Complete Periodic Synchronization of Memristor-Based Neural Networks with Time-Varying Delays

Abstract

This paper investigates the complete periodic synchronization of memristor-based neural networks with time-varying delays. Firstly, under the framework of Filippov solutions, by using M-matrix theory and the Mawhin-like coincidence theorem in set-valued analysis, the existence of the periodic solution for the network system is proved. Secondly, complete periodic synchronization is considered for memristor-based neural networks. According to the state-dependent switching feature of the memristor, the error system is divided into four cases. Adaptive controller is designed such that the considered model can realize global asymptotical synchronization. Finally, an illustrative example is given to demonstrate the validity of the theoretical results.

1. Introduction

Memristor, as the fourth fundamental passive circuit, was firstly postulated by Chua [1] in 1971. On May 1, 2008, the Hewlett-Packard (HP) research team announced their realization of a memristor prototype, with an official publication in Nature [2, 3]. This new circuit element of memristor shares many properties of resistors and shares the same unit of measurement. Recently, memristor has received a great deal of attention because of its potential applications in next generation computer and powerful brain-like “neural” computer. The papers [4–21] have given a detailed introduction on the memristor, so readers can consult [4–21] to get more explanation. As noted in [10], from a systems-theoretic point of view and a mathematical point of view, memristor dynamics strictly obey Bernoulli's nonlinear differential equation, so the mathematical framework and its usefulness are worth studying. The paper [10] by Wu and Zeng discussed the exponential stabilization of memristive neural networks with time delays. The papers [11–14] investigated the synchronization and antisynchronization control of a class of memristor-based recurrent neural networks. A series of results on stability analysis of memristor-based recurrent neural networks were presented in [15–18]. The papers [19–21] dealt with the existence and stability of periodic solution of almost periodic of a class of memristor-based recurrent neural networks.

Different from the previous works, in this paper, we will study complete periodic synchronization of memristor-based neural networks described by the following differential equation: 𝑥̇_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑖(𝑡)𝑓_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)b_𝑖𝑗𝑥_𝑖(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡),𝑡≥0,𝑖=1,2,…,𝑛, (1) where 𝑑_𝑖𝑥_𝑖(𝑡)=𝑑́_𝑖,𝑥_𝑖(𝑡)≤𝑇_𝑖,𝑑̀_𝑖,𝑥_𝑖(𝑡)>𝑇_𝑖,𝑎_𝑖𝑗𝑥_𝑖(𝑡)=𝑎́_𝑖𝑗,𝑥_𝑖(𝑡)≤𝑇_𝑖,𝑎̀_𝑖𝑗,𝑥_𝑖(𝑡)>𝑇_𝑖,𝑏_𝑖𝑗𝑥_𝑖(𝑡)=𝑏́_𝑖𝑗,𝑥_𝑖(𝑡)≤𝑇_𝑖,𝑏̀_𝑖𝑗,𝑥_𝑖(𝑡)>𝑇_𝑖. (2) in which switching jumps 𝑇_𝑖>0, 𝑑́_𝑖>0, 𝑑̀_𝑖>0, 𝑎́_𝑖𝑗, 𝑎̀_𝑖𝑗, 𝑏́_𝑖𝑗, 𝑏̀_𝑖𝑗, 𝑖,𝑗=1,2,…,𝑛, are constants; 𝑓_𝑗 and 𝑔_𝑗 are feedback functions, 𝜏_𝑖𝑗(𝑡) is the time delay with 0≤𝜏_𝑖𝑗(𝑡)≤𝜏 and 𝜏̇_𝑖𝑗≤𝜇_𝑖𝑗<1 (𝜏 and 𝜇_𝑖𝑗 are negative constants). At first glance, one might intuitively believe that the chaotic motion is more complicated compared with the periodic motion, the synchronization of chaotic oscillators is also complicated than those of periodic oscillators [22]. However, this is not always true, just as indicated in [23, 24], where an opposite result was given.

The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced. In Section 3, the proof of the existence of periodic solutions is presented. Complete periodic synchronization is discussed in Section 4. In Section 5, a numerical example is presented to demonstrate the validity of the proposed results. Some conclusions are drawn in Section 6.

Notation 1. ℝ denotes the set of real numbers, ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space, and ℝ^𝑚×𝑛 denotes the set of all 𝑚×𝑛 real matrices. Given the vectors 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇, 𝑦=(𝑦₁,…,𝑦_𝑛)^𝑇∈ℝ^𝑛, ‖𝑥‖=(∑^𝑛_𝑖=1𝑥²_𝑖)^1/2, 𝑥^𝑇𝑦=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖. For 𝑟>0, 𝐶([−𝑟,0];ℝ^𝑛) denotes the family of continuous function 𝜑 from [−𝑟,0] to ℝ^𝑛 with the norm ‖𝜑‖=sup_{−𝑟≤𝑠≤0}|𝜑(𝑠)|. [⋅,⋅] represents the interval. co(𝑄) denotes the closure of the convex hull of 𝑄. 𝐸_𝑛 denotes the identity matrix of size 𝑛. A vector or matrix 𝐴≥0 means that all entries of 𝐴 are greater than or equal to zero; 𝐴>0 can be defined similarly. For vectors or matrices 𝐴 and 𝐵, 𝐴≥𝐵 (or 𝐴>𝐵) means that 𝐴−𝐵≥0 (or 𝐴−𝐵>0). 𝐾(ℝ^𝑛) denotes the collection of all nonempty compact subsets of ℝ with the Hausdorff metric 𝜌 defined by 𝜌(𝐴,𝐵)=max{𝛽(𝐴,𝐵),𝛽(𝐵,𝐴)}, 𝐴,𝐵∈𝐾(ℝ^𝑛), and 𝜌(𝐴,𝐵)=sup{dist(𝑥,𝐵)∶𝑥∈𝐴}, 𝜌(𝐵,𝐴)=sup{dist(𝑦,𝐴)∶𝑦∈𝐵}, 𝐾𝜐(ℝ^𝑛)={𝐴∈𝐾(ℝ^𝑛)∶𝐴 is convex}.

2. Preliminaries

In this section, we give some definitions and properties, which are needed later.

Definition 1 (see [25]). Suppose 𝐸⊆ℝ^𝑛; then 𝑥→𝐹(𝑥) is called a set-valued map from 𝐸→ℝ^𝑛, if for each point 𝑥∈𝐸, there exists a nonempty set 𝐹(𝑥)⊆ℝ^𝑛. A set-valued map 𝐹 with nonempty values is said to be upper semicontinuous (USC) at 𝑥₀∈𝐸, if for any open set 𝑁 containing 𝐹(𝑥₀), there exists a neighborhood 𝑀 of 𝑥₀ such that 𝐹(𝑀)⊆𝑁. The map 𝐹(𝑥) is said to have a closed (convex, compact) image if for each 𝑥∈𝐸, 𝐹(𝑥) is closed (convex, compact).

