STABILITY ANALYSIS FOR ONE CLASS OF COUPLED SYSTEM OF FRACTIONAL-ORDER DIFFERENTIAL EQUATIONS ON NETWORK

GAO Yang,ZHAO Wei

(Department of Teaching Education,Daqing Normal University,Daqing 163712,China)

Abstract:In this paper,the stability problem of the new coupled model constructed by two fractional-order differential equations for every vertex 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 is globally Mittag-Leffler stable in R2nare derived.An example is given to illustrate the applications of main results.

Keywords: Mittag-Leffler stable;coupled model;global stability;Caputo derivative

1 Introduction

The global-stability problem of equilibria was 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 coupled system of fractional-order differential equations on network.Recently,Li[7]investigated the global Mittag–Leffler stability of the following coupled system of fractional-order differential equations on network(CSFDEN)

where D denoted Caputo fractional derivative,α∈(0,1).t0was the inial time,n(n≥2)denoted the number of vertices in the network.(x(t))T=(x1(t),x2(t),···,xn(t))Tdenoted the state variable of the system where xi(t) ∈ R. αiwas positive constant.Constantrepresented the in fluence of vertex j on vertex i withif i≠j.Function fiwas 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 authors’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(1.1)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 R2nare derived by using the method of constructing Lyapunov functions based on graphtheoretical approach for coupled systems.

Remark 1.1 The generalization of model(1.1)is important and meanful.Because a lot of ecological model can be seen as high-dimensional coupled system.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 Section 2.In Section 3,main results are obtained.In the sequel,an example is presented in Section 4.Finally,the conclusions and outlooks are drawn in Section 5.

2 Preliminaries

In this section,we will 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,···,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.aij＞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 i1,i2,···,imsuch that its set of arcs is{(ik,ik+1):k=1,2,···,m}.If im=i1,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=(aij)n×ncan be defined by their entry aijequals the weight of arc(j,i)if it exists,and 0 otherwise.For our purpose,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 cidenote the cofactor of the i-th diagonal element of L.The following results are listed.

Lemma 2.1[6]Assume n≥2.Then

where Tiis 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 ci＞0 for 1≤i≤n.

Lemma 2.2[6]Assume n≥2.Let cibe given in Lemma 2.1.Then the following identity holds

here Fij(xi,xj),1≤i,j≤n,are arbitrary functions,Q is the set of all spanning unicyclic graphs of(G,A),w(Q)is the weight of Q,and CQdenotes the directed cycle of Q.

If(G,A)is balanced,then

Definition 2.3[9]The Caputo fractional derivative of order α ∈ (n?1,n)for a continuous function f:R+→R is given by

3 Main Results

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

here D denotes Caputo fractional derivative,α∈(0,1),t0is the inial time,n(n≥2)denotes the number of vertices in the network.z(t)=(x(t),y(t))T=(x1(t),x2(t),···,xn(t),y1(t),y2(t),···,yn(t))Tdenotes the state variable of the system where xi(t)∈ R and yi(t)∈R. αi,βi,θi= εi=l are all positive constants.Constantrepresents the in fluence of xjon xiwith=0,= ?,if i≠j.Constantrepresents the in fluence of yjon yiwithThe following assumptions are given for system(3.1).

(H1)Function fi,giare Lipschtiz-continuous on R with Lipschitz constant＞0,＞0,respectively,i.e.,

for all u,v∈R.

(H2)There exists a constant λ such that

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 βijrepresent the in fluence of vertex j on vertex i with

A digraph(G,A)with n vertices for system(3.1)can be constructed as follows.Each vertex represents a patch and(j,i) ∈ E(G)if and only if≠0 or≠0.Here E(G)denotes the set of arcs(i,j)leading from inial vertex i to terminal vertex j.At each vertex of G,the vertex dynamics are described by the following system(3.2),

The coupling among system(3.1)is provided by the network.The G is strongly connected if and only if the matrix A=(|βij|)n×nis 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(3.1)is globally Mittag-Leffler stable in R2nare derived.

We obtain main theorem as follows.

Theorem 3.1 Assume the following conditions hold

1.diagraph(G,A)is balanced;

3.condition(H1)and(H2)hold;

4.there exists constant p ≥ 0 such that=for i,j=1,2,···,n.

Then system(3.1)is globally Mittag-Leffler stable.

Two case will be discussed about p.

Case I 0≤p≤1.

Case II p＞1.

For Case I,It is easy to obtain thatTherefore,A=Ax.From the condition of theorem,we obtain Axis irreducible.Furthermore,(G,A)is strongly connected.Let cidenote the cofactor of the ith diagonal element of Laplacian matrix of(G,A).Then we have ci＞0.Let

Calculating the fractional-order derivative of V(t,e(t))along the solution of system(3.1),we have

Furthermore,we obtain that

In the sequel,we have

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

By the definition of V(t,e(t)),we obtain that system(3.1)is globally Mittag-Leffler stable.

With the similar arguments to Case I,we can prove system(3.1)is globally Mittag-Leffler stable for Case II.Then the proof is completed.

By Theorem 3.1,we obtain that the following corollary naturally.

Corollary 3.2 Consider the model

Assume that(G,A)is balanced andis irreducible,condition(H1)and(H2)*hold.Then system(3.3)is globally Mittag-Leffler stable.Here,condition(H2)*is denoted as follows.

(H2)*There is a constant λ such that

4 An Example

In this section,a numerical example is presented to illustrate the Theorem 3.1.

Consider the following system of fractional equations on network

where

Therefore,we have

Then we obtain that c1=c2=6.Obviously,(G,A)is strongly connected and balanced.It is easy to obtain that condition(H1),(H2)hold.According to Theorem 3.1,system(4.1)has an equilibrium point(0,0,0,0)which is globally Mittag-Leffler stable.

5 Conclusions and Outlooks

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 R2nare derived.Finally,an example is given to illustrate the applications of main results.

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