Periodic solutions of discrete time periodic time-varying coupled systems on networks

Chaos, Solitons and Fractals 103 (2017) 246–255

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Frontiers

Periodic solutions of discrete time periodic time-varying coupled systems on networks Shang Gao a, Songsong Li b, Boying Wu a,∗ a b

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China School of Management, Harbin Institute of Technology, Harbin, 150001, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 9 September 2016 Revised 17 March 2017 Accepted 13 June 2017

In this paper, we consider the existence of periodic solutions for discrete time periodic time-varying coupled systems on networks (DPTCSN). Some novel sufficient conditions are obtained to guarantee the existence of periodic solutions for DPTCSN, which have a close relation to the topology property of the corresponding network. Our approach is based on the continuation theorem of coincidence degree theory, generalized Kirchhoff’s matrix tree theorem in graph theory, Lyapunov method and some new analysis techniques. The approach is applied to the existence of periodic solutions for discrete time Cohen– Grossberg Neural Networks (CGNN). Finally, an example and numerical simulations are provided to illustrate the effectiveness of our theoretical results.

Keywords: Periodic solutions Discrete time Time-varying coupled systems Networks

© 2017 Published by Elsevier Ltd.

1. Introduction Complex networks are all around us in the real world nowdays. Discrete time coupled systems on networks as a mathematical framework could describe many complex networks in science and engineering, such as complex biological systems, neural networks, chemical systems, etc. [1–4]. On the other hand, periodic phenomena exist widely in biological systems such as the seasonal effects of weather, food supplies, etc., as well as electronic systems and neural networks. In this paper, we investigate the model of discrete time periodic time-varying coupled systems on networks (DPTCSN) as follows.





xi ( n + 1 ) = f i xi ( n ), n +

l 





bi j (n )Hi j x j (n ) , i ∈ L, n ∈ N,

(1)

j=1

where fi : Rmi × N → Rmi are continuous functions satisfying fi (·, n ) = fi (·, n + ω ) (ω ∈ Z+ ), bij (n) represent the ω-periodic time  varying coupling strength, i.e. bi j (n ) = bi j (n + ω ) and Hi j x j : Rm j → Rmi stand for normalized interference functions. Exploring the global dynamics of DPTCSN is generally a challenging and difficult task as the following list of possible complications illustrates. Firstly, structural complexity: the global dynamics of DPTCSN do not only depend on every vertex system but also rely on the topology property of the networks structural. Secondly,

∗

Corresponding author. E-mail addresses: [email protected] (S. Gao), [email protected] (B. Wu).

http://dx.doi.org/10.1016/j.chaos.2017.06.012 0960-0779/© 2017 Published by Elsevier Ltd.

dynamical complexity: every vertex could be described by nonlinear difference equations. It is well known that, compared with the continuous time systems, the discrete time ones are more difficult to deal with. Usually, some simple difference equations can always produce very complicated dynamics. Thirdly, connection diversity: the links between vertices could have different weights and the weight could be changing over time. Among the global dynamics behavior of DPTCSN, periodicity is the main one. In fact, the existence of a periodic solution as a similar character played by the equilibrium of the autonomous systems is a very basic and important issue in the study of DPTCSN. Motivated by the above discussions, investigated the existence of periodic solutions for DPTCSN is a splendid work and some new methods should be recommended. Lots of approaches are used to investigate periodic solutions including many fixed point theorems, the upon and lower solution method, etc. Recently, with the help of a powerful technique which is the continuation theorem of coincidence degree theory, a lot of good results concerned with the existence of periodic solutions for discrete time systems and continuous time systems are obtained (see Refs. [5–10] and the references therein). However, how to acquire the priori estimate of unknown solutions to the equation Lx = λNx is still a difficult issue and many scholars obtain the priori bounds by employing the inequality |x(t )| ≤  |x(t0 )| + 0ω |x˙ (t )|dt and matrix’s spectral theory in the previous literatures. But, applying these previous approaches to acquire the priori estimate of unknown solutions to the equation Lx = λNx for DPTCSN is very difficult due to its inherently difficulties discussed above. Fortunately, Li and Shuai in [11], use graph theory

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247

method to investigate the global-stability problem for coupled systems of differential equations on networks. Based on the method in [11], many scholars have acquired lots of results about stability for different coupled systems, such as discrete time coupled systems [4,12], stochastic coupled systems [13,14], and delay coupled systems [15,16], etc. However, to the authors’ knowledge, few people use this technique to investigate the existence of periodic solutions for DPTCSN. The main contribution and novelties of the current work are as follows. 1. Some new techniques based on Lyapunov method, generalized Kirchhoff’s matrix tree theorem in graph theory and analysis skills for the priori estimate of unknown solutions to the equation Lx = λNx are provided. 2. By employing the continuation theorem of coincidence degree theory, Lyapunov method and generalized Kirchhoff’s matrix tree theorem in graph theory, some sufficient conditions are obtained which have a close relation to the topology property of the network’s structure. The tree of this paper is the following. In Section 2, some useful notations and basic preliminaries are given. In Section 3, some sufficient conditions for the existence of periodic solutions of DPTCSN are obtained. In Section 4, our approach is applied to discrete time Cohen–Grossberg Neutral Networks (CGNN). In Section 5, an example and numerical simulations are given to show the effectiveness and feasibility of our results. Finally, conclusions are presented in Section 6. 2. Preliminaries In this section, we shall summarize some useful notations, basic concepts and lemmas in the following which will be used throughout this paper.

