Multistability and Multiperiodicity for a General Class of Delayed Cohen-Grossberg Neural Networks with Discontinuous Activation Functions

Abstract

A general class of Cohen-Grossberg neural networks with time-varying delays, distributed delays, and discontinuous activation functions is investigated. By partitioning the state space, employing analysis approach and Cauchy convergence principle, sufficient conditions are established for the existence and locally exponential stability of multiple equilibrium points and periodic orbits, which ensure that 𝑛-dimensional Cohen-Grossberg neural networks with 𝑘-level discontinuous activation functions can have 𝑘^𝑛 equilibrium points or 𝑘^𝑛 periodic orbits. Finally, several examples are given to illustrate the feasibility of the obtained results.

1. Introduction

In recent years, great attention has been paid to the neural networks due to their applications in many areas such as signal processing, associative memory, pattern recognition, parallel computation, and optimization. It should be pointed out that the successful applications heavily rely on the dynamic behaviors of neural networks. Stability, as one of the most important properties for neural networks, is crucially required when designing neural networks. As models of human brains, neural networks have memory function that is, the state of the present time is related to the state of the past. Time delay provides information of history. In addition, neural networks have recently been implemented on electronic chips. In electronic implementation of neural networks, time delays are unavoidably encountered during the processing and transmission of signals. As in many other dynamical systems, it is well known that delays may result in oscillation and instability. Hence, to study delayed neural networks, one must address the problem of how to remove this destabilizing effect. There have been a great number of results on stability of delayed neural networks reported in the past two decades. Most of them focus on the uniqueness and global stability (or attractiveness) of the equilibrium, and periodic (or almost-periodic) trajectory of the delayed neural networks. Readers can refer to [1–6] and many others. On the other hand, in the applications of associative memory storage, pattern recognition, and decision making, the addressable memories or patterns or decisions are stored as stable equilibria or stable periodic orbits. So it is necessary that there exist multiple stable equilibria or periodic orbits for neural networks, which are usually referred to as multistability or multiperiodicity, respectively. In the last few years, the multistability and multiperiodicity of neural networks have been reported in [7–26] and the references therein. In particular, [14–21] investigated the multistability or multiperiodicity of neural networks with sigmoidal activation functions or nondecreasing saturated activation functions. In [22], sufficient conditions of the multistability were presented for a class of neural networks with Mexican-hat-type activation functions. [23] studied some multistability properties for a class of bidirectional associative memory recurrent neural networks with unsaturating piecewise linear transfer functions. In order to increase storage capacity, [24, 25] investigated the multistability of two classes of neural networks with piecewise linear activation functions. Stability of multiple equilibria of neural networks with time-varying delays and concave-convex characteristics was formulated in [26]. Note that most of the results above were based on the assumption that the activation functions are continuous, whereas, as mentioned in [27] the neural networks with discontinuous activation functions are important and frequently encountered in applications when dealing with dynamical systems possessing high-slope nonlinear elements. For this reason, much effort has been devoted to analyzing the dynamic behavior of neural networks with discontinuous activation functions (see [28–31]). In [29], the authors considered a recurrent neural network with a special class of discontinuous activation functions which is piecewise constants in the state space. In [30, 31], the multistability issues were discussed for two classes of recurrent neural networks with 𝑘-level discontinuous activation functions. Sufficient conditions are established to ensure the existence of 𝑘^𝑛 locally exponentially stable equilibria.

We point out that the majority of the references mentioned above studied the Hopfield neural networks. As ones of the important neural networks, Cohen-Grossberg neural networks (CGNNs) include a lot of famous ecological systems and neural networks as special cases such as the Lotka-Volterra system, the Gilpia-Analg competition system, the Eingen-Schuster system, and the Hopfield neural networks, cellular neural networks, and other neural networks. CGNNs have aroused a tremendous surge of investigation in these years. On the other hand, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. It is desired to model them by introducing continuously distributed delays over a certain duration of time such that the distant past has less influence compared with the recent behavior of the state, whereas, to the best of our knowledge, the multistability and multiperiodicity issues are seldom considered for the CGNNs with distributed delays and discontinuous activation functions.

Motivated by the works of [30, 31] and the discussions above, the purpose of this paper is to explore the multistability and multiperiodicity of CGNNs with time-varying delays, distributed delays, and discontinuous activation functions. To this end, we consider the following differential equations: 𝑢̇_𝑖(𝑡)=−𝛼_𝑖𝑢_𝑖(𝑡)𝛽_𝑖𝑢_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗×𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗×^𝑡_𝑡−𝜎𝑘_𝑖𝑗(𝑡−𝑠)𝑓_𝑗𝑢_𝑗(𝑠)𝑑𝑠−𝐽_𝑖,𝑖=1,2,…,𝑛, (1) where 𝑢_𝑖(𝑡) denotes the state of the 𝑖th neuron at time 𝑡; 𝛼_𝑖(𝑢_𝑖(𝑡)) denotes the amplification function; 𝛽_𝑖>0; 𝑓_𝑗(𝑢_𝑗(𝑡)) denotes the activation function; 𝜏_𝑗(𝑡) denotes the time-varying delay associated with the 𝑗th neuron, which is variable with time due to the finite switching speed of amplifiers and satisfies 0≤𝜏_𝑗(𝑡)≤𝜏; 𝑘_𝑖𝑗(𝑡)>0 represents the delay kernel function, which is a real-valued continuous function; 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, and 𝑐_𝑖𝑗 denote the connection strengths; 𝐽_𝑖 is the external input bias.

The characteristics of activation functions have a major impact on the existence and stability of stationary patterns for neural networks. In this paper, we consider the following 𝑘-level discontinuous activation functions: 𝑓_𝑗(𝜉)=𝑑_𝑗1,𝜉∈−∞,𝑝_𝑗1,𝑑_𝑗2,𝜉∈𝑝_𝑗1,𝑝_𝑗2,𝑑⋮⋮_{𝑗,𝑘−1},𝜉∈𝑝_{𝑗,𝑘−2},𝑝_{𝑗,𝑘−1},𝑑_𝑗𝑘,𝜉∈𝑝_{𝑗,𝑘−1},𝑝,+∞undefined,𝜉∈_𝑗1,𝑝_𝑗2,…,𝑝_{𝑗,𝑘−1}, (2) where 𝑗=1,2,…,𝑛; 𝑘 is an integer satisfying 𝑘≥1; 𝑑_𝑗1,𝑑_𝑗2,…,𝑑_𝑗𝑘, 𝑝_𝑗1,𝑝_𝑗2,…,𝑝_{𝑗,𝑘−1} are constants; and 𝑝_𝑗1<𝑝_𝑗2<⋯<𝑝_{𝑗,𝑘−1}.