Definition 2 (see [26]). For the system 𝑥̇(𝑡)=𝑓(𝑡,𝑥),𝑥∈ℝ^𝑛, with discontinuous right-hand sides, a set-valued map is defined as 𝐹(𝑡,𝑥)=_{𝛿>0𝜇(𝑁)=0}co𝑓(𝐵(𝑥,𝛿)⧵𝑁), (3) where co[𝐸] is the closure of the convex hull of set 𝐸, 𝐵(𝑥,𝛿)={𝑦∶‖𝑦−𝑥‖≤𝛿} and 𝜇(𝑁) is Lebesgue measure of set 𝑁. A solution in Filippov's sense of the Cauchy problem for this system with initial condition 𝑥(0)=𝑥₀ is an absolutely continuous function 𝑥(𝑡),𝑡∈[0,𝑇], which satisfies 𝑥(0)=𝑥₀ and differential inclusion 𝑥̇(𝑡)∈𝐹(𝑡,𝑥),fora.a𝑡∈[0,𝑇]. (4)

The initial value associated with system (1) is 𝑥_𝑖(𝑡)=𝜙(𝑡)∈𝐶([−𝜏,0];ℝ), 𝑖=1,2,…,𝑛. Let 𝑑_𝑖=min{𝑑́_𝑖,𝑑̀_𝑖}, 𝑑_𝑖=max{𝑑́_𝑖,𝑑̀_𝑖}, 𝑎_𝑖𝑗=min{𝑎́_𝑖𝑗,𝑎̀_𝑖𝑗}, 𝑎_𝑖𝑗=max{𝑎́_𝑖𝑗,𝑎̀_𝑖𝑗}, 𝑏_𝑖𝑗=min{𝑏́_𝑖𝑗,𝑏̀_𝑖𝑗}, 𝑏_𝑖𝑗=max{𝑏́_𝑖𝑗,𝑏̀_𝑖𝑗}, 𝑎_𝑖𝑗=max{|𝑎_𝑖𝑗|,|𝑎_𝑖𝑗|}, and 𝑏̃_𝑖𝑗=max{|𝑏_𝑖𝑗|,|𝑏_𝑖𝑗|}. We define co𝑑_𝑖𝑥_𝑖(𝑡)=𝑑́_𝑖,𝑥_𝑖(𝑡)<𝑇_𝑖,𝑑_𝑖,𝑑_𝑖,𝑥_𝑖(𝑡)=𝑇_𝑖,𝑑̀_𝑖,𝑥_𝑖(𝑡)>𝑇_𝑖,co𝑎_𝑖𝑗𝑥_𝑖(𝑡=𝑎́)^𝑖𝑗,𝑥_𝑖(𝑡)<𝑇_𝑖,𝑎_𝑖𝑗,𝑎_𝑖𝑗,𝑥_𝑖(𝑡)=𝑇_𝑖,𝑎̀_𝑖𝑗,𝑥_𝑖(𝑡)>𝑇_𝑖,co𝑏_𝑖𝑗𝑥_𝑖(𝑡)=𝑏́_𝑖𝑗,𝑥_𝑖(𝑡)<𝑇_𝑖,𝑏_𝑖𝑗,𝑏_𝑖𝑗,𝑥_𝑖(𝑡)=𝑇_𝑖,𝑏̀_𝑖𝑗,𝑥_𝑖(𝑡)>𝑇_𝑖. (5) Clearly, system (1) is a differential equation with discontinuous right-hand side; its solution in the conventional sense does not exist. Inspired by [10, 20, 21, 27–30], we adopt the following definition of the solution in the sense of Filippov for system (1).

Definition 3. Suppose that 𝜙(𝑠)=(𝜙₁(𝑠),𝜙₂(𝑠),…,𝜙_𝑛(𝑠))^𝑇∈𝐶([−𝜏,0];ℝ^𝑛) is a continuous function. An absolutely continuous function 𝑥(𝑡) is said to be a solution with initial data 𝜙(𝑠) of system (1), if 𝑥(𝑡) satisfies the differential inclusion 𝑥̇_𝑖(𝑡)∈−co𝑑_𝑖𝑥_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1co𝑎_𝑖𝑗𝑥_𝑖(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1co𝑏_𝑖𝑗𝑥_𝑖(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡), (6) for all 𝑖=1,2,…,𝑛, or equivalently, there exist 𝑑_𝑖(𝑡)∈co(𝑑_𝑖(𝑥_𝑖(𝑡))), 𝑎_𝑖𝑗(𝑡)∈co(𝑎_𝑖𝑗(𝑥_𝑖(𝑡))), and 𝑏_𝑖𝑗(𝑡)∈co(𝑏_𝑖𝑗(𝑥_𝑖(𝑡))), such that 𝑥̇_𝑖(𝑡)=−𝑑_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡). (7)

Remark 4. From the theoretical point of view, the above parameters 𝑑_𝑖(𝑡), 𝑎_𝑖𝑗(𝑡), and 𝑏_𝑖𝑗(𝑡) in (7) are measurable functions and depend on the state 𝑥_𝑖(𝑡) and time 𝑡.

Definition 5. We say that real matrix 𝑀=(𝑚_𝑖𝑗)∈ℝ^𝑛×𝑛 is an 𝑀-matrix, if and only if we have 𝑚_𝑖𝑗≤0, 𝑖,𝑗=1,2,…,𝑛, 𝑖≠𝑗, and all successive principal minors of 𝑀 are positive.

Lemma 6 (see [31]). Let 𝑀 be an 𝑛×𝑛 matrix with nonpositive off-diagonal elements. Then 𝑀 is an 𝑀-matrix if and only if one of the following conditions hold:(1)there exists a vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)>(0,0,…,0) such that 𝜉𝑀>0;(2)there exists a vector 𝜂=(𝜂₁,𝜂₂,…,𝜂_𝑛)^𝑇>(0,0,…,0)^𝑇 such that 𝑀𝜂>0.

Lemma 7 (Mawhin-Like Coincidence Theorem [32]). Suppose that 𝐹: ℝ×ℝ^𝑛→𝐾𝜐(ℝ^𝑛) is USC and 𝜔-periodic in 𝑡. If the following conditions hold:(1) there exists a bounded open set Ω⊆𝐶_𝜔, the set of all continuous, 𝜔-periodic functions: ℝ→ℝ^𝑛, such that for any 𝜆∈(0,1), each 𝜔-periodic function 𝑥(𝑡) of the inclusion 𝑥̇(𝑡)∈𝜆𝐹(𝑡,𝑥) (8) satisfies 𝑥∉𝜕Ω if it exists;(2) each solution 𝑢∈ℝ^𝑛 of the inclusion 10∈𝜔^𝜔₀𝐹(𝑡,𝑢)𝑑𝑡=𝜛(𝑢) (9) satisfies 𝑢∉𝜕Ω⋂ℝ^𝑛;(3)deg(𝜛,Ω⋂ℝ^𝑛,0)≠0,  then differential inclusion (4) has at least one 𝜔-periodic solution 𝑥(𝑡) with 𝑥∈Ω.

To proceed with our analysis, we need the following assumptions for system (1).(A₁)𝐼_𝑖(𝑡) and 𝜏_𝑖𝑗(𝑡) are continuous 𝜔-periodic functions.(A₂) For 𝑖=1,2,…,𝑛, for all 𝑠₁,𝑠₂∈ℝ, 𝑠₁≠𝑠₂, the neural activation function 𝑔_𝑖 is bounded and 𝑓_𝑖,𝑔_𝑖 satisfy Lipschitz condition; that is, there exist 𝜚_𝑖>0, 𝜌_𝑖>0 and 𝐺_𝑖, such that 𝑓_𝑖𝑠₁−𝑓_𝑖𝑠₂≤𝜚_𝑖𝑠₁−𝑠₂,𝑔_𝑖𝑠₁−𝑔_𝑖𝑠₂≤𝜌_𝑖𝑠₁−𝑠₂,𝑔_𝑖(⋅)≤𝐺_𝑖. (10)

3. Existence of Periodic Solution

In this section, we will give a sufficient condition which ensures the existence of periodic solution of memristor-based neural network (1).

