Stability and Bifurcation in a Neural Network Model with Two Delays

Transcription

1 Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy of Informaion Science and Technology, Nanjing 144, P.R. China hugp@nuis.edu.cn Absrac This paper is concerned wih a wo-neurons nework model wih wo discree delays. By regarding he sum of wo discree ime delay as he bifurcaion parameer, he sabiliy of he equilibrium and Hopf bifurcaions are invesigaed. Finally, o verify our heoreical predicions, some numerical simulaions are also included. Mahemaics Subjec Classificaion: 34K18; 34K; 9B Keywords: Time delay; Neuron nework; Sabiliy; Hopf bifurcaion 1 Inroducion Recenly, a large number of neural neworks models have been proposed and sudied exensively since Hopfield consruced a simplified neural nework. In mos neworks however, i is usually expeced ha ime delays exis during he processing and ransmission of signals. In general, delay-differenial equaions exhibi much more complicaed dynamics han ordinary differenial equaions since a ime delay could cause a sable equilibrium o become unsable [1]. Recenly, ime delays have been incorporaed ino neural nework models by many auhors [1,, 4, 5, 6], here has been grea ineres in dynamical characerisics of neural nework model wih delay. In presen paper, we consider a simplified Hopfield-ype neural nework model wih wo delays () = u 1 ()+a 11 f(u 1 ()) + a 1 f(u ( τ 1 )), (1) u () = u ()+a 1 f(u 1 ( τ )) + a f(u ()), where τ i (i = 1, ) are non-negaive consans, and f(x) is C funcion. Throughou his paper we also assume ha f() =.

2 176 GuangPing Hu and XiaoLing Li In general, he delays appearing in differen erms of a neural nework model are no equal each oher. Therefore, i is more realisic o consider he dynamics of a dynamical sysem wih differen delays. Based on his idea, in his paper, we consider he dynamical behaviors of sysem (1), ha is, by aking τ 1 + τ as he bifurcaion parameer, we invesigae he sabiliy and Hopf bifurcaions of sysem (1) induced by he delays. This paper is organized as follows. In Secion, we shall consider he sabiliy of he zero equilibrium and he exisence of local Hopf bifurcaion. In order o verify our heoreical predicion, some numerical simulaions are included in Secion 3. Sabiliy analysis and Hopf bifurcaion Taking he following variable change: v 1 () =u 1 ( τ ), v () =u (), hen he sysem (1) can be rewrien as v1 () = v 1 ()+a 11 f(v 1 ()) + a 1 f(v ( τ)), () v () = v ()+a 1 f(v 1 ()) + a f(v ()), where τ = τ 1 + τ. I is obvious ha he origin (, ) is an equilibrium of sysem (). Linearizing sysem () abou he origin (, ) yields he following linear sysem v1 () = v 1 ()+α 11 v 1 ()+α 1 v ( τ), (3) v () = v ()+α 1 v 1 ()+α v (), where α ij = a ij f (), i, j=1,. The associaed characerisic equaion of sysem (3) is λ + pλ + q α 1 α 1 e λτ =, (4) where p = α 11 α,q=(1 α 11 )(1 α ). The sabiliy of he origin (, ) of sysem () depends on he locaions on he complex plane of he roos of he characerisic equaion (3). When all roos of Eq. (3) locae on he lef half-plane of complex plane, he origin (, ) of sysem () is sable; oherwise, i is unsable. Noe ha when τ =, (4) becomes solve Eq. (5), hen he roos of (5) are given by λ + pλ + q α 1 α 1 =, (5) λ 1, = p ± p 4(q α 1 α 1 ). Thus, one can immediaely obain he following resul.

3 Neural nework model wih wo delays 177 Lemma.1 Assume ha (H1) p>, q α 1 α 1 <. Then all he roos of Eq. () wih τ =have always negaive real pars. Nex, we shall invesigae he disribuion of roos of Eq.(4) wih τ>. Firs noe ha, under condiion (H1), Eq.(4) has no zero roo. Nex we shall look for he possibiliy of occurrence of a pair pure imaginary roos. Obviously, iω(ω >) is a roo of (4) if and only if ω saisfies he following equaion ω + ωpi + q α 1 α 1 (cos ωτ i sin ωτ) =. (6) Separaing he real and imaginary pars of (6) gives he following equaions ω + q = α 1 α 1 cos ωτ, (7) ωp = α 1 α 1 sin ωτ. By some simple calculaions, i is easy o obain ω 4 +(p q)ω + q (α 1 α 1 ) =, (8) and an ωτ = pω q ω. (9) I is easy o see ha he firs equaion of (8) has only one posiive roo ω = [ q p + ] 1 (q p ) 4[q (α 1 α 1 ) ] (1) provided ha he following assumpion (H) q p <, q (α 1 α 1 ) < is saisfied. From equaion (9), we define ) τ j = 1 ω (arcan pω + πj,j =, 1,,, (11) q ω hen Eq. (4) wih τ = τ j has a pair of purely imaginary roos ±iω. Since he roos of Eq. (4) coninuously depend on he parameer τ, summarizing he above remarks and combining Lemma.1, he following resul holds. Lemma. Suppose ha (H1) and (H) hold, hen (i) If τ [,τ ), hen all roos of Eq. (4) have sricly negaive real pars. (ii) If τ = τ, hen Eq. (4) has a pair of purely imaginary roos ±iω and oher roos have sricly negaive real pars.