Throughout this paper, we denote ω a positive integer, and Iω = {0, 1, · · · , ω − 1}. Let  f (n ) = f (n + 1 ) − f (n ), L = {1, 2, · · · , l }, N = {0, 1, 2, · · · }, R1+ = [0, +∞ ), and Z+ = {1, 2, · · · }. Write f =   max f (n ), f = min f (n ) and fˆ = 1 ω−1 f (n ). Let m = l mi , n∈Iω

ω

n=0

i=1

mi ∈ Z+ . Set Rn and Rn×m denote n–dimensional real space and n × m-dimensional real matrix space, respectively. The transpose of vectors and matrices is denoted by superscript “T”. For vector y = (y1 , y2 , · · · , yn )T ∈ Rn , |y| denotes the Euclidean norm |y| =  ( ni=1 y2i )1/2 . Denote by C (Rd × N; R1+ ) the family of all real-valued nonnegative functions V(x, n) denoted on Rd × N such that they are continuously in x and n. Other notations will be explained where they first appear. 2.2. Graph theory We introduce some basic concepts on graph theory [17,18]. A directed graph or digraph G = (H, E ) contains a set H = {1, 2, · · · , l } 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 (n), for n ∈ N. In our convention, aij (n) > 0 if and only if there exists an arc from vertex j to vertex 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 , , im } such that its set of arcs is (ik , ik+1 ) : k = 1, 2, · · · , m − 1 . 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 l vertices, define the weight matrix A(k ) = ai j (n ) whose entry aij (n) equals the l×l weight of arc (j, i) if it exists, and 0 otherwise. Denote the directed  graph with weight matrix A(n) as G, A(k ) . A digraph G is strongly connected if for any pair of distinct vertices, there  existsa directed path from one to the other. A weighted digraph G, A(n ) is said to be balanced if W (C ) = W (−C ) for all directed cycles C and n ∈ N. Here, −C denotes the reverse of C and is constructed by reversing the direction of all arcs in C. For example, consider a weighted digraph with 7 vertices (see Fig. 1), where A(n) is a 7 × 7 matrix as

⎛

2.1. Notations

n∈Iω



Fig. 1. A balanced digraph G, A(n ) with 7 vertices.

0

a12 (n )

a13 (n )

a15 (n )

0

a16 (n )

0

⎞

⎜a21 (n ) 0 a23 (n ) 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ 0 a34 (n ) 0 0 0 ⎟ ⎜a31 (n ) a32 (n ) ⎜ ⎟ ⎜ 0 0 0 0 a45 (n ) 0 0 ⎟ ⎜ ⎟, n ∈ N. ⎜a (n ) ⎟ 0 0 0 0 a56 (n ) 0 ⎟ ⎜ 51 ⎜ ⎟ ⎝a61 (n ) 0 0 0 a65 (n ) 0 a67 (n )⎠ 0 a72 (n ) 0 0 0 0 0 This digraph is balanced if and only if a31 (n )a23 (n )a12 (n ) = a13 (n )a21 (n )a32 (n ) hods for any n ∈ N. For a unicyclic graph Q with cycle CQ , let Q˜ be the unicyclic graph obtained by replacing CQ with −CQ . Suppose that G, A(n ) is balanced, then W (Q ) = W (Q˜ ). The Laplacian matrix of (G, A(n )) is defined as

⎛ 

a1 j ( n )

⎜ j=1 ⎜ ⎜ −a (n ) ⎜ 21 ⎜ L (n ) = ⎜ ⎜ .. ⎜ ⎜ . ⎜ ⎝ −a (n ) l1

−a12 (n ) 

a2 j ( n )

··· ···

j=2

.. .

..

−al2 (n )

···

.

−a1l (n )

⎞

⎟ ⎟ ⎟ −a2l (n ) ⎟ ⎟ ⎟. ⎟ .. ⎟ ⎟ . ⎟  ⎠ al j ( n ) j=l

The following result is standard in graph theory, and customarily called Kirchhoff’s matrix tree theorem [19]. Lemma 1. [19] Assume that l ≥ 2. Let ci (n) denote the cofactor of the ith diagonal element of L(n ), for any n ∈ N. Then

ci ( n ) =



T ∈T i

W ( T ), i ∈ L,

248

S. Gao et al. / Chaos, Solitons and Fractals 103 (2017) 246–255





where Ti is the set of all spanning trees T of G, A(n ) that are rooted





at vertex i, and W (T ) is the weight of T . In particular, if G, A(n ) is strongly connected, then ci (n) > 0 for i ∈ L, n ∈ N. Lemma 2. Assume that l ≥ 2. Let ci (n) denote the cofactor of the ith diagonal element of L(n ), for any n ∈ N. Then the following identity holds: l  i, j=1

=



ci (n )ai j (n )Fi j xi (n ), x j (n ) 

l−2  h=1





∗  B(n ) −det a B(n ) +   B(n ) −e1   B (n ) B (n ) +  a∗ a∗





det



− det

+ det



Frs xr (n ), xs (n ) .

 a∗  B (n ) − B (n ) +  e r B ( n ) a∗ h

r=1

∗

a  B (n ) − B (n ) +  a∗



h+1 

 B (n ) er 



r=1

 a  B (n ) −  B (n ) B (n ) +  er  a∗ l−1

∗





 − det

 a∗

r=1

(s,r )∈E (CQ )

Q∈Q

+

 a∗







W (Q )





= det

a∗

B (n )

 

< 0,

Here Fi j (xi , x j ) : Rmi × Rm j → R, 1 ≤ i, j ≤ l are arbitrary functions,   Q is the set of all spanning unicyclic graphs of G, A(n ) , W (Q ) is the weight of Q, and CQ denotes the directed cycle of Q.

where er represents (l − 1 ) × (l − 1 ) matrix. And the rth diagonal element of er is 1 and the else is 0. This completed the proof of Lemma 3. 

