Mittag–Leffler stability for a new coupled system of fractional-order differential equations on network

Abstract

In this paper, the stability problem of a new coupled model constructed by two fractional-order differential equations for every vertex is studied. The coupled relationship is hybrid. By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, sufficient conditions that the coexistence equilibrium of the coupling model is globally Mittag–Leffler stable in $R^{2n}$ are derived. An example is given to illustrate the main results.

1 Introduction

The global-stability problem of equilibria has been investigated for coupled systems of differential equations on networks for many years [1–6]. For example, Li and Shuai developed a systematic approach that allowed one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems by using results from graph theory. The approach was applied to several classes of coupled systems in engineering, ecology, and epidemiology. Although there exist many results about stability of coupled systems on networks (CSNs), most efforts have been devoted to CSNs whose nodes are constructed by integer-order differential equations. In fact, it is more valuable and practical to investigate a coupled system of fractional-order differential equations on the network. Recently, Li [7] investigated the global Mittag–Leffler stability of the following coupled system of fractional-order differential equations on network (CSFDEN):

$$ \textstyle\begin{cases} _{t_{0}}D_{t}^{\alpha }x_{i}=-\alpha_{i}x_{i}(t)+f_{i}(x_{i}(t))+ \sum_{j=1}^{n}\beta^{x}_{ij}(x_{j}(t)-x_{i}(t)), \\ x_{i}(t_{0})=x_{it_{0}},\quad i=1, \ldots, n, \end{cases} $$

(1)

where D denoted Caputo fractional derivative, $\alpha \in (0, 1)$. $t_{0}$ was the initial time, n ($n\geq 2$) denoted the number of vertices in the network. $(x(t))^{T}=(x_{1}(t), x_{2}(t), \ldots, x _{n}(t))^{T}$ denoted the state variable of the system where $x_{i}(t)\in R$. $\alpha_{i}$ was a positive constant. Constant $\beta^{x}_{ij}$ represented the influence of vertex j on vertex i with $\beta^{x}_{ii}=0, \beta^{x}_{ij}=-\beta^{x}_{ji}$, if $i\neq j$. Function $f_{i}$ was Lipschitz continuous. Several sufficient conditions were obtained to ensure the Mittag–Leffler stability of CSFDEN by using graph theory and the Lyapunov method.

Furthermore, Li [8] investigated a coupled system of fractional-order differential equations on network with feedback controls (CSFDENFCs). By using the contraction mapping principle, Lyapunov method, graph theoretic approach, and inequality techniques, some sufficient conditions were derived to ensure the existence, uniqueness, and global Mittag–Leffler stability of the equilibrium point of CSFDENFCs.

As far as we know, most of researchers are interested in CSNs constructed by only one fractional-order differential equation for every vertex. To the best of author’s knowledge, there are less results about CSNs constructed by two or many fractional-order differential equations for every vertex. In this paper, the coupled model (1) is generalized to the more complicated model. The vertex’s dynamical character is presented by the two-dimensional system. The coupled relationship is constructed by two components of the vertex. The coupled system of fractional differential equations on network is studied. Sufficient conditions that the coexistence equilibrium of the coupling model is globally Mittag–Leffler stable in $R^{2n}$ are derived by using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems.

Remark 1.1

In fact, the generalization of model (1) is important and meaningful. Because a lot of ecological models can be seen as high-dimensional coupled systems. Every node is constructed by two or many differential equations in integer-order systems. For example, predator–prey models with patches and dispersal are studied by a lot of researchers [1–6].

This paper is organized as follows. Preliminary results are introduced in Sect. 2. In Sect. 3, the main results are obtained. In the sequel, an example is presented in Sect. 4. Finally, the conclusions and outlooks are drawn in Sect. 5.

2 Preliminaries

In this section, we list some definitions and theorems which will be used in the later sections.

A directed graph or digraph $G=(V, E)$ contains a set $V=\{1, 2, \ldots, n\}$ of vertices and a set E of arcs $(i, j)$ leading from initial vertex i to terminal vertex j. A subgraph H of G is said to be spanning if H and G have the same vertex set. A digraph G is weighted if each arc $(j, i)$ is assigned a positive weight. $a_{ij}>0$ if and only if there exists an arc from vertex j to i in G.

