# Chaos on Discrete Neural Network Loops with Self-Feedback

Authors: Yuanlong Chen; Xiaoying Wu
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, the complex dynamical behaviors in a discrete neural network loop with self-feedback are studied. Specifically, an invariant closed set of the system of neural network loops is built and the subsystem restricted on this invariant closed set is topologically conjugate to a two-sided symbolic dynamical system which has two symbols. In the end, some illustrative numerical examples are given to demonstrate our theoretical results.


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## Body

## 1. Introduction

In recent years, researchers have found various chaotic phenomena in the nervous system and that chaotic neural networks play an important role in neural activities. Chaos in neural networks systems have been applied to all kinds of practical problems such as combinatorial optimizations, associative recognition memory, deep learning, and biotechnology (see [1–5]). In fact, some nervous systems consist of large-scale and complex nonlinear dynamics. At present, neuroscience has provided abundant evidence to prove that the central nervous system has complex nonlinear dynamic behavior at all levels [6]. So how to analyze the dynamical behavior of neural networks plays an important role in practical applications. In order to obtain a deep and clear understanding of complex neural networks, there are increasing studies on bifurcations and chaotic behaviors of neural network systems [7].Recently, Huang and Zou in [8] showed the discrete network system consisting of two identical neurons with a uniform delay demonstrates snapback repeller chaotic behaviors near an equilibrium point. For the Hopfield networks with two different neurons [9–11], the conditions that the systems exhibit chaos are obtained. In [12], Wu et al. analyzed the chaotic behaviors of the parameterized discrete dynamics of recurrent m-neuron networks evoked by external inputs and obtained some conditions which the subsystem is topologically conjugate to symbolic dynamical system. In this paper, we will devote to analysis of the chaotic behaviors of the following discrete neural network loops with multiple delays and self-feedback:(1)x1n+1=β1x1n+α11f1x1n−k11+α1mfmxmn−k1m,x2n+1=β2x2n+α21f1x1n−k21+α22f2x2n−k22,⋮xmn+1=βmxmn+αmm−1fm−1xmn−kmm−1+αmmfmxmn−kmm,n∈ℤ,kij≥1,where n∈ℤ, for i=1,2,…,m, βi∈0,1 is the internal decay rate of the neurons, αij is the self-feedback strength or the connection strength of the ith neuron to the next neuron, and the transmission delay kij≥1 is a positive integer.For the case of the neural network with m-identical neurons, Cheng constructed a snapback repeller in [13] and then justified chaos in neural networks. When the discrete neural network with m-different neurons has multiple time delays and self-feedback, it is challenging to rigorously analyze the dynamical behaviors. In this paper, we consider the chaotic behaviors of model (1). To this end, we first rewrite the model (1) as a system of difference equations without delay by a novel way. Especially, this transformation requires a little skill. Then, we find an invariant set for the transformed system by projection and show that the system restricted on this set is topologically conjugate to the full shift map on the symbolic dynamical system. This implies that the system has chaotic behaviors. The obtained results extend the related ones in [10, 11, 13]. Also, we provide some numerical simulations to verify the theoretical results.


## 2. Invariant Subsystem of Model (1)

Letl∞ denote the Banach space of bounded sequences of real numbers with the supremum norm defined on it. The norm is denoted by ⋅. Let σ:l∞⟶l∞ be shift map defined by σξn=ξn+1,n∈ℤ, for ξ=…,ξ−n,…,ξ−1,ξ˜0,ξ1,…,ξn,…∈l∞. That is,(2)σ…,ξ−n,…,ξ−1,ξ˜0,ξ1,…,ξn,…=…,ξ−n+1,…,ξ0,ξ˜1,ξ2,…,ξn+1,….Clearly, the shift mapσ on l∞ is continuously invertible, and its inverse σ−1 is being defined by σ−1ξn=ξn−1,n∈ℤ. The ith iterate of σ, σσ°⋅σ°︷itimes, is denoted as σi. Let Σk=…i−1i0i1…|in∈1,2,…,k,n∈ℤ denote a symbolic space with k symbols. Endowing it with the metric(3)ds,t=max2−n|tn≠sn,n∈ℤ,t=…t−1t0t1…,s=…s−1s0s1…∈Σk,Σk becomes a compact and totally disconnected metric space. The shift map σ:Σk⟶Σk is defined by σtn=tn+1. Then, Σk,σ is a two-sided symbolic system. To proceed, let m,l≥2,i,j be positive integers.Lemma 1.