Theorem 8. Under assumptions (𝐴₁) and (𝐴₂), if 𝐸_𝑛−𝑄 is an 𝑀-matrix, where 𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛 and 𝑞_𝑖𝑗=1𝑑_𝑖𝜚_𝑗𝑎_𝑖𝑗+𝜌_𝑗𝑏̃_𝑖𝑗1−𝜇_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛, (11) then system (1) has at least one 𝜔-periodic solution.

Proof. Set 𝐶_𝜔={𝑥(𝑡)∈𝐶(ℝ,ℝ^𝑛)∶𝑥(𝑡+𝜔)=𝑥(𝑡)}, ‖𝑥(𝑡)‖_𝐶𝜔=∑^𝑛_𝑖=1max_{𝑡∈[0,𝜔]}|𝑥_𝑖(𝑡)|. Then 𝐶_𝜔 is a Banach space with the norm ‖⋅‖_𝐶𝜔. Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝐶_𝜔, 𝐹(𝑡,𝑥)=(𝐹₁(𝑡,𝑥),𝐹₂(𝑡,𝑥),…,𝐹_𝑛(𝑡,𝑥))^𝑇, where 𝐹_𝑖(𝑡,𝑥)=−co𝑑_𝑖𝑥_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1co𝑎_𝑖𝑗𝑥_𝑖(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1co𝑏_𝑖𝑗𝑥_𝑖(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡), (12)𝑖=1,2,…,𝑛. It is obvious that the set-valued map 𝐹(𝑡,𝑥) has nonempty compact convex values. Futhermore, it is USC.
Based on the conditions of Lemma 7, the proof will be divided into three steps.
Step  1. We need to search for appropriate open bounded subset Ω. Assume that 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is an arbitrary 𝜔-periodic solution of differential inclusion 𝑥̇(𝑡)∈𝜆𝐹(𝑡,𝑥) for a certain 𝜆∈(0,1). Then, one has 𝑥̇_𝑖(𝑡)∈𝜆−co𝑑_𝑖𝑥_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1co𝑎_𝑖𝑗𝑥_𝑖(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1co𝑏_𝑖𝑗𝑥_𝑖(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡), (13) or equivalently, there exist measurable functions 𝑑_𝑖(𝑡)∈co(𝑑_𝑖(𝑥_𝑖(𝑡))), 𝑎_𝑖𝑗(𝑡)∈co(𝑎_𝑖𝑗(𝑥_𝑖(𝑡))), and 𝑏_𝑖𝑗(𝑡)∈co(𝑏_𝑖𝑗(𝑥_𝑖(𝑡))), such that 𝑥̇_𝑖(𝑡)=𝜆−𝑑_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡), (14) for 𝑖=1,2,…,𝑛. Multiplying both sides of (14) by 𝑥_𝑖(𝑡) and integrating over the interval [0,𝜔], one has ^𝜔₀𝑑_𝑖(𝑡)𝑥²_𝑖(𝑡)=𝑑𝑡_𝜔0𝑥_𝑖(𝑡)^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡)𝑑𝑡. (15) Noting that ^𝜔₀𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)=𝑑𝑡_{𝜔−𝜏𝑖𝑗(𝜔)−𝜏𝑖𝑗(0)}𝑥_𝑗(𝑡)1−𝜏̇_𝑖𝑗𝜅⁻¹_𝑖𝑗(𝑡)𝑑𝑡=^𝜔₀𝑥_𝑗(𝑡)1−𝜏̇_𝑖𝑗𝜅⁻¹_𝑖𝑗(𝑡)1𝑑𝑡≤1−𝜇_𝑖𝑗^𝜔₀𝑥_𝑗(𝑡)𝑑𝑡, (16) where 𝜅⁻¹_𝑖𝑗 is the inverse function of 𝜅_𝑖𝑗(𝑡)=𝑡−𝜏_𝑖𝑗(𝑡), 𝑖,𝑗=1,2,…,𝑛, from (15) and (16), it yields 𝑑_𝑖^𝜔₀𝑥_𝑖(𝑡)²≤𝑑𝑡_𝑛𝑗=1𝑎_𝑖𝑗^𝜔₀𝑥_𝑖(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+𝑑𝑡_𝑛𝑗=1𝑏̃_𝑖𝑗^𝜔₀𝑥_𝑖(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼̃𝑑𝑡_𝑖𝜔0𝑥_𝑖(𝑡≤)𝑑𝑡_𝑛𝑗=1𝑎_𝑖𝑗𝜚_𝑗^𝜔₀𝑥_𝑖(𝑡)𝑥_𝑗(𝑡+)𝑑𝑡_𝑛𝑗=1𝑏̃_𝑖𝑗𝜌_𝑗^𝜔₀𝑥_𝑖(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼̃𝑑𝑡_𝑖𝜔0𝑥_𝑖(𝑡≤)𝑑𝑡_𝑛𝑗=1𝑎_𝑖𝑗𝜚_𝑗+𝑏̃_𝑖𝑗𝜌_𝑗1−𝜇_𝑖𝑗^𝜔₀𝑥_𝑖(𝑡)𝑥_𝑗(𝑡+𝐼̃)𝑑𝑡_𝑖𝜔0𝑥_𝑖(𝑡≤)𝑑𝑡_𝑛𝑗=1𝑎_𝑖𝑗𝜚_𝑗+𝑏̃_𝑖𝑗𝜌_𝑗1−𝜇_𝑖𝑗^𝜔₀𝑥_𝑖(𝑡)²𝑑𝑡^1/2×^𝜔₀𝑥_𝑗(𝑡)²𝑑𝑡^1/2+𝐼_𝑖̃√𝜔^𝜔₀𝑥_𝑖(𝑡)²𝑑𝑡^1/2, (17) where 