, a pinning control strategy is designed. Some pinning synchronization criteria are established for nonlinearly coupled complex dynamical networks on time scales, which guarantee the whole network can be pinned to some desired state. The model investigated in this paper generalizes the continuous-time and discrete-time nonlinearly coupled complex dynamical networks to a unique and general framework. Moreover, two numerical examples are given for illustration and verification of the obtained results.


Introduction
Complex networks are an important part of our daily life and in nature, due to many systems in the real world which can be modeled by complex dynamical networks, such as the Internet, World Wide Web, and food webs [1]. There have been a lot of researches on complex networks and neural networks (see, for example, [2][3][4][5][6][7][8] and references therein). Synchronization of complex dynamical network has been a hot topic in the past decades [9][10][11][12][13][14][15][16]. When considering synchronization problem of complex dynamical network, the controlled synchronization problem of complex dynamical network is significant. In recent years, the pinning synchronization of complex dynamical networks, which means the network to achieve desired synchronization by applying control to a small fraction of network nodes, has become a topic of great interest; see [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In particular, Liu and Chen [16] investigated the global synchronization for nonlinearly coupled complex networks. Further, they investigated pinning synchronization for continuous-time nonlinearly coupled networks; see [32]. In [33], a pinning control scheme was developed for continuous-time nonlinearly coupled complex dynamical network, while the results were extended to discrete-time case. In [34], the synchronization of continuous-time dynamical networks with nonlinearly coupling function was considered.
In real life, the time domains do not always match the known continuous-time intervals or discrete integer time domains. From practical point of view, it is important to study complex dynamic networks on general time domains. This is the starting point of the present investigation. Recently, synchronization of complex dynamical networks on time scales has attracted considerable attention [35][36][37][38][39][40][41][42][43], which contains not only synchronization of continuoustime and discrete-time complex dynamical networks but also some continuous-time intervals accompanying some discrete moments.
Motivated by the aforementioned discussions, the synchronization of nonlinearly coupled complex dynamical networks on time scales by applying pinning control scheme will be investigated. The objective in this paper is driving the whole network to some desired state by pinning control strategy. By investigating pinning controlled networks on time scales, some sufficient conditions are presented to guarantee the realization of pinning synchronization for nonlinearly coupled complex dynamical networks on time scales. The main contributions of this paper are listed as follows: (i) The model investigated in this paper generalizes the continuous-time and discrete-time nonlinearly coupled complex dynamical networks to a unique and general framework. Therefore, the obtained results include continuous-time and discrete-time nonlinearly coupled complex dynamical networks as special cases. Moreover, the model investigated in this paper is more general. Our results can be applied to investigate pinning synchronization of nonlinearly coupled complex dynamical networks on a mixed time domain (ii) Linearly coupled complex dynamical networks on time scale are included as a special case of the present work The rest of this paper is organized as follows. Some foundational knowledge about time scales and some notations and supporting lemmas are simply outlined in Section 2. In Section 3, the pinning synchronization problem of nonlinearly coupled complex dynamical networks on time scales is formulated. In Section 4, the main theorems and some corollaries are established. In Section 5, two numerical examples are given to verify the effectiveness of our results. Finally, conclusions are provided in Section 6.