Letq≤m be a positive integer. aq1,aq2,…,aql are l different real numbers with l≥2 and ai is a real number with 1≤i≠q≤m. Λ=ξn∈l∞|ξmn+i=ai,ξmn+q=aqj be a subset of l∞. Then, Λ,σm is topological conjugate to Σl,σ.Proof.
Defineg:Λ⟶Σl by gξ=…,ξ−mn+q,…,ξ−m+q,ξq,…,ξmn+q,…, for ξ=ξn∈Λ. In fact, gξ is defined by deleting the elements whose indexes are congruent i modulo m in W, where 1≤i≠q≤m. It is not difficult to see that g is a homeomorphism. By definition of g, we have g∘σm=σ∘g. So Λ,σm and Σl,σ are topological conjugacy.Lemma 2 (see [14]).
LetX and Y be Banach spaces, L is an invertible linear map from X to Y, and S is a bounded linear map from X to Y. If S<L−1−1, then L+S is an invertible linear map from X to Y.Lemma 3 (see [15]).
LetΛ,d be a metric space, Y and X be Banach spaces, and U⊂Λ×Y be open. Suppose that F:U⟶X is a continuous map and that there exists a point λ0,y0∈U with the following conditions:(i)
Fλ0,y0=0.(ii)
DFyλ,y is continuous at λ0,y0, where DFyλ,y is Fre´chet partial derivative of Fλ,y with respect to y.(iii)
DFyλ0,y0:Y⟶X is an invertible linear map.Then, there exist open ballsBδ0y0=y:y−y0<δ0 and Br0λ0=λ:dλ,λ0<r0, where δ0>0,r0>0 such that, for any λ∈Br0λ0, the equation Fλ,y=0 has a unique continuous solution y=hλ∈Bδ0y0 with hλ0=y0.For convenience, we seti−1=m when i−1=0. Let α=α11,Cij=αij/αi∈1,2,…,m,j=i−1ori. Without losing generality, we may suppose that kmm≥k1m≥km−1m−1≥kmm−1⋯≥k22≥k32≥k11≥k21. In the other cases, we can discuss it in a similar way. The activation functions fii=1,…,m have the following conditions (G1):(G1) For every i∈1,2,…,m,fi is a continuously differentiable function from ℝ to ℝ. f1 has two distinct zero points x^q1,x^q2, satisfying f1x^q1=f1x^q2=0,f1′x^q1≠0,andf1′x^q2≠0, and fii∈2,…,m has a zero point x¯i,satisfying fix¯i=0andfi′x¯i≠0.Letp1=mk11,pl=mk11+∑i=2lm−i+1kii−ki−1i−1,2≤l≤m,p=pm+m and define ηn=η1n,…,ηpn, where(4)ηmj+in=xin−kii+j,0≤j≤k11,1≤i≤m,ηp1+jm−1+in=xin−kii+k11+j,1≤j≤k22−k11,2≤i≤m,ηp2+jm−2+in=xin−kii+k22+j,1≤j≤k33−k22,3≤i≤m,⋮ηpm−1+m+jn=xin−kii+km−1m−1+j,1≤j≤kmm−km−1m−1,i=m.