# Multistability and Multiperiodicity for a General Class of Delayed

Authors: Yanke Du; Yanlu Li; Rui Xu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## 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 thatn-dimensional Cohen-Grossberg neural networks with k-level discontinuous activation functions
can have kn equilibrium points or kn periodic orbits. Finally, several examples are given to
illustrate the feasibility of the obtained results.


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## Body

## 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 kn 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:
(1)u˙i(t)=-αi(ui(t))[βiui(t)-∑j=1n‍aijfj(uj(t))-∑j=1n‍bij×fj(uj(t-τj(t)))-∑j=1n‍cij×∫t-σt‍kij(t-s)fj(uj(s))ds-Ji∑j=1n‍],i=1,2,…,n,
where ui(t) denotes the state of the ith neuron at time t; αi(ui(t)) denotes the amplification function; βi>0; fj(uj(t)) denotes the activation function; τj(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≤τj(t)≤τ; kij(t)>0 represents the delay kernel function, which is a real-valued continuous function; aij, bij, and cij denote the connection strengths; Ji 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 followingk-level discontinuous activation functions:
(2)fj(ξ)={dj1,ξ∈(-∞,pj1),dj2,ξ∈(pj1,pj2),⋮⋮dj,k-1,ξ∈(pj,k-2,pj,k-1),djk,ξ∈(pj,k-1,+∞),undefined,ξ∈{pj1,pj2,…,pj,k-1},
where j=1,2,…,n; k is an integer satisfying k≥1; dj1,dj2,…,djk, pj1,pj2,…,pj,k-1 are constants; and pj1<pj2<⋯<pj,k-1.The organization of this paper is as follows. In Section2, 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

Denoteδ=max{τ,σ}. Let 𝒞([t0-δ,t0],Rn) be the space of all continuous functions mapping [t0-δ,t0] into Rn with the norm defined by ∥Φ∥=max1≤i≤n{supt0-δ≤s≤t0|ϕi(s)|}, where Φ=(ϕ1,ϕ2,…,ϕn)∈𝒞([t0-δ,t0],Rn). Denote ∥u∥=max1≤i≤n{|ui|} as the vector norm of u=(u1,u2,…,un)∈Rn. For any function u(·)∈𝒞([t0-δ,t0],Rn), we define ut(s)=u(t+s), s∈[t0-δ,t0], and t≥t0. We denote by uΦ(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 functionαi(ui(t)) is continuous, and there exist constants α_i and α¯i such that
(3)0<α_i≤αi(ui(t))≤α¯i<+∞,i=1,2,…,n.(H2)
Each kernel functionkij(s):[0,+∞)→[0,+∞) is continuous and satisfies
(4)λij(σ)=∫t-σt‍kij(t-s)ds=∫0σ‍kij(s)ds≥0,i,j=1,2,…,n.Moreover, when σ=+∞, λij(+∞)=1.Sincefj(·) 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 functionu(t)=(u1(t),u2(t),…,un(t))T:[t0-δ,T)→Rn, 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 functionγ(t)=(γ1(t),γ2(t),…,γn(t))T:[t0-δ,T)→Rn, such that γi(t)∈K[fi(ui(t))] for almost all (a.a.) t∈[t0-δ,T), and
(5)u˙i(t)=-αi(ui(t))[βiui(t)-∑j=1n‍aijγj(t)-∑j=1n‍bij×γj(t-τj(t))-∑j=1n‍cij×∫t-σt‍kij(t-s)γj(s)ds-Ji∑j=1n‍]for a.a. t∈[t0,T), where K[fi(·)] represents the closure of the convex hull of fi(·).Definition 2 (Forti et al. [33]).
