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Existence and Global Stability of a Periodic Solution for Discrete-Time Cellular Neural Networks
A novel sufficient condition is developed to obtain the discrete-time analogues of cellular neural network (CNN) with periodic coefficients in the three-dimensional space. Existence and global stability of a periodic solution for the discrete-time cellular neural network (DT-CNN) are analysed by utilizing continuation theorem of coincidence degree theory and Lyapunov stability theory, respectively. In addition, an illustrative numerical example is presented to verify the effectiveness of the proposed results.
Cellular neural networks (CNNs) are the basis of both discrete-time cellular neural networks (DT-CNNs)  and the cellular neural networks universal machine (CNNs-UM). The dynamical behaviour of Chua and Yang cellular neural network (CY-CNN) is given by the state equation where , , , and denotes input bias, input, output, and state variable of each cell, respectively. is the t-neighbourhood of cell as , and denote the position of the cell in the network, and denote the position of the neighbour cell relative to the cell in consideration. is the nonlinear weights template matrices for input feedback and is the corresponding template matrices for the outputs of neighbour cells. Non-linearity means that templates can change over time.
A large number of cellular neural networks (CNNs) models have appeared in the literature [2–4], and these models differ in cell complexity, parameterization, cell dynamics, and network topology. Various generalizations of cellular neural networks have attracted attention of scientific community due to their promising potential for tasks of classification, associative memory, parallel computation [5–9], pattern recognition, computer vision, and solving any optimization problem [10–13]. Such applications rely on the existence of equilibrium points and the qualitative properties of cellular neural networks.
Discrete-time cellular neural networks (DT-CNNs) have been studied both in theory and applications. Previous results introduced many properties of DT-CNN in the two dimensional plane. For instance,  has been successfully applied to investigate the discrete-time analogues of cellular neural network (CNN) with variable coefficients in the two-dimensional plane. However, three-dimensional structure is more accurate, specific, and closer to real structures of CNN. Based on the above discussion, this paper proposes some effective results of DT-CNN in the three-dimensional space.
The rest of the paper is organized as follows: in Section 2, system description and preliminaries are developed in detail and some definitions, assumptions, and lemmas are stated. Section 3 gives sufficient conditions for a periodic solution for DT-CNN in three-dimensional space by utilizing continuation theorem of coincidence degree theory. Section 4 proposes global stability of a periodic solution for the DT-CNN. A numerical simulation is given to show correctness of our analysis in Section 5 and concluded in Section 6.
2. System Preliminaries and Description
Consider the following model which is equivalent to the (1.3): where , for all , are -periodic sequences, that is, .
Throughout the paper, the following definitions and lemmas will be introduced.
Definition 2.1 (Fredholm operator). Let and be a Banach space, an operator is called Fredholm operator if is a bounded linear operator between and whose kernel and cokernel are finite-dimensional and whose range is closed. Equivalently, an operator is Fredholm if it is invertible modulo compact operator, that is, if there exists a bounded linear operator such that , are compact operators on and , respectively, where and are the identity operator.
Definition 2.2 (-compact). An operator will be called -compact on if the open bounded set is bounded and is compact, where is the inverse operator of . Since is isomorphic to , there exists an isomorphism .
The index of a Fredholm operator is , then operator will be called a Fredholm operator of index zero if and is closed in . Then a following abstract equation in Banach space is defined by
Let be linear operator, and be a continuous operator. If is a Fredholm operator of index zero, there must exist continuous projectors and , such that:
In other words, is invertible, and the inverse of the operator is denoted by .
Lemma 2.3 (Gaines and Mawhin ). Let be a Banach space, L be a Fredholm operator of index zero, and let be L-compact on , where is an open bounded set, suppose: Then has at least one solution in .
Lemma 2.4. If a and b are some certain nonnegative vectors, then there exists a positive constant β, such that .
Proof. Assuming and are some certain non-negative vectors, is a positive constant, then Thus, the proof of Lemma 2.4 is completed.
Assumption 2.5. ) are N-periodic sequence of . For the sake of convenience, we use the following notations: . For each operator and any , , such that: where is the spherical centre of with a radius length , , then it is easy to obtain: .
Assumption 2.6. There is a positive constant , such that , for all .