∀n∈ℤ.For any1≤i≤m, there exists 1≤li≤m−1 such that klili<ki−1i−1−kii−1≤kli+1li+1. Then, we transform system (1) into the discrete dynamical system without delays on ℝp:(5)ηn+1=Fαηn,n∈ℤ,where Fα:ℝp⟶ℝp is defined as(6)Fαη1nη2n⋮ηp1+1nηp1+2n⋮ηpi+inηpi+i+1n⋮ηp−1nηpn=ηm+1nηm+2n⋮β1ηp1+1n+C11αf11η1n+C1mαf1mηpl1+kmm−k1m−kl1l1m−l1+mnηp1+m+1n⋮βiηpi+in+Cii−1αfii−1ηp˜lin+Ciiαfiiηinηpi+m+1n⋮ηpnβmηpn+Cmm−1αfmm−1ηp˜lmn+Cmmαfmmηmn,where p˜li=pli+ki−1i−1−kii−1−klilim−li+i−1,2≤i≤m.To investigate chaos in System (1), we only consider the chaotic behavior of the system ℝp,Fα. Next, by the projection approach, we are going to find the invariant set Λα of Fα such that the subsystem Fα,Λα has chaotic behavior for α being sufficiently large.We consider a family of mapsΦλ,⋅:l∞⟶l∞ depending on a parameter λ∈ℝ, and the class of maps is defined by(7)Φλ,ξmn+1+1=λ−ξmn+1+1+β1ξmn+1+C11f1ξmn−k11+1+C1mfmξmn−k1m+m,⋮Φλ,ξmn+1+i=λ−ξmn+1+i+βiξmn+i+Cii−1fi−1ξmn−kii−1+i−1+Ciifiξmn−kii+i,∀ξ=ξn∈l∞,2≤i≤m.It is easy to see that ifξ=ξnn∈ℤ∈l∞ satisfies Φ1/α,ξ=0, then the sequence x1n,x2n,…,xmnn∈ℤ with xin=ξmn+i satisfies (1). On the contrary, if the sequence x1n,x2n,…,xmnn∈ℤ satisfies (1), then ξ=ξnn∈ℤ∈l∞ with ξmn+i=xin satisfies Φ1/α,ξ=0.Let(8)Γ=ξ=ξn∈l∞|ξmn+i=x¯i,ξmn+1=x^q1orx^q2,2≤i≤m,n∈ℤ,b11=|C11maxf1′x^q1,f1′x^q1,b21=C21minf1′x^q1,f1′x^q2,bii=Ciifi′x¯i,i∈2,…,m,bii−1=Cii−1fi−1′x¯i,i∈1,3,4,…,m,b≜1maxb11−1,bii−1+bii−1bii−1bi−1i−1−1+⋯+bii−1bii−1bi−1i−1−1…b21b11−1,∀2≤i≤m.Lemma 4.
Under the assumptionG1, if b11>b1m and bii>b1m+bii−12≤i≤m, then we have the following:(i)
There exist positive real numbersr0 and δ0 such that, for any ξ¯∈Γ and −r0≤λ≤r0, there exists a unique point ξλ∈Bδ0ξ¯, satisfying Φλ,ξλ=0.(ii)
For every0<δ<δ0, there exists 0<r<r0 such that, for any −r≤λ≤r and ξ¯∈Γ, there is a unique point ξλ, satisfying ξλ−ξ¯≤δ and Φλ,ξλ=0.Bδ0ξ¯ is the open ball in l∞ centered at ξ¯ with radius δ0.Proof.
For a given sequenceξ¯∈Γ, we have Φ0,ξ¯=0. By the assumption (G1) and the definition of Φλ,ξ, this can ensure the continuous differentiability of Φλ,ξ. The Fréchet derivative of Φ0,ξ with respect to ξ at the point 0,ξ¯ be denoted as DΦξ0,ξ¯ which is represented as(9)DΦξ0,ξ¯ξmn+1+1=C11f1′ξ¯mn−k11+1ξmn−k11+1+C1mfm′ξ¯mn−k1m+mξmn−k1m+m,DΦξ0,ξ¯ξmn+1+i=Cii−1fi−1′ξ¯mn−kii−1+i−1ξmn−kii−1+i−1+Ciifi′ξ¯mn−kii+iξmn−kii+i,n∈ℤ,2≤i≤m.