Remark 1. The proof of this lemma is similar to which appeared in Theorem 2.2 in [11]. It just needs some obvious modifications, so we omit it here for brevity.

2.3. Coincidence degree theory

Lemma   3. Suppose the time-varying weighted matrix A(n ) = ai j ( n ) of digraph G is irreducible for n ∈ N, where aij (n) l×l are continuous ω-periodic functions. Then we have ci (n + 1 ) < (a∗ /a∗ )l−1 ci (n ), where ci (n) denote the cofactor of the ith diagonal element of the Laplacian matrix L(n ), a∗ = max{ai j }, and a∗ = i, j∈L

max{ai j }. i, j∈L

Proof. Let

 a (n ) = a (n + 1 ) −  ij ij

a∗ ai j ( n ), a∗

∀i, j ∈ L, n ∈ N.

 a (n ) < 0. Denote Thus, it is easy to see that  ij

⎛

a1 j ( n ) · · ·

⎜ j=1 ⎜ ⎜ . ⎜ . . ⎜ ⎜ ⎜−ai−1,1 (n ) ⎜ ⎜ B (n ) = ⎜ ⎜−a ⎜ i+1,1 (n ) ⎜ ⎜ ⎜ . ⎜ . ⎜ . ⎜ ⎝ −al1 (n )

..

.

···

−a1,i−1 (n )

−a1,i+1 (n )

. . .

. . .



ai−1, j (n )

j=i−1

···

−ai+1,i−1 (n )

···

ai+1, j (n ) · · ·

j=i+1

···

−a1l (n )

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ −ai−1,l (n )⎟ ⎟ ⎟ ⎟, ⎟ −ai+1,l (n )⎟ ⎟ ⎟ ⎟ . ⎟ . ⎟ . ⎟  ⎠ al j ( n ) . . .

−ai−1,i+1 (n ) 

···

. . .

. . .

..

−al,i−1 (n )

−al,i+1 (n )

···

.

j=l

and

⎛  a 1 j ( n ) ⎜ j=1 ⎜ ⎜ . ⎜ . ⎜ . ⎜ ⎜− a i−1,1 (n ) ⎜ ⎜  ⎜ B ( n ) = ⎜ a ⎜− i+1,1 (n ) ⎜ ⎜ ⎜ . ⎜ . ⎜ . ⎜ ⎝  −al,1 (n )

···

a − 1,i−1 (n )

a − 1,i+1 (n )

..

. . .

. . .

.

···



···

a · · · − i+1,i−1 (n )



a  i+1, j (n ) · · ·

j=i+1

···

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ a − ( n ) i−1,l ⎟ ⎟ ⎟. ⎟ a − ( n ) ⎟ i+1,l ⎟ ⎟ ⎟ . ⎟ . ⎟ . ⎟   a (n ) ⎠  lj . . .

 a  i−1, j (n ) −ai−1,i+1 (n ) · · ·

j=i−1

 a (n ) − 1l

. . .

. . .

..

a − l,i−1 (n )

a − l,i+1 (n )

···

.

j=l

By the definition of ci (n), and matrix A(n) is irreducible for n ∈ N, we have

ci ( n + 1 ) − = det

 a∗ l−1

 a∗ a∗

a∗

ci ( n )



 B(n ) − det B (n ) + 

 a∗ a∗

B (n )



Now we shall summarize below some concepts and a lemma from [20] that will be basic for this paper. Let X, Z be normed vector spaces, L: DomL ⊂ X → Z be a linear mapping, and N: X → Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim KerL = codim ImL < +∞ and ImL is closed in Z. If L is a Fredholm mapping of index zero, and there exist continuous projectors P: X → X and Q: Z → Z such that ImP = KerL, ImL = KerQ = Im(I − Q ). It follows that L|DomL ∩ KerP : (I − P )X → ImL is invertible. We denote the inverse of that map by Kp . If  is an open bounded subset of X, the mapping N will be called ¯ if QN () ¯ is bounded and K p (I − Q )N :  ¯ → X is L-compact on  compact. Since ImQ is isomorphic to KerL, there exist isomorphism J: ImQ → KerL. Lemma 4 [21] (Continuation Theorem). Let L be a fredholm map¯ . Suppose that the ping of index zero and let N be L-compact on  following conditions hold. (P1) For each λ ∈ (0, 1), very solution x of Lx = λNx is such that x ∈ ∂ ; (P2) For each x ∈ KerL ∩ ∂ , QNx = 0, and

deg{JQN,  ∩ kerL, 0} = 0, where deg is the Brouwer degree. ¯. Then the equation Lx = Nx has at least one solution in DomL ∩  3. The existence of periodic solutions for DPTCSN (1) The objective of this section is to obtain some sufficient conditions for the existence of periodic solution of DPTCSN. We establish DPTCSN based on a digraph G with l(l ≥ 2) vertices, firstly. More precisely, the ith vertex assigns a system of difference equations as follows:





xi ( n + 1 ) = f i xi ( n ), n , where xi ∈ Rmi and fi : Rmi × N → Rmi are continuous functions satisfying fi (·, n ) = fi (·, n + ω ). Moreover, assumption that bij (n) are periodic time-varying coupled strength (i.e. bi j (n ) = bi j (n +   ω )), and Hi j x j : Rm j → Rmi stand for normalized interference functions from vertex j to vertex i. Hence, we can acquire DPTCSN (1) (see Fig. 2. as an example). The initial value of system (1) is



T

given by x(0 ) = xT1 (0 ), · · · , xTl (0 ) ∈ Rm . In the end of this section, we state the main results of this paper in the following which are sufficient criteria of the existence of periodic solutions for DPTCSN (1). Theorem 1. Suppose that there exist positive constants α i satisfying 1 + (a∗ /a∗ )l−1 (min αi − 1 ) > 0 and the following assumptions H1– i∈L

H3 holding for any i ∈ L, n ∈ N.