For any continuous functionϕ(θ)=(ϕ1(θ),ϕ2(θ),…,ϕn(θ))T:[t0-δ,t0]→Rn and any measurable selection λ(θ)=(λ1(θ),λ2(θ),…,λn(θ))T:[t0-δ,t0]→Rn, such that λi(θ)∈K[fi(ϕi(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)→Rn×Rn, 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
(6)u˙i(t)=-αi(ui(t))[βiui(t)-∑j=1n‍aijγj(t)-∑j=1n‍bij×γj(t-τj(t))-∑j=1n‍cij×∫t-σt‍kij(t-s)γj(s)ds-Ji∑j=1n‍],fora.a.t∈[t0,T),ui(θ)=ϕi(θ),foranyθ∈[t0-δ,t0],γi(θ)=λi(θ),fora.a.θ∈[t0-δ,t0],
where K[fi(·)] represents the closure of the convex hull of fi(·).Definition 3.
LetD be a subset of Rn. 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 uΦ(t)∈D for all t≥t0.Definition 4.
An equilibriumu* of system (1) is said to be locally exponentially stable in D∈Rn, if there exist constants T≥t0, ε>0 and M>0 such that
(7)∥uΦ(t)-u*∥≤M∥Φ-u*∥exp(-ε(t-t0)),t≥T,
where initial condition Φ∈𝒞([t0-δ,t0],D). When D=Rn, u* is said to be globally exponentially stable.Definition 5.
A periodic orbitu*(t) of system (1) is said to be locally exponentially stable in D∈Rn, if there exist constants T≥t0, ε>0 and M>0 such that
(8)∥uΦ(t)-u*(t)∥≤M∥Φ-ut0*∥exp(-ε(t-t0)),t≥T,
where initial condition Φ∈𝒞([t0-δ,t0],D) and ut0*(s)=u*(t0+s), s∈[t0-δ,t0]. When D=Rn, u*(t) is said to be globally exponentially stable.Denote(9)(-∞,pi1)=(-∞,pi1)1×(pi1,pi2)0×⋯×(pi,k-1,+∞)0,(pi1,pi2)=(-∞,pi1)0×(pi1,pi2)1×⋯×(pi,k-1,+∞)0,⋮(pi,k-1,+∞)=(-∞,pi1)0×(pi1,pi2)0×⋯×(pi,k-1,+∞)1,Ω={∏i=1n‍(-∞,pi1)δi1×(pi1,pi2)δi2×⋯×(pi,k-1,+∞)δik∏i=1n‍},
where δil equals 1 or 0  (l=1,2,…,k) and ∑l=1k‍δil=1. It is easy to see that Ω is composed of kn 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 regionΛ∈Ω, Λ is a positively invariant set of system (1), if the following conditions are satisfied:
(10)-βipil+(aii+bii+ciiλii(σ))dil+Ki+Ji<0,l=1,2,…,k-1,-βipi,m-1+(aii+bii+ciiλii(σ))dim-Ki+Ji>0,m=2,3,…,k,
where i=1,2,…,n, Ki=∑j=1,j≠in(|aij|+|bij|+|cij|λij(σ))di, and di=max1≤j≤n,j≠i,1≤l≤k{|djl|}.Proof.
From (10), it is readily seen that there exists a small constant 0<ϵ0<min1≤j≤n,2≤l≤k-1{(pjl-pj,l-1)/2} such that for all 0<ϵ<ϵ0,
(11)-βi(pil-ϵ)+(aii+bii+ciiλii(σ))dil+Ki+Ji<0,l=1,2,…,k-1,-βi(pi,m-1+ϵ)+(aii+bii+ciiλii(σ))dim-Ki+Ji>0,m=2,3,…,k.
Denote
(12)(-∞,pi1-ϵ)=(-∞,pi1-ϵ)1×(pi1+ϵ,pi2-ϵ)0×⋯×(pi,k-1+ϵ,+∞)0,(pi1+ϵ,pi2-ϵ)=(-∞,pi1-ϵ)0×(pi1+ϵ,pi2-ϵ)1×⋯×(pi,k-1+ϵ,+∞)0,⋮(pi,k-1+ϵ,+∞)=(-∞,pi1-ϵ)0×(pi1+ϵ,pi2-ϵ)0×⋯×(pi,k-1+ϵ,+∞)1,Ω~={∏i=1n‍(-∞,pi1-ϵ)δi1×(pi1+ϵ,pi2-ϵ)δi2×⋯×(pi,k-1+ϵ,+∞)δik∏i=1n‍},
where δil equals 1 or 0(l=1,2,…,k) and ∑l=1k‍δil=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 eachΛ~∈Ω~, we consider any solution uΦ(t)=(u1Φ(t),u2Φ(t),…,unΦ(t)) of system (1) with initial condition Φ∈𝒞([t0-δ,t0],Λ~). We claim that the solution uΦ(t)∈Λ~ for all t≥t0. If this is not true, we need to discuss the following three cases.
