Research Article Bifurcation and Hybrid Control for A Simple Hopfield Neural Networks with Delays

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1 Mahemaical Problems in Engineering Volume 23, Aricle ID 35367, 8 pages hp://dx.doi.org/5/23/35367 Research Aricle Bifurcaion and Hybrid Conrol for A Simple Hopfield Neural Neworks wih Delays Zisen Mao, Hao Wang, Dandan Xu, and Zhoujin Cui College of Science, PLA Universiy of Science and Technology, Nanjin 2, China Correspondence should be addressed o Zisen Mao; maozisen@26.com Received 3 March 23; Acceped 6 May 23 Academic Edior: Guanghui Wen Copyrigh 23 Zisen Mao e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. A deailed analysis on he Hopf bifurcaion of a delayed Hopfield neural nework is given. Moreover, a new hybrid conrol sraegy is proposed, in which ime-delayed sae feedback and parameer perurbaion are used o conrol he Hopf bifurcaion of he model. Numerical simulaion resuls confirm ha he new hybrid conroller using ime delay is efficien in conrolling Hopf bifurcaion.. Inroducion I is well known ha neural neworks are complex and largescale nonlinear dynamical sysem. In he las decade, he dynamical characerisics (including sable, unsable, oscillaory, and chaoic behavior) of Hopfield neural neworks (HNNs) wih ime delays have become a subjec of inense research aciviies. Many sabiliy crieria are obained. We refer he reader o [ 8] and he references cied herein. However, he periodic naure of neural impulses is of fundamenal significance in he conrol of regular dynamical funcions such as breahing and hear beaing. Neural neworks involving persisen oscillaions such as limi cycle may be applied o paern recogniion and associaive memory. In differenial equaions wih delays, periodic oscillaory behavior can arise hrough he Hopf bifurcaion. Therefore, i is also very significan o sudy he class of problem. Olien and Bélair [9] invesigaed he bifurcaion of he following HNNs sysem: x = a x () +b f (x ( τ )) +b 2 f 2 (x 2 ( τ 2 )), x 2 = a 2 x 2 () +b 2 f (x ( τ 3 )) +b 22 f 2 (x 2 ( τ )), () in which τ =τ 2 =τ 3 =τ, a i =and f i () =, i=,2,and Huang e al. [] sudy furher he bifurcaion and periodic naure of sysem ()wih2τ =τ 2 +τ 3, a i =and f i () =, i=,2. Moreover, many auhors also consider discree form of sysem (); we can see [ 3]. In recen years, bifurcaion conrol has araced many researchers from various disciplines. The aim of bifurcaion conrol is o design a conroller o modify he bifurcaion properies of a given nonlinear sysem, hereby o achieve some desirable dynamical behaviors. Afer he pioneering work iniiaed by O e al. [4],herehavebeenmanyideas and mehods of bifurcaion conrol [5 2]. However, from he conrol heory poin of view, we may classify he curren mehods ino wo main caegories: he firs one is feedback conrol where sae feedback is applied o conrol bifurcaion or chaos, and he oher is nonfeedback mehods. Recenly, Luo e al. [2] proposed a new conrol sraegy for perioddoubling bifurcaions in a discree nonlinear dynamical sysem. Moreover, Liu and Chung [22] invesigaed he same conrol sraegy in a coninuous dynamical sysem wihou ime delays. Now, we exend his sraegy o deal wih bifurcaion conrol in HNNs sysem (). In he paper, we will propose a new hybrid conrol sraegy in which he parameer perurbaion and ime-delayed sae feedback are combined and used o conrol Hopf bifurcaion in sysem (). Simulaion resuls demonsrae he correcness of our heoreical analysis. The comparison shows