Preliminaries
In this section, we will present some foundational knowledge about time scales and some notations and lemmas which are needed later.
2.1. Foundational Knowledge on Time Scales. Throughout this paper, ℕ, ℤ, and ℝ denote the sets of positive integers, integers, and real numbers, respectively. A time scale is defined as a nonempty closed subset of ℝ and denoted by T .
Definition 1 (see [44,45]). Let t ∈ T . Define the forward jump operator σ : T ⟶ T by σðtÞ = inf fτ ∈ T : τ > tg, while the backward jump operator ρ : T ⟶ T is defined by ρðtÞ = sup fτ ∈ T : τ < tg. In this definition, we put inf ∅ = sup T and sup∅ = inf T , where ∅ denotes the empty set. If σðtÞ > t, we say that t is right-scattered, while if ρðtÞ < t, we say that t is left-scattered. Also, if t < sup T and σðtÞ = t, then t is called right-dense, and if t > inf T and ρðtÞ = t, then t is called left-dense. The graininess function μ : T ⟶ ½0, +∞Þ is defined by μðtÞ ≔ σðtÞ − t. The set T κ is derived from the time scale T as follows: if T has a left-scattered maximum m, then T κ = T \ fmg. Otherwise, T κ = T .
Definition 2 (see [44]). Let f : T ⟶ ℝ. Define the function f σ : T ⟶ ℝ by f σ ðtÞ = f ðσðtÞÞ for all t ∈ T, i.e., f σ = f ∘ σ: Definition 3 (see [44,45]). Assume that f : T ⟶ ℝ and t ∈ T κ , then f is called Δ-differentiable at the point t if there exists θ ∈ ℝ such that for any given ε > 0, there is an open neighborhood U of the point t such that In this case, θ is called the Δ-derivative of f at the point t and we denote it by θ = f Δ ðtÞ. Moreover, we say that f is Δ-differentiable (or in short: differentiable) on T κ provided f Δ ðtÞ exists for all t ∈ T κ . The function f Δ : T κ ⟶ ℝ is called the Δ-derivative of f on T κ . If F Δ ðtÞ = f ðtÞ, t ∈ T κ , then for any a, b ∈ T , the integral is defined as follows: Lemma 5 (see [44,45]). If f , g : Lemma 6 (see [44,45]). If f : T ⟶ ℝ is differentiable at t ∈ T κ , then f σ ðtÞ = f ðtÞ + μðtÞf Δ ðtÞ: Definition 7 (see [44,45]). A function f : T ⟶ ℝ is called rd -continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T . The set of these functions is denoted by C rd ðT ; ℝÞ.
Definition 9 (see [44]). If p ∈ RðT ; ℝÞ, then we define the exponential function by where the cylinder transformation ξ h ðzÞ is defined by where Log is the principal logarithm function.

Notations and Supporting Lemmas.
For each interval I of ℝ, I ∩ T is denoted by I T . ℝ n denotes the n-dimensional Euclidean space with the Euclidean norm ∥·∥. ℝ m×n denotes the set of all m × n real matrices. Let I N ∈ ℝ N×N be the Ndimensional identity matrix, diag ðd 1 , d 2 , ⋯, d N Þ indicate the diagonal matrix with diagonal entries d 1 to d N , A T be the transpose of matrix A, and λ min ð·Þ and λ max ð·Þ represent the minimum eigenvalue and the maximum eigenvalue of a real symmetric matrix. For a symmetric matrix P ∈ ℝ n×n , write P > 0 ðP < 0, P ≥ 0, and P ≤ 0, respectivelyÞ if P is positive definite (negative definite, positive semidefinite, and negative semidefinite, respectively). For square matrices A and B, the notation A ≥ B ðA ≤ BÞ means that A − B is a positive semidefinite (negative semidefinite) matrix. The symbol ⊗ denotes the Kronecker product.
⋯, NÞ. If U = U T , and each row sum of U is zero, then Lemma 16 (see [26].
Then, all the eigenvalues of B are less than 0.

Problem Formulations
Throughout the rest of the paper, let T be a time scale with 0 ∈ T and sup T = +∞. In this section, the nonlinearly coupled complex dynamical network on the time scale T will be introduced. In general, the dynamic for each isolated (uncoupled) node of the dynamical network can be described as where sðtÞ = ðs 1 ðtÞ, s 2 ðtÞ, ⋯, s n ðtÞÞ T ∈ ℝ n , s Δ is the Δ-derivative of s on ½0, +∞Þ T , f : ℝ n ⟶ ℝ n is continuous and of such a nature that existence and uniqueness of solutions to dynamic equation (10) subject to sð0Þ = s 0 ðs 0 ∈ ℝ n Þ as well as their dependence on initial values is guaranteed. Suppose that the dynamical network consists of N identical nodes, with each node being an n-dimensional dynamical system. Then, the nonlinearly coupled dynamical network can be described by where x i ðtÞ = ðx i1 ðtÞ, x i2 ðtÞ, ⋯, x in ðtÞÞ T ∈ ℝ n is the state vector of the ith node at time t; the constant c > 0 represents the 3 Advances in Mathematical Physics coupling strength of network; the nonlinearly coupled function h : ℝ n ⟶ ℝ n is continuous, which satisfies standard assumptions on existence and uniqueness of solutions to dynamic equation (11) subject to xð0Þ = x 0 ðx 0 ∈ ℝ n Þ as well as their dependence on initial values; the coupling configuration matrix G = ðg ij Þ ∈ ℝ N×N represents the topological structure of the complex network and is defined as follows: if there exists a connection between the node i and the node j (i ≠ j), then g ij = g ji = 1; otherwise, g ij = g ji = 0, and (11) is connected in the sense of having no isolated clusters, which means that the coupling configuration matrix G is irreducible.
Suppose that sðtÞ is a solution of the uncoupled system (10). In order to synchronize the network (11) to the objective state sðtÞ, we will design a pinning control scheme, if the network (11) cannot synchronize to the objective state s ðtÞ without control. Without loss of generality, we add the controllers to the first l nodes. Hence, we have the pinningcontrolled network as follows: with the feedback controllers given by where Definition 18 (see [30]). The network (12) is said to be synchronized by pinning control, if We get the following error dynamical network by letting z i ðtÞ = x i ðtÞ − sðtÞ ∈ ℝ n ði = 1, 2, ⋯, NÞ: It follows from Lemma 16 that the symmetric matrix G − D is negative definite, and so the maximal eigenvalue λ max ðG − DÞ < 0. From Lemma 17((1)-(3)), it is easy to prove that the matrix ðG − DÞ 2 is symmetric positive definite.

