- Email: [email protected]
Bifurcation analysis in a limit cycle oscillator with delayed feedback
Chaos, Solitons and Fractals 23 (2005) 817–831 www.elsevier.com/locate/chaos
Bifurcation analysis in a limit cycle oscillator with delayed feedback q Weihua Jiang, Junjie Wei
*
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China Accepted 5 May 2004
Abstract A kind of limit cycle oscillator with delayed feedback is considered. Firstly, the linear stability is investigated. According to the analysis results, the bifurcation diagram is drawn in the parameter plane. It is found that there are stability switches for time delay, and Hopf bifurcations when time delay crosses through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results found. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Recently, there has been extensive interest in studying the effect of time delay on the collective dynamics of coupled models [1–10]. It is well know that time delay is ubiquitous in most physical, chemical, biological, neural, and other natural system due to finite propagation speeds of signals, finite processing times in synapses, and finite reaction times. We refer the reader to [11–13]. Time delayed coupling introduces interesting new features in the collective dynamics, for example, simultaneous existence of several different synchronized states [4–7], regions of amplitude death even among identical oscillators [8,9], and bi-stability between synchronized and incoherent states [9,10]. One of the markable aspects of this cooperative dynamics is that many of its salient features can be observed even in a system consisting of just two coupled oscillators [1–3,9,10]. The temporal behavior of either of the two oscillators in such a case reveals a great deal about the collective aspects of larger systems. In fact, a useful point of view to adopt is to regard each oscillator as being driven autonomously by a source term that represents the collective feedback of the rest of the system. Based on the consideration, Reddy et al. [1] proposed the following model system of an autonomously driven single limit cycle oscillator: z_ ðtÞ ða þ ix jzðtÞj2 ÞzðtÞ ¼ k1 zðt sÞ k2 z2 ðt sÞ;
ð1Þ
where z ¼ x þ iy is a complex quantity, x, the frequency of oscillation, a, real constant, s P 0 is the time delay of the autonomous feedback term, and k1 and k2 represent the strengths of the linear and nonlinear contributions of the feedback. Using a combination of analytical methods and numerical analysis, Reddy et al. investigated the temporal dynamics of system (1) in various regimes characterized by the natural parameters of the oscillator (e.g. its frequency x, linear growth rate a), strengths of the feedback components ðk1 ; k2 Þ and the time delay parameter s. Their principal
q
Supported by the National Natural Science Foundation of China. Corresponding author. E-mail address: [email protected] (J. Wei).
*
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.05.028
818
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
results are presented in the form of bifurcation diagrams in these parameter spaces, where the bifurcation diagrams are given in numerical simulation. The goal of the present paper is to study system (1) also. We shall regard a, k1 and the delay s as parameters to study the linear stability to system (1). According to the analysis results, the bifurcation diagram in ða; k1 Þ-plane is drawn. We get four curves which divide the ða; k1 Þ-plane into six regions, where one is an absolutely stable region, two regions are unstable for all s P 0, another two regions are conditionally stable, and the other is a conditionally stable region in which it is found that there exist stability switches. Hopf bifurcation occurs when stability is lost. The nature questions are what is the direction of the bifurcation, and how about the stability of the bifurcating periodic solutions? It is important to answer these questions for studying the dynamics of differential systems. The normal form theory and center manifold theorem, introduced by Hassard et al. [14], are very useful for answering these questions. Recently, there are several papers, for example [15–19], to investigate the properties of Hopf bifurcation in delayed differential equations employing the method. Here, we also use the normal form method and center manifold theorem due to Hassard et al. [14] to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. We would like to mention that, for system (1), the directions of the Hopf bifurcations and the stability of the bifurcating periodic solutions on the center manifolds are given precisely. The remainder of the paper is organized as follows. In Section 2, the linear stability is studied, and the bifurcation set is drawn on ða; k1 Þ-plane. In Section 3, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions on the center manifold are determined, using normal form method and center manifold introduced by Hassard et al. [14]. Finally, some numerical simulations are performed to illustrate the analytical results found.
2. Local stability and Hopf bifurcation In this section, we shall employ the result due to Ruan and Wei [20] to study the distribution of the eigenvalues of the linearization of system (1), and hence, get the stability of the fixed point and the existence of Hopf bifurcations. Let z ¼ x þ iy. Then Eq. (1) becomes x_ ðtÞ ¼ ða x2 ðtÞ y 2 ðtÞÞxðtÞ xyðtÞ k1 xðt sÞ k2 ðx2 ðt sÞ y 2 ðt sÞÞ; ð2Þ _ ¼ xxðtÞ þ ða x2 ðtÞ y 2 ðtÞÞyðtÞ k1 yðt sÞ 2k2 xðt sÞyðt sÞ: yðtÞ The linearization of Eq. (2) around the origin ð0; 0Þ is given by x_ ðtÞ ¼ axðtÞ xyðtÞ k1 xðt sÞ; _ ¼ xxðtÞ þ ayðtÞ k1 yðt sÞ: yðtÞ
ð3Þ
Its characteristic equation is ðk a þ k1 eks Þ2 þ x2 ¼ 0;
ð4Þ
that is k a þ k1 eks ¼ ix:
ð5Þ
Clearly, we have the following for Eq. (5). Lemma 2.1. Assume s ¼ 0. Then the couple of roots of Eq. (5) have negative real parts when k1 > a, and have positive real parts when k1 < a, as well as are purely imaginary when k1 ¼ a. Let ib ðb > 0Þ be a root of Eq. (5), then b solves a ¼ k1 cosðbsÞ; b x ¼ k1 sinðbsÞ: It follows that a2 þ ðb xÞ2 ¼ k12 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and hence b ¼ x k12 a2 .