S. Gao et al. / Chaos, Solitons and Fractals 103 (2017) 246–255

249

Fig. 2. The discrete time periodic time-varying coupled systems on networks with l vertices.

H1. There exist functions Vi (xi , n ) ∈ C (Rmi × N; R1+ ), Fij (xi , xj ), pos  itive constants β i and a matrix A(n ) = ai j (n ) (aij (n) ≥ 0), such l×l that

Vi (xi , n ) = Vi (xi , n + ω ),

lim Vi (xi , n ) = ∞,

(2)

|xi |→∞

and for any λ ∈ (0, 1),

    Vi xi (n ), n ≤ −λαiVi xi (n ), n +λ

l 





ai j (n )Fi j xi (n ), x j (n ) + λβi ,

(3)

where xi (n) satisfies





x i ( n ) = λ f i x i ( n ) , n − x i ( n ) + 

l 



bi j (n )Hi j x j (n )

n∈Iω

.

H2. The digraph G, A(n ) is strongly connected and along each







Frs xr (n ), xs (n ) ≤ 0.

for any x = {x(n ), n ∈ N} ∈ ω . Thus, it is not difficult to show that ω is a finite-dimensional Banach Space. Let



directed cycle C of the weighted digraph G, A(n ) ,



Let ω ⊂ m denote the subspace of all ω periodic sequences equipped with the usual supremum norm · , i.e.

x = max |x(n )|,





j=1



Proof. In order to apply Lemma 4 to DPTCSN (1), we start with some useful function spaces and their norms, which are used throughout this paper. Denote

m = {x = {x(n )} : x(n ) ∈ Rm , n ∈ N}.

j=1



Then DPTCSN (1) has at least one ω-periodic solution.

(4)

 ω 0 =

x = {x ( n )}

H3. Suppose that

⎞

l 

bˆ 1 j H1 j (x j )⎟ ⎜ fˆ1 (x1 , n ) − x1 + ⎜ ⎟ j=1 ⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ = G(x )x, . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ l ⎝ fˆ (x , n ) − x +  bˆ H (x ) ⎠ l

l

l

lj

lj

j

j=1

where G(x)m × m is a nonsingular matrix.

ω −1 

 x (n ) = 0 ,

n=0

(s,r )∈E (CQ )

⎛

∈ ω :

and





ω m ω c = x = {x ( n )} ∈ : x ( n ) = h ∈ R , n ∈ N .

It is easy to see that ω and ω c are both closed linear subspaces of 0 ω and

ω = ω 0



ω ω c , dim c = m.

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S. Gao et al. / Chaos, Solitons and Fractals 103 (2017) 246–255

      V x(n ), n  V x(n + 1 ), n + 1 − V x(n ), n

Now, let us define X = Z = ω , (Lx )(n ) = x(n + 1 ) − x(n ), and

⎛

⎞  b1 j (n )H1 j x j (n ) ⎟ ⎜ f 1 x1 ( n ), n − x1 ( n ) + ⎜ ⎟ j=1 ⎜ ⎟ ⎜ ⎟ .. ⎟, (Nx )(n ) = ⎜ . ⎜ ⎟ ⎜ ⎟ ⎜  ⎟ l ⎝ f x ( n ) , n  − x ( n ) +  b ( n ) H x ( n )  ⎠ 

l





l 

l

l

lj

lj

= =

j

+

l 



ci (n )Vi xi (n ), n

l 

k=0

ω−1 1 

ω

i=1

z (s ) −

s=0

1

ω

ω −1 

k=0

x i ( n ) = λ f i x i ( n ) , n − x i ( n ) +





,

j=1

i ∈ L, n ∈ N.

ai j (n )Fi j xi (n ), x j (n ) + βi

a∗

a∗

i=1

where α = min{αi }. In view of Lemma 2, condition H2 and fact



l 

ci (n )ai j (n )Fi j xi (n ), x j (n )

i, j=1



(5)

Let







W (Q )



Fi j xi (n ), x j (n ) ≤ 0.

(i, j )∈E (CQ )

Q∈Q

bi j (n )Hi j x j (n )



j=1

i, j=1

=





i∈L

s=0

l 

ci (n ) −αiVi xi (n ), n +



W (Q ) > 0, we can get

( ω − s )z ( s ).



l  

 a∗ l−1  l l    × ci (n )ai j (n )Fi j xi (n ), x j (n ) + λ ci (n )βi , (6)

open bounded set  ⊂ X. Now we are at the point to search for an appropriate open, bounded subset  for the application of the continuation theorem. Corresponding to the operator equation Lx = λNx, λ ∈ (0, 1), we have





a∗

z(k ), z ∈ Z.

Obviously, QN and K p (I − Q )N are continuous. Since X is a finitedimensional Banach space and K p (I − Q )N is continuous, one can ¯ is uniformly bounded for any open show that K p (I − Q )N () bounded set  ⊂ X. On the other hand, x ∈  is an ω-periodic ¯ is equicontinufunction. So it is easy to see that K p (I − Q )N () ous. Applying the Arzela–Ascoli theorem, we can see that K p (I − ¯ is sequentially compact for any open bounded set  ⊂ X. Q )N () ¯ is compact for any open bounded set  ⊂ X. Thus, K p (I − Q )N () ¯ is bounded. Thus, N is L-compact on  ¯ with any Moreover, QN ()





 ∗ l−1   a∗ l−1   a ≤ −λ ( α − 1 ) + 1 V x ( n ), n + λ

Furthermore, the generalized inverse (to L)Kp : ImL → KerP ∩ DomL exists and is given by ω −1 



i=1

ImL = KerQ = Im(I − Q ).

and



ai j (n )Fi j xi (n ), x j (n ) + βi

l 

+λ

It is not difficult to show that P and Q are continuous projectors such that

ImP = KerL







j=1

Then it follows that L is a Fredholm mapping of index zero. Define

K p (z ) =

ci (n + 1 ) − ci (n ) Vi xi (n + 1 ), n + 1

a∗

dim KerL = m = codim ImL < +∞.