Firstly, we have to show the invertibility ofDΦξ0,ξ¯. We denote that DΦξ0,ξ¯=L1ξ¯+L2ξ¯, where(10)L1ξ¯ξmn+1+1=C11f1′ξ¯mn−k11+1ξmn−k11+1,L1ξ¯ξmn+1+i=Cii−1fi−1′ξ¯mn−kii−1+i−1ξmn−kii−1+i−1+Ciifi′ξ¯mn−kii+iξmn−kii+i,n∈ℤ,2≤i≤m,L2ξ¯ξmn+1+1=C1mfm′ξ¯mn−k1m+mξmn−k1m+m,L2ξ¯ξmn+1+i=0,n∈ℤ.
Let(11)aii−1l=Ci−l+1i−lfi−l′ξ¯mn+kii−kii−1+⋯+ki−l+1i−l+1−ki−li−l−1+i−l−10≤l≤i−1,aiil=Ci−li−lfi−l′ξ¯mn+kii−kii−1+ki−1i−1−⋯+ki−li−l+i−l0≤l≤i−1.
It follows from (G1) that the linear operator L1ξ¯ is invertible. By directing calculation, the inverse operator L1ξ¯−1 is(12)L1ξ¯−1ξmn−1+1=1a110ξmn+k11+1,L1ξ¯−1ξmn−1+i=1aii0ξmn+kii+i−1aii0aii−101aii1ξmn+kii−kii−1+ki−1i−1+i−1+,⋯+−1i−11aii0aii−101aii1,…,aii−1i−21aiii−1ξmn+kii−kii−1+⋯−k21+k11+1.n∈ℤ,2≤i≤m.
Sinceξ¯∈Γ, ξ¯mn+1=x^q1orx^q2, ξ¯mn+i=x¯i,2≤i≤m. This implies that(13)L1ξ¯−1≤1b,L2ξ¯=b1m,so(14)L1ξ¯−1−1>L2ξ¯,by the fact that b11>b1m and bii>b1m+bii−12≤i≤m. This shows the invertibility of DΦξ0,ξ¯ by Lemma 2.
Therefore, according the implicit function theorem, there exist positive constantsrξ¯,δξ¯ such that, for every −rξ¯≤λ≤rξ¯, there is a unique point ξ=ξλ∈Bδξ¯ξ¯ with Φλ,ξλ=0.
To complete the proof of (i), it only needs to prove that there exist two positive constantsr0,δ0 which are independent of ξ¯∈Γ such that the conclusion is satisfied in (i). From the proof of the implicit function theorem, for the given ξ¯∈Γ, the constants rξ¯ and δξ¯ above are chosen such that, for −rξ¯≤λ≤rξ¯ and ξ∈Bδξ¯ξ¯, we have(15)DΦξλ,ξ−DΦξ0,ξ¯≤12Mξ¯,Φλ,ξ¯≤δξ¯2Mξ¯.
Here,Mξ¯ is the constant such that DΦξ0,ξ¯−1≤Mξ¯.
We now give the above estimates which are independent ofξ¯∈Γ. Firstly, we have, for any ξ¯∈Γ,(16)DΦξ0,ξ¯−1≤1L1ξ¯−1−1−L2ξ¯≤1b−b1m≜M,where b is given by (8). Secondly, by assumption G1, there exists δ1 such that(17)maxC11,C21f1′x−f1′x^q1≤18M,for x∈Bδ1x^q1,(18)maxC11,C21f1′x−f1′x^q2≤18M,for x∈Bδ1x^q2, and(19)maxcii−1,ci−1i−1fi−1′x−fi−1′x¯i−1≤14M,for x∈Bδ1x¯i−1,1≤i≠2≤m. Note that(20)DΦξλ,ξ−DΦξ0,ξ¯ξmn+1+1=λ−ξmn+1+1+β1ξmn+1+1+C11f1′ξmn−k11+1−f1′ξ¯mn−k11+1ξmn−k11+1+C1mfm′ξmn−k1m+m−fm′ξ¯mn−k1m+mξmn−k1m+m,DΦξλ,ξ−DΦξ0,ξ¯ξmn+1+i=λ−ξmn+1+i+βiξmn+1+i+Cii−1fi−1′ξmn−kii−1+i−1−f1′ξ¯mn−kii−1+i−1ξmn−kii−1+i−1+Ciifi′ξmn−kii+i−fi′ξ¯mn−kii+iξmn−kii+i,2≤i≤m.
