1 Introduction

Complex network field has made great progress in recent years, in which the nodes denote individuals of network and the part of connection is called as the edges. In general, the dynamical systems are abstracted as nodes, which interacts to constitute a complex network^[1]. At present, scientists in different fields have researched on complex networks from physical to biological and social sciences^[2-4].

Complex network model can explain some complex phenomena, for example, power grid^[5] metabolic networks^[6], gene networks^[7], and so on. Synchronization is vital in complex networks, and it has raised the concern of many academic researchers since it has a significant impact on many field, such as, in the field of secure communication, information science and biological systems. Many varieties of synchronizations including complex complete synchronization^[8-9], cluster synchronization^[10-12] and pinning adaptive synchronization^[13] etc. have been studied.

Compared with integer order, fractional calculus is its extension, which is used to study the mathematical properties of arbitrary order calculus, and it has been of great importance in the field of engineering and physics because some applications of fractional calculus have obtained great progress in last few years^[14-15]. Modeling and simulating some systems by using the concept of fractional order may be closer to reality because the actual phenomenon are generally fractional^[16]. So, compared to the integer order method, modeling method of fractional order calculus can make us more accurately describe and simulate the real process. fractional-order complex networks′ synchronization issue has gained the attention of the researchers in recent years ^[17-20]. Many fractional-order differential systems are certified to be chaotic or hyperchaotic, for example, Rössler system^[21]. After these discoveries, owing to great value in the field of control processing and information science, hyper-chaotic differential systems′ synchronization is a central issue.

With regard to synchronization, inner synchronization is happening in a complex network and outer synchronization is appearing in two coupled networks^[22]. In general, as the complex network's typical synchronization research, inner synchronization has been broadly studied. Wang et al.^[23] and Chen et al.^[24] have respectively investigated single complex network's inner synchronization. Outer synchronization has also been studied. Qin et al.^[25]and Wang et al.^[26] have studied two complex networks′ outer synchronization respectively. However, the above-mentioned inner or outer synchronization was not discussed at the same time. Today, complex dynamical networks′ hybrid synchronization is everywhere^[27], for example, infectious disease transmits between different communities and within the group at the same time. Therefore, it is very meaningful that how to reach the two coupled networks′ hybrid synchronization. Recently, the complex networks′ hybrid synchronization gives rise to attention greatly, which was respectively studied in Refs.[28-30] in two complex networks with fractional-order. According to our understanding, the above articles were focused on the hybrid synchronization by different control methods, few researches have been made on the hybrid synchronization via fractional order control.

Due to the flexibility and completeness of fractional-order control, more and more attention has been paid^{[15, 31-33]}. The well-known fractional-order controllers have PI^λD^μ controller^[15], the TID controller^[32], and the CRONE controller^[33]. Compared to traditional controllers, it has been shown in above articles that the fractional-order controller implements easily.

Under the above discussion, we use a fractional order controller to control two complex networks to reach hybrid synchronization in this article, where hyper-chaotic systems with fractional-order as nodes in complex network give a hybrid synchronization's law via the Lasalle invariance principle and Lyapunov stability theorem. In the practical application, the inner and outer synchronization are discussed at the same time, which is more reasonable. In this article, the given controller is ordinary and can be applied to fractional order complex networks with chaotic (Chen system^[26], Lorenz system^[26], Liu system^[34]) or hyperchaotic type (hyperchaotic Chen system^[34]).

Next is the organization of this article. Mathematical preliminaries of fractional derivative and network model are provided in the second part. In the third part, we give hybrid synchronization definitions. Then in the fourth part, according to a controller with fractional-order, the two coupled networks' hybrid synchronization is discussed. Examples are given in the fifth part. This paper concludes with a summary of this work in the sixth part.

Expression of some notations in this paper, such as ‖·‖, A^T, I, $\otimes $ and A^s are in Ref.[28].

2 Background Knowledge 2.1 Definition of Fractional Derivative^[28]

$ \mathit{\boldsymbol{D}}_t^{\rm{q}}x\left( t \right) = \left\{ \begin{array}{l} \frac{1}{{\mathit{\Gamma }\left( {n - q} \right)}}\int_a^t {{{\left( {t - \tau } \right)}^{n - q - 1}}{x^{\left( n \right)}}\left( \tau \right){\rm{d}}\tau } ,\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n - 1 < q < n\\ \frac{{{{\rm{d}}^n}}}{{{\rm{d}}{t^n}}}x\left( t \right),\;\;\;\;\;\;\;\;q = n \end{array} \right. $

(1)

in which $\mathit{\Gamma} \left( z \right) = \int_0^\infty {{e^{-z}}{t^{z-1}}{\text{d}}t} $, where Γ is called as the gamma function, n is the lowest integer which is more than or equal to q, the order q∈(0, 1).