Case I. If δi1=1 and there exists a t1>t0 such that uiΦ(t1)=pi1-ϵ, u˙iΦ(t1)≥0 and uiΦ(t)<pi1-ϵ for t0-δ≤t<t1, we get from (1) and (11) that
(13)u˙iΦ(t1)=-αi(uiΦ(t1))[(ujΦ(s))ds-Ji∑j=1n‍βiuiΦ(t1)-∑j=1n‍aijfj(ujΦ(t1))-∑j=1n‍bijfj(ujΦ(t1-τj(t1)))-∑j=1n‍cij×∫t1-σt1‍kij(t1-s)fj(ujΦ(s))ds-Ji∑j=1n‍]≤αi(pi1-ϵ)[∑j=1,j≠in‍-βi(pi1-ϵ)+(aii+bii+ciiλii(σ))×di1+∑j=1,j≠in‍(|aij|+|bij|+|cij|×λij(σ)|aij|)di+Ji∑j=1,j≠in‍]<0,
which yields a contradiction to u˙iΦ(t1)≥0. Therefore, uiΦ(t)∈(-∞,pi1-ϵ) for all t≥t0.
Case II. If δil=1  (l=2,3,…,k-1) and there exists a t1>t0 such that either uiΦ(t1)=pi,l-1+ϵ,  u˙iΦ(t1)≤0 and pi,l-1+ϵ<uiΦ(t)<pil-ϵ for t0-δ≤t<t1 or uiΦ(t1)=pil-ϵ,  u˙iΦ(t1)≥0 and pi,l-1+ϵ<uiΦ(t)<pil-ϵ for t0-δ≤t<t1. For the first subcase, we have
(14)u˙iΦ(t1)=-αi(uiΦ(t1))×[βiuiΦ(t1)-∑j=1n‍aijfj(ujΦ(t1))-∑j=1n‍bijfj(ujΦ(t1-τj(t1)))-∑j=1n‍cij×∫t1-σt1‍kij(t1-s)fj(ujΦ(s))ds-Ji∑j=1n‍]≥αi(pi,l-1+ϵ)×[∑j=1,j≠in‍-βi(pi,l-1+ϵ)+(aii+bii+ciiλii(σ))dil-∑j=1,j≠in‍(|aij|+|bij|+|cij|λij(σ))di+Ji]>0,
which leads to a contradiction with u˙iΦ(t1)≤0. Similarly, one can prove the second subcase. Therefore, uiΦ(t)∈(pi,l-1+ϵ,pil-ϵ) for all t≥t0.
Case III. If δik=1 and there exists a t1>t0 such that uiΦ(t1)=pi,k-1+ϵ, u˙iΦ(t1)≤0 and uiΦ(t)>pi,k-1+ϵ for t0-δ≤t<t1, we obtain that
(15)u˙iΦ(t1)≥αi(pi,k-1+ϵ)×[∑j=1j≠in‍-βi(pi,k-1+ϵ)+(aii+bii+ciiλii(σ))dik-∑j=1,j≠in(|aij|+|bij|+|cij|λij(σ))di+Ji∑j=1j≠in‍]>0,
which is a contradiction. Therefore, uiΦ(t)∈(pi,k-1+ϵ,+∞) for all t≥t0.
From Case I to Case III, we know thatuΦ(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.