The organization of this paper is as follows. In Section 2, some preliminaries are given. In Section 3, through decomposing state space into multiple positively invariant sets, we derive sufficient conditions for the existence and exponential stability of equilibria in any designated region. As an extension, similar results are presented for delayed CGNNs with periodic time-varying delays and external inputs in Section 4. In Section 5, four numerical examples are provided to illustrate the feasibility of the obtained results. A concluding remark is given in Section 6 to end this work.

2. Preliminaries

Denote 𝛿=max{𝜏,𝜎}. Let 𝒞([𝑡₀−𝛿,𝑡₀],𝑅^𝑛) be the space of all continuous functions mapping [𝑡₀−𝛿,𝑡₀] into 𝑅^𝑛 with the norm defined by ‖Φ‖=max_{1≤𝑖≤𝑛}{sup_{𝑡0−𝛿≤𝑠≤𝑡0}|𝜙_𝑖(𝑠)|}, where Φ=(𝜙₁,𝜙₂,…,𝜙_𝑛)∈𝒞([𝑡₀−𝛿,𝑡₀],𝑅^𝑛). Denote ‖𝑢‖=max_{1≤𝑖≤𝑛}{|𝑢_𝑖|} as the vector norm of 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)∈𝑅^𝑛. For any function 𝑢(⋅)∈𝒞([𝑡₀−𝛿,𝑡₀],𝑅^𝑛), we define 𝑢_𝑡(𝑠)=𝑢(𝑡+𝑠), 𝑠∈[𝑡₀−𝛿,𝑡₀], and 𝑡≥𝑡₀. We denote by 𝑢^Φ(𝑡) the solution of system (1) with initial condition 𝑢(𝑠)=Φ(𝑠), 𝑠∈[𝑡₀−𝛿,𝑡₀]. Throughout this paper, we make the following assumptions.  (H1) The amplification function 𝛼_𝑖(𝑢_𝑖(𝑡)) is continuous, and there exist constants 𝛼_𝑖 and 𝛼_𝑖 such that 0<𝛼_𝑖≤𝛼_𝑖𝑢_𝑖(𝑡)≤𝛼_𝑖<+∞,𝑖=1,2,…,𝑛. (3) (H2) Each kernel function 𝑘_𝑖𝑗(𝑠)∶[0,+∞)→[0,+∞) is continuous and satisfies 𝜆_𝑖𝑗(𝜎)=^𝑡_𝑡−𝜎𝑘_𝑖𝑗(𝑡−𝑠)𝑑𝑠=^𝜎₀𝑘_𝑖𝑗(𝑠)𝑑𝑠≥0,𝑖,𝑗=1,2,…,𝑛. (4)Moreover, when 𝜎=+∞, 𝜆_𝑖𝑗(+∞)=1.

Since 𝑓_𝑗(⋅) is allowed to have points of discontinuity, we consider the solution of system (1) in the sense of Filippov [32].

Definition 1 (Forti et al. [33]). A function 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡))^𝑇∶[𝑡₀−𝛿,𝑇)→𝑅^𝑛, 𝑇∈(𝑡₀,+∞) is a solution of system (1) on [𝑡₀−𝛿,𝑇) if (1)𝑢(𝑡) is continuous on [𝑡₀−𝛿,𝑇) and absolutely continuous on [𝑡₀,𝑇); (2)there exists a measurable function 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡),…,𝛾_𝑛(𝑡))^𝑇∶[𝑡₀−𝛿,𝑇)→𝑅^𝑛, such that 𝛾_𝑖(𝑡)∈𝐾[𝑓_𝑖(𝑢_𝑖(𝑡))] for almost all (a.a.) 𝑡∈[𝑡₀−𝛿,𝑇), and 𝑢̇_𝑖(𝑡)=−𝛼_𝑖𝑢_𝑖(𝑡)𝛽_𝑖𝑢_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝛾_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗×𝛾_𝑗𝑡−𝜏_𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗×^𝑡_𝑡−𝜎𝑘_𝑖𝑗(𝑡−𝑠)𝛾_𝑗(𝑠)𝑑𝑠−𝐽_𝑖 (5)for a.a. 𝑡∈[𝑡₀,𝑇), where 𝐾[𝑓_𝑖(⋅)] represents the closure of the convex hull of 𝑓_𝑖(⋅).

Definition 2 (Forti et al. [33]). For any continuous function 𝜙(𝜃)=(𝜙₁(𝜃),𝜙₂(𝜃),…,𝜙_𝑛(𝜃))^𝑇∶[𝑡₀−𝛿,𝑡₀]→𝑅^𝑛 and any measurable selection 𝜆(𝜃)=(𝜆₁(𝜃),𝜆₂(𝜃),…,𝜆_𝑛(𝜃))^𝑇∶[𝑡₀−𝛿,𝑡₀]→𝑅^𝑛, such that 𝜆_𝑖(𝜃)∈𝐾[𝑓_𝑖(𝜙_𝑖(𝜃))] for a.a. 𝜃∈[𝑡₀−𝛿,𝑡₀], by an initial value problem associated to system (1) with initial condition 𝜙(𝜃), 𝜆(𝜃), we mean the following problem: find a couple of functions [𝑢(𝑡),𝛾(𝑡)]∶[𝑡₀−𝛿,𝑇)→𝑅^𝑛×𝑅^𝑛, such that 𝑢(𝑡) is a solution of system (1) on [𝑡₀−𝛿,𝑇) for some 𝑇>𝑡₀, 𝛾(𝑡) is an output associated to 𝑢(𝑡), and 𝑢̇_𝑖(𝑡)=−𝛼_𝑖𝑢_𝑖(𝑡)𝛽_𝑖𝑢_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝛾_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗×𝛾_𝑗𝑡−𝜏_𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗×^𝑡_𝑡−𝜎𝑘_𝑖𝑗(𝑡−𝑠)𝛾_𝑗(𝑠)𝑑𝑠−𝐽_𝑖,fora.a.𝑡∈𝑡₀,𝑢,𝑇_𝑖(𝜃)=𝜙_𝑖(𝜃),forany𝜃∈𝑡₀−𝛿,𝑡₀,𝛾_𝑖(𝜃)=𝜆_𝑖(𝜃),fora.a.𝜃∈𝑡₀−𝛿,𝑡₀, (6) where 𝐾[𝑓_𝑖(⋅)] represents the closure of the convex hull of 𝑓_𝑖(⋅).