𝐼_𝑖̃=sup_{𝑡∈[0,𝜔]}|𝐼_𝑖(𝑡)|. This means ^𝜔₀𝑥_𝑖(𝑡)²𝑑𝑡^1/2≤^𝑛_𝑗=1𝑞_𝑖𝑗^𝜔₀𝑥_𝑗(𝑡)²𝑑𝑡^1/2+√𝜔𝐼_𝑖̃𝑑_𝑖. (18) Define ‖𝑥_𝑖‖^𝜔₂=(∫^𝜔₀|𝑥_𝑖(𝑡)|²𝑑𝑡)^1/2, 𝑥_𝑖∈𝐶(ℝ,ℝ), 𝑖=1,2,…,𝑛, and 𝐼_𝑁=(𝐼₁̃/𝑑₁,𝐼₂̃/𝑑₂,…,𝐼_𝑛̃/𝑑_𝑛)^𝑇. From (18), we have 𝐸_𝑛−𝑄𝑥₁^𝜔₂,𝑥₂^𝜔₂,…,𝑥_𝑛^𝜔₂^𝑇≤√𝜔𝐼_𝑁. (19) Since 𝐸_𝑛−𝑄 is an 𝑀-matrix, we can choose a vector 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)>(0,0,…,0) such that 𝜉^∗=𝜉^∗₁,𝜉^∗₂,…,𝜉^∗_𝑛=𝜉𝐸_𝑛−𝑄>(0,0,…,0). (20) By combining (19) and (20) together, one can derive min𝜉^∗₁,𝜉^∗₂,…,𝜉^∗_𝑛𝑥₁^𝜔₂+𝑥₂^𝜔₂+⋯+𝑥_𝑛^𝜔₂≤𝜉^∗₁𝑥₁^𝜔₂+𝜉^∗₂𝑥₂^𝜔₂+⋯+𝜉^∗_𝑛𝑥_𝑛^𝜔₂=𝜉𝐸_𝑛−𝑄𝑥₁^𝜔₂,𝑥₂^𝜔₂,…,𝑥_𝑛^𝜔₂^𝑇≤𝜉√𝜔𝐼₁̃𝑑₁,𝐼₂̃𝑑₂𝐼̃,…,^𝑛𝑑_𝑛^𝑇=√𝜔^𝑛_𝑖=1𝜉_𝑖𝐼_𝑖̃𝑑_𝑖. (21) Thus, we can easily get that 𝑥_𝑖^𝜔₂=^𝜔₀𝑥_𝑖(𝑡)²𝑑𝑡^1/2≤√𝜔𝑁,𝑖=1,2,…,𝑛, (22) where 𝑁=∑^𝑛_𝑖=1(𝜉_𝑖𝐼_𝑖̃/𝑑_𝑖)/min{𝜉^∗₁,𝜉^∗₂,…,𝜉^∗_𝑛}. Then, there exists 𝑡^∗∈[0,𝜔] such that 𝑥_𝑖𝑡^∗≤𝑁,𝑖=1,2,…,𝑛. (23) Obviously, for 𝑡∈[0,𝜔], 𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡^∗)+∫^𝑡_𝑡∗𝑥̇_𝑖(𝑠)𝑑𝑠. It follows from (23) that 𝑥_𝑖(𝑡)≤𝑁+^𝜔₀𝑥̇_𝑖(𝑡)𝑑𝑡,𝑖=1,2,…,𝑛. (24) On the other hand, from (14), one easily obtains that 𝑥̇^𝜔_0𝑖(𝑡)𝑑𝑡<^𝜔₀𝑑_𝑖(𝑡)𝑥_𝑖(𝑡+)𝑑𝑡_{𝑛𝑗=1𝜔0}𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+𝑑𝑡_{𝑛𝑗=1𝜔0}𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑑𝑡_𝜔0𝐼_𝑖(𝑡≤)𝑑𝑡𝑑_𝑖^𝜔₀𝑥_𝑖(𝑡)𝑑𝑡+^𝑛_𝑗=1𝑎_𝑖𝑗^𝜔₀𝑓_𝑗𝑥_𝑗(𝑡)+𝑑𝑡_𝑛𝑗=1𝑏̃_𝑖𝑗^𝜔₀𝑔_𝑗𝑥_𝑗𝑡−𝜏_{𝑖𝑗(𝑡)}𝑑𝑡+𝜔𝐼_𝑖̃≤𝑑_𝑖^𝜔₀𝑥_𝑖(𝑡)𝑑𝑡+^𝑛_𝑗=1𝑎_𝑖𝑗𝜚_𝑗^𝜔₀𝑥_𝑗(𝑡+)𝑑𝑡_𝑛𝑗=1𝑏̃_𝑖𝑗𝜌_𝑗^𝜔₀𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)𝑑𝑡+𝜔𝐼_𝑖̃≤𝑑_𝑖√𝜔𝑥_𝑖^𝜔₂+^𝑛_𝑗=1𝜚_𝑗𝑎_𝑖𝑗+𝜌_𝑗𝑏̃_𝑖𝑗1−𝜇_𝑖𝑗√𝜔𝑥_𝑗^𝜔₂+𝜔𝐼_𝑖̃≤√𝜔𝑁𝑑_𝑖+𝑑_𝑖^𝑛_𝑗=1𝑞_𝑖𝑗+𝜔𝐼_𝑖̃. (25) Let 𝑀_𝑖=√𝜔𝑁(𝑑_𝑖+𝑑_𝑖∑^𝑛_𝑗=1𝑞_𝑖𝑗)+𝜔𝐼_𝑖̃; combining (24) and (25), we can derive 𝑥_𝑖(𝑡)<𝑁+𝑀_𝑖=𝑅_𝑖,𝑖=1,2,…,𝑛. (26) Clearly, 𝑅_𝑖 is independent of 𝜆. In addition, since 𝐸_𝑛−𝑄 is an 𝑀-matrix, there exists a vector 𝜂=(𝜂₁,𝜂₂,…,𝜂_𝑛)^𝑇>(0,0,…,0)^𝑇 such that (𝐸_𝑛−𝑄)𝜂>(0,0,…,0)^𝑇. Thus, we can choose a sufficiently large constant 𝜃 such that 𝜂^∗=(𝜂^∗₁,𝜂^∗₂,…,𝜂^∗_𝑛)^𝑇=(𝜃𝜂₁,𝜃𝜂₂,…,𝜃𝜂_𝑛)^𝑇=𝜃𝜂, 𝜂^∗_𝑖=𝜃𝜂_𝑖>𝑅_𝑖, and 𝐸_𝑛−𝑄𝜂^∗>𝐼_𝑁. (27) Taking Ω={𝑥(𝑡)∈𝐶_𝜔∣−𝜂^∗<𝑥(𝑡)<𝜂^∗}, then, Ω is an open bounded set of 𝐶_𝜔 and 𝑥∉𝜕Ω for any 𝜆∈(0,1). This proves that condition (1) in Lemma 7 is satisfied.