Takingδ0=δ1,r1=1/4M1+a, where a≜maxβi|i=1,2,…,m, we have that, for ξ¯∈Γ, ξ∈l∞ with ξ−ξ¯≤δ0 and λ≤r1:(21)DΦξλ,ξ−DΦξ0,ξ¯≤λ1+a+14M≤12M.
On the contrary, letr2=δ0/2M1+a, and it follows from the definition of Φλ,⋅ that(22)Φλ,ξ¯≤λ1+b≤δ02M,
whenλ≤r2.
Finally, taker0=minr1,r2 and then the constants r0 and δ0 satisfy (i).
For every0<δ<δ0, (ii) follows by taking r=min1/4M1+a,δ/2M1+a<r0 and the proof of (i).


## 3. Chaos in System (1)

In this section, we shall show that the system (1) exists chaotic behaviors. By Lemma 4, for sufficiently large α>0, we define the map Tα from Γ to l∞ by(23)Tαξ¯=ξ1α,where ξ1/α is the unique solution of Φ1/α,ξ=0, satisfying ξ1/α−ξ¯≤δ. Then, we have the following proposition.Proposition 1.
For sufficiently largeα>0, let Γα=TαΓ, then the map Tα and the shift map σm are commutative, i.e.,(24)σm∘Tα=Tα∘σm.
Moreover,σmΓα=Γα.Proof.
Note that ifξ is a solution of Φ1/α,ξ=0 so is σmξ. Thus, for any ξ¯∈Γ,σmT°αξ¯=σmξ1/α is a solution of Φ1/α,ξ=0. On the contrary, ξ1/α−ξ¯≤δ by Lemma 4, which leads to σmTαξ¯−σmξ¯=σmξ1/α−σmξ¯=ξ1/α−ξ¯≤δ. Hence, by the uniqueness of ξλ in Lemma 4, we have σmTαξ¯=Tασmξ¯. Note that σmΓ=Γ, it follows that σmΓα=Γα.
For everyk∈ℤ, we define the projection Πk:l∞⟶ℝp by(25)Πkξ=ηk,∀ξ∈l∞,where for ξ=ξn∈l∞, ηk=η1k,…,ηpk∈ℝp is given by(26)ηmj+ik=ξmk−kii+j+i,0≤j≤k11,1≤i≤m,ηp1+jm−1+ik=ξmk−kii+k11+j+i,1≤j≤k22−k11,2≤i≤m,ηp2+jm−2+ik=ξmk−kii+k22+j+i,1≤j≤k33−k22,3≤i≤m,⋮ηpm−1+m+jk=ξmk−kmm+km−1m−1+j+m,1≤j≤kmm−km−1m−1.Proposition 2.
LetΛα=Π0Γα, then Λα is invariant for Fα.Proof.
For eachη0∈Λα, then there exists ξ∈Γα such that Π0ξ=η0. Therefore,(27)Fαη0=η1=Π0σmξ∈Π0σmΓα=Π0Γα=Λα.
This provesFαΛα⊂Λα.
On the contrary, by Proposition1, we have σmΓα=Γα. Thus, there exists ξ′∈Γα such that ξ=σmξ′. Thus,(28)η0=Π0ξ=Π0σmξ′=η′1=Fαη′0=FαΠ0ξ′∈FαΛα,which shows that Λα⊂FαΛα. Therefore, FαΛα=Λα.Theorem 1.
Under the assumption ofG1, if b11>b1m and bii>b1m+bii−12≤i≤m, then there exists α0>0 such that, for any α>α0, Λα,Fα is topologically conjugate to the full shift map Σ2,σ, and therefore, the system is chaotic in the sense of Devaney.Proof.