LetuΦ(t) and uΨ(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 uΦ(t), uΨ(t)∈Λ for all t≥t0. For ε<βiα_i, define
(16)Xi(t)=eεt|∫uiΨ(t)uiΦ(t)‍1αi(s)ds|.
Calculating the derivative of Xi(t) along the trajectories of system (1), we obtain that
(17)X˙i(t)=εeεt|∫uiΨ(t)uiΦ(t)‍1αi(s)ds|+eεtsgn(∫uiΨ(t)uiΦ(t)‍1αi(s)ds)×{-βi(uiΦ(t)-uiΨ(t))+∑j=1n‍aij×(fj(ujΦ(t))-fj(ujΨ(t)))+∑j=1n‍bij(fj(ujΦ(t-τj(t)))-fj(ujΨ(t-τj(t))))+∑j=1n‍cij×∫t-σt‍kij(t-s)(fj(ujΦ(s))-fj(ujΨ(s)))ds∑j=1n‍}=εXi(t)-βieεt|uiΦ(t)-uiΨ(t)|.
Since
(18)1α¯ieεt|uiΦ(t)-uiΨ(t)|≤Xi(t)≤1α_ieεt|uiΦ(t)-uiΨ(t)|,
we have
(19)α_ie-εtXi(t)≤|uiΦ(t)-uiΨ(t)|≤α¯ie-εtXi(t).
It follows from (17) and (19) that
(20)X˙i(t)≤(ε-βiα_i)Xi(t)<0.
Hence,
(21)1α¯ieεt|uiΦ(t)-uiΨ(t)|≤Xi(t)≤Xi(t0)≤1α_ieεt0∥Φ-Ψ∥,
which yields that
(22)|uiΦ(t)-uiΨ(t)|≤α¯iα_ie-ε(t-t0)∥Φ-Ψ∥,t≥t0.
Inequality (22) shows that uΦ(t)-uΨ(t) exponentially converges to 0 as t→+∞. In particular, for any fixed Δt, denote uΦ~(t)=uΦ(t+Δt). Then uΦ~(t) is also a solution of system (1). Thus, uΦ(t)-uΦ~(t)→0 as t→+∞ for any Δt>0, which indicates that uΦ(t) is a Cauchy sequence when t is large enough. By Cauchy convergence principle, uΦ(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.Ifαi(ui(t))=1, system (1) reduces to the following Hopfield neural networks:
(23)u˙i(t)=-βiui(t)+∑j=1n‍aijfj(uj(t))+∑j=1n‍bijfj(uj(t-τj(t)))+∑j=1n‍cij∫t-σt‍kij(t-s)fj(uj(s))ds+Ji,i=1,2,…,n.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 regionΛ is kn in Ω. Hence, under conditions of Theorem 7, there exist kn 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 αi(ui(t))=1, βi=1, cij=0 and f1(ξ)=f2(ξ)=⋯=fn(ξ), system (1) reduces to system (1) in [30]. Denote pj0=-∞, pjk=+∞ in (2). Then the inequalities in (10) are equivalent to
(24)(aii+bii+ciiλii(σ))dim+Ki+Ji<βipim,(aii+bii+ciiλii(σ))dim-Ki+Ji>βipi,m-1,
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
(25)di1<di2<⋯<dik,ifaii+bii+ciiλii(σ)>0,di1>di2>⋯>dik,ifaii+bii+ciiλii(σ)<0.Theorem 13.
Under assumptions (H1) and (H2), for each division regionΛ∈Ω, system (1) has a unique equilibrium located in Λ which is globally exponentially stable, if the following conditions are satisfied:
(26)-βipi1+Fi+Ji<0,ifδi1=1,-βipi,l-1-Fi+Ji>0,-βipil+Fi+Ji<0,ifδil=1,l=2,3,…,k-1,-βipi,k-1-Fi+Ji>0,ifδik=1,
where i=1,2,…,n, Fi=∑j=1n(|aij|+|bij|+|cij|λij(σ))dM, and dM=max1≤j≤n,1≤l≤k{|djl|}.Proof.