2 2 Mahemaical Problems in Engineering ha he conrol sraegy is effecive as i mees he purpose of rearding he occurrence of bifurcaion. 2. Sabiliy and Hopf Bifurcaion of Sysem () wihou Conrol In his secion, we will consider sysem () wih2τ =τ 2 + τ 3 2τand f i () =, i=,2.iisobviouslyha(, ) is an equilibrium poin of sysem (). To simplify, here we denoe c =b f (), c 2 =b 2 f 2 (), c 2 = b 2 f (), c 22 = b 22 f 2 (), a = a 2 = b. Consider he linearized sysem of sysem ()a(, ) x = bx () +c x ( τ) +c 2 x 2 ( τ 2 ), x 2 = bx 2 () +c 2 x ( τ 3 )+c 22 x 2 ( τ). The characerisic equaion of he linearized sysem (2)is (λ+b) 2 (c +c 22 ) (λ+b) e λτ +(c c 22 c 2 c 2 )e 2λτ =, (3) which deermines he local sabiliy of he equilibrium soluion. Thus, we will find some condiions which ensure ha all roos of (3) have negaive real pars. To faciliae he calculaion in his paper, we rewrie he characerisic equaion (3) as follows: (2) (λ+b) 2 e 2λτ 2T(λ+b) e λτ +D=, (4) where D=c c 22 c 2 c 2, T = (/2)(c +c 22 ).Obviously,(4) is a quadraic polynomial in he variable (λ + b)e λτ and has roos given by (λ+b) e λτ =T± T 2 D. (5) In he following, we disinguish wo cases o discuss (5). 2.. As T 2 D. In his par, we sae a resul due o [23] as alemmaoanalyze(5), which is, for he convenience of he reader, saed as follows. Lemma. For he ranscendenal equaion λ n +p () λn + +p () n λ+p() n +[p () λn + +p () n λ+p() n ]e λσ + +[p (m) λ n + +p (m) n λ+p(m) n ]e λσ m =, where σ i (i =,...,m) and p (i) j (i =,...,m;j =,...,n) are consans. As (σ,...,σ m ) varies, he sum of he orders of he zeros of (6) in he open righ half-plane can change only if a zero appears on or crosses he imaginary axis. (6) For convenience, we make he following assumpions: (H) b>t± T 2 D. (H2) b 2 >(T± T 2 D) 2. (H3) (T + T 2 D) 2 <b 2 <(T T 2 D) 2. (H4) b 2 <(T± T 2 D) 2. Lemma 2. If (H) and (H2) hold, hen all roos of (3) have negaive real pars for every τ [,+ ). Proof. For (3), when τ =, is roos can be expressed as λ,2 = b+t± T 2 D.Clearly,allroosof(3)arenegaive if (H) holds. We wan o deermine if he real par of some roo increases o reach zero and evenually becomes posiive as τ =.Wecanseehaλ is a roo of (3)ifandonlyifλis a roo of (5). We wrie λ=ρ+iωfor a roo of he characerisic equaions (5), separae he real and imaginary pars of he ensuing equaions (5), and obain e ρτ [(ρ+ b) cos (ωτ) ωsin (ωτ)] =T± T 2 D, e ρτ [(ρ+ b) sin (ωτ) +ωcos (ωτ)] =. A change in he sabiliy of he saionary soluion can only occur when ρ=,hais, By (8), we have b cos (ωτ) ωsin (ωτ) =T± T 2 D, b sin (ωτ) +ωcos (ωτ) =. (7) (8) ω 2 = (T± T 2 D) 2 b 2. (9) By (9), if (H2) holds, we know ha (5) hasnopurelyimaginary roos, and hen applying Lemma one obains ha all roos of (3) have negaive real pars. This complees he proof of lemma. Lemma 3. For (5), one obains he following resuls. () If (H) and (H3) hold, hen (5) have a pair of purely imaginary roos ±iω a τ=τ,j. (2) If (H) and (H4) hold, hen (5) have a pair of purely imaginary roos ±iω + a τ=τ +,j and have anoher pair of purely imaginary roos ±iω a τ=τ,j, where ω 2 ± = (T± T 2 D) 2 b 2, () τ ±,j = arcan ( ω ± jπ )+, j=,,...; ω ± b ω ± τ = min {τ +,,τ, }. () Here one denoes ±ω especiallyasapairofpurely imaginary roos of (5) aτ = τ.toseeifτ, and τ are bifurcaion values, one needs o verify if he ransversaliy condiions hold. In fac, one has he following.