Note thatΓ,σm is an invariant subsystem. By Lemma 1 and Proposition 1, we only need to prove that there is α0>0 such that, for any α>α0, Λα,Fα is topological conjugate to Γ,σm.
LetΩ=Π0Γ, then Ω is a set in ℝp consisting of 2k11+1 elements, denoted by(29)Ω=b1,b2,…,b2k11+1.
Letδ0 and r0 be given as in Lemma 4, and let δ∈0,δ0 be small enough such that the family of closed balls Ai=B¯bi,δi=12k11+1 in ℝp is piecewise disjoint.
For the givenδ and any ξ¯=ξ¯n∈Γ, by (ii) in Lemma 4, there exists an α0=1/r>0 such that, for every α>α0, there exists a unique Tαξ¯=ξ1/α satisfying ξ1/α−ξ¯≤δ and Φ1/α,ξ1/α=0. By the definition of the projections Πk and Γ, we have ΠkΓ=Π0Γ=Ω. So we let(30)S=s=…,s−1,s0,s1,…|si∈1,2,…,2k11+1,ξsi=Πiξ¯,for someξ¯∈Γ.
The setS is a subset of Σ2k11+1. For every s=…,s−1,s0,s1,…∈S, for all i,j>0, we set(31)Vs−i…s0…sj=Fα−jAsj∩⋯∩As0∩⋯∩FαiAsi,Vs=∩i>0,j>0Vs−i…s0…sj.
We claim the following:(1)
For everys∈S, Vs contains a unique point.(2)
∪s∈SVs=Λα.
In fact, for eachs∈S, we note that(32)Vs−i…s0…sj=η∈ℝp|Fα−iη∈Asi,…,Fαjη∈Asj.
Therefore, there exists a uniqueξ¯∈Γ such that, for all i∈ℤ, Πiξ¯=ξsi∈Ω. Then, by the definition of Tα and Lemma 4, there exists a unique Tαξ¯=ξ1/α, satisfying ξ1/α−ξ¯≤δ and Φ1/α,ξ1/α=0. So Πnξ1/α=ηnn∈ℤ is a bounded orbit of Fα, that is, ηn=Fαnη0∈Asn,∀n∈ℤ. Therefore, η0∈Vs, which implies Vs is nonempty.
On the contrary, for anyη′∈Vs, for all n∈ℤ, there are Fαnη′∈Asn. Thus, Fαnη′n∈ℤ is a bounded orbit of Fα. Then, there exists ξ∈l∞ such that Πnξ=Fαnη′. So ξ−ξ¯≤δ, and Φ1/α,ξ=0. Again by Lemma 4 (ii), there is ξ=Tαξ¯, and hence, η′=η0. Claim (1) holds.
For Claim (2), letη∈Λα. Then, there exists a ξ¯∈Γ such that η=Π0Tαξ¯ Let s=…s−1,s0,s1…∈S be the corresponding sequence of ξ¯. Similar to the above argument, we have η∈Vs. Therefore,(33)Λα⊂∪s∈SVs.
From Claim (1), eachVs contains a unique point which belongs to Λα, so the converse inclusion holds. This proves Claim (2).
For everyα>α0, define a map h:Γ⟶Λα by h=Π0∘Tα. We claim that h is a conjugacy from σm to Fα. To prove this, we need to show that both h and h−1 are continuous and(34)h∘σm=Fα∘h,onΓ.