Definition 3. Let 𝐷 be a subset of 𝑅^𝑛. 𝐷 is said to be a positively invariant set of system (1) if and only if for any initial condition Φ∈𝒞([𝑡₀−𝛿,𝑡₀],𝐷), we have 𝑢^Φ(𝑡)∈𝐷 for all 𝑡≥𝑡₀.

Definition 4. An equilibrium 𝑢^∗ of system (1) is said to be locally exponentially stable in 𝐷∈𝑅^𝑛, if there exist constants 𝑇≥𝑡₀, 𝜀>0 and 𝑀>0 such that 𝑢^Φ(𝑡)−𝑢^∗‖≤𝑀‖Φ−𝑢^∗exp−𝜀𝑡−𝑡₀,𝑡≥𝑇, (7) where initial condition Φ∈𝒞([𝑡₀−𝛿,𝑡₀],𝐷). When 𝐷=𝑅^𝑛, 𝑢^∗ is said to be globally exponentially stable.

Definition 5. A periodic orbit 𝑢^∗(𝑡) of system (1) is said to be locally exponentially stable in 𝐷∈𝑅^𝑛, if there exist constants 𝑇≥𝑡₀, 𝜀>0 and 𝑀>0 such that 𝑢^Φ(𝑡)−𝑢^∗(𝑡)‖≤𝑀‖Φ−𝑢^∗_𝑡0exp−𝜀𝑡−𝑡₀,𝑡≥𝑇, (8) where initial condition Φ∈𝒞([𝑡₀−𝛿,𝑡₀],𝐷) and 𝑢^∗_𝑡0(𝑠)=𝑢^∗(𝑡₀+𝑠), 𝑠∈[𝑡₀−𝛿,𝑡₀]. When 𝐷=𝑅^𝑛, 𝑢^∗(𝑡) is said to be globally exponentially stable.

Denote −∞,𝑝_𝑖1=−∞,𝑝_𝑖1¹×𝑝_𝑖1,𝑝_𝑖2⁰×𝑝×⋯_{𝑖,𝑘−1},+∞⁰,𝑝_𝑖1,𝑝_𝑖2=−∞,𝑝_𝑖1⁰×𝑝_𝑖1,𝑝_𝑖2¹×𝑝×⋯_{𝑖,𝑘−1},+∞⁰,⋮𝑝_{𝑖,𝑘−1},+∞=−∞,𝑝_𝑖1⁰×𝑝_𝑖1,𝑝_𝑖2⁰×𝑝×⋯_{𝑖,𝑘−1},+∞¹,Ω=^𝑛_𝑖=1−∞,𝑝_𝑖1^𝛿𝑖1×𝑝_𝑖1,𝑝_𝑖2^𝛿𝑖2×𝑝×⋯_{𝑖,𝑘−1},+∞^𝛿𝑖𝑘, (9) where 𝛿_𝑖𝑙 equals 1 or 0  (𝑙=1,2,…,𝑘) and ∑^𝑘_𝑙=1𝛿_𝑖𝑙=1. It is easy to see that Ω is composed of 𝑘^𝑛 parts.

3. Multistability for the CGNNs

In this section, we consider the existence and exponential stability of equilibrium points for system (1) in each division region.

Theorem 6. Under assumptions (H1) and (H2), for each division region Λ∈Ω, Λ is a positively invariant set of system (1), if the following conditions are satisfied: −𝛽_𝑖𝑝_𝑖𝑙+𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑙+𝐾_𝑖+𝐽_𝑖<0,𝑙=1,2,…,𝑘−1,−𝛽_𝑖𝑝_{𝑖,𝑚−1}+𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑚−𝐾_𝑖+𝐽_𝑖>0,𝑚=2,3,…,𝑘, (10) where 𝑖=1,2,…,𝑛, 𝐾_𝑖=∑^𝑛_{𝑗=1,𝑗≠𝑖}(|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|+|𝑐_𝑖𝑗|𝜆_𝑖𝑗(𝜎))𝑑_𝑖, and 𝑑_𝑖=max_{1≤𝑗≤𝑛,𝑗≠𝑖,1≤𝑙≤𝑘}{|𝑑_𝑗𝑙|}.