2.2 Model Introduction

A complex network containing N nodes that we introduce^[35]:

$ {\mathit{\boldsymbol{D}}^{\rm{q}}}{\mathit{\boldsymbol{x}}_i}\left( t \right) = \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + c\sum\limits_{j = 1}^N {{a_{ij}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}{x_j}\left( t \right)} $

(2)

where the order q∈(0, 1), the ith node's state variable x_i belongs to Rⁿ, continuously differentiable vector function f belongs to Rⁿ, the network topology A is equal to (a_ij)_N×N; the inner coupling matrix Γ=diag(ζ₁, ζ₂, ..., ζ_n) belongs to R^n×n, c represents the network's coupling strength. a_ij is the element of matrix A (the ith row and the jth column): if a_ij satisfies 0 < a_ij, then there is a connection from node i to node j(i≠j), otherwise, a_ij is equal to 0.

$ {a_{ii}} = - \sum\limits_{j = 1,j \ne i}^N {{a_{ij}}} $

(3)

The matrix A's diagonal elements satisfy the above equation.

Model (2) is the driving network. System (2) plus a control function φ_i(u_i) equals the controlled response system, which is:

$ {\mathit{\boldsymbol{D}}^{\rm{q}}}{\mathit{\boldsymbol{y}}_i}\left( t \right) = \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) + c\sum\limits_{j = 1}^N {{a_{ij}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}{\mathit{\boldsymbol{y}}_j}\left( t \right)} + {\mathit{\boldsymbol{\varphi }}_i}\left( {{\mathit{\boldsymbol{u}}_i}} \right) $

(4)

where the ith node's response state vector y_i belongs to Rⁿ, the explanation of f, c, Γ as those in Eq.(2), continuous control functions u_i and φ_i:Rⁿ→Rⁿ.

3 Preparation Knowledge of Mathematics

Definition 1^[30]  We say that the hybrid synchronization in the network (2) and the controlled network (4) is reached when the equations as follows are satisfied:

$ \left\{ \begin{array}{l} \mathop {\lim }\limits_{t \to \infty } \left\| {{x_i}\left( {t,{X_0}} \right) - {x_j}\left( {t,{X_0}} \right)} \right\| = 0\\ \mathop {\lim }\limits_{t \to \infty } \left\| {{y_i}\left( {t,{Y_0}} \right) - {y_j}\left( {t,{Y_0}} \right)} \right\| = 0\\ \mathop {\lim }\limits_{t \to \infty } \left\| {{y_i}\left( {t,{Y_0}} \right) - {x_i}\left( {t,{X_0}} \right)} \right\| = 0 \end{array} \right. $

(5)

Assumption 1^[36]  According to Function class QUAD(P, Δ): Suppose that Δ is equal to diag(ρ₁, ρ₂, ..., ρ_n), matrix P is positive definite and also equal to diag(p₁, p₂, ..., p_n), continuous function f(x) belongs to Rⁿ. When the inequalities as follows are satisfied, then f∈QUAD(P, Δ)

$ \begin{array}{l} - \alpha {\left( {\mathit{\boldsymbol{x}} - \mathit{\boldsymbol{y}}} \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{x}} - \mathit{\boldsymbol{y}}} \right) \ge {\left( {\mathit{\boldsymbol{x}} - \mathit{\boldsymbol{y}}} \right)^{\rm{T}}}P\left( {\mathit{\boldsymbol{f}}\left( \mathit{\boldsymbol{x}} \right) - } \right.\\ \;\;\;\;\;\left. {\mathit{\boldsymbol{f}}\left( \mathit{\boldsymbol{y}} \right) - \Delta \left( {\mathit{\boldsymbol{x}} - \mathit{\boldsymbol{y}}} \right)} \right) \end{array} $

(6)

supports for all x belongs to Rⁿ, y belongs to Rⁿ, t is positive and some α is positive.

Definition 2^[37]  Assume that A∈R^N×N meets the two conditions as follows:

(ⅰ) ${a_{ii}} =-\sum\limits_{j = 1, j \ne i}^N {{a_{ij}}} $ and a_ij(i≠j) is positive;

(ⅱ) A is irreducible, and it satisfies:

1) One of the eigenvalues of A is 0 with the multiplicity 1, and other eigenvalues' real part are negative number.

2) Regarding the eigenvalue 0, (1, 1, ...1)^T is the A's right eigenvector.

3) Regarding the eigenvalue 0, ξ is equal to (ξ₁, ξ₂, ..., ξ_N)^T is the A's left eigenvector, and it satisfies $\sum\limits_{i = 1}^N {{\xi _i} = 1} $, where ξ_i is positive.

Assume that A is irreducible. According to Definition 2, G and L is defined as diag(ξ₁, ξ₂, ..., ξ_N) and G－ξξ^T respectively, in which the matrix L and ξξ^T are symmetric, the matrix G is positive definite and also diagonal matrix.

4 The Hybrid Synchronization Analysis

For the sake of convenience, we describe the specific content of the hybrid synchronization from the following two aspects including the inner synchronization and the outer synchronization.