4 178 GuangPing Hu and XiaoLing Li (iii) If τ = τ j, hen Eq. (4) has a simple pair of purely imaginary roo ±iω, where τ j are defined by (11) and ω is defined by (1). Le λ(τ) =α(τ)+iω(τ) be he roos of Eq. (4) saisfying α(τ j )=,ω(τ j )=ω,j=, 1,,. Lemma.3 The following ransversaliy condiion is saisfied dreλ (τ) >. (1) dτ τ=τ j In fac, differeniaing he wo sides of (4) wih respec o τ, we can obain λλ + pλ + α 1 α 1 λe λτ =, which implies dλ dτ = α 1α 1 λe λτ. λ + p We can direcly compue ha dreλ(τ) dτ τ=τ j = α 1α 1 ( pω sin ω τ j ω cos ω τ j ). p +4ω Under he condiion (H), by he equaion (7), we have dreλ(τ) dτ = ω (p q +ω) >. τ=τ j p +4ω As he mulipliciies of roos wih posiive real pars of Eq. (4) can change only if a roo appears on or crosses he imaginary axis as ime delay τ varies, similar o he proof of he lemma of Wei and Ruan [6], by Lemma.3, we have he following resul. Lemma.4 If τ (τ j,τ j+1 ), hen Eq. (4) has (j + 1)(j =, 1,, ) roos wih posiive real par. By Lemmas.1-.4, we have he following resul on sabiliy and bifurcaion for sysem (). Theorem.5 Assume ha (H1) and (H) hold. (i) If τ [,τ ), hen he zero soluion of sysem () is asympoically sable. (ii) If τ>τ, hen he zero soluion of sysem () is unsable. (iii) τ = τ j (j =, 1,, ) are Hopf bifurcaion values for sysem ().

5 Neural nework model wih wo delays A numerical example In his secion, we give some numerical simulaions o illusrae our resuls. As an example, we consider sysem (1) wih f( ) = anh( ), a 11 =.5, a 1 = 1.8, a 1 =1.3, a =1.7, hen (1) becomes he following sysem () = u 1 ().5 anh u 1 () 1.8 anh(u ( τ 1 )), u () = u ()+1.3 anh(u 1 ( τ ))+1.7 anh u (). (13) Fig. 1. The rajecory graph of (13) wih τ 1 =.3, τ =.8 and u 1 () =u () =., [.11, ]. By direcly calculaing, we may verify ha hypoheses (H1) and (H) hold, and τ =.1. Thus from Theorem.5 we know ha he zero soluion of sysem (13) is asympoically sable when <τ<τ =.1, (see Fig.1-Fig.). The sysem (13) also undergoes a Hopf bifurcaion a he origin (, ) when τ crosses hrough increasingly he criical value τ =.1 (see Fig.3-Fig.5) Fig.. The phase graph of (13) wih τ 1 =.3, τ =.8 and u 1 () =u () =., [.11, ].

6 173 GuangPing Hu and XiaoLing Li Fig. 3. The rajecory graph of (13) wih τ 1 =.4, τ =.9 and u 1 () =u () =., [.13, ] Fig. 4. The rajecory graph of (13) wih τ 1 =.4, τ =.9 and u 1 () =u () =.3, [.13, ]..4.3 ini ==. ini == Fig. 5. The phase graph of sysem (13) wih τ 1 =.4, τ =.9. 4 Conclusions In presen paper, we have already obained ha, under cerain condiions, he sysem (1) can undergo a Hopf bifurcaion a he zero equilibrium when τ 1 + τ akes some criical values τ j (j =, 1,,...). The dynamics of sysems similar o (1) have been invesigaed exensively and many ineresing resuls

7 Neural nework model wih wo delays 1731 have been obained (e.g. [3, 5, 7, 8, 9]). Differ from hese papers menioned, our resul in his paper is general since we do no limi he values of τ 1 and τ. In fac, for he sysem (1), he change of he values of τ 1 and τ will no affec is opological srucure. For example, he phase graph of (13) wih τ 1 =.1, τ =.1 and τ 1 =.7, τ =.6 are same as Fig. and Fig. 5 respecively. ACKNOWLEDGEMENTS. This work was suppored by he Naional Naural Science Foundaion of China (1161), and he Foundaion of Nanjing Universiy of Informaion and Technology. References [1] Cao J, Yuan K, Li H, Global asympoical sabiliy of recurren neural neworks wih muliple discree delays and disribued delays, IEEE Trans Neural Neworks, 17 (6), [] Hideaki Masunaga, Sabiliy swihches in a sysem of linear differenial equaions wih diagonal dela, Appl. Mah. Compu. 1 (9), [3] Hu G, Li W, Yan X, Hopf Bifurcaions in a Predaor-Prey Sysem wih Muliple Delays, Chaos, Solions and Fracals, 4 (9), [4] Ruan S, Wei J, On he zeros of ranscendenal funcions wih applicaions o sabiliy of delay differenial equaions wih wo delays, Dyn Coninuous Discree Impuls Sys Ser A, 1 (3), [5] Song Y, Han M, Wei J, Sabiliy and Hopf bifurcaion analysis on a simplified BAM neural nework wih delays, Physica D, (5), [6] Wei J, Ruan S, Sabiliy and bifurcaion in a neural nework model wih wo delays, Physica D. 13 (1999), [7] Hideaki Masunaga, Sabiliy swihches in a sysem of linear differenial equaions wih diagonal delay, Appl. Mah. Compu. 1 (9), [8] Hu H, Huang L, Sabiliy and Hopf bifurcaion analysis on a ring of four neurons wih delays, Appl. Mah. Compu. 13 (9), [9] Marcus C.M, Weservel R.M, Sabiliy of analog neural neworks wih delay, Physical Review A, 39 (1989), Received: December, 1