Proof. From (10), it is readily seen that there exists a small constant 0<𝜖₀<min_{1≤𝑗≤𝑛,2≤𝑙≤𝑘−1}{(𝑝_𝑗𝑙−𝑝_{𝑗,𝑙−1})/2} such that for all 0<𝜖<𝜖₀, −𝛽_𝑖𝑝_𝑖𝑙−𝜖+𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑙+𝐾_𝑖+𝐽_𝑖<0,𝑙=1,2,…,𝑘−1,−𝛽_𝑖𝑝_{𝑖,𝑚−1}+𝜖+𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑚−𝐾_𝑖+𝐽_𝑖>0,𝑚=2,3,…,𝑘. (11) Denote −∞,𝑝_𝑖1−𝜖=−∞,𝑝_𝑖1−𝜖¹×𝑝_𝑖1+𝜖,𝑝_𝑖2−𝜖⁰×⋯×𝑝_{𝑖,𝑘−1}+𝜖,+∞⁰,𝑝_𝑖1+𝜖,𝑝_𝑖2−𝜖=−∞,𝑝_𝑖1−𝜖⁰×𝑝_𝑖1+𝜖,𝑝_𝑖2−𝜖¹×⋯×𝑝_{𝑖,𝑘−1}+𝜖,+∞⁰,⋮𝑝_{𝑖,𝑘−1}+𝜖,+∞=−∞,𝑝_𝑖1−𝜖⁰×𝑝_𝑖1+𝜖,𝑝_𝑖2−𝜖⁰×⋯×𝑝_{𝑖,𝑘−1}+𝜖,+∞¹,Ω=^𝑛_𝑖=1−∞,𝑝_𝑖1−𝜖^𝛿𝑖1×𝑝_𝑖1+𝜖,𝑝_𝑖2−𝜖^𝛿𝑖2×⋯×𝑝_{𝑖,𝑘−1}+𝜖,+∞^𝛿𝑖𝑘, (12) where 𝛿_𝑖𝑙 equals 1 or 0(𝑙=1,2,…,𝑘) and ∑^𝑘_𝑙=1𝛿_𝑖𝑙=1. Obviously, there exists a bijection between Ω and Ω, and Ω is the limit set of Ω as 𝜖→0. Consequently, we only need to prove that for all Λ∈Ω, it is a positively invariant set of system (1).
For each Λ∈Ω, we consider any solution 𝑢^Φ(𝑡)=(𝑢^Φ₁(𝑡),𝑢^Φ₂(𝑡),…,𝑢^Φ_𝑛(𝑡)) of system (1) with initial condition Φ∈𝒞([𝑡₀−𝛿,𝑡₀],Λ). We claim that the solution 𝑢^Φ(𝑡)∈Λ for all 𝑡≥𝑡₀. If this is not true, we need to discuss the following three cases.
Case I. If 𝛿_𝑖1=1 and there exists a 𝑡₁>𝑡₀ such that 𝑢^Φ_𝑖(𝑡₁)=𝑝_𝑖1−𝜖, 𝑢̇^Φ_𝑖(𝑡₁)≥0 and 𝑢^Φ_𝑖(𝑡)<𝑝_𝑖1−𝜖 for 𝑡₀−𝛿≤𝑡<𝑡₁, we get from (1) and (11) that 𝑢̇^Φ_𝑖𝑡₁=−𝛼_𝑖𝑢^Φ_𝑖𝑡₁𝛽_𝑖𝑢^Φ_𝑖𝑡₁−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑢^Φ_𝑗𝑡₁−^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢^Φ_𝑗𝑡₁−𝜏_𝑗𝑡₁−^𝑛_𝑗=1𝑐_𝑖𝑗×^𝑡1_𝑡1−𝜎𝑘_𝑖𝑗𝑡₁−𝑠𝑓_𝑗𝑢^Φ_𝑗(𝑠)𝑑𝑠−𝐽_𝑖≤𝛼_𝑖𝑝_𝑖1−𝜖−𝛽_𝑖𝑝_𝑖1−𝜖+𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)×𝑑_𝑖1+^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗×𝜆_𝑖𝑗(𝜎)𝑑_𝑖+𝐽_𝑖<0, (13) which yields a contradiction to 𝑢̇^Φ_𝑖(𝑡₁)≥0. Therefore, 𝑢^Φ_𝑖(𝑡)∈(−∞,𝑝_𝑖1−𝜖) for all 𝑡≥𝑡₀.
Case II. If 𝛿_𝑖𝑙=1  (𝑙=2,3,…,𝑘−1) and there exists a 𝑡₁>𝑡₀ such that either 𝑢^Φ_𝑖(𝑡₁)=𝑝_{𝑖,𝑙−1}+𝜖,  𝑢̇^Φ_𝑖(𝑡₁)≤0 and 𝑝_{𝑖,𝑙−1}+𝜖<𝑢^Φ_𝑖(𝑡)<𝑝_𝑖𝑙−𝜖 for 𝑡₀−𝛿≤𝑡<𝑡₁ or 𝑢^Φ_𝑖(𝑡₁)=𝑝_𝑖𝑙−𝜖,  𝑢̇^Φ_𝑖(𝑡₁)≥0 and 𝑝_{𝑖,𝑙−1}+𝜖<𝑢^Φ_𝑖(𝑡)<𝑝_𝑖𝑙−𝜖 for 𝑡₀−𝛿≤𝑡<𝑡₁. For the first subcase, we have 𝑢̇^Φ_𝑖𝑡₁=−𝛼_𝑖𝑢^Φ_𝑖𝑡₁×𝛽_𝑖𝑢^Φ_𝑖𝑡₁−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑢^Φ_𝑗𝑡₁−^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢^Φ_𝑗𝑡₁−𝜏_𝑗𝑡₁−^𝑛_𝑗=1𝑐_𝑖𝑗×^𝑡1_𝑡1−𝜎𝑘_𝑖𝑗𝑡₁−𝑠𝑓_𝑗𝑢^Φ_𝑗(𝑠)𝑑𝑠−𝐽_𝑖≥𝛼_𝑖𝑝_{𝑖,𝑙−1}×+𝜖−𝛽_𝑖𝑝_{𝑖,𝑙−1}+𝜖+𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑙−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗𝜆_𝑖𝑗(𝜎)𝑑_𝑖+𝐽_𝑖>0, (14) which leads to a contradiction with 𝑢̇^Φ_𝑖(𝑡₁)≤0. Similarly, one can prove the second subcase. Therefore, 𝑢^Φ_𝑖(𝑡)∈(𝑝_{𝑖,𝑙−1}+𝜖,𝑝_𝑖𝑙−𝜖) for all 𝑡≥𝑡₀.
Case III. If 𝛿_𝑖𝑘=1 and there exists a 𝑡₁>𝑡₀ such that 𝑢^Φ_𝑖(𝑡₁)=𝑝_{𝑖,𝑘−1}+𝜖, 𝑢̇^Φ_𝑖(𝑡₁)≤0 and 𝑢^Φ_𝑖(𝑡)>𝑝_{𝑖,𝑘−1}+𝜖 for 𝑡₀−𝛿≤𝑡<𝑡₁, we obtain that 𝑢̇^Φ_𝑖𝑡₁≥𝛼_𝑖𝑝_{𝑖,𝑘−1}×+𝜖−𝛽_𝑖𝑝_{𝑖,𝑘−1}+𝜖+𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑘−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗𝜆_𝑖𝑗(𝜎)𝑑_𝑖+𝐽_𝑖>0, (15) which is a contradiction. Therefore, 𝑢^Φ_𝑖(𝑡)∈(𝑝_{𝑖,𝑘−1}+𝜖,+∞) for all 𝑡≥𝑡₀.
From Case I to Case III, we know that 𝑢^Φ(𝑡)∈Λ for all 𝑡≥𝑡₀ that is, Λ is a positively invariant set of system (1). The proof is complete.

Theorem 7. Assume that assumptions (H1)and (H2) and the inequalities in (10) hold. For each division region Λ∈Ω, system (1) has a unique equilibrium located in Λ which is locally exponentially stable.