Step  2. Suppose that there exists a solution 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈𝜕Ω⋂ℝ^𝑛 of the inclusion 0∈(1/𝜔)∫^𝜔₀𝐹(𝑡,𝑢)𝑑𝑡=𝜛(𝑢); then 𝑢 is a constant vector on ℝ^𝑛 such that |𝑢_𝑖|=𝜂^∗_𝑖 for some 𝑖∈{1,2,…,𝑛}. Therefore, one has 0∈(𝜛(𝑢))_𝑖𝑢=−^𝑖𝜔^𝜔₀co𝑑_𝑖𝑥_𝑖(𝑡)+𝑑𝑡_𝑛𝑗=11𝜔^𝜔₀co𝑎_𝑖𝑗𝑥_𝑖(𝑡)𝑓_𝑗𝑢_𝑗+𝑑𝑡_𝑛𝑗=11𝜔^𝜔₀co𝑏_𝑖𝑗𝑥_𝑖(𝑡)𝑔_𝑗𝑢_𝑗+1𝑑𝑡𝜔^𝜔₀𝐼_𝑖(𝑡)𝑑𝑡, (28) or equivalently, there exist 𝑑_𝑖(𝑡)∈co(𝑑_𝑖(𝑥_𝑖(𝑡))), 𝑎_𝑖𝑗(𝑡)∈co(𝑎_𝑖𝑗(𝑥_𝑖(𝑡))), and 𝑏_𝑖𝑗(𝑡)∈co(𝑏_𝑖𝑗(𝑥_𝑖(𝑡))), such that 𝑢0=−^𝑖𝜔^𝜔₀𝑑_𝑖(𝑡)𝑑𝑡+^𝑛_𝑗=1𝑓_𝑗𝑢_𝑗𝜔^𝜔₀𝑎_𝑖𝑗(𝑡)+𝑑𝑡_𝑛𝑗=1𝑔_𝑗𝑢_𝑗𝜔^𝜔₀𝑏_𝑖𝑗(𝑡)1𝑑𝑡+𝜔^𝜔₀𝐼_𝑖(𝑡)𝑑𝑡. (29) Then, there exists 𝑡_∗∈[0,𝜔] such that 0=−𝑢_𝑖𝑑_𝑖𝑡_∗+^𝑛_𝑗=1𝑎_𝑖𝑗𝑡_∗𝑓_𝑗𝑢_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗𝑡_∗𝑔_𝑗𝑢_𝑗+𝐼_𝑖𝑡_∗. (30) It follows from (30) that 𝜂^∗_𝑖=𝑢_𝑖≤1𝑑_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝜚_𝑗+𝑏̃_𝑖𝑗𝜌_𝑗𝑢_𝑗+𝐼_𝑖̃≤^𝑛_𝑗=1𝑞_𝑖𝑗𝑢_𝑗+𝐼_𝑖̃𝑑_𝑖=^𝑛_𝑗=1𝑞_𝑖𝑗𝜂^∗_𝑗+𝐼_𝑖̃𝑑_𝑖. (31) This means (𝐸_𝑛−𝑄)𝜂^∗≤𝐼_𝑁, which contradicts (27).
Step  3. We define a homotopic set-valued map 𝜙∶Ω⋂ℝ^𝑛×[0,1]→𝐶_𝜔 by 𝜙(𝑢,ℎ)=−diag(ℎ𝑑₁,ℎ𝑑₂,…,ℎ𝑑_𝑛)𝑢+(1−ℎ)𝜛(𝑢).
If 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈𝜕Ω⋂ℝ^𝑛, then 𝑢 is a constant vector on ℝ^𝑛 such that |𝑢_𝑖|=𝜂^∗_𝑖 for some 𝑖∈{1,2,…,𝑛}. It follows that 𝜙(𝑢,ℎ)_𝑖=−ℎ𝑑_𝑖𝑢_𝑖+(1−ℎ)−co𝑑_𝑖𝑥_𝑖(𝑡)𝑢_𝑖+^𝑛_𝑗=1𝑓_𝑗𝑢_𝑗𝜔^𝜔₀co𝑎_𝑖𝑗𝑥_𝑖(𝑡)+𝑑𝑡_𝑛𝑗=1𝑔_𝑗𝑢_𝑗𝜔^𝜔₀co𝑏_𝑖𝑗𝑥_𝑖(𝑡)+1𝑑𝑡𝜔^𝜔₀𝐼_𝑖(𝑡)𝑑𝑡. (32) In fact, we have 0∉(𝜙(𝑢,ℎ))_𝑖, 𝑖∈{1,2,…,𝑛}. If 0∈(𝜙(𝑢,ℎ))_𝑖, that is, 0∈−ℎ𝑑_𝑖𝑢_𝑖+(1−ℎ)−co𝑑_𝑖𝑥_𝑖(𝑡)𝑢_𝑖+^𝑛_𝑗=1𝑓_𝑗𝑢_𝑗𝜔^𝜔₀co𝑎_𝑖𝑗𝑥_𝑖(𝑡+)𝑑𝑡_𝑛𝑗=1𝑔_𝑗𝑢_𝑗𝜔^𝜔₀co𝑏_𝑖𝑗𝑥_𝑖(𝑡)+1𝑑𝑡𝜔^𝜔₀𝐼_𝑖(𝑡)𝑑𝑡, (33) or 0=−ℎ𝑑_𝑖𝑢_𝑖+(1−ℎ)−𝑑_𝑖(𝑡)𝑢_𝑖+^𝑛_𝑗=1𝑓_𝑗𝑢_𝑗𝜔^𝜔₀𝑎_𝑖𝑗(𝑡)+𝑑𝑡_𝑛𝑗=1𝑔_𝑗𝑢_𝑗𝜔^𝜔₀𝑏_𝑖𝑗(𝑡)1𝑑𝑡+𝜔^𝜔₀𝐼_𝑖(𝑡)𝑑𝑡. (34) There exists 𝑡₀∈[0,𝜔], such that 0=−ℎ𝑑_𝑖𝑢_𝑖+(1−ℎ)−𝑑_𝑖𝑡₀𝑢_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝑡₀𝑓_𝑗𝑢_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗𝑡₀𝑔_𝑗𝑢_𝑗+𝐼_𝑖𝑡₀, (35) that is, ℎ𝑑_𝑖−𝑑_𝑖𝑡₀𝑢_𝑖+𝑑_𝑖𝑡₀𝑢_𝑖=(1−ℎ)^𝑛_𝑗=1𝑎_𝑖𝑗𝑡₀𝑓_𝑗𝑢_𝑗+𝑏_𝑖𝑗𝑡₀𝑔_𝑗𝑢_𝑗+𝐼_𝑖𝑡₀. (36)
Therefore, we have 𝜂^∗_𝑖=𝑢_𝑖≤1−ℎℎ𝑑_𝑖−𝑑_𝑖𝑡₀+𝑑_𝑖𝑡₀×^𝑛_𝑗=1𝑎_𝑖𝑗𝑡₀𝑓_𝑗𝑢_𝑗+𝑏_𝑖𝑗𝑡₀𝑔_𝑗𝑢_𝑗+𝐼_𝑖𝑡₀≤1𝑑_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝜚_𝑗+𝑏̃_𝑖𝑗𝜌_𝑗𝑢_𝑗+𝐼_𝑖̃≤^𝑛_𝑗=1𝑞_𝑖𝑗𝑢_𝑗+𝐼_𝑖̃𝑑_𝑖=^𝑛_𝑗=1𝑞_𝑖𝑗𝜂^∗_𝑖+𝐼_𝑖̃𝑑_𝑖. (37) This means that (𝐸_𝑛−𝑄)𝜂^∗≤𝐼_𝑁, which contradicts (27). Thus, 0∉(𝜙(𝑢,ℎ))_𝑖, 𝑖=1,2,…,𝑛. It follows that (0,0,…,0)^𝑇∉𝜙(𝑢,ℎ), for any 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈𝜕Ω⋂ℝ^𝑛, ℎ∈[0,1]. Therefore, by the homotopy invariance and the solution properties of the topological degree, one has deg𝜛,Ωℝ^𝑛,0=deg𝜙(𝑢,0),Ωℝ^𝑛𝜙()ℝ,0=deg𝑢,1,Ω^𝑛−,0=deg𝑑₁𝑢₁,−𝑑₂𝑢₂,…,−𝑑_𝑛𝑢_𝑛^𝑇,Ωℝ^𝑛,(0,0,…,0)^𝑇−=sign𝑑₁⋯0⋮⋱⋮0…−𝑑_𝑛=(−1)^𝑛≠0, (38) where deg(⋅,⋅,⋅) denotes the topological degree for USC set-valued maps with compact convex values.
Up to now, we have proved that Ω satisfies all the conditions in Lemma 7, then system (1) has at least one 𝜔-periodic solution. This completes the proof.