The Eq.(2) can be rewritten as^[28]:

$ {\mathit{\boldsymbol{D}}^{\rm{q}}}\mathit{\boldsymbol{X}}\left( t \right) = \mathit{\boldsymbol{F}}\left( {\mathit{\boldsymbol{X}}\left( t \right)} \right) + \left( {c\mathit{\boldsymbol{A}} \otimes \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}} \right)\mathit{\boldsymbol{X}}\left( t \right) $

(7)

where

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{X}}\left( t \right) = {{\left( {\mathit{\boldsymbol{x}}_1^{\rm{T}}\left( t \right),\mathit{\boldsymbol{x}}_2^{\rm{T}}\left( t \right), \cdots ,\mathit{\boldsymbol{x}}_N^{\rm{T}}\left( t \right)} \right)}^{\rm{T}}}}\\ {\mathit{\boldsymbol{F}}\left( {\mathit{\boldsymbol{X}}\left( t \right)} \right) = \left( {{\mathit{\boldsymbol{f}}^{\rm{T}}}\left( {{\mathit{\boldsymbol{x}}_1}\left( t \right)} \right),{\mathit{\boldsymbol{f}}^{\rm{T}}}\left( {{\mathit{\boldsymbol{x}}_2}\left( t \right)} \right), \cdots ,{\mathit{\boldsymbol{f}}^{\rm{T}}}{{\left( {{\mathit{\boldsymbol{x}}_N}\left( t \right)} \right)}^{\rm{T}}}} \right.} \end{array} $

When s(t) is equal to $\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{\xi }}_i}{\mathit{\boldsymbol{x}}_i}\left( t \right)} $ which is used to describe the network (2)'s inner synchronization state, then δ_i(t) is equal to x_i(t)－s(t) that represents the network (2)'s inner synchronization error, e_i(t) is equal to y_i(t)－x_i(t) that shows the network (2) and the controlled network (4)'s outer synchronization error.

Therefore, e_i(t) can be rewritten as:

$ \begin{array}{l} {\mathit{\boldsymbol{D}}^q}{\mathit{\boldsymbol{e}}_i}\left( t \right) = \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right)} \right) - \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + c\sum\limits_{j = 1}^N {{a_{ij}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}{\mathit{\boldsymbol{e}}_j}} + \\ \;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{\varphi }}_i}\left( {{\mathit{\boldsymbol{u}}_i}} \right) = \mathit{\boldsymbol{B}}\left( {{\mathit{\boldsymbol{y}}_i}\left( t \right),{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right){\mathit{\boldsymbol{e}}_i} + c\sum\limits_{j = 1}^N {{a_{ij}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}{\mathit{\boldsymbol{e}}_j}} + \\ \;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{\varphi }}_i}\left( {{\mathit{\boldsymbol{u}}_i}} \right) \end{array} $

(8)

where

$ {\mathit{\boldsymbol{\varphi }}_i}\left( {{\mathit{\boldsymbol{u}}_i}} \right) = - {\mathit{\boldsymbol{k}}_i}{\mathit{\boldsymbol{u}}_i} $

(9)

where k_i is equal to diag(k_i1, k_i2, ..., k_in), where k_ij is more than or equal to 0, u_i is defined by:

$ {D^{1 - q}}{\mathit{\boldsymbol{u}}_i} = {\mathit{\boldsymbol{\omega }}_i}{\mathit{\boldsymbol{e}}_i} $

(10)

with ω_i is equal to diag(ω_i1, ω_i2, ..., ω_in), where ω_ij is more than or equal to 0. Due to Property 1, 2 in Ref.[26] and Eq.(10), we can obtain:

$ {{\mathit{\boldsymbol{\dot u}}}_i} = {\mathit{\boldsymbol{D}}^1}{\mathit{\boldsymbol{u}}_i} = {\mathit{\boldsymbol{D}}^q}{\mathit{\boldsymbol{D}}^{1 - q}}{\mathit{\boldsymbol{u}}_i} = {\mathit{\boldsymbol{\omega }}_i}{\mathit{\boldsymbol{D}}^q}{\mathit{\boldsymbol{e}}_i} $

(11)

And according to Property 3 in Ref.[26], there is a positive number λ_i>0 for Eq.(10), such that ‖ω(y_i－x_i)‖≤λ_i‖u_i‖, we can get:

$ \left\| {{\mathit{\boldsymbol{e}}_i}} \right\| \le {\lambda _i}/\omega _i^ * \left\| {{\mathit{\boldsymbol{u}}_i}} \right\| $

(12)

where ω_i^* is equal to{ω_ij, j=1, 2, ...n}'s minimum value, and y_i is equal to x_i if u_i=0.

The assumptions as follows are proposed to get some synchronization's sufficient condition.

Assumption 2^[26]  Assume that Γ meets ‖Γ‖= η, in which η is a positive constant.

Assumption 3^[26]  Assume that ‖B(y_i, x_i)‖(i=1, 2, ..., N) is finite, namely, ‖B(y_i, x_i)‖≤M_i, where M_i is a constant.

Theorem 1  Assume that Assumptions 1, 2 and 3 support, the network (2) and the controlled network (4) will reach the hybrid synchronization when the control parameters ω_i and k_i meet the inequalities as follows, and the constants ρ_j and ζ_j are present, where j=1, 2, ..., n.