The weight $w(H)$ of a subgraph H is the product of the weights on all its arcs. A directed path P in G is a subgraph with distinct vertices ${i_{1}, i_{2}, \ldots, i_{m}}$ such that its set of arcs is $\{(i_{k}, i_{k+1}):k =1, 2, \ldots, m\}$. If $i_{m}=i_{1}$, we call P a directed cycle.

A connected subgraph T is a tree if it contains no cycles, directed or undirected.

A tree T is rooted at vertex i, called the root if i is not a terminal vertex of any arcs, and each of the remaining vertices is a terminal vertex of exactly one arc. A subgraph Q is unicyclic if it is a disjoint union of rooted trees whose roots form a directed cycle.

Given a weighted digraph G with n vertices, the weight matrix $A=(a_{ij})_{n\times n}$ can be defined by their entry $a_{ij}$ equals the weight of arc $(j, i)$ if it exists, and 0 otherwise. We denote a weighted digraph as $(G, A)$. A digraph G is strongly connected if, for any pair of distinct vertices, there exists a directed path from one to the other. A weighted digraph $(G, A)$ is strongly connected if and only if the weight matrix A is irreducible.

The Laplacian matrix of $(G, A)$ is denoted by L. Let $c_{i}$ denote the cofactor of the ith diagonal element of L. The following results are listed.

Lemma 2.1

([6])

Assume $n\geq 2$. Then

$$ c_{i}=\sum_{\mathbf{T}\in T_{i}}w(\mathbf{T}), $$

where $T_{i}$ is the set of all spanning trees T of $(G, A)$ that are rooted at vertex i, and $w(T)$ is the weight of T. In particular, if $(G, A)$ is strongly connected, then $c_{i} > 0$ for $1\leq i\leq n$.

Lemma 2.2

([6])

Assume $n\geq 2$. Let $c_{i}$ be given in Lemma 2.1. Then the following identity holds:

$$ \sum_{i, j=1}^{n} c_{i}a_{ij}F_{ij}(x_{i}, x_{j})=\sum_{Q\in \mathrm{Q}} w(Q)\sum _{(s, r)\in E(C_{\mathrm{Q}})}F_{rs}(x_{r}, x_{s}). $$

Here, $F_{ij}(x_{i}, x_{j}), 1\leq i, j\leq n$, are arbitrary functions, Q is the set of all spanning unicyclic graphs of $(G, A)$, $w(Q)$ is the weight of Q, and $C_{\mathrm{Q}}$ denotes the directed cycle of Q.

If $(G, A)$ is balanced, then

$$ \sum_{i, j=1}^{n} c_{i}a_{ij}F_{ij}(x_{i}, x_{j})=\frac{{1}}{{2}} \sum_{Q\in \mathrm{Q}}w(Q) \sum_{(j, i)\in E(C_{\mathrm{Q}})} \bigl[F_{ij}(x _{i}, x_{j})+F_{ji}(x_{j}, x_{i}) \bigr]. $$

Definition 2.3

([9])

The Caputo fractional derivative of order $\alpha \in (n-1, n)$ for a continuous function $f:R^{+}\rightarrow R$ is given by

$$ _{t_{0}}D_{t}^{\alpha }f(t)=\frac{{1}}{{\Gamma (n-\alpha )}} \int_{t_{0}} ^{t}\frac{{f^{(n)}(s)}}{{(t-s)^{\alpha +1-n}}}\,ds. $$

Definition 2.4

([7, 9])

The solution of the system

$$ _{t_{0}}D_{t}^{\alpha }x(t)=f(t, x) $$

is said to be Mittag–Leffler stable if

$$ \bigl\Vert x(t) \bigr\Vert \leq \bigl\{ m \bigl[x(t_{0}) \bigr]E_{\alpha } \bigl(-\lambda (t-t_{0})^{\alpha } \bigr) \bigr\} ^{b}. $$

Here, $t_{0}$ is the initial time, $\alpha \in (0, 1), \lambda >0, b>0, m(0)=0, m(x)\geq 0$. $m(x)$ is locally Lipschitz on $x\in B\subseteq R^{n}$ with Lipschitz constant $m_{0}$. $E_{\alpha }(t)$ is a Mittag–Leffler function. Moreover, the domain of the function $f(t, x)$ is $[t_{0}, +\infty ) \times \Omega$, and the function $f(t, x)$ is piecewise continuous in t and locally Lipschitz in x.