3 Mahemaical Problems in Engineering 3 Lemma 4. The following ransversaliy condiions: are saisfied. Proof. By (5), we have Hence, Re ( dλ dτ ) τ ±,j > (2) dλ dτ eλτ + (λ+b) (τe λτ dλ dτ +λeλτ )=. (3) dλ (+(λ+b) τ) = λ(λ+b). (4) dτ Obviously, we have hen ( dλ dτ ) (+(λ+b) τ) = λ (λ+b) = λ (λ+b) τ λ, (5) sign (Re ( dλ dτ ) τ ±,j )= sign (Re ( dλ dτ ) τ ±,j ) We complee he proof of Lemma 4. = sign (Re ( iω (iω + b) )) = sign (Re ( ω 2 +ibω )) = sign ( ω2 ω 4 +b 2 ω 2 ) = sign ( ω 2 +b 2 )>. (6) From Lemmas 2 4, we can obain he following heorem abou he disribuion of he characerisic roos of (3). Theorem 5. Le τ,, τ be defined by (). (i) If (H) and (H2) hold, hen all roos of (3) have negaive real pars for all τ. (ii) If (H) and (H3)((H4)) hold, hen when τ<τ, (τ < τ ),allroosof (3) have negaive real pars, when τ= τ, (τ = τ ), (3) has a pair of purely imaginary roos ±iω (±iω ),andwhenτ>τ, (τ > τ ), (3) has a leas one roo wih posiive real par. (ii) If (H3)((H4)) holds, here is a criical value τ = τ, (τ = τ ) of he discree delay so ha if τ<τ, (τ < τ ), hen he equilibrium poin (, ) is asympoically sable; if τ>τ, (τ > τ ),hen(, ) is unsable; Hopf bifurcaion occurs when τ=τ, (τ = τ ) As T 2 <D. For convenience, we have he following assumpions: (H5) b>t. (H6) b 2 >D. (H7) b 2 <D. Similar o he deducion of Lemma 2,wehavehefollowing resul. Lemma 7. If (H5) and (H6) hold, hen all roos of (3) have negaive real pars for every τ [,+ ). Proof. For (3), when τ =, is roos can be expressed as λ,2 = b+t± T 2 D.Clearly,allroosof(3)havenegaive real pars if (H5) holds. We wan o deermine if he real par of some roo increases o reach zero and evenually becomes posiive as τ =.Wecanseehaλ is a roo of (3)ifandonly if λ is a roo of (5). We wrie λ = ρ + iω for a roo of he characerisic equaion (3), separae he real and imaginary pars of he ensuing equaions (5), and obain e ρτ [(ρ+ b) cos (ωτ) ωsin (ωτ)] =T, (7) e ρτ [(ρ+ b) sin (ωτ) +ωcos (ωτ)] =± D T 2. A change in he sabiliy of he equilibrium poin can only occur when ρ=,hais, b cos (ωτ) ωsin (ωτ) =T, (8) b sin (ωτ) +ωcos (ωτ) =± D T 2. Hence, we have ω 2 =D b 2. (9) By (9), if (H6) holds, we know ha (5) hasnopurelyimaginary roos, and hen applying Lemma one obains ha all roos of (3) have negaive real pars. This complees he proof of lemma. Lemma 8. For (3), one obains he following resuls. If (H5) and (H7) hold, hen (3) have a pair of purely imaginary roos ±iω a τ=τ ±,j,where ω 2 =D b 2, (2) By using Theorem 5, he sabiliy and bifurcaion of sysem () can be summarized as he following heorem. τ ±,j = ω arccos (bt ± ω D T 2 b 2 +ω 2 )+ 2jπ ω, j=,,...; Theorem 6. For sysem (), le (H) hold and le τ,,τ be defined by (). (i) If (H2) holds, hen he equilibrium poin (, ) is asympoically sable for discree delays τ. τ = min {τ +,,τ, }. (2) According o Lemma 4, oneknowshaτ is bifurcaion values.