Proof. Let 𝑢^Φ(𝑡) and 𝑢^Ψ(𝑡) be any two solutions of system (1) with Φ, Ψ∈𝒞([𝑡₀−𝛿,𝑡₀],Λ), respectively. By the positive invariance of Λ, we get that 𝑢^Φ(𝑡), 𝑢^Ψ(𝑡)∈Λ for all 𝑡≥𝑡₀. For 𝜀<𝛽_𝑖𝛼_𝑖, define 𝑋_𝑖(𝑡)=𝑒^{𝜀𝑡𝑢Φ𝑖}_𝑢^(𝑡)_Ψ𝑖(𝑡)1𝛼_𝑖(𝑠)𝑑𝑠. (16) Calculating the derivative of 𝑋_𝑖(𝑡) along the trajectories of system (1), we obtain that 𝑋̇_𝑖(𝑡)=𝜀𝑒^{𝜀𝑡𝑢Φ𝑖}_𝑢^(𝑡)_Ψ𝑖(𝑡)1𝛼_𝑖(𝑠)𝑑𝑠+𝑒^𝜀𝑡sgn^𝑢Φ𝑖_𝑢^(𝑡)_Ψ𝑖(𝑡)1𝛼_𝑖(𝑠)×𝑑𝑠−𝛽_𝑖𝑢^Φ_𝑖(𝑡)−𝑢^Ψ_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗×𝑓_𝑗𝑢^Φ_𝑗(𝑡)−𝑓_𝑗𝑢^Ψ_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢^Φ_𝑗𝑡−𝜏_𝑗(𝑡)−𝑓_𝑗𝑢^Ψ_𝑗𝑡−𝜏_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗×^𝑡_𝑡−𝜎𝑘_𝑖𝑗(𝑡−𝑠)𝑓_𝑗𝑢^Φ_𝑗(𝑠)−𝑓_𝑗𝑢^Ψ_𝑗(𝑠)𝑑𝑠=𝜀𝑋_𝑖(𝑡)−𝛽_𝑖𝑒^𝜀𝑡𝑢^Φ_𝑖(𝑡)−𝑢^Ψ_𝑖(𝑡). (17) Since 1𝛼_𝑖𝑒^𝜀𝑡𝑢^Φ_𝑖(𝑡)−𝑢^Ψ_𝑖(𝑡)≤𝑋_𝑖(𝑡)≤1𝛼_𝑖𝑒^𝜀𝑡𝑢^Φ_𝑖(𝑡)−𝑢^Ψ_𝑖(𝑡), (18) we have 𝛼_𝑖𝑒^−𝜀𝑡𝑋_𝑖(𝑡)≤𝑢^Φ_𝑖(𝑡)−𝑢^Ψ_𝑖(𝑡)≤𝛼_𝑖𝑒^−𝜀𝑡𝑋_𝑖(𝑡). (19) It follows from (17) and (19) that 𝑋̇_𝑖(𝑡)≤𝜀−𝛽_𝑖𝛼_𝑖𝑋_𝑖(𝑡)<0. (20) Hence, 1𝛼_𝑖𝑒^𝜀𝑡𝑢^Φ_𝑖(𝑡)−𝑢^Ψ_𝑖(𝑡)≤𝑋_𝑖(𝑡)≤𝑋_𝑖𝑡₀≤1𝛼_𝑖𝑒^𝜀𝑡0‖Φ−Ψ‖, (21) which yields that 𝑢^Φ_𝑖(𝑡)−𝑢^Ψ_𝑖(𝑡)≤𝛼_𝑖𝛼_𝑖𝑒^{−𝜀(𝑡−𝑡0)}‖Φ−Ψ‖,𝑡≥𝑡₀. (22) Inequality (22) shows that 𝑢^Φ(𝑡)−𝑢^Ψ(𝑡) exponentially converges to 0 as 𝑡→+∞. In particular, for any fixed Δ𝑡, denote 𝑢Φ(𝑡)=𝑢^Φ(𝑡+Δ𝑡). Then 𝑢Φ(𝑡) is also a solution of system (1). Thus, 𝑢^Φ(𝑡)−𝑢Φ(𝑡)→0 as 𝑡→+∞ for any Δ𝑡>0, which indicates that 𝑢^Φ(𝑡) is a Cauchy sequence when 𝑡 is large enough. By Cauchy convergence principle, 𝑢^Φ(𝑡) approaches a constant vector 𝑢^∗ which is a solution of system (1). From (22) and the invariance of Λ, we know that 𝑢^∗ is the unique equilibrium of system (1) which is locally exponentially stable in Λ. The proof is complete.

If 𝛼_𝑖(𝑢_𝑖(𝑡))=1, system (1) reduces to the following Hopfield neural networks: 𝑢̇_𝑖(𝑡)=−𝛽_𝑖𝑢_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗^𝑡_𝑡−𝜎𝑘_𝑖𝑗(𝑡−𝑠)𝑓_𝑗𝑢_𝑗(𝑠)𝑑𝑠+𝐽_𝑖,𝑖=1,2,…,𝑛. (23)

Corollary 8. Assume that assumption (H2) and the inequalities in (10) hold. For each division region Λ∈Ω, system (23) has a unique equilibrium located in Λ which is locally exponentially stable.

Remark 9. It is noted that the number of such region Λ is 𝑘^𝑛 in Ω. Hence, under conditions of Theorem 7, there exist 𝑘^𝑛 locally exponentially stable equilibrium points for system (1) with activation functions (2). Neural networks with this class of activation functions can store many more patterns than those with sigmoidal activation functions and nondecreasing activation functions with saturation (see, e.g., [14–21]) in practical applications.

Remark 10. An important application of multistability of recurrent neural networks is to implement pattern memory (see [34, 35]). A recalling probe, which is sufficiently close to the pattern to be retrieved, is set as an initial state and the state variables converge to the locally stable equilibrium point, which corresponds to the pattern to be retrieved. Hence, our results are useful for associative memories since they provide new criteria to guarantee the coexistence of encoded patterns and their local attractivity.

Remark 11. In [30], the authors investigated the multistability for a class of Hopfield neural networks with time-varying delays and discontinuous activation functions. By applying the principle of compressed mapping, sufficient conditions are established for the existence of multiple equilibrium points. In contrast, our multistability results are derived by means of analysis approach and Cauchy convergence principle. When 𝛼_𝑖(𝑢_𝑖(𝑡))=1, 𝛽_𝑖=1, 𝑐_𝑖𝑗=0 and 𝑓₁(𝜉)=𝑓₂(𝜉)=⋯=𝑓_𝑛(𝜉), system (1) reduces to system (1) in [30]. Denote 𝑝_𝑗0=−∞, 𝑝_𝑗𝑘=+∞ in (2). Then the inequalities in (10) are equivalent to 𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑚+𝐾_𝑖+𝐽_𝑖<𝛽_𝑖𝑝_𝑖𝑚,𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑚−𝐾_𝑖+𝐽_𝑖>𝛽_𝑖𝑝_{𝑖,𝑚−1}, (24) where 𝑖=1,2,…,𝑛 and 𝑚=1,2,…,𝑘. We see that the main result Theorem  2 in [30] is a special case of Theorem 7 in our paper. Therefore, the obtained results in this paper improve and generalize those in [30].