3 Main results

A coupled system of fractional differential equations on network is constructed as follows:

$$ \textstyle\begin{cases} _{t_{0}}D_{t}^{\alpha }x_{i}=-\alpha_{i}x_{i}(t)+\theta_{i}y_{i}(t)+f _{i}(x_{i}(t))+\sum_{j=1}^{n}\beta^{x}_{ij}(y_{j}(t)-x_{i}(t)), \\ _{t_{0}}D_{t}^{\alpha }y_{i}=-\beta_{i}y_{i}(t)-\varepsilon_{i}x_{i}(t)+g _{i}(y_{i}(t))+\sum_{j=1}^{n}\beta^{y}_{ij}(x_{j}(t)-y_{i}(t)), \\ x_{i}(t_{0})=x_{it_{0}}, \qquad y_{i}(t_{0})=y_{it_{0}}, \quad i=1, \ldots, n. \end{cases} $$

(2)

Here, D denotes the Caputo fractional derivative, $\alpha \in (0, 1)$. $t_{0}$ is the initial time, n($n\geq 2$) denotes the number of vertices in the network. $z(t)=(x(t), y(t))^{T}=(x_{1}(t), x_{2}(t), \ldots, x_{n}(t), y_{1}(t), y_{2}(t), \ldots, y_{n}(t))^{T}$ denotes the state variable of the system where $x_{i}(t)\in R$ and $y_{i}(t)\in R$. $\alpha_{i}$, $\beta_{i}$, $\theta_{i}$, $\varepsilon_{i}$ are all positive constants. Constant $\beta^{x}_{ij}$ represents the influence of $y_{j}$ on $x_{i}$ with $\beta^{x}_{ii}=0, \beta^{x}_{ij}=- \beta^{x}_{ji}$, if $i\neq j$. Constant $\beta^{y}_{ij}$ represents the influence of $x_{j}$ on $y_{i}$ with $\beta^{y}_{ii}=0, \beta^{y}_{ij}=- \beta^{y}_{ji}$, if $i\neq j$.

The following assumptions are given for system (2).

$(H_{1})$ :

Functions $f_{i}$, $g_{i}$ are Lipschitz-continuous on R with Lipschitz constant $L^{x}_{i}>0, L ^{y}_{i}>0$, respectively, i.e.,

$$\begin{aligned} & \bigl\vert f_{i}(u)-f_{i}(v) \bigr\vert \leq L^{x}_{i}\vert u-v \vert , \\ & \bigl\vert g_{i}(u)-g_{i}(v) \bigr\vert \leq L^{y} _{i}\vert u-v \vert \end{aligned}$$

for all $u, v \in R$.

$(H_{2})$ :

There exists a constant λ such that

$$ \lambda =\min \Biggl\{ 2 \Biggl(\alpha_{i}+\sum _{j=1}^{n}\beta^{x}_{ij}-L^{x}_{i} \Biggr), 2 \Biggl( \beta_{i}+\sum_{j=1}^{n} \beta^{y}_{ij}-L^{y}_{i} \Biggr)\Bigm| i=1, 2, \ldots, n \Biggr\} >0. $$

A mathematical description of a network is a directed graph consisting of vertices and directed arcs connecting them. At each vertex, the local dynamics are given by a system of differential equations called the vertex system. The directed arcs indicate inter-connections and interactions among vertex systems.