4 4 Mahemaical Problems in Engineering From Lemmas and 2, onecanobainhefollowing heorem abou he disribuion of he characerisic roos of (3). Theorem 9. Le τ be defined by (2). (i) If (H5) and (H6) hold, hen all roos of (3) have negaive real pars for all τ. (ii) If (H5) and (H7) hold, hen when τ<τ,allroosof (3) have negaive real pars, when τ=τ, (3) has a pair of purely imaginary roos ±iω, andwhenτ>τ, (3) hasaleasoneroowihposiiverealpar. By using Theorem 9, he sabiliy and bifurcaion of sysem () can be summarized as he following heorem. Theorem. For sysem (), le (H5) hold and le τ be defined by he following: (2). (i) If (H6) holds, hen he equilibrium poin (, ) is asympoically sable for discree delays τ. (ii) If (H7) holds, here is a criical value τ = τ of he discree delay so ha if τ < τ, hen he equilibrium poin (, ) is asympoically sable; if τ>τ,hen(, ) is unsable; Hopf bifurcaion occurs when τ=τ. 3. Sabiliy Analysis and Bifurcaion wih Hybrid Conrol In his secion, we will consider sysem () wihhybridconrol described by he following differenial equaion: x =α( a x () +b f (x ( τ )) + b 2 f 2 (x 2 ( τ 2 ))) +βx ( τ ), x 2 =α( a 2 x 2 () +b 2 f (x ( τ 3 )) + b 22 f 2 (x 2 ( τ ))) +βx 2 ( τ ), (22) where α > and β R.Obviously,(, ) is also an equilibrium poin of sysem (22). Linearizing he sysem (22)aheequilibriumpoin(, ), we obain x = αbx () +(c α+β)x ( τ )+c 2 αx 2 ( τ 2 ), x 2 = αbx 2 () +c 2 αx ( τ 3 )+(c 22 α+β)x 2 ( τ ). (23) Then he characerisic equaion for he linearized sysem around (, ) T is given by (λ+αb) 2 e 2λτ (α(c +c 22 )+2β)(λ+αb) e λτ +((αc +β)(αc 22 +β) α 2 c 2 c 2 )=, (24) which is a quadraic polynomial in he variable (λ + αb)e λτ and has roos given by (λ+αb) e λτ =T ± (T ) 2 D, (25) where D =α 2 D+2αβT+β 2, T =αt+β. (26) By (26), we know ha (T ) 2 D =α 2 (T 2 D),andhus, (T ) 2 D ((T ) 2 <D ) holds if and only if T 2 D(T 2 <D) holds. Inhefollowing,wealsodisinguishwocasesodiscuss (25). 3.. As T 2 D. Corresponding o Par I of Secion 2, we make he following assumpions for convenience: (H) b>t± T 2 D+(β/α). (H2) b 2 >(T± T 2 D+(β/α)) 2. (H3) (T+ T 2 D+(β/α)) 2 <b 2 <(T T 2 D+(β/α)) 2. (H4) b 2 <(T± T 2 D+(β/α)) 2. Denoe (ω ± )2 =(T ± (T ) 2 2 D ) α 2 b 2, (27) τ ±,j = ω± arcan ( αb )+ jπ ω ± ω±, j=,,..., (28) τ = min {τ +,,τ, }. Similarly, we can obain he following heorem. Theorem. For sysem (22),le(H) hold and le τ ±,, τ be defined by (28). (i) If (H2) holds, hen he equilibrium poin (, ) is asympoically sable for discree delays τ. (ii) If (H3) ((H4) ) holds, here is a criical value τ = τ, (τ = τ ) of he discree delay so ha if τ<τ, (τ < τ ), hen he equilibrium poin (, ) is asympoically sable; if τ>τ, (τ > τ ),hen(, ) is unsable; Hopf bifurcaion occurs when τ=τ, (τ = τ ) As T 2 <D. Similar o deducion of Secion 2.2, wehave he following assumpions: (H5) b > T + (β/α), (H6) b 2 > D + (2β/α)T + (β/α) 2, (H7) b 2 < D + (2β/α)T + (β/α) 2. In his par, we denoe (ω ) 2 =D α 2 b 2, (29) τ ±,j = ω arccos (αbt ±ω D (T ) 2 α 2 b 2 +(ω ) 2 ) + 2jπ ω, j=,, ; τ = min {τ +,j,τ,j }. Hence, we can obain he following heorem. (3)