LetuΦ(t) be any solution of (1) with initial condition Φ∈𝒞([t0-δ,t0],Rn). If δi1=1 and uiΦ(t)≥pi1, we obtain from (1) and (26) that
(27)u˙iΦ(t)=αi(uiΦ(t))[-βiuiΦ(t)+∑j=1n‍aijfj(ujΦ(t))+∑j=1n‍bijfj(ujΦ(t-τj(t)))+∑j=1n‍cij×∫t-σt‍kij(t-s)fj(ujΦ(s))ds+Ji∑j=1n‍]≤αi(uiΦ(t))[-βipi1+∑j=1n‍(|aij|+|bij|+|cij|×λij(σ))dM+Ji∑j=1n‍]<0.
If δil=1  (l=2,3,…,k-1) and uiΦ(t)≤pi,l-1 or uiΦ(t)≥pil, we have
(28)u˙iΦ(t)≥αi(uiΦ(t))×[-βipi,l-1-∑j=1n‍(|aij|+|bij|+|cij|λij(σ))dM+Ji]>0,
or
(29)u˙iΦ(t)≤αi(uiΦ(t))×[-βipil+∑j=1n‍(|aij|+|bij|+|cij|λij(σ))dM+Ji]<0.
If δik=1 and uiΦ(t)≤pi,k-1, we get
(30)u˙iΦ(t)≥αi(uiΦ(t))×[-βipi,k-1-∑j=1n‍(|aij|+|bij|+|cij|λij(σ))dM+Ji]>0.
Therefore, uΦ(t) will go into and stay in Λ; that is, there exists a T1>0 such that uΦ(t)∈Λ for all t≥T1. For any other solution uΨ(t) of system (1) with initial condition Ψ∈𝒞([t0-δ,t0],Rn), there exists a T2>0 such that uΨ(t)∈Λ for all t≥T2. Similar to the proof of Theorem 7, we can show that
(31)|uiΦ(t)-uiΨ(t)|≤α¯iα_ie-ε(t-t0)∥Φ-Ψ∥,t≥max{T1,T2},
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:(32)u˙i(t)=-αi(ui(t))[βiui(t)-∑j=1n‍aijfj(uj(t))-∑j=1n‍bijfj(uj(t-τj(t)))-∑j=1n‍cij∫t-σt‍kij(t-s)fj(uj(s))ds-Ji(t)∑j=1n‍],i=1,2,…,n,
where Ji(t+ω)=Ji(t) and τj(t+ω)=τj(t) for all t≥t0-δ.Theorem 14.
Under assumptions (H1) and (H2), for each division regionΛ∈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:
(33)-βipil+(aii+bii+ciiλii(σ))dil+Gi+supt∈[0,ω]Ji(t)<0,l=1,2,…,k-1,-βipi,m-1+(aii+bii+ciiλii(σ))dim-Gi+inft∈[0,ω]Ji(t)>0,m=2,3,…,k,
where i=1,2,…,n, Gi=∑j=1,j≠in(|aij|+|bij|+|cij|λij(σ))di, and di=max1≤j≤n,j≠i,1≤l≤k{|djl|}.Proof.
Similar to the proof of Theorem6, we can prove that each division region Λ is a positively invariant set of system (32). Hence, for any initial condition Φ=(ϕ1,ϕ2,…,ϕn)∈𝒞([t0-δ,t0],Λ), we have utΦ(s)∈𝒞([t0-δ,t0],Λ) for any t≥t0. Define a Poincare´ mapping ℋ:𝒞([t0-δ,t0],Λ)→𝒞([t0-δ,t0],Λ) by ℋ(Φ)=uωΦ. Then ℋ(𝒞([t0-δ,t0],Λ))⊂𝒞([t0-δ,t0],Λ) and ℋN(Φ)=uNωΦ. 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 ∥ℋN(Φ)-ℋN(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 ℋN(Φ*)=Φ*. Note that ℋN(ℋ(Φ*))=ℋ(ℋN(Φ*))=ℋ(Φ*). It means that ℋ(Φ*) is also a fixed point of ℋN; then ℋ(Φ*)=Φ*, that is, uωΦ*=Φ*. Obviously, if uΦ*(t) is a solution of system (32), uΦ*(t+ω) is also a solution of system (32) and ut+ωΦ*=utuωΦ*=utΦ* for all t≥t0. Therefore, uΦ*(t+ω)=uΦ*(t) for all t≥t0. This shows that uΦ*(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 Sections3 and 4.Example 1.