$ \left( {\rm{i}} \right)0 \ge {\rho _j}L + {\left( {c{\zeta _j}GA} \right)^s} $

(13)

$ \left( {{\rm{ii}}} \right)\omega _i^{ * * }{\lambda _i}\left( {{M_i} + c{a_{ii}}\eta } \right) \le k_i^ * \omega _i^{ * 2},1 \le i \le N $

(14)

$ \begin{array}{l} \left( {{\rm{iii}}} \right)\frac{{{{\left( {N - 1} \right)}^2}{c^2}{\eta ^2}{{\left( {{a_{ij}}\omega _i^{ * * }{\lambda _j}\omega _i^ * + {a_{ji}}\omega _j^{ * * }{\lambda _i}\omega _j^ * } \right)}^2}}}{{4\omega _i^ * \omega _j^ * }} \le \\ \;\;\;\;\;\;\left( {k_i^ * \omega _i^{ * 2} - \omega _i^{ * * }{\lambda _i}\left( {{M_i} + c{a_{ii}}\eta } \right)} \right)\left( {k_j^ * \omega _j^{ * 2} - } \right.\\ \;\;\;\;\;\;\left. {\omega _j^{ * * }{\lambda _j}\left( {{M_j} + c{a_{jj}}\eta } \right)} \right),1 \le i \le j \le N \end{array} $

(15)

where ω_i^* and ω_i^** is equal to {ω_im, m=1, 2, ..., n}'s minimum and maximum values respectively, k_i^* is equal to {k_im, m=1, 2, ..., n}'s minimum value.

Based on Lemma 1, Lemma 2 and Lemma 3 in Ref.[28], the following is the Theorem 1's proof.

Proof  V₁ is a Lyapunov function that we select, in which:

$ {\mathit{\boldsymbol{V}}_1} = \frac{1}{2}\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{\xi }}_i}{\mathit{\boldsymbol{\delta }}_i}{{\left( t \right)}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{\delta }}_i}\left( t \right)} $

(16)

according to Ref.[28], V₁ is rewritten by:

$ {\mathit{\boldsymbol{V}}_1} = \frac{1}{2}\mathit{\boldsymbol{X}}{\left( t \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} \otimes \mathit{\boldsymbol{P}}} \right)\mathit{\boldsymbol{X}}\left( t \right) $

(17)

There is a unitary matrix Q due to the matrix $\mathit{\boldsymbol{L}} \otimes \mathit{\boldsymbol{P}}$ is real symmetric, such that

$ \mathit{\boldsymbol{L}} \otimes \mathit{\boldsymbol{P}} = \mathit{\boldsymbol{Q \boldsymbol{\varLambda} }}{\mathit{\boldsymbol{Q}}^{\rm{T}}} $

(18)

where Λ=diag(λ₁, λ₂, ..., λ_nN), i=1, 2, ..., nN, where λ_i is positive, and Q^TQ is equal to I.

We introduce a function Y(t)=Q^TX(t), so that

$ {\mathit{\boldsymbol{V}}_1} = \frac{1}{2}\mathit{\boldsymbol{Y}}{\left( t \right)^T}\mathit{\boldsymbol{ \boldsymbol{\varLambda} Y}}\left( t \right) $

(19)

There is a (1/2)minλ_i≥α₁>0 and (1/2)maxλ_i≤α₂, that

$ {\alpha _2}\mathit{\boldsymbol{Y}}{\left( t \right)^2} \ge {\mathit{\boldsymbol{V}}_1} \ge {\alpha _1}\mathit{\boldsymbol{Y}}{\left( t \right)^2} $