5 Mahemaical Problems in Engineering 5 Theorem 2. For sysem (22), le(h5) hold and le τ be defined by (3). (i) If (H6) holds, hen he equilibrium poin (, ) is asympoically sable for discree delays τ. (ii) If (H7) holds, here is a criical value τ = τ of he discree delay so ha if τ < τ hen he equilibrium poin (, ) is asympoically sable; If τ>τ,hen(, ) is unsable; Hopf bifurcaion occurs when τ=τ. Remark 3. When α= γand β=γin he sysem (22), hen we obain he same hybrid conrol wih [22]; however, a conrol model based on delayed feedback is proposed in his paper; i is well know ha conrol heory should conain delay since any conrol acion akes effec only afer a cerain delay. Hence, our hybrid conrol is more helpful han [22]. Remark 4. When α=in he sysem (23), hen we obain a conrol model only based on delayed feedback in his paper, i is clear ha our hybrid conrol is more general han conrol sraegy proposed by [7]. Remark 5. In [], auhors invesigaed he Hopf bifurcaion of following HNNs wih α=and β=: x =α( x () ( 2 )f (x ( τ )) +b 2 f 2 (x 2 ( τ 2 )) ) + βx ( τ ), x 2 =α( x 2 () +b 2 f (x ( τ 3 )) ( 2 )f 2 (x 2 ( τ ))) + βx 2 ( τ ), (3) By choosing τ =3π/4, τ 2 =π/8, τ 3 = π/8, heauhors obained ha Hopf bifurcaion occurs when b 2 b 2 =. However, if choosing he parameers α =.8 and β=,by he hybrid conrol sraegy of his paper, he Hopf bifurcaion in [] will be eliminaed. We can see Figures and 2. Remark 6. I is known o all ha neural neworks are a special case of complex neworks. Thus, i is ineresing and imporan o furher sudy how o expand he applicaion of heoreical resuls in [24 27] and any oher complex neworks. 4. Examples In his secion, we give wo examples o illusrae our resuls. Example. Consider he following HNNs sysem wih hybrid conrol: x =α( a x () +b f (x ( τ )) + b 2 f 2 (x 2 ( τ 2 ))) +βx ( τ ), x 2 =α( a 2 x 2 () +b 2 f (x ( τ 3 )) + b 22 f 2 (x 2 ( τ ))) +βx 2 ( τ ), (32) x () x () Wihou conrol Wih conrol Figure : The rajecory of x () versus ime in he sysem (3)wih τ=3π/4. x 2 () x 2 () Wihou conrol Wih conrol Figure 2: The rajecory of x 2 () versus ime in he sysem (3)wih τ=3π/4. where τ =τ 2 =τ 3 =τ, a i =, b ij =.3,andf i (x) = anh(x), i, j =, 2. Iisobviousha(, ) is an equilibrium poinofsysem(32). Choosing α=,β=,bycalculaion, he periodic oscillaory behavior can arise hrough he Hopf bifurcaion as τ = 7.763; wecanseefigures3 and 5 (τ = 7.763, τ = 7.763). However, when α/5 < 2β < α, wih complicaed calculaion, (H2) holds; by Theorem (i), he equilibrium poin (, ) is asympoically sable for any discree delays τ τ. For he convenience of numerical

6 6 Mahemaical Problems in Engineering x (), x 2 () x () x 2 () Figure 3: The rajecory of x () and x 2 () versus ime in he sysem (32)wihouconrol(τ = 7.763). x (), x 2 () x () x 2 () Figure 7: The rajecory of x () and x 2 () versus ime in he sysem (33) wihou conrol (τ =.292). x (), x 2 () x () x 2 () x (), x 2 () x () x 2 () Figure 4: The rajecory of x () and x 2 () versus ime in he sysem (32)wihconrol(τ = 7.763). Figure 8: The rajecory of x () and x 2 () versus ime in he sysem (33) wih conrol (τ =.292). x (), x 2 () x () x 2 () Figure 5: The rajecory of x () and x 2 () versus ime in he sysem (32)wihouconrol(τ > 7.763). x (), x 2 () x () x 2 () Figure 6: The rajecory of x () and x 2 () versus ime