Let $\beta_{ij}$ represent the influence of vertex j on vertex i, with

$$ \beta_{ij}=\textstyle\begin{cases} \beta^{x}_{ij}\theta^{-1}_{j},& \mbox{if }\vert \beta^{x}_{ij}\theta^{-1}_{j} \vert \geq \vert \beta^{y}_{ij}\varepsilon^{-1}_{j} \vert , \\ \beta^{y}_{ij}\varepsilon^{-1}_{j},& \mbox{if }\vert \beta^{x}_{ij}\theta^{-1}_{j} \vert < \vert \beta^{y}_{ij}\varepsilon^{-1}_{j} \vert . \end{cases} $$

Let $A=(\vert \beta_{ij} \vert )_{n\times n}$, $A^{x}=(\vert \beta^{x}_{ij} \vert )_{n\times n}$, $A^{y}=(\vert \beta^{y}_{ij} \vert )_{n\times n}$.

A digraph $(G, A)$ with n vertices for system (2) can be constructed as follows. Each vertex represents a patch and $(j, i)\in E(G)$ if and only if $\beta^{x}_{ij}\neq 0$ or $\beta^{y} _{ij}\neq 0$. Here, $E(G)$ denotes the set of arcs $(i, j)$ leading from initial vertex i to terminal vertex j. At each vertex of G, the vertex dynamics are described by the following system (3):

$$ \textstyle\begin{cases} {}_{t_{0}}D_{t}^{\alpha }x_{i}=-\alpha_{i}x_{i}(t)+\theta_{i}y_{i}(t)+f_{i}(x_{i}(t)), \\ {}_{t_{0}}D_{t}^{\alpha }y_{i}=-\beta_{i}y_{i}(t)-\varepsilon_{i}x_{i}(t)+g_{i}(y_{i}(t)). \end{cases} $$

(3)

The coupling among system (2) is provided by the network. The G is strongly connected if and only if the matrix $A=(\vert \beta_{ij} \vert )_{n \times n}$ is irreducible.

In this section, the coupled system of fractional differential equations on network is studied. By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, sufficient conditions that the coexistence equilibrium of the coupling model (2) is globally Mittag–Leffler stable in $R^{2n}$ are derived.

We obtain the main theorem as follows.

Theorem 3.1

Assume that the following conditions hold:

• 1. Diagraph $(G, A)$ is balanced;

• 2. $A^{x}=(\vert \beta^{x}_{ij} \vert )_{n\times n}, A^{y}=(\vert \beta^{y}_{ij} \vert )_{n\times n}$ are irreducible;

• 3. Conditions $(H_{1})$ and $(H_{2})$ hold;

• 4. The formula $\beta^{x}_{ij}\theta^{-1}_{j}=\beta ^{y}_{ij}\varepsilon^{-1}_{j}$ holds for $i, j=1, 2, \ldots, n$.

Then system (2) is globally Mittag–Leffler stable.

Proof

Let $E^{*}=(x^{*}, y^{*})^{T}=(x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*}, y_{1}^{*}, y_{2}^{*}, \ldots, y_{n} ^{*})^{T}$ be an equilibrium of (2). Assume that $e^{x}_{i}(t)=x_{i}(t)-x _{i}^{*}, e^{y}_{i}(t)=y_{i}(t)-y_{i}^{*}$ ($i=1, 2, \ldots, n$). After calculating, we obtain that

$$\begin{aligned} {}_{t_{0}}D_{t}^{\alpha }e^{x}_{i}(t)&=- \alpha_{i}e^{x}_{i}(t)+\theta_{i}e^{y}_{i}(t)+f_{i} \bigl(x^{*}_{i}+e^{x} _{i}(t) \bigr)-f_{i} \bigl(x_{i}^{*} \bigr) \\ &\quad {}+\sum_{j=1}^{n} \beta^{x}_{ij} \bigl(y^{*}_{j}+e^{y}_{j}(t)-x^{*}_{i}-e^{x} _{i}(t) \bigr)-\sum_{j=1}^{n} \beta^{x}_{ij} \bigl(y^{*}_{j}-x^{*}_{i} \bigr), \\ {}_{t_{0}}D_{t}^{\alpha }e^{y}_{i}(t)&=- \beta_{i}e^{y}_{i}(t)-\varepsilon_{i}e^{x}_{i}(t)+g_{i} \bigl(y^{*}_{i}+e ^{y}_{i}(t) \bigr)-g_{i} \bigl(y_{i}^{*} \bigr) \\ &\quad {}+\sum_{j=1}^{n} \beta^{y}_{ij} \bigl(x^{*}_{j}+e^{x}_{j}(t)-y^{*}_{i}-e^{y} _{i}(t) \bigr)-\sum_{j=1}^{n} \beta^{y}_{ij} \bigl(x^{*}_{j}-y^{*}_{i} \bigr). \end{aligned}$$