ω

 

i=1

+

Qz =



 ∗ l−1  l     a <λ −1 ci (n ) −(αi − 1 )Vi xi (n ), n

as well as

x(k ), x ∈ X,



ci (n )Vi xi (n ), n

i=1

l  

ImL = ω 0,

ω−1 1 

l 

i=1

for any x ∈ X and n ∈ N. Obviously, L is a bounded linear operator and

Px =



ci (n + 1 )Vi xi (n + 1 ), k + 1 −

i=1

j=1

KerL = ω c ,



l 

Substituting this into (6) we obtain that

 ∗ l−1    a V x(n ), n < −λ ( α − 1 ) + 1 V x ( n ), n 



a∗

 a∗ l−1  l

+λ

a∗

ci (n )βi .

i=1

By (2), it is easy to see that V(x(n), n) < 0, for |x| sufficiently large. This contradicts with V(x(n), n) is ω-periodic function. So, there exists H > 0 which is independent of the choice of λ, such that the solutions of equation Lx = λNx satisfy x < H. Denote  = {x ∈ X : x < H }, then we know that Lx = λNx for x ∈ KerL ∩ ∂  and λ ∈ (0, 1). By condition H3, we have

⎛

⎜ fˆ1 (x1 , n ) − x1 +

l 

⎞

bˆ 1 j H1 j (x j )⎟

⎜ j=1 ⎜ ⎜ .. QNx = ⎜ . ⎜ ⎜ ⎜ l ⎝ fˆ (x , n ) − x +  bˆ H (x ) j l l l lj lj

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

j=1





V x, n =

l 





ci (n )Vi xi , n ,

i=1

where ci (n) is the cofactor of the ith diagonal element of the Laplacian matrix L(n ). The property of strong connectedness of digraph G, A(n ) implies that ci (n) > 0 for any i ∈ L, n ∈ N. Let



xT1 (n ), · · ·

T , xTl (n )

x (n ) = ∈ X be a solution of system (5). Using (3) and Lemma 3, we can get

= G(x )x, x ∈ KerL ∩ ∂ . Hence, we can get QNx = 0 has a unique solution x∗ = (0, · · · , 0 )T . It implies that QNx = 0, for x ∈ KerL ∩ ∂ . Finally, a straight forward calculation and condition H3 shows that

deg{JQN,  ∩ KerL, 0} = sgn(det G(0 )) = 0. Hence, from Lemma 4, DPTCSN (1) has at least one ω-periodic so¯ . The proof is complete.  lution in DomL ∩ 

S. Gao et al. / Chaos, Solitons and Fractals 103 (2017) 246–255

Remark 2. In Theorem 1, we let digraph





G, A(n ) , n ∈ N be

strongly connected. Moreover, if we suppose digraph n ∈ N is balanced, then l 



ci (n )ai j (n )Fi j xi (n ), x j (n )





G, A(n ) ,



i, j=1

=

1 W (Q ) 2 Q∈Q

 







Fi j xi (n ), x j (n ) + Fji x j (n ), xi (n )



.

( j,i )∈E (CQ )

 





Fi j xi (n ), x j (n ) + Fji x j (n ), xi (n )



≤ 0.

(7)

( j,i )∈E (CQ )

So, we get the following corollary from Theorem 1.





Corollary 1. Suppose that digraph G, A(n ) , n ∈ N is balanced. Then the conclusion of Theorem 1 holds if (4) is replaced by (7). Remark 3. We also note that if there exist functions Ti and Tj for every i, j ∈ L, such that









the DPTCSN with time delay will be the topic of future investigation. On the other hand, the condition H2 in Theorem 1 desire the network is strongly connected which is related to topological property of the network anatomy. However, this property might not be always satisfied in practice and some scholars [22] have explored global dynamics of coupled systems on networks when the network is not strongly connected. It is worth discussing the existence of periodic solutions for DPTCSN when the network is not strongly connected in the future. 4. An application to discrete time Cohen–Grossberg Neural Networks

Hence, (4) in condition H2 could be replaced by



251



Fi j xi (n ), x j (n ) ≤ Ti xi (n )) − T j x j (n ) , n ∈ N.

(8)

Then (4) clearly holds. So we can easily obtain the Corollary 2 as follows.

Since Cohen and Grossberg initiated study of CGNN in 1983 [23], the dynamics of CGNN have been studied widely over the past decades. There are lots of results have been reported, see [24– 27] and references therein. On the other hand, some discrete time CGNN are also investigated widely due to its’ application in pattern recognition and image processing, etc., which are the versions of the continuous time CGNN. In this section, we consider the discrete time CGNN. By using our results in Section 3, some sufficient criteria of the existence of periodic solutions for discrete time CGNN are presented. Now, let us give the discrete time CGNN as follows



xi (n + 1 ) = xi (n ) − αi xi (n )



Corollary 2. The conclusion of Theorem 1 holds if (4) is replaced by (8).