in he sysem (32)wihconrol(τ > 7.763). simulaion, herewechooseα =.75, β =.25, andτ = as an example. i can be seen in Figure 4 (τ = 7.763). Fix all coefficiens of sysem (32) and le τ vary, and he waveforms x (), x 2 () wihou and wih conrol are shown, respecively. Obviously, we obain ha he Hopf bifurcaion in (32) wihou hybrid conrol could be eliminaed by hybrid conrol; we can see Figure 6 (τ = > τ ). Example 2. Consider he following HNNs sysem wih hybrid conrol: x =α( a x () +b f (x ( τ )) + b 2 f 2 (x 2 ( τ 2 ))) +βx ( τ ), x 2 =α( a 2 x 2 () +b 2 f (x ( τ 3 )) + b 22 f 2 (x 2 ( τ ))) +βx 2 ( τ ), (33) where τ =τ 2 =τ 3 =τ, a i =, b ij =,andf i (x) = anh(x), i, j =, 2. Iisobviousha(, ) is an equilibrium poin of sysem (33). Choosing α=,β=,bycalculaion,weknow τ =.292.However,when α < β < α, a family of periodic soluions bifurcaes from (, ) a τ (see Figure 7). Choosing α =.75 and β =.25, wih complicaed calculaion, we know τ = (see Figure 8). Fix all coefficiens of sysem (33)andleτ=τ ;hewaveformsx (), x 2 () wihou andwihconrolareshown,respecively.however,hehopf

7 Mahemaical Problems in Engineering 7 x (), x 2 () x () x 2 () Figure 9: The rajecory of x () and x 2 () versus ime in he sysem (33)wihconrol(τ = 2.243). bifurcaion in (33)couldbedelayedbyhybridconrolwhich couldbeseenbyfigure Conclusions In his paper, he bifurcaion and he bifurcaion conrol problems have furher been invesigaed for a HNNs model wih delays. For he model, hybrid conrol sraegy in which he parameer perurbaion and ime-delayed sae feedback are combined and used o conrol various bifurcaions in a coninuous nonlinear dynamical sysem. I should be poined ou ha, alhough Liu also have deal wih hybrid conrol, he ime delayed feedback conrol used in our paper is more helpful han he conroller in [22]. On he oher hand, using parameer perurbaion in his paper, our conrol sraegy is more general han he oher feedback conrol. Numerical simulaions are given o jusify he validiy of hybrid conroller in bifurcaion conrol. Acknowledgmens This research was suppored by he Pre-research Foundaion of PLA Universiy of Science and Technology. Youh Research Foundaion of College of Science of PLA Universiy of Science and Technology. References [] J. J. Hopfield, Neurons wih graded response have collecive compuaional properies like hose of wo-sae neurons, Proceedings of he Naional Academy of Sciences of he Unied Saes of America, vol. 8, no., pp , 984. [2] C. Li, X. Liao, and K.-W. Wong, Delay-dependen and delayindependen sabiliy crieria for cellular neural neworks wih delays, Inernaional Bifurcaion and Chaos in Applied Sciences and Engineering,vol.6,no.,pp , 26. [3] V. Singh, On global robus sabiliy of inerval Hopfield neural neworks wih delay, Chaos, Solions and Fracals,vol.33,no.4, pp , 27. [4] L. Wan and J. Sun, Mean square exponenial sabiliy of sochasic delayed Hopfield neural neworks, Physics Leers A, vol. 343, no. 4, pp , 25. [5] Z.Wang,Y.Liu,K.Fraser,andX.Liu, Sochasicsabiliyof uncerain Hopfield neural neworks wih discree and disribued delays, Physics Leers A, vol. 354, no. 4, pp , 26. [6] D.-Y. Xu and H.