Let

$$ e(t)= \bigl(e^{x}_{1}(t), e^{y}_{1}(t) , e^{x}_{2}(t), e^{y}_{2}(t), \ldots, e^{x}_{n}(t), e^{y}_{n}(t) \bigr) $$

and

$$ V_{i} \bigl(e^{x}_{i}(t), e^{y}_{i}(t) \bigr)=\frac{{1}}{{2}} \bigl[\varepsilon_{i} \bigl(e^{x} _{i}(t) \bigr)^{2}+\theta_{i} \bigl(e^{x}_{i}(t) \bigr)^{2} \bigr]. $$

From condition 2 of Theorem 3.1, we have the matrix A is irreducible. Furthermore, $(G, A)$ is strongly connected. Let $c_{i}$ denote the cofactor of the ith diagonal element of Laplacian matrix of $(G, A)$. Then we have $c_{i}>0$. Let

$$ V \bigl(t, e(t) \bigr)=\sum_{i=1}^{n}c_{i} V_{i} \bigl(e^{x}_{i}(t), e^{y}_{i}(t) \bigr). $$

The α-derivative of V along the trajectories of system (2) is

$$\begin{aligned} &{}_{t_{0}}D_{t}^{\alpha }V \bigl(t, e(t) \bigr) \\ &\quad = \frac{1}{2}\sum_{i=1}^{n}c_{i} {}_{t_{0}}D_{t}^{\alpha } \bigl[\varepsilon _{i} \bigl(e^{x}_{i}(t) \bigr)^{2}+ \theta_{i} \bigl(e^{y}_{i}(t) \bigr)^{2} \bigr] \\ &\quad \leq \sum_{i=1}^{n} \bigl[c_{i} \varepsilon_{i} e^{x}_{i}(t)_{t_{0}}D_{t}^{\alpha }e^{x}_{i}(t)+c_{i} \theta_{i}e^{y}_{i}(t)_{t_{0}}D_{t}^{\alpha }e ^{y}_{i}(t) \bigr] \\ &\quad \leq \sum_{i=1}^{n}c_{i}e^{x}_{i}(t) 2 \Biggl(-\alpha_{i}-\sum_{j=1}^{n} \beta^{x} _{ij}+L^{x}_{i} \Biggr) \varepsilon_{i}e^{x}_{i}(t)+\sum _{i=1}^{n}c_{i}e^{y} _{i}(t)2 \Biggl(-\beta_{i}-\sum _{j=1}^{n}\beta^{y}_{ij}+L^{y}_{i} \Biggr)\theta_{i}e ^{y}_{i}(t) \\ &\quad \quad {}+ c_{i}\varepsilon_{i} \theta_{i}e^{y}_{i}(t)e^{x}_{i}(t)-c_{i} \theta _{i}\varepsilon_{i}e^{x}_{i}(t)e^{y}_{i}(t)+ \sum_{i=1}^{n}c_{i} \varepsilon_{i}\beta^{x}_{ij}\theta^{-1}_{j} \theta_{j}e_{i}^{x}e_{j} ^{y}+\sum_{i=1}^{n}c_{i} \theta_{i}\beta^{y}_{ij}\varepsilon^{-1}_{j} \varepsilon_{j}e_{i}^{y}e_{j}^{x} \\ &\quad = \sum_{i=1}^{n}c_{i}e^{x}_{i}(t) 2 \Biggl(-\alpha_{i}-\sum_{j=1}^{n} \beta^{x} _{ij}+L^{x}_{i} \Biggr) \varepsilon_{i}e^{x}_{i}(t)+\sum _{i=1}^{n}c_{i}e^{y} _{i}(t)2 \Biggl(-\beta_{i}-\sum _{j=1}^{n}\beta^{y}_{ij}+L^{y}_{i} \Biggr)\theta_{i}e ^{y}_{i}(t) \\ &\quad \quad {}+ c_{i}\varepsilon_{i} \theta_{i}e^{y}_{i}(t)e^{x}_{i}(t)-c_{i} \theta _{i}\varepsilon_{i}e^{x}_{i}(t)e^{y}_{i}(t)+ \sum_{i=1}^{n}c_{i}a_{ij}F ^{x}_{ij}(t, x, y) +\sum_{i=1}^{n}c_{i}a_{ij}F^{y}_{ij}(t, x, y). \end{aligned}$$