Based on Lemma 2 in Ref.[28], V₁ changes into

$ \begin{array}{l} {\mathit{\boldsymbol{D}}^{\rm{q}}}{\mathit{\boldsymbol{V}}_1} \le \mathit{\boldsymbol{Y}}{\left( t \right)^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}{\mathit{\boldsymbol{D}}^{\rm{q}}}\mathit{\boldsymbol{Y}}\left( t \right) = \mathit{\boldsymbol{X}}{\left( t \right)^{\rm{T}}}\mathit{\boldsymbol{Q \boldsymbol{\varLambda} }}{\mathit{\boldsymbol{Q}}^{\rm{T}}}{\mathit{\boldsymbol{D}}^{\rm{q}}}\mathit{\boldsymbol{X}}\left( t \right) = \\ \;\;\;\mathit{\boldsymbol{X}}{\left( t \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} \otimes \mathit{\boldsymbol{P}}} \right)\left[ {\mathit{\boldsymbol{F}}\left( {\mathit{\boldsymbol{X}}\left( t \right)} \right) + \left( {c\mathit{\boldsymbol{A}} \otimes \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}} \right)\mathit{\boldsymbol{X}}\left( t \right)} \right] = \\ \;\;\;\mathit{\boldsymbol{X}}{\left( t \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} \otimes \mathit{\boldsymbol{P}}} \right)\left[ {\mathit{\boldsymbol{F}}\left( {\mathit{\boldsymbol{X}}\left( t \right)} \right) - \left( {{\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}} \right)\mathit{\boldsymbol{X}}\left( t \right)} \right] + \\ \;\;\;\mathit{\boldsymbol{X}}{\left( t \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} \otimes \mathit{\boldsymbol{P}}} \right)\left[ {\left( {{\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}} \right)\mathit{\boldsymbol{X}}\left( t \right) + \left( {c\mathit{\boldsymbol{A}} \otimes } \right.} \right.\\ \;\;\;\left. {\left. \mathit{\boldsymbol{ \boldsymbol{\varGamma} }} \right)\mathit{\boldsymbol{X}}\left( t \right)} \right] = - \sum\limits_{i = 1}^{N - 1} {\sum\limits_{j = i + 1}^N {{L_{ij}}{{\left( {{\mathit{\boldsymbol{x}}_j}\left( t \right) - {\mathit{\boldsymbol{x}}_i}\left( t \right)} \right)}^{\rm{T}}}\mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{x}}_j}\left( t \right) - } \right.} \right.} } \\ \;\;\;f\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right) - \Delta \left( {{\mathit{\boldsymbol{x}}_j}\left( t \right) - {\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) + \mathit{\boldsymbol{X}}{{\left( t \right)}^{\rm{T}}}\left( {\mathit{\boldsymbol{L}}{\mathit{\boldsymbol{I}}_N} \otimes } \right.} \right.\\ \;\;\;\left. {\mathit{\boldsymbol{P \boldsymbol{\varDelta} }}} \right)\mathit{\boldsymbol{X}}\left( t \right) + \mathit{\boldsymbol{X}}{\left( t \right)^{\rm{T}}}\left( {c\mathit{\boldsymbol{LA}} \otimes \mathit{\boldsymbol{P \boldsymbol{\varGamma} }}} \right)\mathit{\boldsymbol{X}}\left( t \right) \le \\ \;\;\; - \alpha \sum\limits_{j > i}^N {{L_{ij}}{{\left( {{\mathit{\boldsymbol{x}}_j}\left( t \right) - {\mathit{\boldsymbol{x}}_i}\left( t \right)} \right)}^{\rm{T}}}\left( {{\mathit{\boldsymbol{x}}_j}\left( t \right) - {\mathit{\boldsymbol{x}}_i}\left( t \right)} \right)} + \\ \;\;\;\sum\limits_{j = 1}^n {{p_j}{{\mathit{\boldsymbol{\tilde x}}}_j}{{\left( t \right)}^{\rm{T}}}{\rho _j}{L_j}{{\mathit{\boldsymbol{\tilde x}}}_j}\left( t \right)} + \sum\limits_{j = 1}^n {{p_j}{{\mathit{\boldsymbol{\tilde x}}}_j}{{\left( t \right)}^{\rm{T}}}c{\zeta _j}\mathit{\boldsymbol{LA}}{{\mathit{\boldsymbol{\tilde x}}}_j}\left( t \right)} = - \\ \;\;\;\alpha \mathit{\boldsymbol{X}}{\left( t \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)\mathit{\boldsymbol{X}}\left( t \right) + \sum\limits_{j = 1}^n {{p_j}{{\mathit{\boldsymbol{\tilde x}}}_j}{{\left( t \right)}^{\rm{T}}}\left( {{\rho _j}\mathit{\boldsymbol{L + }}} \right.} \\ \;\;\;\left. {c{\zeta _j}\mathit{\boldsymbol{LA}}} \right){{\mathit{\boldsymbol{\tilde x}}}_j}\left( t \right) \le - \frac{\alpha }{{\max {p_i}}}\mathit{\boldsymbol{X}}{\left( t \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} \otimes \mathit{\boldsymbol{P}}} \right)\mathit{\boldsymbol{X}}\left( t \right) + \\ \;\;\;\sum\limits_{j = 1}^n {{p_j}{{\mathit{\boldsymbol{\tilde x}}}_j}{{\left( t \right)}^{\rm{T}}}\left[ {{\rho _j}\mathit{\boldsymbol{L}} + {{\left( {\mathit{\boldsymbol{c}}{\mathit{\boldsymbol{\zeta }}_\mathit{\boldsymbol{j}}}\mathit{\boldsymbol{LA}}} \right)}^s}} \right]{{\mathit{\boldsymbol{\tilde x}}}_j}\left( t \right)} = \\ \;\;\; - \frac{\alpha }{{\max {p_i}}}\mathit{\boldsymbol{Y}}{\left( t \right)^T}\mathit{\boldsymbol{ \boldsymbol{\varLambda} Y}}\left( t \right) + \sum\limits_{j = 1}^n {{p_j}{{\mathit{\boldsymbol{\tilde x}}}_j}{{\left( t \right)}^{\rm{T}}}\left[ {{\rho _j}\mathit{\boldsymbol{L}} + } \right.} \\ \;\;\;\left. {{{\left( {\mathit{c}{\mathit{\zeta }_\mathit{j}}\mathit{\boldsymbol{LA}}} \right)}^s}} \right]{{\mathit{\boldsymbol{\tilde x}}}_j}\left( t \right) \end{array} $

If

$ {\left( {\mathit{c}{\mathit{\zeta }_\mathit{j}}\mathit{\boldsymbol{LA}}} \right)^s} + {\rho _j}\mathit{\boldsymbol{L}} \le 0 $

(20)

we can obtain:

$ {\mathit{\boldsymbol{D}}^{\rm{q}}}{\mathit{\boldsymbol{V}}_1} \le - \alpha \frac{{\min {\mathit{\boldsymbol{\lambda }}_\mathit{\boldsymbol{i}}}{{\left\| {\mathit{\boldsymbol{Y}}\left( t \right)} \right\|}^2}}}{{\max {p_i}}} $

(21)

where ${\mathit{\boldsymbol{\tilde x}}_j}$ is equal to (x_1j(t), x_2j(t), ..., x_Nj(t))^T. Therefore, the network (2)'s inner synchronization is reached.