Remark 12. We relax the restriction on the discontinuous activation functions adopted in [30] which are required to be nondecreasing. Theoretically, the activation functions (2) in our paper may be nondecreasing or nonincreasing. More precisely, we can deduce from (10) that 𝑑_𝑖1<𝑑_𝑖2<⋯<𝑑_𝑖𝑘,if𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑>0,_𝑖1>𝑑_𝑖2>⋯>𝑑_𝑖𝑘,if𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)<0. (25)

Theorem 13. Under assumptions (H1) and (H2), for each division region Λ∈Ω, system (1) has a unique equilibrium located in Λ which is globally exponentially stable, if the following conditions are satisfied: −𝛽_𝑖𝑝_𝑖1+𝐹_𝑖+𝐽_𝑖<0,if𝛿_𝑖1=1,−𝛽_𝑖𝑝_{𝑖,𝑙−1}−𝐹_𝑖+𝐽_𝑖>0,−𝛽_𝑖𝑝_𝑖𝑙+𝐹_𝑖+𝐽_𝑖<0,if𝛿_𝑖𝑙=1,𝑙=2,3,…,𝑘−1,−𝛽_𝑖𝑝_{𝑖,𝑘−1}−𝐹_𝑖+𝐽_𝑖>0,if𝛿_𝑖𝑘=1, (26) where 𝑖=1,2,…,𝑛, 𝐹_𝑖=∑^𝑛_𝑗=1(|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|+|𝑐_𝑖𝑗|𝜆_𝑖𝑗(𝜎))𝑑_𝑀, and 𝑑_𝑀=max_{1≤𝑗≤𝑛,1≤𝑙≤𝑘}{|𝑑_𝑗𝑙|}.

Proof. Let 𝑢^Φ(𝑡) be any solution of (1) with initial condition Φ∈𝒞([𝑡₀−𝛿,𝑡₀],𝑅^𝑛). If 𝛿_𝑖1=1 and 𝑢^Φ_𝑖(𝑡)≥𝑝_𝑖1, we obtain from (1) and (26) that 𝑢̇^Φ_𝑖(𝑡)=𝛼_𝑖𝑢^Φ_𝑖(𝑡)−𝛽_𝑖𝑢^Φ_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑢^Φ_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢^Φ_𝑗𝑡−𝜏_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗×^𝑡_𝑡−𝜎𝑘_𝑖𝑗(𝑡−𝑠)𝑓_𝑗𝑢^Φ_𝑗(𝑠)𝑑𝑠+𝐽_𝑖≤𝛼_𝑖𝑢^Φ_𝑖(𝑡)−𝛽_𝑖𝑝_𝑖1+^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗×𝜆_𝑖𝑗(𝜎)𝑑_𝑀+𝐽_𝑖<0. (27) If 𝛿_𝑖𝑙=1  (𝑙=2,3,…,𝑘−1) and 𝑢^Φ_𝑖(𝑡)≤𝑝_{𝑖,𝑙−1} or 𝑢^Φ_𝑖(𝑡)≥𝑝_𝑖𝑙, we have 𝑢̇^Φ_𝑖(𝑡)≥𝛼_𝑖𝑢^Φ_𝑖(𝑡×)−𝛽_𝑖𝑝_{𝑖,𝑙−1}−^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗𝜆_𝑖𝑗(𝜎)𝑑_𝑀+𝐽_𝑖>0, (28) or 𝑢̇^Φ_𝑖(𝑡)≤𝛼_𝑖𝑢^Φ_𝑖(𝑡×)−𝛽_𝑖𝑝_𝑖𝑙+^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗𝜆_𝑖𝑗(𝜎)𝑑_𝑀+𝐽_𝑖<0. (29) If 𝛿_𝑖𝑘=1 and 𝑢^Φ_𝑖(𝑡)≤𝑝_{𝑖,𝑘−1}, we get 𝑢̇^Φ_𝑖(𝑡)≥𝛼_𝑖𝑢^Φ_𝑖(𝑡×)−𝛽_𝑖𝑝_{𝑖,𝑘−1}−^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗𝜆_𝑖𝑗(𝜎)𝑑_𝑀+𝐽_𝑖>0. (30) Therefore, 𝑢^Φ(𝑡) will go into and stay in Λ; that is, there exists a 𝑇₁>0 such that 𝑢^Φ(𝑡)∈Λ for all 𝑡≥𝑇₁. For any other solution 𝑢^Ψ(𝑡) of system (1) with initial condition Ψ∈𝒞([𝑡₀−𝛿,𝑡₀],𝑅^𝑛), there exists a 𝑇₂>0 such that 𝑢^Ψ(𝑡)∈Λ for all 𝑡≥𝑇₂. Similar to the proof of Theorem 7, we can show that 𝑢^Φ_𝑖(𝑡)−𝑢^Ψ_𝑖(𝑡)≤𝛼_𝑖𝛼_𝑖𝑒^{−𝜀(𝑡−𝑡0)}‖Φ−Ψ‖,𝑡≥max𝑇₁,𝑇₂, (31) and region Λ is a positively invariant set. Hence, system (1) has a unique equilibrium located in Λ which is globally exponentially stable. The proof is complete.

4. Multiperiodicity for the CGNNs

Consider the following CGNNs in which the time-varying delays and external inputs are periodic: 𝑢̇_𝑖(𝑡)=−𝛼_𝑖𝑢_𝑖(𝑡)𝛽_𝑖𝑢_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗^𝑡_𝑡−𝜎𝑘_𝑖𝑗(𝑡−𝑠)𝑓_𝑗𝑢_𝑗(𝑠)𝑑𝑠−𝐽_𝑖(𝑡),𝑖=1,2,…,𝑛, (32) where 𝐽_𝑖(𝑡+𝜔)=𝐽_𝑖(𝑡) and 𝜏_𝑗(𝑡+𝜔)=𝜏_𝑗(𝑡) for all 𝑡≥𝑡₀−𝛿.