βi xi (n ) −

×



l 







ti j (n )s j x j (n ) + Ji (n ) ,

j=1

Remark 4. In Theorem 1, we give a systematic approach for checking the existence of periodic solutions of DPTCSN and some novel techniques are given to estimate the unknown solutions of equation Lx = λNx which combines the generalized Kirchhoff’s matrix tree theorem in graph theory, Lyapunov method and some analysis skills. In [11], authors investigated the global stability problem for coupled systems on networks by using graph theory and Lyapunov method. Based on this approach, Su et al. [4] investigated the stability for discrete-time systems on networks. Moreover, Su et al. [21] generalize this approach to investigated the stability for discrete-time coupled systems with multi-dispersal. However, the coupling strength considered in the previous literatures is constant. In this paper, we consider the coupling strength is periodic timevarying. Remark 5. We construct the global Lyapunov function for DPTCSN  (1) as V (x, n ) = li=1ci (n )Vi (xi , n ), which is close related to the weighted of digraph G, A(n ) and Lyapunov function Vi (xi , n) of the ith vertex system. Thus, finding Lyapunov function Vi (xi , n) of each vertex system is very important in practice. In fact, due to the complexity of DPTCSN, many large-scale discrete time coupled systems are coupled together by the simple and nearly identical discrete time systems in regular ways to make some improvements. So, our approach can be used to investigate the DPTCSN which coupling some classical discrete time systems whose Lyapunov functions Vi (xi , n) are known or constructed without difficulty. Remark 6. In practice, time delay unavoidably exist due to the finite speed of switching and transmission of signal. So, we can consider the following DPTCSN with delay.





xi ( n + 1 ) = f i xi ( n ), xi ( n − v ), n +

l 





bi j (n )Hi j x j (n ) ,

j=1

i ∈ L, n ∈ N, Z+

where v ∈ represents the time delay. How to use this method in this paper to efficiently solve the existence of periodic solutions for

n ∈ N, i ∈ L,

(9)

where l ≥ 2 is the number of neurons in the network, xi (n ) ∈ R1 denotes the state variable associated with the ith neuron at time n, α i and β i are amplification functions and appropriately behaved functions, respectively, the connection matrices ω-periodic functions (tij (n))l × l show how the neurons are connected in the network and ω-periodic functions Ji (n) represent the external input. In order to establish the sufficient criteria of the existence of periodic solutions for system (9), we provided some standard and usual assumptions which can be found in the present literature, one can see [24]. Assumption 1. For any i ∈ L, functions β i (x) and si (x) are Lipschitz continuous with Lipschitz constants 1 and 2 . Moreover βi (0 ) = si ( 0 ) = 0. Assumption 2. There exist constants q1 and q2 such that 0 < q1 ≤ α i (x) ≤ q2 for any i ∈ L. Now, we give the main result of this section in the following. Theorem 2. Suppose that Assumptions 1 and 2 and the following conditions hold. U1. For every i ∈ L, there exists a γ i , such that xβ i (x) ≥ γ i x2 . U2. For every i, j ∈ L and n ∈ N, there is

!

0 < t∗ ≤ t i j < ti j (n ) < t i j ≤ t ∗ ≤ min and

0<

 t ∗ 2l−2 

t∗ 1 q2 < min{γi }. 12q1 i∈L

min i∈L

"5 3

q1 γi −

q22 21 1 , , 12l 2 12l 2

#



217 2 2 q − 1 + 1, 144 2 1

Then system (9) has at least one ω-periodic solution. Proof. Let







f i x i ( n ) , n = −α i x i ( n )









    βi xi (n ) + Ji (n ) ,

bi j ( n ) =

ti j (n ) and Hi j x j (n ) = s j x j (n ) for any i, j ∈ L, n ∈ N. The vector

252

S. Gao et al. / Chaos, Solitons and Fractals 103 (2017) 246–255

spaces X, Z and mappings L, N, P, Q, Kp are constructed as the same form in the proof of Theorem 1. Obviously, L is a Fred¯ with any holm mapping of index zero and N is L-compact on  open bounded set  ⊂ X. Corresponding to the operator equation Lx = λNx, λ ∈ (0, 1), we have



    xi (n ) = −λαi xi (n ) βi xi (n ) −

l 







ti j (n )s j x j (n ) + Ji (n ) ,

n ∈ N, i ∈ L.



Let x(n ) = x1 (n ), · · · , xl (n )

T

+

+αi xi (n )



∈ X be a solution of system (10) for

j=1





βi xi (n ) −

l 



− 2xi (n )αi xi (n )







βi xi (n ) −







+ q22 1 2







= −2αi xi (n ) xi (n )βi xi (n ) +2



Ji2 (n )



ti j (n )xi (n )αi xi (n ) s j x j (n )

j=1



(n )

21 6l 2 (t ∗ )2 + t 2 (n )x2j (n ) 2 ∗ 2 ij 6 l 1 2 (t )

t2 t∗ i j

(n )x2j (n ) + t ∗ x2i (n )









q22

 

ti j (n )|xi (n )||x j (n )|

1 ≤ q22 21 + q1 γi + q22 1 2 lt ∗ x2i (n ) 6 +

 

l   1 j=1



j=1



l  

γ

j=1

(11)

l  j=1

6 21 q42 2 J q1 i i

q22 Ji2 (n ) +

+ q22 2

ti j (n )s j x j (n ) + Ji (n )

l 

ti j (n )|x j (n )| + 2q22 1 2 l

2 !



ti j (n )s j x j (n ) + Ji (n )

l 

l 



 1 q1 γi )x2i (n ) + q22 l 22 ti2j (n )x2j (n ) 6

≤ (q22 21 + +

j=1



q22 1



On the other hand, by Assumptions 1 and 2, condition U1 and U2, we can get



6 1 q22 2 q1 γi 2 x ( n ) + J (n ) i q1 γi i 6 1 q22

(10)

      Vi xi (n ), n  Vi xi (n + 1 ), n + 1 − Vi xi (n ), n   l        ≤ λ − 2xi (n )αi xi (n ) βi xi (n ) − ti j (n )s j x j (n ) + Ji (n )  2



j=1

some λ ∈ (0, 1) and Vi (xi , n ) = x2i . Now we can compute





ti2j (n )s2j x j (n ) + q22 Ji2 (n )

j=1



+ 2q22 |Ji (n )| 2

j=1

l 

≤ q22 21 x2i (n ) + q22 l

+

6 2 q4 6q2 l 2 2 (t ∗ )2 2 + 1 2 + 2 22 Ji (n ) q1 γi 1

q22 l 22 +

q22 21 q2 1 2 + 2 ∗ ∗ 2 t 6l (t )



l 

ti2j (n )x2j (n ).