-Y. Zhao, Invarian and aracing ses of Hopfieldneuralneworkswihdelay, Inernaional Sysems Science,vol.32,no.7,pp ,2. [7] Z.Yuan,D.Hu,L.Huang,andG.Dong, Onheglobalasympoic sabiliy analysis of delayed neural neworks, Inernaional Bifurcaion and Chaos in Applied Sciences and Engineering,vol.5,no.2,pp ,25. [8]H.Zhao, GlobalasympoicsabiliyofHopfieldneuralnework involving disribued delays, Neural Neworks, vol. 7, no., pp , 24. [9] L. Olien and J. Bélair, Bifurcaions, sabiliy, and monooniciy properies of a delayed neural nework model, Physica D, vol. 2, no. 3-4, pp , 997. [] C.Huang,Y.He,L.Huang,andY.Zhaohui, Hopfbifurcaion analysis of wo neurons wih hree delays, Nonlinear Analysis. Real World Applicaions,vol.8,no.3,pp.93 92,27. [] W. He and J. Cao, Sabiliy and bifurcaion of a class of discreeime neural neworks, Applied Mahemaical Modelling,vol.3, no.,pp.2 222,27. [2] Z.Yuan,D.Hu,andL.Huang, Sabiliyandbifurcaionanalysis on a discree-ime sysem of wo neurons, Applied Mahemaics Leers,vol.7,no.,pp ,24. [3] H. Zhao, L. Wang, and C. Ma, Hopf bifurcaion and sabiliy analysis on discree-ime Hopfield neural nework wih delay, Nonlinear Analysis. Real World Applicaions, vol.9,no.,pp. 3 3, 28. [4] E.O,C.Grebogi,andJ.A.Yorke, Conrollingchaos, Physical Review Leers, vol. 64, no., pp , 99. [5] E.H.Abed,H.O.Wang,andR.C.Chen, Sabilizaionofperiod doubling bifurcaions and implicaions for conrol of chaos, Physica D,vol.7,no.-2,pp.54 64,994. [6] Y. Braiman and I. Goldhirsch, Taming chaoic dynamics wih weak periodic perburbaions, Physical Review Leers, vol.66, no. 2, pp , 99. [7] G. Chen, On some conrollabiliy condiions for chaoic dynamics conrol, Chaos, Solions and Fracals, vol. 8, no. 9, pp , 997. [8] R. Lima and M. Peini, Suppression of chaos by resonan parameric perurbaions, Physical Review A,vol.4,no.2,pp , 99. [9] X. F. Wang and G. Chen, Pinning conrol of scale-free dynamical neworks, Physica A,vol.3,no.3-4,pp.52 53,22. [2] L.Yang,Z.Liu,andJ.Mao, Conrollinghyperchaos, Physical Review Leers,vol.84,pp.67 7,2. [2] X.S.Luo,G.Chen,B.H.Wang,andJ.Q.Fang, Hybridconrol of period-doubling bifurcaion and chaos in discree nonlinear dynamical sysems, Chaos, Solions and Fracals, vol.8,no.4, pp , 23.

8 8 Mahemaical Problems in Engineering [22] Z. Liu and K. W. Chung, Hybrid conrol of bifurcaion in coninuous nonlinear dynamical sysems, Inernaional Bifurcaion and Chaos in Applied Sciences and Engineering,vol. 5, no. 2, pp , 25. [23] S. Ruan and J. Wei, On he zeros of ranscendenal funcions wih applicaions o sabiliy of delay differenial equaions wih wo delays, Dynamics of Coninuous, Discree & Impulsive Sysems A,vol.,no.6,pp ,23. [24] W. Yu and J. Cao, Synchronizaion conrol of sochasic delayed neural neworks, Physica A, vol. 373, pp , 27. [25] W. Yu,G.Chen, and J.Lü, On pinning synchronizaion of complex dynamical neworks, Auomaica, vol. 45, no. 2, pp , 29. [26] G. Wen, Z. Duan, W. Yu, and G. Chen, Consensus in muliagen sysems wih communicaion consrains, Inernaional Robus and Nonlinear Conrol,vol.22,no.2,pp.7 82, 22. [27] G.Wen,Z.Duan,W.Yu,andG.Chen, Consensusofsecondorder muli-agen sysems wih delayed nonlinear dynamics and inermien communicaions, Inernaional Conrol,vol.86,no.2,pp ,23.

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