Theorem 14. Under assumptions (H1) and (H2), for each division region Λ∈Ω, Λ is a positively invariant set of system (32), and system (32) has a unique periodic orbit located in Λ which is locally exponentially stable, if the following conditions are satisfied: −𝛽_𝑖𝑝_𝑖𝑙+𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑙+𝐺_𝑖+sup_{[]𝑡∈0,𝜔}𝐽_𝑖(𝑡)<0,𝑙=1,2,…,𝑘−1,−𝛽_𝑖𝑝_{𝑖,𝑚−1}+𝑎_𝑖𝑖+𝑏_𝑖𝑖+𝑐_𝑖𝑖𝜆_𝑖𝑖(𝜎)𝑑_𝑖𝑚−𝐺_𝑖+inf_{[]𝑡∈0,𝜔}𝐽_𝑖(𝑡)>0,𝑚=2,3,…,𝑘, (33) where 𝑖=1,2,…,𝑛, 𝐺_𝑖=∑^𝑛_{𝑗=1,𝑗≠𝑖}(|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|+|𝑐_𝑖𝑗|𝜆_𝑖𝑗(𝜎))𝑑_𝑖, and 𝑑_𝑖=max_{1≤𝑗≤𝑛,𝑗≠𝑖,1≤𝑙≤𝑘}{|𝑑_𝑗𝑙|}.

Proof. Similar to the proof of Theorem 6, we can prove that each division region Λ is a positively invariant set of system (32). Hence, for any initial condition Φ=(𝜙₁,𝜙₂,…,𝜙_𝑛)∈𝒞([𝑡₀−𝛿,𝑡₀],Λ), we have 𝑢^Φ_𝑡(𝑠)∈𝒞([𝑡₀−𝛿,𝑡₀],Λ) for any 𝑡≥𝑡₀. Define a Poincar𝑒́ mapping ℋ∶𝒞([𝑡₀−𝛿,𝑡₀],Λ)→𝒞([𝑡₀−𝛿,𝑡₀],Λ) by ℋ(Φ)=𝑢^Φ_𝜔. Then ℋ(𝒞([𝑡₀−𝛿,𝑡₀],Λ))⊂𝒞([𝑡₀−𝛿,𝑡₀],Λ) and ℋ^𝑁(Φ)=𝑢^Φ_𝑁𝜔. We can choose a positive integer 𝑁 such that (𝛼_𝑖/𝛼_𝑖)𝑒^{−𝜀𝑁𝜔}≤𝜈<1 for all 1≤𝑖≤𝑛. From (22), we have ‖ℋ^𝑁(Φ)−ℋ^𝑁(Ψ)‖≤𝜈‖Φ−Ψ‖, where Φ,Ψ∈𝒞([𝑡₀−𝛿,𝑡₀],Λ). This inequality implies that there exists a unique fixed point Φ^∗∈𝒞([𝑡₀−𝛿,𝑡₀],Λ) such that ℋ^𝑁(Φ^∗)=Φ^∗. Note that ℋ^𝑁(ℋ(Φ^∗))=ℋ(ℋ^𝑁(Φ^∗))=ℋ(Φ^∗). It means that ℋ(Φ^∗) is also a fixed point of ℋ^𝑁; then ℋ(Φ^∗)=Φ^∗, that is, 𝑢^Φ∗_𝜔=Φ^∗. Obviously, if 𝑢^Φ∗(𝑡) is a solution of system (32), 𝑢^Φ∗(𝑡+𝜔) is also a solution of system (32) and 𝑢^Φ∗_𝑡+𝜔=𝑢^{𝑢Φ∗𝜔}_𝑡=𝑢^Φ∗_𝑡 for all 𝑡≥𝑡₀. Therefore, 𝑢^Φ∗(𝑡+𝜔)=𝑢^Φ∗(𝑡) for all 𝑡≥𝑡₀. This shows that 𝑢^Φ∗(𝑡) is a periodic orbit of (32) located in Λ. The locally exponential stability can be similarly proved by Theorem 7. This completes the proof.

5. Illustrative Examples

In this section, four examples are presented to illustrate our main results derived in Sections 3 and 4.

Example 1. Consider a 2-neuron delayed CGNN with 3-level discontinuous activation function; that is, in system (1), we choose 𝐴=𝑎_𝑖𝑗=,𝑏10.20.52𝐵=_𝑖𝑗=,𝑐20.60.72𝐶=_𝑖𝑗=,𝛽20.60.31₁=2,𝛽₂=3,𝜎=50,𝐽₁𝐽=0.1,₂=0.2,𝛼_𝑖𝑢_𝑖(𝑡)=2+sin𝑢_𝑖(𝑡),𝜏_𝑗(𝑡)𝑒=1−^−𝑡2,𝑘_𝑖𝑗(𝑡)=𝑡𝑒^−𝑡,𝑖,𝑗=1,2. (34)

For simplicity, we choose the activation functions as 𝑓₁(𝜉)=𝑓₂(𝜉)=(),(),().−1,𝜉∈−∞,−10,𝜉∈−1,11,𝜉∈1,+∞ (35)

It is not difficult to verify that system (1) with (34) and (35) satisfies assumptions (H1) and (H2) and the inequalities in (10). According to Theorems 6 and 7, system (1) has nine locally exponentially stable equilibrium points: (),(),(),(),(),(),(),(),().−1.7464,1.23210.7484,1.73163.2433,2.2311−2.4448,−0.43280.05,0.06672.5448,0.5661−3.1433,−2.0978−0.6484,−1.59831.8464,−1.0988 (36)

The dynamics of this system are depicted in Figure 1(a), where evolutions of 120 initial conditions have been tracked.

(a)

(b)

Figure 1 

Transient behaviors of the state variables 𝑢1(𝑡) and 𝑢2(𝑡) with different initial values in Example 1.

In system (32), let 𝐽₁(𝑡)=0.1+0.4sin𝑡, 𝐽₂(𝑡)=0.4cos𝑡, 𝜏_𝑗(𝑡)=1, and 𝑗=1,2, and let the other parameters and functions be the same as those in (34) and (35). We can verify that system (32) satisfies assumptions (H1) and (H2) and the inequalities in (33). According to Theorem 14, system (32) has nine locally exponentially stable periodic orbits. Figure 1(b) shows the phase view of state variable (𝑢₁,𝑢₂) of system (32).