By Claim (2) and the definition ofh, it is easy see that h is surjective. From Claim (1) and Lemma 4, it follows that h is injective. Therefore, h is bijective. Since h is a map from a compact metric space Γ to a Hausdorff space Λα, to prove that h is homeomorphic, we just need to show the continuity of h. Let the corresponding subindex sequence of ξ¯∈Γ be s¯=…s−1,s0,s1…∈S. It follows from Claim (1) that(35)limi,j⟶+∞diamVs−i…s0…sj=0,where diamVs−i…s0…sj denotes the diameter of the set Vs−i…s0…sj. Thus, for any ε>0, there exists a positive integer n such that diamVs¯−n…s¯0…s¯n<ε. Take δ1=1/2mn+kmm+1. Then, for any ξ˜∈Γ with dξ˜,ξ¯<δ1, it follows that ξ˜ agrees with ξ¯ in those terms with lower indices from i=−mn+kmm+1 to i=mn+kmm+1. Let s^,s¯∈S be the symbolic sequences corresponding to ξ^ and ξ¯, respectively. We have s^ agrees with s¯ in those terms with subscripts from i=−n to i=n+kmm+1. Thus, hξ^,hξ¯∈Vs¯−n…s¯0…s¯n and hξ^−hξ¯<ε. This shows the continuity of h. Hence, we conclude that h is a homeomorphism.
Finally, for anyξ¯∈Γ, we have(36)hξ¯=Π0∘Tαξ¯=η0=η10,…,ηq0T.
Thus,(37)Fαhξ¯=η11,η21,…,ηq1T=Π0∘σm∘Tαξ¯by1=Π0∘Tα∘σmξ¯,by Proposition1=h∘σmξ¯.
The Theorem1 holds.


## 4. Some Simulations

In this section, we will give some numerical simulation results to verify our theoretical results. We chooseβ1=β3=1/4,β2=β4=3/43/4,f1t=sint,f2t=tanht,f3t=cost,f4t=tanht,α11=0.5α,α14=α,α21=−0.4α,α22=2α,k11=1,k21=2,k14=3,k22=4,k32=3,k33=1,k43=2,andk44=4. In this case, system (1) becomes(38)x1n+1=14x1n+1.5αsinx1n−1+αtanhx4n−3,x2n+1=34x2n−0.4αsinx1n−2+2αtanhx2n−4,x3n+1=14x3n+αtanhx2n−3+1.5αcosx3n−1,x4n+1=34x4n−0.4αcosx3n−2+2αtanhx4n−4,∀n≥5.In Figure1, for every α value, the initial values were reset to x11=−0.1,x12=0.1,x13=0.12,x14=−0.2,x15=0.9, x21=0.11,x22=−0.2,x23=−0.1,x24=0.2,x25=0.1, x31=0.12,x32=0.15,x33=−0.2,x34=0.22,x35=1.1, and x41=−0.1,x42=−0.23,x43=−0.1,x44=0.2,x45=0.11. After 104 time steps being iterated, we plot the data consisting of 500 points for per α value. The plotting is for x1,x3 vs the parameter α. The bifurcation figures illustrate that the fixed point of x1 loses stability and period bifurcation occurs when α≈0.95, and the fixed point of x3 loses stability and period bifurcation occurs when α≈1.1. Making the bifurcation figures of the x2 vs α and the x4 vs α similar, they are omitted.Figure 1
Bifurcation diagram.In Figure2, we show the largest Lyapunov exponent diagram for α∈0,6. For every α value, the initial values were the same as Figure 1. From the simulation results in Figure 2, we can find that the largest Lyapunov exponent is negative when α∈0,1 and is positive when α>2.8. Thus, the figures illustrate that the system (38) has chaotic behaviors when α is large enough.Figure 2
Largest Lyapunov exponents diagram.In Figure3, we show the chaotic figures. For each α value, after 6×105 time steps being iterated, plot the 6000 data points. The figure illustrates that there are no chaos for small α (e.g., α=0.72,2.1) and chaotic behavior occurs when α is larger (e.g., α=1.68,3.0,6.7,100). Those numerical simulations support the theoretical results in Section 2.Figure 3
Chaos diagram.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 5. Conclusion

In this paper, the chaos of a discrete neural network loops with self-feedback is studied. The discrete neural network loops with multiple delays and self-feedback can demonstrate chaotic behavior when the interconnection strengths are large enough. Numerical simulations support the theoretical results. The theoretical results are to provide some new methods for the design of chaotic neural networks.



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