Owing to the controlled network (4) meets Assumptions 2 and 3, we can get ‖Γ‖=η and ‖B(y_i, x_i)‖≤M_i. V₂ is a Lyapunov function that we construct.

$ \begin{array}{l} {\mathit{\boldsymbol{V}}_2} = \frac{1}{2}\left( {\mathit{\boldsymbol{u}}_1^{\rm{T}},\mathit{\boldsymbol{u}}_2^{\rm{T}}, \cdots ,\mathit{\boldsymbol{u}}_N^{\rm{T}}} \right){\left( {\mathit{\boldsymbol{u}}_1^{\rm{T}},\mathit{\boldsymbol{u}}_2^{\rm{T}}, \cdots ,\mathit{\boldsymbol{u}}_N^{\rm{T}}} \right)^{\rm{T}}} = \\ \;\;\;\;\;\;\;\frac{1}{2}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{u}}_i^{\rm{T}}{u_i}} \end{array} $

(22)

Differentiating V₂ about t, then we can get:

$ \begin{array}{l} {{\mathit{\boldsymbol{\dot V}}}_2} = \sum\limits_{i = 1}^N {\mathit{\boldsymbol{u}}_i^{\rm{T}}{{\mathit{\boldsymbol{\dot u}}}_i}} = \sum\limits_{i = 1}^N {\mathit{\boldsymbol{u}}_i^{\rm{T}}{\omega _i}{\mathit{\boldsymbol{D}}^{\rm{q}}}{e_i}} = \sum\limits_{i = 1}^N {\mathit{\boldsymbol{u}}_i^{\rm{T}}{\omega _i}B\left( {{y_i}\left( t \right),} \right.} \\ \;\;\;\;\left. {{x_i}\left( t \right)} \right){e_i} + c\sum\limits_{i = 1}^N {\sum\limits_{j = 1,j \ne i}^N {{a_{ij}}\mathit{\boldsymbol{u}}_i^{\rm{T}}{\omega _i}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}{e_j}} } + c\sum\limits_{i = 1}^N {{a_{ii}}\mathit{\boldsymbol{u}}_i^{\rm{T}}{\omega _i}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}{e_i}} - \\ \;\;\;\;\sum\limits_{i = 1}^N {\mathit{\boldsymbol{u}}_i^{\rm{T}}{\omega _i}{k_i}{\mathit{\boldsymbol{u}}_i}} \le \sum\limits_{i = 1}^N {\left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\left\| {{\omega _i}} \right\|} \\ \;\;\;\;\left\| {B\left( {{y_i}\left( t \right),{x_i}\left( t \right)} \right)} \right\|\left\| {{e_i}} \right\| + c\sum\limits_{i = 1}^N {\sum\limits_{j = 1,j \ne i}^N {{a_{ij}}\left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|} } \times \\ \;\;\;\;\left\| {{\omega _i}} \right\|\left\| \mathit{\boldsymbol{ \boldsymbol{\varGamma} }} \right\|\left\| {{e_j}} \right\| + c\sum\limits_{i = 1}^N {{a_{ii}}\left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\left\| {{\omega _i}} \right\|\left\| \mathit{\boldsymbol{ \boldsymbol{\varGamma} }} \right\|} \\ \;\;\;\;\left\| {{e_i}} \right\| - \sum\limits_{i = 1}^N {\mathit{\boldsymbol{u}}_i^{\rm{T}}{\omega _i}{k_i}{\mathit{\boldsymbol{u}}_i}} \le \sum\limits_{i = 1}^N {\omega _i^{ * * }{M_i}{\lambda _i}} /\\ \;\;\;\;\omega _i^ * \left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\left\| {{\mathit{\boldsymbol{u}}_i}} \right\| + c\sum\limits_{i = 1}^N {\sum\limits_{j = 1,j \ne i}^N {{a_{ij}}\omega _i^{ * * }\eta {\lambda _j}} } /\\ \;\;\;\;\omega _j^ * \left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\left\| {{\mathit{\boldsymbol{u}}_j}} \right\| + c\sum\limits_{i = 1}^N {{a_{ii}}\omega _i^{ * * }\eta {\lambda _j}} /\omega _i^ * \left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\left\| {{\mathit{\boldsymbol{u}}_i}} \right\| - \\ \;\;\;\;\sum\limits_{i = 1}^N {\omega _i^ * k_i^ * {M_i}\mathit{\boldsymbol{u}}_i^{\rm{T}}{\mathit{\boldsymbol{u}}_i}} = \sum\limits_{i = 1}^N {\left( {\omega _i^{ * * }{M_i}{\lambda _i}/\omega _i^ * } \right.} + \\ \;\;\;\;\left. {c{a_{ii}}\omega _i^{ * * }\eta {\lambda _i}/\omega _i^ * - \omega _i^ * k_i^ * } \right]\left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\left\| {{\mathit{\boldsymbol{u}}_i}} \right\| + \\ \;\;\;\;c\sum\limits_{i = 1}^N {\sum\limits_{j = 1,j \ne i}^N {{a_{ij}}\omega _i^{ * * }\eta {\lambda _j}} } /\omega _j^ * \left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\left\| {{\mathit{\boldsymbol{u}}_j}} \right\| = \\ \;\;\;\;\sum\limits_{i = 1}^N {\left( {\omega _i^{ * * }{M_i}{\lambda _i}/\omega _i^ * + c{a_{ii}}\omega _i^{ * * }\eta {\lambda _i}/\omega _i^ * - \omega _i^ * k_i^ * } \right)} \\ \;\;\;\;\left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\left\| {{\mathit{\boldsymbol{u}}_i}} \right\| + c\sum\limits_{i < j} {\left( {{a_{ij}}\omega _i^{ * * }\eta {\lambda _j}/\omega _j^ * + } \right.} \\ \;\;\;\;\left. {{a_{ji}}\omega _j^{ * * }\eta {\lambda _i}/\omega _i^ * } \right)\left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\left\| {{\mathit{\boldsymbol{u}}_j}} \right\| \le - \sum\limits_{i < j} {\left[ {\frac{2}{{N - 1}}} \right.} \\ \;\;\;\;\sqrt {\omega _i^ * k_i^ * - \omega _i^{ * * }{M_i}{\lambda _i}/\omega _i^ * - c{a_{ii}}\omega _i^{ * * }\eta {\lambda _i}/\omega _i^ * } \times \\ \;\;\;\;\sqrt {\omega _j^ * k_j^ * - \omega _j^{ * * }{M_j}{\lambda _j}/\omega _j^ * - c{a_{jj}}\omega _j^{ * * }\eta {\lambda _j}/\omega _j^ * } - \\ \;\;\;\;\left. {c\left( {{a_{ij}}\omega _i^{ * * }\eta {\lambda _j}/\omega _j^ * + {a_{ji}}\omega _j^{ * * }\eta {\lambda _i}/\omega _i^ * } \right)} \right]\left\| {\mathit{\boldsymbol{u}}_i^{\rm{T}}} \right\|\\ \;\;\;\;\left\| {{\mathit{\boldsymbol{u}}_j}} \right\| \end{array} $