Consider a 2-neuron delayed CGNN with 3-level discontinuous activation function; that is, in system (1), we choose
(34)A=(aij)=(10.20.52),B=(bij)=(20.60.72),C=(cij)=(20.60.31),β1=2,β2=3,σ=50,J1=0.1,J2=0.2,αi(ui(t))=2+sin(ui(t)),τj(t)=1-e-t2,kij(t)=te-t,i,j=1,2.For simplicity, we choose the activation functions as(35)f1(ξ)=f2(ξ)={-1,ξ∈(-∞,-1),0,ξ∈(-1,1),1,ξ∈(1,+∞).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:
(36)(-1.7464,1.2321),(0.7484,1.7316),(3.2433,2.2311),(-2.4448,-0.4328),(0.05,0.0667),(2.5448,0.5661),(-3.1433,-2.0978),(-0.6484,-1.5983),(1.8464,-1.0988).The dynamics of this system are depicted in Figure1(a), where evolutions of 120 initial conditions have been tracked.Transient behaviors of the state variablesu1(t) and u2(t) with different initial values in Example 1.
(a)
(b)In system (32), let J1(t)=0.1+0.4sint, J2(t)=0.4cost, τj(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
(37)A=(aij)=(10.20.30.12-0.1-0.1-0.21),B=(bij)=(20.1-0.10.320.1-0.2-0.22),C=(cij)=(20.30.20.320.10.2-0.11),fj(ξ)={-1,ξ∈(-∞,1),1,ξ∈(1,+∞),β1=β2=β3=1,σ=10,J1=0.2,J2=0.4,J3=-0.5,αi(ui(t))=2+sin(ui(t)),τj(t)=1-e-t2,kij(t)=te-t,i,j=1,2.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:
(38)B1=(-5.7982,-6.3983,-3.8992),B2=(-5.3982,-6.3983,2.1008),B3=(-4.5987,5.5989,-4.8991),B4=(-4.1987,5.5989,1.1009),B5=(4.5986,-4.7989,-2.1009),B6=(4.9986,-4.7989,3.8991),B7=(5.7982,7.1983,-3.1008),B8=(6.1982,7.1983,2.8992).
The dynamics of this system are depicted in Figure 2(a).Transient behaviors of the state variablesu1(t), u2(t), and u3(t) with different initial values in Example 2.
(a)
(b)In system (32), let J1(t)=0.6sint, J2(t)=0.1+0.4cost, J3(t)=0.4cost, τj(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
(39)A=(aij)=(2.50.20.13),B=(bij)=(30.30.22.5),C=(cij)=(20.10.32),β1=β2=5,σ=10,J1=0.3,J2=0.4,αi(ui(t))=2+sin(ui(t)),τj(t)=1-e-t2,kij(t)=te-t,i,j=1,2.The activation functions are described by(40)f1(ξ)=f2(ξ)={-1,ξ∈(-∞,-1),0,ξ∈(-1,1),1,ξ∈(1,2),2,ξ∈(2,+∞).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)).Transient behaviors of the state variablesu1(t) and u2(t) with different initial values in Example 3.
(a)
(b)In system (32), let J1(t)=0.3sint, J2(t)=0.4cost, τj(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 patternsAi  (1≤i≤8) shown in Figure 4 as stable memories (white = −1 and black = 1).Figure 4
Eight desired memories patterns for Example4.We consider system (1) with (37) in Example 2. Transform
(41)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.ThenBi  (1≤i≤8) are given in (38). According to Example 2, the corresponding memory patterns Bi  (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 arekn 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 kn 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.



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