Notice that a constant function can be regarded as a special periodic function with arbitrary period or zero amplitude. Hence, we can obtain the following result.

Corollary 9. Suppose assumption (𝐴₂) holds and 𝐼_𝑖(𝑡)=𝐼_𝑖, 𝜏_𝑖𝑗(𝑡)=𝜏_𝑖𝑗, if 𝐸_𝑛−𝑄 is an 𝑀-matrix, where 𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛 and 𝑞_𝑖𝑗=1𝑑_𝑖𝜚_𝑗𝑎_𝑖𝑗+𝜌_𝑗𝑏̃_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛, (39) then system (1) exists at least one equilibrium point.

Remark 10. By employing the method based on the 𝑀-matrix theory, our results can be easily verified and are much different from these in the literature [20, 21]. It is also worth mentioning that the 𝑀-matrix theory is one of the effective and important methods to deal with the existence of periodic solution and equilibrium point for large-scale dynamical neuron systems.

4. Complete Periodic Synchronization

In this paper, we consider model (1) as the master system, and a slave system for (1) can be described by the following equation: 𝑦̇_𝑖(𝑡)=−𝑑_𝑖𝑦_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑦_𝑖(𝑡)𝑓_𝑗𝑦_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗𝑦_𝑖(𝑡)𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡)+𝑢_𝑖(𝑡),𝑡≥0,𝑖=1,2,…,𝑛, (40) where 𝑢_𝑖(𝑡) is the controller to be designed, and 𝑑_𝑖𝑦_𝑖(𝑡)=𝑑́_𝑖,𝑦_𝑖(𝑡)≤𝑇_𝑖,𝑑̀_𝑖,𝑦_𝑖(𝑡)>𝑇_𝑖,𝑎_𝑖𝑗𝑦_𝑖(𝑡)=𝑎́_𝑖𝑗,𝑦_𝑖(𝑡)≤𝑇_𝑖,𝑎̀_𝑖𝑗,𝑦_𝑖(𝑡)>𝑇_𝑖,𝑏_𝑖𝑗𝑦_𝑖(𝑡)=𝑏́_𝑖𝑗,𝑦_𝑖(𝑡)≤𝑇_𝑖,𝑏̀_𝑖𝑗,𝑦_𝑖(𝑡)>𝑇_𝑖. (41) Let 𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝑥_𝑖(𝑡), 𝑖=1,2,…,𝑛; one can obtain the following result.

Theorem 11. Suppose that all the conditions of Theorem 8 are satisfied; then the salve system (40) can globally synchronize with the master system (1) under the following adaptive controller: 𝑢_𝑖(𝑡)=−𝛼_𝑖(𝑡)𝑒_𝑖(𝑡)−𝛿_𝑖𝛽_𝑖(𝑡)sign𝑒_𝑖(𝑡),𝛼̇_𝑖(𝑡)=−𝜀_𝑖𝑒²_𝑖(𝑡)𝛽̇,𝑖=1,2,…,𝑛,_𝑖(𝑡)=−𝜂_𝑖𝑒_𝑖(𝑡),𝑖=1,2,…,𝑛, (42) where 𝛼_𝑖, 𝜂_𝑖 are arbitrary positive constants and 𝛿_𝑖>1.