Here, $a_{ij}=\vert \beta_{ij} \vert =\vert \beta^{x}_{ij}\theta^{-1}_{j} \vert =\vert \beta^{y}_{ij}\varepsilon^{-1}_{j} \vert$ and $F^{x}_{ij}(t, x, y)=\operatorname{sgn}(\beta_{ij})\varepsilon_{i}\theta_{j}e^{x}_{i}e^{y}_{j}$, $F^{y}_{ij}(t, x, y)= \operatorname{sgn}( \beta_{ij})\varepsilon_{j}\theta_{i}e^{y}_{i}e^{x}_{j}$. From $(G, A)$’s balanced and strongly connected character, it follows that

$$\begin{aligned}& \sum_{i=1}^{n}c_{i}a_{ij}F^{x}_{ij}(t, x, y) = \frac{{1}}{{2}}\sum_{Q\in \mathrm{Q}} w(Q)\sum _{(j, i)\in E(C_{ \mathrm{Q}})}\bigl[F^{x}_{ij}(t, x, y)+F^{x}_{ji}(t, x, y)\bigr], \\& \sum_{i=1}^{n}c_{i}a_{ij}F^{y}_{ij}(t, x, y) = \frac{{1}}{{2}}\sum_{Q\in \mathrm{Q}} w(Q)\sum _{(j, i)\in E(C_{ \mathrm{Q}})}\bigl[F^{y}_{ij}(t, x, y)+F^{y}_{ji}(t, x, y)\bigr]. \end{aligned}$$

Furthermore, we obtain that

$$\begin{aligned} &\sum_{i=1}^{n}c_{i}a_{ij} \bigl[F^{x}_{ij}(t, x, y)+F^{y}_{ij}(t, x, y)\bigr] \\ &\quad =\frac{{1}}{{2}}\sum_{Q\in \mathrm{Q}} w(Q) \sum _{(j, i)\in E(C_{\mathrm{Q}})}\bigl[F^{x}_{ij}(t, x, y)+F^{y}_{ji}(t, x, y)\bigr] \\ &\quad \quad {}+ \frac{{1}}{{2}}\sum_{Q\in \mathrm{Q}} w(Q) \sum_{(j, i)\in E(C_{ \mathrm{Q}})}\bigl[F^{y}_{ij}(t, x, y)+F^{x}_{ji}(t, x, y)\bigr] \\ &\quad = \frac{{1}}{{2}}\sum_{Q\in \mathrm{Q}} w(Q)\sum _{(j, i)\in E(C_{ \mathrm{Q}})}\bigl[\operatorname{sgn}(\beta_{ij}) \theta_{i}\varepsilon_{j}e_{i}^{x}e_{j} ^{y}+\operatorname{sgn}(\beta_{ji})\theta_{i} \varepsilon_{j}e_{i}^{x}e_{j}^{y} \bigr] \\ &\quad \quad {}+ \frac{{1}}{{2}}\sum_{Q\in \mathrm{Q}} w(Q) \sum_{(j, i)\in E(C_{ \mathrm{Q}})}\bigl[\operatorname{sgn}( \beta_{ij})\varepsilon_{j}\theta_{i}e^{y}_{i}e^{x} _{j}+\operatorname{sgn}(\beta_{ij})\varepsilon_{j} \theta_{i}e^{y}_{i}e^{x}_{j} \bigr] \\ &\quad = \frac{{1}}{{2}}\sum_{Q\in \mathrm{Q}} w(Q)\sum _{(j, i)\in E(C_{ \mathrm{Q}})}\bigl[\operatorname{sgn}(\beta_{ij}) \theta_{i}\varepsilon_{j}e_{i}^{x}e_{j} ^{y}-\operatorname{sgn}(\beta_{ij})\theta_{i} \varepsilon_{j}e_{i}^{x}e_{j}^{y} \bigr] \\ &\quad \quad {}+ \frac{{1}}{{2}}\sum_{Q\in \mathrm{Q}} w(Q) \sum_{(j, i)\in E(C_{ \mathrm{Q}})}\bigl[\operatorname{sgn}( \beta_{ij})\varepsilon_{j}\theta_{i}e^{y}_{i}e^{x} _{j}-\operatorname{sgn}(\beta_{ij})\varepsilon_{j} \theta_{i}e^{y}_{i}e^{x}_{j} \bigr] \\ &\quad = 0+0 \\ &\quad =0. \end{aligned}$$