Example 2. Consider a 3-neuron delayed CGNN with 2-level discontinuous activation function. In system (1), we choose 𝐴=𝑎_𝑖𝑗=,𝑏10.20.30.12−0.1−0.1−0.21𝐵=_𝑖𝑗=,𝑐20.1−0.10.320.1−0.2−0.22𝐶=_𝑖𝑗=,𝑓20.30.20.320.10.2−0.11_𝑗(𝜉)=(),(),𝛽−1,𝜉∈−∞,11,𝜉∈1,+∞₁=𝛽₂=𝛽₃𝐽=1,𝜎=10,₁=0.2,𝐽₂𝐽=0.4,₃=−0.5,𝛼_𝑖𝑢_𝑖(𝑡)=2+sin𝑢_𝑖(𝑡),𝜏_𝑗(𝑡)𝑒=1−^−𝑡2,𝑘_𝑖𝑗(𝑡)=𝑡𝑒^−𝑡,𝑖,𝑗=1,2. (37)

We can verify that system (1) with (37) satisfies assumptions (H1) and (H2) and the inequalities in (10). According to Theorems 6 and 7, system (1) has eight locally exponentially stable equilibrium points: 𝐵₁=(),𝐵−5.7982,−6.3983,−3.8992₂=(),𝐵−5.3982,−6.3983,2.1008₃=(),𝐵−4.5987,5.5989,−4.8991₄=(),𝐵−4.1987,5.5989,1.1009₅=(),𝐵4.5986,−4.7989,−2.1009₆=(),𝐵4.9986,−4.7989,3.8991₇=(),𝐵5.7982,7.1983,−3.1008₈=(6.1982,7.1983,2.8992). (38) The dynamics of this system are depicted in Figure 2(a).

(a)

(b)

Figure 2 

Transient behaviors of the state variables 𝑢1(𝑡), 𝑢2(𝑡), and 𝑢3(𝑡) with different initial values in Example 2.

In system (32), let 𝐽₁(𝑡)=0.6sin𝑡, 𝐽₂(𝑡)=0.1+0.4cos𝑡, 𝐽₃(𝑡)=0.4cos𝑡, 𝜏_𝑗(𝑡)=1, and 𝑗=1,2,3, and let the other parameters and functions be the same as those in (37). Then system (32) satisfies assumptions (H1) and (H2) and the inequalities in (33). According to Theorem 14, system (32) has eight locally exponentially stable periodic orbits (see Figure 2(b)).

Example 3. Consider a 2-neuron delayed CGNN with 4-level discontinuous activation function. In system (1), we choose 𝐴=𝑎_𝑖𝑗=,𝑏2.50.20.13𝐵=_𝑖𝑗=,𝑐30.30.22.5𝐶=_𝑖𝑗=,𝛽20.10.32₁=𝛽₂=5,𝜎=10,𝐽₁=0.3,𝐽₂𝛼=0.4,_𝑖𝑢_𝑖(𝑡)=2+sin𝑢_𝑖(𝑡),𝜏_𝑗(𝑡)𝑒=1−^−𝑡2,𝑘_𝑖𝑗(𝑡)=𝑡𝑒^−𝑡,𝑖,𝑗=1,2. (39)

The activation functions are described by 𝑓₁(𝜉)=𝑓₂(𝜉)=(),(),(),().−1,𝜉∈−∞,−10,𝜉∈−1,11,𝜉∈1,22,𝜉∈2,+∞ (40)

We can verify that system (1) with (39) and (40) satisfies assumptions (H1) and (H2) and the inequalities in (10). According to Theorems 6 and 7, system (1) has sixteen locally exponentially stable equilibrium points (see Figure 3(a)).

(a)

(b)

Figure 3 

Transient behaviors of the state variables 𝑢1(𝑡) and 𝑢2(𝑡) with different initial values in Example 3.

In system (32), let 𝐽₁(𝑡)=0.3sin𝑡, 𝐽₂(𝑡)=0.4cos𝑡, 𝜏_𝑗(𝑡)=1, and 𝑗=1,2, and let the other parameters and functions be the same as those in (39) and (40). We can verify that system (32) satisfies assumptions (H1) and (H2) and the inequalities in (33). According to Theorem 14, system (32) has sixteen locally exponentially stable periodic orbits (see Figure 3(b)).

Example 4. For associative memories of neural networks, the desired memory patterns are usually represented by binary vectors. Design a CGNN to store eight patterns 𝐴_𝑖  (1≤𝑖≤8) shown in Figure 4 as stable memories (white = −1 and black = 1).

Figure 4 

Eight desired memories patterns for Example 4.

We consider system (1) with (37) in Example 2. Transform 𝐵₁=diag{5.7982,6.3983,3.8992}𝐴₁,𝐵₂=diag{5.3982,6.3983,2.1008}𝐴₂,𝐵₃=diag{4.5987,5.5989,4.8991}𝐴₃,𝐵₄=diag{4.1987,5.5989,1.1009}𝐴₄,𝐵₅=diag{4.5986,4.7989,2.1009}𝐴₅,𝐵₆=diag{4.9986,4.7989,3.8991}𝐴₆,𝐵₇=diag{5.7982,7.1983,3.1008}𝐴₇,𝐵₈=diag{6.1982,7.1983,2.8992}𝐴₈. (41)

Then 𝐵_𝑖  (1≤𝑖≤8) are given in (38). According to Example 2, the corresponding memory patterns 𝐵_𝑖  (1≤𝑖≤8) are stored as stable equilibria of system (1) with (37) which can be retrieved when the initial patterns contain sufficient information (should fall in the attracting domains).

6. Conclusions

In this paper, we performed multistability and multiperiodicity analyses for a general class of CGNNs with time-varying delays, distributed delays, and discontinuous activation functions. Sufficient conditions were established to ensure that there are 𝑘^𝑛 locally exponential stable equilibrium points for the 𝑛-dimensional CGNNs with 𝑘-level discontinuous activation functions. Also, we derived sufficient conditions for the existence and global exponential stability of equilibrium point for the system. As an extension of multistability, sufficient conditions were established to ensure that the system has 𝑘^𝑛 locally exponentially stable periodic orbits when time-varying delays and external inputs are periodic. Numerical simulations demonstrated the main results. The existing multistability results in [30] were improved and extended. Our results provide new criteria to guarantee the coexistence of encoded patterns and their local attractivity and are useful for associative memories.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11071254), the Natural Science Foundation of Hebei Province (A2013506012), and the Science Foundation of Mechanical Engineering College (YJJXM11004).