Proof. Consider the following Lyapunov functional: 𝑉(𝑡)=12^𝑛_𝑖=1𝑒²_𝑖(𝑡)+^𝑛_𝑖,𝑗=1121−𝜇_𝑖𝑗^{𝑡𝑡−𝜏}_𝑖𝑗^(𝑡)𝑒²_𝑖(𝑠)+𝑑𝑠_𝑛𝑖=112𝜀_𝑖𝛼_𝑖(𝑡)−𝐾_𝑖²+^𝑛_𝑖=112𝜂_𝑖𝑀_𝑖−𝛽_𝑖(𝑡)², (43) where 𝐾_𝑖≥max−𝑑́_𝑖+^𝑛_𝑗=1𝜚²_𝑗𝑎́²_𝑖𝑗+𝜌²_𝑗𝑏́²_𝑖𝑗+12+^𝑛_𝑗=1121−𝜇_𝑗𝑖,−𝑑̀_𝑖+^𝑛_𝑗=1𝜚²_𝑗𝑎̀²_𝑖𝑗+𝜌²_𝑗𝑏̀²_𝑖𝑗+12+^𝑛_𝑗=1121−𝜇_𝑗𝑖,𝑀_𝑖≥𝑑̀_𝑖−𝑑́_𝑖𝑇_𝑖+^𝑛_𝑗=1𝜚_𝑗𝑎́_𝑖𝑗−𝑎̀_𝑖𝑗𝑇_𝑖+^𝑛_𝑗=1𝑏́_𝑖𝑗−𝑏̀_𝑖𝑗𝐺_𝑗. (44)
The master system (1) and the slave system (40) are state-dependent switching systems; then, the four cases may appear in the following at time 𝑡.
Case  1. If |𝑥_𝑖(𝑡)|≤𝑇_𝑖, |𝑦_𝑖(𝑡)|≤𝑇_𝑖 at time 𝑡, then the master system (1) and the slave system (40) reduce to the following systems, respectively, 𝑥̇_𝑖(𝑡)=−𝑑́_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎́_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡+_{𝑛𝑖=𝑗})𝑏́_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡),𝑦̇_𝑖(𝑡)=−𝑑́_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎́_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏́_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡)+𝑢_𝑖(𝑡). (45)(46) Correspondingly, the error system can be written as 𝑒_𝑖̇(𝑡)=−𝑑́_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑎́_𝑖𝑗𝑓_𝑗𝑒_𝑗(𝑡+_𝑛𝑗=1)𝑏́_𝑖𝑗𝑔_𝑗𝑒_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑢_𝑖(𝑡), (47) where 𝑓_𝑗(𝑒_𝑗(𝑡))=𝑓_𝑗(𝑦_𝑗(𝑡))−𝑓_𝑗(𝑥_𝑗(𝑡)), 𝑔_𝑗(𝑒_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))=𝑔_𝑗(𝑦_𝑗(𝑡−𝜏_𝑖𝑗(𝑡)))−𝑔_𝑗(𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡))). Under assumption (A₂), evaluating the upper right derivation 𝐷⁺𝑉(𝑡) of 𝑉(𝑡) along the trajectory of (47) gives 𝐷⁺𝑉(𝑡)≤^𝑛_𝑖=1𝑒_𝑖(𝑡)−𝑑́_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑎́_𝑖𝑗𝑓_𝑗𝑒_𝑗(𝑡)+^𝑛_𝑗=1𝑏́_𝑖𝑗𝑔_𝑗𝑒_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝛼_𝑖(𝑡)𝑒_𝑖(𝑡)−𝛿_𝑖𝛽_𝑖(𝑡)sign𝑒_𝑖(𝑡)+^𝑛_𝑗=1121−𝜇_𝑖𝑗𝑒²_𝑗(𝑡)−12𝑒²_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝛼_𝑖(𝑡)−𝐾_𝑖𝑒²_𝑖(𝑡)−𝑀_𝑖−𝛽_𝑖(𝑡)𝑒_𝑖(𝑡≤_𝑛𝑖=1)−𝑑́_𝑖𝑒²_𝑖(𝑡)+^𝑛_𝑗=1𝜚_𝑗𝑎́_𝑖𝑗𝑒_𝑖(𝑡)𝑒_𝑗(𝑡+_𝑛𝑗=1)𝜌_𝑗𝑏́_𝑖𝑗𝑒_𝑖(𝑡)𝑒_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝛿_𝑖𝛽_𝑖(𝑡)𝑒_𝑖(𝑡+_𝑛𝑗=1)121−𝜇_𝑖𝑗𝑒²_𝑗(𝑡)−12𝑒²_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝐾_𝑖𝑒²_𝑖(𝑡)−𝑀_𝑖𝑒_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑒_𝑖(𝑡≤_𝑛𝑖=1)−𝐾_𝑖+𝑑́_𝑖𝑒²_𝑖(𝑡)+^𝑛_𝑗=112𝜚²_𝑗𝑎́²_𝑖𝑗𝑒_𝑖(𝑡)²+12𝑒_𝑗(𝑡)²+^𝑛_𝑗=112𝜌²_𝑗𝑏́²_𝑖𝑗𝑒_𝑖(𝑡)²+12𝑒²_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝛿_𝑖−1𝛽_𝑖(𝑡)𝑒_𝑖(𝑡+_𝑛𝑗=1)121−𝜇_𝑖𝑗𝑒²_𝑗(𝑡)−12𝑒²_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑀_𝑖𝑒_𝑖(𝑡=_𝑛𝑖=1)−𝐾_𝑖−𝑑́_𝑖+^𝑛_𝑗=1𝜚²_𝑗𝑎́²_𝑖𝑗+𝜌²_𝑗𝑏́²_𝑖𝑗+12+^𝑛_𝑗=1121−𝜇_𝑗𝑖𝑒²_𝑖(𝑡)−𝛿_𝑖−1𝛽_𝑖(𝑡)𝑒_𝑖(𝑡)−𝑀_𝑖𝑒_𝑖(𝑡). (48) Considering the definition of 𝐾_𝑖, and 𝛿_𝑖>1, 𝑀_𝑖>0, one has 𝐷⁺𝑉(𝑡)≤0. (49)Case  2. If |𝑥_𝑖(𝑡)|>𝑇_𝑖, |𝑦_𝑖(𝑡)|>𝑇_𝑖 at time 𝑡, then the master system (1) and the slave system (40) reduce to the following systems, respectively, 𝑥̇_𝑖(𝑡)=−𝑑̀_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎̀_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡+_{𝑛𝑖=𝑗})𝑏̀_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡),𝑦̇_𝑖(𝑡)=−𝑑̀_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎̀_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏̀_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡)+𝑢_𝑖(𝑡). (50)(51) Correspondingly, the error system can be rewritten as 𝑒_𝑖̇(𝑡)=−𝑑̀_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑎̀_𝑖𝑗𝑓_𝑗𝑒_𝑗(𝑡+_𝑛𝑗=1)𝑏̀_𝑖𝑗𝑔_𝑗𝑒_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑢_𝑖(𝑡). (52) Arguing as in the proof of Case  1, we can obtain 𝐷⁺𝑉(𝑡)≤^𝑛_𝑖=1−𝐾_𝑖−𝑑̀_𝑖+^𝑛_𝑗=1𝜚²_𝑗𝑎̀²_𝑖𝑗+𝜌²_𝑗𝑏̀²_𝑖𝑗+12+^𝑛_𝑗=1121−𝜇_𝑗𝑖𝑒²_𝑖(𝑡)−𝛿_𝑖−1𝛽_𝑖(𝑡)𝑒_𝑖(𝑡)−𝑀_𝑖𝑒_𝑖(𝑡)≤0. (53)Case  3. If |𝑥_𝑖(𝑡)|>𝑇_𝑖, |𝑦_𝑖(𝑡)|≤𝑇_𝑖 at time 𝑡, then the master system (1) and the slave system (40) reduce to (50) and (46). Correspondingly, the error system can be rewritten as 𝑒_𝑖̇(𝑡)=−𝑑̀_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑎̀_𝑖𝑗𝑓_𝑗𝑒_𝑗(𝑡)+^𝑛_𝑗=1𝑏̀_𝑖𝑗𝑔_𝑗𝑒_𝑗𝑡−𝜏_𝑖𝑗(𝑡+𝑑̀_𝑖)−𝑑́_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎́_𝑖𝑗−𝑎̀_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏́_𝑖𝑗−𝑏̀_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑢_𝑖(𝑡). (54) Similarly, evaluating the upper right derivation 𝐷⁺𝑉(𝑡) of 𝑉(𝑡) along the trajectory of (54), we have 𝐷⁺𝑉(𝑡)≤^𝑛_𝑖=1−𝐾_𝑖−𝑑̀_𝑖+^𝑛_𝑗=1𝜚²_𝑗𝑎̀²_𝑖𝑗+𝜌²_𝑗𝑏̀²_𝑖𝑗+12+^𝑛_𝑗=1121−𝜇_𝑗𝑖𝑒²_𝑖(𝑡)+𝑒_𝑖(𝑡)𝑑̀_𝑖−𝑑́_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎́_𝑖𝑗−𝑎̀_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏́_𝑖𝑗−𝑏̀_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝛿_𝑖−1𝛽_𝑖(𝑡)𝑒_𝑖(𝑡)−𝑀_𝑖𝑒_𝑖(𝑡). (55) Note that |𝑦_𝑖(𝑡)|≤𝑇_𝑖; by using assumption (A₂), one has 𝐷⁺𝑉(𝑡)≤^𝑛_𝑖=1−𝐾_𝑖−𝑑̀_𝑖+^𝑛_𝑗=1𝜚²_𝑗𝑎̀²_𝑖𝑗+𝜌²_𝑗𝑏̀²_𝑖𝑗+12+^𝑛_𝑗=1121−𝜇_𝑗𝑖𝑒²_𝑖(𝑡)+𝑒_𝑖(𝑡)𝑑̀_𝑖−𝑑́_𝑖𝑇_𝑖+^𝑛_𝑗=1𝜚_𝑗𝑎́_𝑖𝑗−𝑎̀_𝑖𝑗𝑇_𝑖+^𝑛_𝑗=1𝑏́_𝑖𝑗−𝑏̀_𝑖𝑗𝐺_𝑗−𝛿_𝑖−1𝛽_𝑖(𝑡)𝑒_𝑖(𝑡)−𝑀_𝑖𝑒_𝑖(𝑡). (56) According to the definition of 𝐾_𝑖, 𝑀_𝑖, and 𝛿_𝑖>1, one has 𝐷⁺𝑉(𝑡)≤0. (57)Case  4. If |𝑥_𝑖(𝑡)|≤𝑇_𝑖, |𝑦_𝑖(𝑡)|>𝑇_𝑖 at time 𝑡, then the master system (1) and the slave system (40) reduce to (45) and (51). Correspondingly, the error system can be rewritten as 𝑒_𝑖̇(𝑡)=−𝑑̀_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑎̀_𝑖𝑗𝑓_𝑗𝑒_𝑗(𝑡)+^𝑛_𝑗=1𝑏̀_𝑖𝑗𝑔_𝑗𝑒_𝑗𝑡−𝜏_𝑖𝑗(𝑡+𝑑́_𝑖)−𝑑̀_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎̀_𝑖𝑗−𝑎́_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏̀_𝑖𝑗−𝑏́_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑢_𝑖(𝑡). (58) By using |𝑥_𝑖(𝑡)|≤𝑇_𝑖, we can also have 𝐷⁺𝑉(𝑡)≤^𝑛_𝑖=1−𝐾_𝑖−𝑑̀_𝑖+^𝑛_𝑗=1𝜚²_𝑗𝑎̀²_𝑖𝑗+𝜌²_𝑗𝑏̀²_𝑖𝑗+12+^𝑛_𝑗=1121−𝜇_𝑗𝑖𝑒²_𝑖(𝑡)+𝑒_𝑖(𝑡)𝑑̀_𝑖−𝑑́_𝑖𝑇_𝑖+^𝑛_𝑗=1𝜚_𝑗𝑎́_𝑖𝑗−𝑎̀_𝑖𝑗𝑇_𝑖+^𝑛_𝑗=1𝑏́_𝑖𝑗−𝑏̀_𝑖𝑗𝐺_𝑗−𝛿_𝑖−1𝛽_𝑖(𝑡)𝑒_𝑖(𝑡)−𝑀_𝑖𝑒_𝑖(𝑡)≤0. (59) The above proving procedures clearly imply that one always has 𝐷⁺𝑉(𝑡)≤0 at time 𝑡. Therefore, the salve system (40) globally synchronizes with the master system (1) under the adaptive controller (42). This completes the proof.