Pinning Synchronization Criteria for Nonlinearly Coupled Complex Dynamical Networks on Time Scales
In order to derive the synchronization criteria for the pinning-controlled network (12), we make the following assumptions: (A1) (see [18]). The function f : ℝ n ⟶ ℝ n is assumed to satisfy Lipschitz condition, that is, there exists a constant l 1 > 0 such that (A2) (see [9,21,49]). There exists a constant l 2 > 0, such that (A4) (see [18]). The function h : ℝ n ⟶ ℝ n is assumed to satisfy Lipschitz condition, that is, there exists a constant l 3 > 0 such that We have the following theorems and corollaries, which give some sufficient conditions to guarantee synchronization of the network (12) by pinning control.
Proof. Lemma 11 yields lim t→∞ e α ðt, 0Þ = 0. Choose the Lyapunov function VðtÞ = ∑ N i=1 z T i ðtÞz i ðtÞ, t ∈ ½0, +∞Þ T : Note that The rest proof of the theorem is analogy with the proof of Theorem 19.
Next, let us consider the particular case when the function hðxÞ is linear: hðxÞ = Γx, where Γ ∈ ℝ n×n . The network (11) can be written as
By Theorem 21 and Corollary 23, we have the following corollary.

Corollary 24.
Let μðtÞ ≤ μ * for all t ∈ T and Γ be symmetric positive definite. Suppose that assumptions (A1) and (A3 ′ ) hold, and K is symmetric in assumption (A3 ′ ). The network (38) is synchronized by pinning control if γ = λ max ð2l 1 I nN + 2 cλ min ðΓÞ½ðG − DÞ ⊗ I n + μ * ½l 2 1 I nN + 2cððG − DÞ ⊗ KÞ + c 2 ðλ max ðΓÞÞ 2 λ max ððG − DÞ 2 ÞI nN Þ < 0 and γ ∈ R + ðT ; ℝÞ: Obviously, the research of synchronization problem for nonlinearly coupled complex dynamical networks on time scales is more general. It contains continuous-time and discrete-time nonlinearly coupled complex dynamical networks. In addition, it contains linearly coupled complex dynamical networks on time scales. Remark 25. According to the previous works [12,26,31,35,39,40], the synchronization problem of complex dynamical networks with delay on time scales by pinning control strategy can be investigated, but there are some challenges for the effects of time delays and the various forms of time scales.

Advances in Mathematical Physics
Note that each isolated node of network (42) is a 2dimensional nonlinear system described by The objective here is to synchronize the network (42) to the solution s = 0 (2-dimensional zero vector) of (45) by pinning control. From Figure 1, we see the complex  Figure 2 also shows that the synchronization is realized.
Example 2. Consider 2-dimensional nonlinearly star-shaped network with five nodes on time scale T , which is described by Note that each isolated node of network (47) is a 2dimensional nonlinear system described by The objective here is to synchronize the network (47) to the solution s = 0 (2-dimensional zero vector) of (49) by pinning control. From Figure 3, we see the complex dynamical network (47) without control cannot reach synchronization with s = 0. Now, we apply pinning control to the network (47) with matrix D = diag ð2:5,2:5,2:5,2:5,0Þ. By calculations, all conditions of Corollary 24 are fulfilled. By Corollary 24, the network (47) can achieve synchronization under the above pinning control strategy. Figure 4 also shows that the synchronization is realized.

Conclusions
In this paper, we have investigated the synchronization problem of nonlinearly coupled complex dynamical networks on time scales by pinning control strategy. Some pinning synchronization criteria have been established which guarantee that the nonlinearly coupled complex dynamical networks on time scales can be pinned to some desired state. Two numerical examples have been given to verify the effectiveness of the obtained results.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The author declares that there is no conflict of interests regarding the publication of this paper.