(13)

j=1

j=1

− 2xi (n )αi (xi (n ))Ji (n ) l 

≤ −2q1 γi x2i (n ) + 2q2 2

Substituting (12) and (13) into (11) and using condition U2, we obtain that

ti j (n )|xi (n )||x j (n )| + 2q2 |xi (n )||Ji (n )|

j=1

≤ −2q1 γi x2i (n ) + q2 2

j=1



q1 γi 2 6q2 2 x (n ) + J (n ) 6q2 i q1 γi i

+ q2

 11

≤−

 l   1 t ∗ x2i (n ) + ∗ ti2j (n )x2j (n ) t





+



l 6q22 2 q2 2  2 ti j (n )x2j (n ) + J ( n ), ∗ t q1 γi i

q1 γi − q2 2 lt ∗ x2i (n ) +

6

j=1

(12) and



 2

αi xi (n )



 ≤ q22





βi xi (n ) −  2





βi xi (n ) +

l 

2







2 

ti j (n )s j x j (n ) + Ji (n )

j=1 l 

ti j ( n )s j x j ( n )



+ Ji2 (n ) + 2βi xi (n ) Ji (n ) − 2Ji (n ) ×

l 



ti j ( n )s j x j ( n ) − 2

j=1

≤





q22 21 x2i

(n ) +

q22

l 

 



ti j (n )βi xi (n ) s j x j (n )



ti j ( n )s j x j ( n )



2 

j=1

+ q22 Ji2 (n ) + 2q22 1 |Ji (n )||xi (n )| + 2q22 |Ji (n )| ×

l  j=1

ti j (n )|s j (x j (n ))| + 2q22

l  j=1

3

6q22

q1 γi



+

q22

6 2 q4 6q2 l 2 2 (t ∗ )2 2 + 1 2 + 2 22 Ji (n ) q1 γi 1

q22 21 q2 1 2 q2 2 + q22 l 22 + + 2 ∗ ∗ ∗ 2 t t 6l (t )

$ 

ti j (n )$βi xi (n )

$$  $ $$s j x j (n ) $



l 

!

ti2j (n )x2j (n )

j=1

 5 7 ≤ −λ q1 γi − q22 21 − 2q2 2 lt ∗ − 2q22 1 2 lt ∗ − q22 l 2 22 (t ∗ )2 3 6   2 6q2 6 21 q42 6q22 l 2 22 (t ∗ )2 2 2 2 × xi ( n ) + λ + + q2 + Ji (n ) q1 γi q1 γi 21   q22 21 q22 1 2 q2 2 2 2 +λ + q l + + 2 2 t∗ t∗ 6l (t ∗ )2 l 



ti2j (n ) x2j (n ) − x2i (n )

j=1



j=1 l 

 +

×

j=1



  Vi xi (n ), n 5  ≤λ − q1 γi − q22 21 − q2 2 lt ∗ − q22 1 2 lt ∗ x2i (n )

5

≤ −λ

3

q1 γi −





217 2 2 2 q x (n ) 144 2 1 i

  l  q22 21 q22 1 2  2 q2 2 2 2 +λ + q2 l 2 + + ti j (n ) x2j (n ) t∗ t∗ 6l (t ∗ )2 j=1  2  2 4 2  6q2 6 1 q2 6q2 l 2 22 (t ∗ )2 2 2 2 − xi ( n ) + λ + + q2 + J¯i q1 γi q1 γi 21 



≤ − λαiVi xi (n ), n + λζ

l  j=1





ai j (n ) x2j (n ) − x2i (n ) + λβi ,

S. Gao et al. / Chaos, Solitons and Fractals 103 (2017) 246–255

αi = 53 q1 γi −

where 6q2

6 21 q42 q1 γi

βi = ( q1 γ2i +

+ q22 +

6q22 l 2 22 (t ∗ )2 21

Let





l 

V x, n =

ζ=

217 2 2 144 q2 1 ,



q2 2 t∗

+ q22 l 22 +

q22 21

6l (t ∗ )2

+

q22 1 2 t∗ ,

Here J can be chosen as the identity map since ImQ = KerL. Then, for any (x, μ) ∈  × [0, 1],

⎛ ⎞

)J¯i2 , and ai j (n ) = ti2j (n ).

x1

F (x, μ ) = μ⎝ .. ⎠



ci (n )Vi xi , n ,

⎛

      V x(n ), n  V x(n + 1 ), n + 1 − V x(n ), n =



ci (n + 1 )Vi xi (n + 1 ), k + 1 −

i=1

ci (n )Vi xi (n ), n

 

ci (n + 1 ) − ci (n ) Vi xi (n + 1 ), n + 1



+



ci (n )Vi xi (n ), n

−



+ζ

i=1









ci (n ) −αiVi xi (n ), n

+ζ

l 







ai j (n )Fi j xi (n ), x j (n ) + βi

 ∗ 2l−2   t ∗ 2l−2   t ≤ −λ ( α − 1 ) + 1 V x ( n ), n + λ t∗

×

ζ

t∗

i=1

(14) where α = min{αi }. i∈L

In view of Lemma 2, condition U2 and fact W (Q ) > 0, we can get



ci (n )ai j (n )Fi j xi (n ), x j (n )

i, j=1



=

W (Q )









Fi j xi (n ), x j (n ) ≤ 0.