If the control parameters ω_i and k_i meet the inequalities as follows, the conclusion ${\dot V_2} \leqslant 0$ would be obtained:

$ \omega _i^{ * * }{\lambda _i}\left( {{M_i} + c{a_{ii}}\eta } \right) \le k_i^ * \omega _i^{ * 2},1 \le i \le N $

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$ \begin{array}{l} \frac{{{{\left( {N - 1} \right)}^2}{c^2}{\eta ^2}\left( {{a_{ij}}\omega _i^{ * * }{\lambda _j}\omega _i^ * + {a_{ji}}\omega _j^{ * * }{\lambda _i}\omega _j^ * } \right)}}{{4\left( {\omega _i^ * \omega _j^ * } \right)}} \le \\ \left( {k_i^ * \omega _i^{ * 2} - \omega _i^{ * * }{\lambda _i}\left( {{M_i} + c{a_{ii}}\eta } \right)} \right)\left( {k_j^ * \omega _j^{ * 2} - } \right.\\ \left. {\omega _j^{ * * }{\lambda _j}\left( {{M_j} + c{a_{jj}}\eta } \right)} \right),1 \le i \le j \le N \end{array} $

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Therefore we get ${\dot V_2} \leqslant 0$. Clearly, if and only if (u₁, u₂, ..., u_N) or (e₁, e₂, ..., e_N) is equal to 0, ${\dot V_2} = 0$ is established. Then, regarding the augment system, the largest invariant set E={((u₁, u₂, ..., u_N), (e₁, e₂, ..., e_N))|${\dot V_2} = 0$}={0}. The error system (8) with the proposed controller via LaSalle invariance principle tends to 0, namely, the network (2) and the controlled network (4) are synchronized. This proof has been completed.

5 Simulation

In this part, two complex networks with 5 nodes, where each node is hyper-chaotic Lorenz type with fractional order, are provided as an example to verify above theoretical analysis.

The following is the hyper-chaotic Lorenz's dynamic equation with fractional-order:

$ \left\{ \begin{array}{l} {D^q}{x_1} = {x_4} + v\left( {{x_2} - {x_1}} \right)\\ {D^q}{x_2} = e{x_1} - {x_2} - {x_1}{x_3}\\ {D^q}{x_3} = {x_1}{x_2} - d{x_3}\\ {D^q}{x_4} = s{x_4} - {x_2}{x_3} \end{array} \right. $

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where we set v, d, e and s as 10, 8/3, 28 and －1 respectively. The fractional-order hyper-chaotic Lorenz's system is chaotic with initial value (2, -2, 1, -1) as shown in Fig. 1.