In the sequel, we have

$$ _{t_{0}}D_{t}^{\alpha }V \bigl(t, e(t) \bigr)\leq -\lambda V \bigl(t, e(t) \bigr). $$

Let

$$ _{t_{0}}D_{t}^{\alpha }V \bigl(t, e(t) \bigr)+M(t)= - \lambda V \bigl(t, e(t) \bigr). $$

Using the Laplace transform for the equation above, we obtain that

$$ s^{\alpha }w(s)-w(0)s^{\alpha -1}+M(s)=-\beta w(s), $$

where $w(s)$, $M(s)$ are the Laplace transform of $V(t, e(t))$ and $M(t)$, respectively. Using the inverse Laplace transform for the formula above, we have

$$ V \bigl(t, e(t) \bigr)\leq V \bigl(0, e(0) \bigr)E_{\alpha } \bigl(-\beta t^{\alpha } \bigr). $$

By the definition of $V(t, e(t))$, we obtain that system (2) is globally Mittag–Leffler stable. Then the proof is completed. □

By Theorem 3.1, we obtain the following corollary naturally.

Corollary 3.2

Consider the model

$$ \textstyle\begin{cases} {}_{t_{0}}D_{t}^{\alpha }x_{i}=-\alpha_{i}x_{i}(t)+\theta_{i}y_{i}(t)+f_{i}(x_{i}(t))+\sum_{j=1}^{n}\beta^{x}_{ij}(y_{j}(t)-x_{i}(t)), \\ {}_{t_{0}}D_{t}^{\alpha }y_{i}=-\beta_{i}y_{i}(t)-\varepsilon_{i}x_{i}(t)+g_{i}(y_{i}(t))+\sum_{j=1}^{n}\beta^{x}_{ij}(x_{j}(t)-y_{i}(t)), \\ x_{i}(t_{0})=x_{it_{0}}, \qquad y_{i}(t_{0})=y_{it_{0}},\quad i=1, \ldots, n. \end{cases} $$

(4)

Assume that $(G, A)$ is balanced and $A=A^{x}=(\vert \beta^{x}_{ij} \vert )_{n \times n}$ is irreducible, $\theta_{i}=\varepsilon_{i}$ for any $i=1, 2, \ldots, n$, conditions $(H_{1})$ and $(H_{2})$ hold. Then system (4) is globally Mittag–Leffler stable.

4 An example

In this section, an example is presented to illustrate Theorem 3.1. Consider the following system of fractional equations on network:

$$ \textstyle\begin{cases} {}_{t_{0}}D_{t}^{\alpha }x_{1}(t)=-\alpha_{1}x_{1}(t)+\theta_{1}y_{1}(t)+f _{1}(x_{1}(t))+\sum_{j=1}^{n}\beta^{x}_{1j}(y_{j}(t)-x_{1}(t)), \\ {}_{t_{0}}D_{t}^{\alpha }y_{1}(t)=-\beta_{1}y_{1}(t)-\varepsilon_{1}x_{1}(t)+g_{1}(y_{1}(t))+\sum_{j=1}^{n}\beta^{y}_{1j}(x_{j}(t)-y_{1}(t)), \\ {}_{t_{0}}D_{t}^{\alpha }x_{2}(t)=-\alpha_{2}x_{2}(t)+\theta_{2}y_{2}(t)+f _{2}(x_{2}(t))+\sum_{j=1}^{n}\beta^{x}_{2j}(y_{j}(t)-x_{2}(t)), \\ {}_{t_{0}}D_{t}^{\alpha }y_{2}(t)=-\beta_{2}y_{2}(t)-\varepsilon_{2}x _{2}(t)+g_{2}(y_{2}(t))+\sum_{j=1}^{n}\beta^{y}_{2j}(x_{j}(t)-y_{2}(t)), \end{cases} $$

(5)

where, $\alpha =0. 5$, $\alpha_{1}=\alpha_{2}=5$, $\beta_{1}=\beta_{2}=9$, $2\theta_{1}=2\theta_{2}=\varepsilon_{1}=\varepsilon_{2}=1$, $f_{1}(x_{1}(t))=\sin(x_{1}(t))$, $f_{2}(x_{2}(t))=\sin(x_{2}(t))$, $g_{1}(y_{1}(t))=\sin(2y_{1}(t))$, $g_{2}(y_{2}(t))=\sin(2y_{2}(t))$, and $\beta^{x}_{11}=\beta^{x}_{22}=\beta^{y}_{11}=\beta^{y}_{22}=0$, $\beta^{x}_{12}=-\beta^{x}_{21}=3$, $\beta^{y}_{12}=-\beta^{y}_{21}=6$. Therefore, we have

$$ A=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 0 & 6 \\ 6 & 0 \end{array}\displaystyle \right ),\qquad L=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 6 & -6 \\ -6 & 6 \end{array}\displaystyle \right ). $$

Then we obtain that $c_{1}=c_{2}=6$. Obviously, $(G, A)$ is strongly connected and balanced. It is easy to obtain that conditions $H_{1}$, $H_{2}$ hold. According to Theorem 3.1, system (5) has an equilibrium point $(0, 0, 0, 0)$ which is globally Mittag–Leffler stable. The solution and Lyapunov function $V(t, e(t))$ for system (5) with initial value $x_{0}=(0, 0, 0, 0.1)$ are shown in Figs. 1 and 2.

Figure 1

The solution

Figure 2

The Lyapunov function

Model (5) can be regarded as the ecology model with two patches and dispersal. Some kind of rate of change for species can be denoted by fractional order differential for the species. This kind of rate of change of species $x_{i}$ is related to $y_{j}$ through the dispersal. Moreover, the rate of change of species $y_{i}$ is related to $x_{j}$ through the dispersal. The model with patches and dispersal is important in application [1–6]. In the sequel, model (5) is universal and useful in reality.

5 Conclusions and utlooks

In this paper, the new coupled model constructed by two fractional-order differential equations for every vertex is studied. The coupled relationship is constructed by two components of the vertex. By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, sufficient conditions that the coexistence equilibrium of the coupling model is globally Mittag–Leffler stable in $R^{2n}$ are derived. Finally, an example is given to illustrate the main results.

Theorem 3.1 is the main result of this paper. This result is different from the previous studies. Firstly, Theorem 3.1 is a generalization of the main result of Li [7]. The model in this paper is more complicated for every vertex and the conditions of Theorem 3.1 are different from the result in Li [7]. Secondly, Theorem 3.1 is different from Shuai’s results [6] with the integer order differential. The fractional order differential is the main character for this paper. Moreover, the hybrid coupled relation is new and different from the previous papers [1–7]. There are also many difficulties in proving Theorem 3.1. One is the proof of $\sum_{i=1}^{n}c_{i}a_{ij}[F^{x}_{ij}(t, x, y)+F^{y}_{ij}(t, x, y)]=0$; the other is the construction of the Lyapunov function for the coupled system.

Further studies on this subject are being carried out by the presenting author in the two aspects: one is to study the model with time delay; the other is to discuss the method to design control terms.