3. Existence of a Periodic Solution with respect to (2.1)
In many cases, many proposed results are not ideal and therefore it is necessary to formulate a novel and effective result for DT-CNN in the three-dimensional space. Can we obtain the result about the existence and stability of a periodic solution for DT-CNN in three-dimensional space? This is the topic we wish to address in this paper. The aim of the present work is to develop a strategy to determine the existence and global stability of a periodic solution with respect to (2.1) in the three-dimensional space. Consequently, we processed with the following result.
Proof. In this section, by means of using Mawhin’s continuation theorem of coincidence degree theory, we will study the existence of at least one periodic solution with respect to (2.1), for convenience, some following notations will be used:
where is any function. Let : , and be the subspace of all N-periodic sequence; equip it with the norm . For any , , there exists and , such that . Thus, is a Cauchy sequence in and is the spherical centre of , . By utilizing the meaning of and Bolzano-Weierstrass theorem (Each bounded sequence in has a convergent subsequence, here , ), it is easy to know that is a Banach space.
Set that is where + , for all . Then we will learn that , it is easy to prove that is a bounded linear operator, and are two continuous operators such that , , and , that is that is where is a constant, which is only depended on variables , , and .
Obviously, employing the Lebesgue’s convergence theorem, we can easily learn that is bounded, is compact for any open bounded set by using Ascoli-Arzela’s theorem (A subset of () is compact if and only if it is closed, bounded and equi-continuous). Thus, is L-compact on a closed set with any open bounded set .
Suppose that is a solution with respect to (2.1), for certain . Then the following equation can be derived by (2.2):
Then, the following results can be derived by utilizing (3.7): where . Therefore, the solution with respect to (2.1) is bounded for certain . In other words,
Then the open bounded set is presented as follows:
Thus for any , the satisfies condition (i) in Lemma 2.3.
In Figure 2, the nonlinear weights template matrices and the boundary of are shown, respectively. Then for any two dimensional plane of any spherical neighbourhood is denoted. Thus, for any , , it is easy to learn that is a constant vector in with ; Thus, we have where . Furthermore, we can calculate the bound of as follows: where , , , . Thus for any , , this proves the condition (ii) in Lemma 2.3.
In order to prove the condition (iii) is satisfied with respect to (2.1), we only need to prove that . Define by where , for all .
Now we will prove that , ≠. If this is not true, then , , thus, for constant vector , we have:
Equivalently, (3.14) can be written as the following form:
Combining (3.12) and (3.15), the following results are obtained:
Thus, the following result is derived by calculating the (3.16):
Obviously, (3.17) is a contradiction since , then for any , , , . Thus, . Therefore, (2.1) has at least one N-periodic solution, thus the proof of Theorem 3.1 is completed.
Proof. Similar to the proof of Theorem 3.1, so it is omitted.
4. Globally Stability of a Periodic Solution with respect to (2.1)
The existence of a periodic solution for the system (2.1) is derived in the Theorem 3.1. Then global stability of a periodic solution with respect to (2.1) in the three-dimensional space is presented in the following.
Proof. It follows from the Theorem 3.1 that (2.1) has at least a periodic solution, without loss of generality, the periodic solution can be described by:
Then we can define the following formula:
Now, we show that the a periodic solution is globally stable, and the following inequality is obtained by utilizing (2.1) and (4.3):
We design the following Lyapunov-type sequence by
Then, we can calculate the by combining (2.1) and (4.5):
Thus, it is easy to obtain by the meaning of the (4.6), and furthermore, where Obviously, from the proof of Theorem 4.1, the globally stable of a periodic solution with respect to (2.1) is derived. Then, existence and global stability of a periodic solution for DT-CNNs are obtained by utilizing the conditions of the proposed theorems in an arbitrary diameter plane of a convex space. Thus the proof of Theorem 4.1 is completed.
5. Numerical Simulation
In this section, we give an example to show the effectiveness and improvement of the derived results. Consider the following continuous cellular neural networks: for , where = . Then, state trajectories of are denoted in Figure 3.
From Figure 3, it is easy to know that a -periodic solution of the continuous cellular neural networks is globally stable. Compared to the system (5.1), we design the discrete-time analogue of the continuous cellular neural network as follows: for , by using Assumptions 2.5 and 2.6 in Section 2, each variable is denoted as: The derived results of this paper are verified by the following steps.
Then, , is calculated below,
Thus, the subset of function is derived by the following:
(2) We will verify the condition of Theorem 3.1 if we want to utilize Theorem 4.1. After strictly calculating the condition of Theorem 3.1, it is easy to obtain that the function , , , ; therefore, the condition of the Theorem 3.1 is critically satisfied as well.