$ \left\{ \begin{array}{l} {D^q}{x_i}\left( t \right) = f\left( {{x_i}\left( t \right)} \right) + c\sum\limits_{j = 1}^5 {{a_{ij}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}{x_j}\left( t \right)} \\ {D^q}{y_i}\left( t \right) = f\left( {{y_i}\left( t \right)} \right) + c\sum\limits_{j = 1}^5 {{a_{ij}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}{y_j}\left( t \right)} + {\varphi _i}\left( {{u_i}} \right) \end{array} \right. $

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The coupling matrix A is:

$ \left( {\begin{array}{*{20}{c}} { - 3}&1&1&1&1\\ 1&{ - 3}&1&0&1\\ 1&1&{ - 3}&0&1\\ 1&1&1&{ - 4}&1\\ 1&1&1&1&{ - 4} \end{array}} \right) $

The matrix A's eigenvalues are λ₁=-4.236 1, λ₂=0.236 1, λ₃=－5, λ₄=－4, λ₅=－4. A does not satisfy Definition 2. We can see that the network (2) and the network (4) cannot achieve the inner synchronization from Figs. 2-3. For the uncontrolled complex network model, we get that the outer synchronization error system does not converges to zero (see Fig. 4).

We should satisfy the Assumption 1 to reach the two networks' hybrid synchronization. As in Ref.[30], we can obtain Δ=3I₃ and P=diag(1, 1, 1).

Assume that Γ=diag(1, 1, 1), and A is:

$ \left( {\begin{array}{*{20}{c}} { - 3}&1&1&0&1\\ 1&{ - 3}&1&0&1\\ 1&1&{ - 3}&0&1\\ 1&1&1&{ - 4}&1\\ 1&1&1&1&{ - 4} \end{array}} \right) $

The matrix A's eigenvalues are λ₁=0, λ₂=－4, λ₃=－4, λ₄=－4, λ₅=－5. ξ=[0.25, 0.25, 0.25, 0.05, 0.2]^T is the A's left eigenvector regarding the eigenvalue 0. Therefore G and L is equal to diag(0.25, 0.25, 0.25, 0.05, 0.2) and G－ξξ^T respectively. When c is selected as 1, the eigenvalues of ρ_jL+(cζGA)^s are -0.401 0, -0.25, -0.25, -0.039 0, 0, therefore, the network (2) and the controlled network (4) can reach the inner synchronization respectively. When the control parameters ω_i(1≤i≤5) and k_i(1≤i≤5) meet the Theorem 1's inequalities, the system (8) can converge to 0, namely, the network (2) and the controlled network (4) can reach the outer synchronization. We choose the parameters ω_i=I, k_i=50*I, x_i(0)=(0.1i+0.2, 0.1i－0.2, 0.1i+0.1, 0.1i－0.1)^T, and y_i(0)=(0.5i+4, 0.5i, 0.5i+3, 0.5i+1)^T so as to calculate easily.

The equations as follows are used to explain the hybrid synchronization^[28]:

$ \left\{ \begin{array}{l} {E_x}\left( t \right) = \frac{1}{{N - 1}}\sum\limits_{i = 1}^N {{{\left\| {{\mathit{\boldsymbol{x}}_{i + 1}}\left( t \right) - {\mathit{\boldsymbol{x}}_i}\left( t \right)} \right\|}^2}} \\ {E_y}\left( t \right) = \frac{1}{{N - 1}}\sum\limits_{i = 1}^N {{{\left\| {{\mathit{\boldsymbol{y}}_{i + 1}}\left( t \right) - {\mathit{\boldsymbol{y}}_i}\left( t \right)} \right\|}^2}} \\ {E_o}\left( t \right) = \frac{1}{N}\sum\limits_{i = 1}^N {{{\left\| {{\mathit{\boldsymbol{y}}_i}\left( t \right) - {\mathit{\boldsymbol{x}}_i}\left( t \right)} \right\|}^2}} \end{array} \right. $

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where driving-response networks' outer synchronization error is represented by E_o(t). The each network's inner synchronization errors are represented by E_x(t) and E_y(t) respectively. Clearly, if the above three errors are stabilized to 0 as time tends to infinity at the same time, the hybrid synchronization can be reached.

Based on Figs. 5-7, we can see that the network (2) and the controlled network (4) can reach hybrid synchronization.

If the control parameters ω_i and k_i meet Theorem 1, the value of ω_i and k_i can influence the synchronization speed. Parameter c is set as 1 and inner coupling matrix A=I₃, larger control parameters ω_i and k_i can make outer synchronization faster (see Figs. 8-9).

6 Conclusions

The hyper-chaotic complex networks' hybrid synchronization law by a controller with fractional order is studied via the LaSalle invariance principle and the Lyapunov stability theorem in this article. When the state of driving-response networks is outer synchronization and each network is in inner synchronization, two coupled dynamics complex networks' hybrid synchronization could be reached. Two fractional-order hyper-chaotic Lorenz complex networks' simulations explain the validity of the theoretical analysis.