(i, j )∈E (CQ )

Q∈Q



%

i∈L

l q2 (1 − μ )H max{Jˆi } > 0, i∈L

5. Example and numerical simulations In this section, in order to illustrate the efficiency of our theoretical result, we consider the following example. Example 1. Let l = 3. Consider discrete time CGNN (9) with the following parameters:

α1 (x1 ) = 0.55 + 0.05 sin x1 , α2 (x2 ) = 0.55 + 0.05 cos x2 , α3 (x3 ) = 0.55 + 0.05 sin 2x3 , βi (x ) = si (x ) = x, (i = 1, 2, 3 ), 

Substituting this into (14) we get

 ∗ 2l−2    t V x(n ), n < −λ ( α − 1 ) + 1 V x ( n ), n 



t∗

 t ∗ 2l−2  l   ci (n )ai j (n )Fi j xi (n ), x j (n ) + λ ci (n )βi ,

i, j=1

l 

l

Remark 7. In recent several decades, Neural Networks have attracted the attention of many scholars and lots of results have been reported, see [28–36] and the references therein. It should be pointed out that Cao et.al [28,31] investigated the almost periodic solutions of Neural Networks with some analysis techniques. Thus, how to use our method in this paper to investigate the almost periodic solutions of DPTCSN will be our future work.

j=1

l 

j

By Lemma 4, system (9) has at least one ω-periodic solution. The proof is complete. 



i=1

j

= deg{−F (x, 1 ),  ∩ kerL, 0} = 0.

ai j (n )Fi j xi (n ), x j (n ) + βi

l 

lj

j=1

deg{JQN,  ∩ kerL, 0} = deg{−F (x, 0 ),  ∩ kerL, 0}



j=1

+λ

l

for sufficiently large H. It implies that F(x, μ) = 0, for all (x, μ) ∈ KerL ∩ ∂  × [0, 1]. Thus, an application of the homotopy invariance theorem results in

i=1

l 

l

⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

xT F (x, μ ) ≥ μH 2 + (1 − μ ) q1 max{γi } − lq2 2 t ∗ H 2

 ∗ 2l−2  l     t <λ −1 ci (n ) −(αi − 1 )Vi xi (n ), n t∗

l

Then we can get



⎞

l 

n=0

i=1 l 



l



i=1

l  

=

l 



−α1 (x1 ) β1 (x1 ) − t1 j (n )s j (x j ) + J1 (n ) ⎜ ⎜ n=0 j=1 ⎜ 1 − μ⎜ .. − . ω ⎜ ⎜    ⎜ ω −1 l  ⎝ −α (x ) β (x ) − t ( n )s ( x ) + J ( n )

where ci (n) is the cofactor of the ith diagonal element of the Laplacian  matrix  L(n ). The property of strong connectedness of digraph G, A(n ) implies that ci (n) > 0 for any i ∈ L, n ∈ N. By condition U2 and Lemma 3, we have



. xl

ω −1 

i=1

l 

253

ti j ( n )

⎛



t∗

 t ∗ 2l−2  l +λ ci (n )βi . t∗

i=1

Thus, it is easy to see that V(x(n), n) < 0, for |x| sufficiently large. This contradicts with that V(x(n), n) is ω-periodic function. So, there exists H > 0 which is independent of the choice of λ, such that the solution of equation Lx = λNx satisfy x < H. Denote  = {x ∈ X : x < H }, then we know that Lx = λNx for x ∈ KerL ∩ ∂  and λ ∈ (0, 1). Moreover, we define

F (x, μ ) = μx − (1 − μ )J QN x,

∀(x, μ ) ∈  × [0, 1].

 3×3

⎞

0.01e0.55+0.02 sin 0.1nπ

0.01e0.56+0.02 sin 0.1nπ

0.01e0.57+0.02 sin 0.1nπ

= ⎝0.01e0.55+0.02 cos 0.1nπ

0.01e0.56+0.02 cos 0.1nπ

0.01e0.57+0.02 cos 0.1nπ ⎠,

0.58+0.02 sin 0.1nπ

0.58+0.02 cos 0.1nπ

0.01e

0.01e

0.01e0.6+0.02 sin 0.1nπ

and J1 (n ) = 2 + sin(0.1nπ ), J2 (n ) = 2.5 + sin(0.1nπ ), J3 (n ) = 3 + sin(0.1nπ ). Then, it is easy to verify that Assumptions 1 and 2 hold with γi = 1 = 2 = 1 (i = 1, 2, 3 ), and q1 = 0.5, q2 = 0.6. Moreover, we can easy to see that conditions U1 and U2 are hold. Take the initial value as



x1 ( 0 ), x2 ( 0 ), x3 ( 0 )

T

= (−1.3, −1.8, −2.3 )T .

Thus, by Theorem 2, we can get system (9) has a periodic solution (see Fig. 3.).

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S. Gao et al. / Chaos, Solitons and Fractals 103 (2017) 246–255

Fig. 3. The dynamical behavior of system (9) with l = 3.

6. Conclusions In this paper, we attempt to study the existence of periodic solutions for DPTCSN. With the help of the continuation theorem of coincidence degree theory, generalized Kirchhoff’s matrix tree theorem in graph theory, and Lyapunov method, a systematic approach to explore the existence of periodic solutions for DPTCSN is introduced. By applying our approach, the existence of periodic solutions for discrete time CGNN has been acquired. Finally, a numerical example has been given to illustrate the effectiveness of our results.

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