Remark 12. In the literature, some results on stability analysis of periodic solution (or equilibrium point) or synchronization (or antisynchronization) control of memristor-based neural network were obtained [11–13, 16, 17, 20, 21]. A typical assumption is that 𝑑_𝑖,𝑑_𝑖𝑥−𝑑_𝑖,𝑑_𝑖𝑦⊆𝑑_𝑖,𝑑_𝑖𝑥−𝑦,…. (60) However, We can prove that this assumption holds only when 𝑥 and 𝑦 have different sign, or 𝑥=0, or 𝑦=0. Without this assumption, we divide the error system into four cases in this paper. Under the adaptive controller (42), globally periodic synchronization criterion between system (1) and (40) is derived. The synchronization criterion of this paper which does not solve any inequality or linear matrix inequality is easily verified.

Remark 13. As far as we know, there is no work on the periodic synchronization of memristor-based neural network via adaptive control. Thus, our outcomes are brand new and original compared to the existing results ([11–14]). In addition, the obtained results in this paper are also applicable to the common systems without memristor or the memductance of the memristor equals a constant since they are special cases of memristor-based neural networks.

5. Numerical Example

In this section, one example is offered to illustrate the effectiveness of the results obtained in this paper.

Example 1. Consider the second-order memristor-based neural network (1) with the following system parameters: 𝑑₁𝑥₁(𝑡)=6.5,𝑥₁(𝑡≤1)4,6,𝑥₁(𝑡>1)4,𝑑₂𝑥₂(𝑡)=6,𝑥₂(𝑡≤1)4,6.5,𝑥₂(𝑡>1)4,𝑎₁₁𝑥₁(𝑡)=−2,𝑥₁(𝑡≤1)4,2,𝑥₁(𝑡>1)4,𝑎₁₂𝑥₁(𝑡=𝑥)−1,₁(𝑡≤1)4,1,𝑥₁(𝑡>1)4,𝑎₂₁𝑥₂(𝑡)=1,𝑥₂(𝑡≤1)4,−1,𝑥₂(𝑡>1)4,𝑎₂₂𝑥₂(𝑡)=−2,𝑥₂(𝑡≤1)4,2,𝑥₂(𝑡>1)4,𝑏₁₁𝑥₁(𝑡)=1,𝑥₁(𝑡≤1)4,−1,𝑥₁(𝑡>1)4,𝑏₁₂𝑥₁(𝑡)=−1,𝑥₁(𝑡≤1)4,1,𝑥₁(𝑡>1)4,𝑏₂₁𝑥₂(𝑡=1𝑥)²(𝑡≤1)4,−1,𝑥₂(𝑡>1)4,𝑏₂₂𝑥₂(𝑡)=−1,𝑥₂(𝑡≤1)4,1,𝑥₂(𝑡>1)4, (61) and the activation functions are taken as follows: 𝑓₁(𝑠)=𝑓₂(𝑠)=𝑠,𝑔₁(𝑠)=𝑔₂(𝑠)=12tanh𝑠. (62) It can be verified that 𝑑₁=𝑑₂=6, 𝑎₁₁=𝑎₂₂=2, 𝑎₁₂=𝑎₂₁=1, 𝑏̃_𝑖𝑗=1, 𝑖,𝑗=1,2, 𝜚₁=𝜚₂=1, and 𝜌₁=𝜌₂=1/2.
We take 𝜏_𝑖𝑗(𝑡)=3/4−(1/4)sin3𝑡, 𝑖,𝑗=1,2. A straightforward calculation gives 𝜏=1 and 𝜇_𝑖𝑗=3/4. Then, we get 𝐸₂−𝑄=^1/2−1/3_−1/31/2. Obviously, 𝐸₂−𝑄 is an 𝑀-matrix. Thus, the conditions required in Theorem 8 are satisfied. When 𝐼(𝑡) is a periodic function, in the view of Theorem 8, this neural network has at least one periodic solution.
For numerical simulations, we choose the external input (𝐼₁(𝑡),𝐼₂(𝑡))^𝑇=(0.65sin3𝑡−2,0.65cos3𝑡+2)^𝑇. The periodic dynamic behavior of the master system (1) and the slave system (40) with 𝑢_𝑖(𝑡)=0 is given in Figures 1, 2, and 3, with the initial states chosen as 𝑥(𝑡)=(0.6sin2𝑡,0.3cos𝑡)^𝑇 and 𝑦(𝑡)=(−0.2cos2𝑡,−0.8sin2𝑡)^𝑇 for 𝑡∈[−1,0].
In order to demonstrate the adaptive controller (42) can realize complete periodic synchronization of memristor-based neural networks, some initial parameters are taken as 𝛼₁(𝑡)=𝛼₂(𝑡)=0.2, 𝛽₁(𝑡)=𝛽₂(𝑡)=0.1 for 𝑡∈[−1,0], 𝛿_𝑖=10, 𝜀_𝑖=𝜂_𝑖=0.1, 𝑖=1,2. We get the simulation results shown in Figures 4–6. Figure 4 describes the time responses of synchronization errors 𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝑥_𝑖(𝑡), 𝑖=1,2, which turn to zero quickly as time goes. Figure 5 shows the time response of 𝛼(𝑡)=(𝛼₁(𝑡),𝛼₂(𝑡))^𝑇. Figure 6 depicts the time response of 𝛽(𝑡)=(𝛽₁(𝑡),𝛽₂(𝑡))^𝑇. From Figures 5 and 6 one can see that the control parameters 𝛼_𝑖(𝑡) and 𝛽_𝑖(𝑡), 𝑖=1,2, turn out to be some constants eventually.

Figure 1 

Time-domain behavior of the state variables 𝑥1(𝑡) and 𝑦1(𝑡) with 𝑢𝑖(𝑡)=0.

Figure 2 

Time-domain behavior of the state variables 𝑥2(𝑡) and 𝑦2(𝑡) with 𝑢𝑖(𝑡)=0.

Figure 3 

Phase plane behavior of the master system (1) and the salve system (40) with 𝑢𝑖=0.

Figure 4 

The synchronization errors 𝑒1(𝑡) and 𝑒2(𝑡) under the adaptive controller (42).

Figure 5 

Trajectories of control parameters 𝛼1(𝑡) and 𝛼2(𝑡).

Figure 6 

Trajectories of control parameters 𝛽1(𝑡) and 𝛽2(𝑡).

6. Conclusion

In this paper, complete periodic synchronization of a class of memristor-based neural networks has been investigated. The master system synchronizes with the slave system by using adaptive control. The obtained results are novel since there are few works about complete periodic synchronization issue of memristor-based neural networks via adaptive control. In addition, the easily testable condition which ensures the existence of periodic solution of a class of memristor-based recurrent neural network is also much different from the existing work. The obtained results are also applicable to the continuous systems without switching jumps. Finally, a numerical example has been given to illustrate the validity of the present results.

Acknowledgments

This work was supported by the Natural Science Foundation of Hebei Province of China (A2011203103) and the Hebei Province Education Foundation of China (2009157).