ð6Þ
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
819
From the above discussion and Lemma 2.1, we can easily obtain the following results. Lemma 2.2. (1) Suppose k12 < a2 is satisfied. Then all the roots of Eq. (5) have negative real parts when a < k1 < a, and Eq. (5) has exactly a couple of roots with positive real parts when a < k1 < a. (2) Suppose k12 > a2 is satisfied. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ii(i) If x < k12 a2 , then Eq. (5) has a pair of imaginary roots ib when s ¼ s j , respectively, where b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þ k12 a2 , and i 8 h 1 a > 2jp þ arccos ; k1 > a P 0; > k1 > i > b h > > 1 a < ð2j þ 1Þp þ arccos k1 ; k1 < a 6 0; b h i s ð7Þ j ¼ 1 > ð2j þ 1Þp arccos ka1 ; k1 > a > 0; > > b > h i > > : 1 2ðj þ 1Þp arccos a ; k < a < 0; j ¼ 0; 1; . . . b
k1
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i(ii) If xp >ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 ffi a2 , then Eq. (5) has a pair 2 x k1 a2 , sþ j is defined by (7), and i 8 h 1 a > ; k1 2ðj þ 1Þp arccos > b k1 > > > h i > > > < b1 ð2j þ 1Þp arccos ka1 ; k1 ¼ s h i j > 1 > ð2j þ 1Þp þ arccos ka1 ; k1 > b > > > h i > > : 1 2jp þ arccos a ; k1 b k1
of imaginary roots ib when s ¼ s j , respectively, where b ¼ > a P 0; < a 6 0; ð8Þ > a > 0; < a < 0; j ¼ 0; 1; . . .
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 and (iii) If x ¼ k12 a2 , then Eq. (5) has a pair of imaginary roots ibþ when s ¼ sþ j , where bþ ¼ 2x ¼ 2 þ sj is defined by (7). (3) If k12 ¼ a2 , then Eq. (5) has a pair of imaginary roots ib (b ¼ x) when s ¼ sj , where sj ¼ 2jp ðj ¼ 0; 1; . . .Þ when x k1 ¼ a, and sj ¼ ð2jþ1Þp ðj ¼ 0; 1; . . .Þ when k1 ¼ a, respectively. x Let k ¼ aðsÞ þ ibðsÞ be the root of Eq. (5) satisfying aðs j Þ ¼ 0, and bðsj Þ ¼ b .
Lemma 2.3. If k12 > a2 , then a0 ðsþ j Þ > 0; and a0 ðs j Þ > 0; a0 ðs j Þ < 0;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi when x > k12 a2 ; j ¼ 0; 1; . . . when x <
Proof. Substituting kðsÞ into Eq. (5) and taking the derivative with respect to s, we get dkðsÞ k1 keks : ¼ 1 k1 seks ds Then " # da dkðsÞ ik1 b cosðbsÞ þ k1 b sinðbsÞ k1 b sinðbsÞ þ ik1 b sinðbsÞ ¼ Re ¼ Re ¼ Re ds a¼0 ds a¼0 1 k1 s cosðbsÞ þ ik1 s sinðbsÞ ð1 k1 s cosðbsÞÞ2 þ ðk1 s sinðbsÞÞ2 ¼
k1 b sinðbsÞ ; D
where D ¼ ð1 k1 s cosðbsÞÞ2 þ ðk1 s sinðbsÞÞ2 .
820
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
From (6), we have sinðbþ sþ j Þ and
b x ¼ þ ¼ k1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 k1
8 pffiffiffiffiffiffiffiffiffi < b þx ¼ k12 a2 ; k1 pk1ffiffiffiffiffiffiffiffiffi sinðb s j Þ ¼ : b x k12 a2 ¼ ; k1 k1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x > k12 a2 ; j ¼ 0; 1; . . .
x<
Hence, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi da bþ k12 a2 ¼ >0 ds s¼sþ D j
and
8 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < b k12 a2 da > 0; when x < k12 a2 ; Dpffiffiffiffiffiffiffiffiffi ¼ ffi ds s¼s : b k12 a2 < 0; when x > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 ; j ¼ 0; 1; . . . j D
This completes the proof.
For the convenience, we make the following assumptions. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a p k12 a2 ; < ðH1 Þ a2 þ x2 < k1 < a 6 0; and arccos k1 x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a ðH2 Þ a2 þ x2 < k1 < a < 0; and x k12 a2 < arccos : p k1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lemma 2.4. (1) If a2 þ x2 6 k1 , then all roots of Eq. (5) have negative real parts when s 2 ð0; sþ 0 Þ, and Eq. (5) has at least þ a pair of roots with positive real parts when s > sþ , as well as all roots of Eq. (5) with s have negative real parts except the 0 0 pair of imaginary roots ibþ . pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ (2) If jaj < k1 < a2 þ x2 , then there exists a integer m P 0 such that 0 < sþ 0 < s0 < < sm1 < þ þ þ sm1 < sm < smþ1 < sm , and all roots of Eq. (5) have negative real parts when s 2 ðsj1 ; sj Þ, for j ¼ 0; 1; . . . ; m with þ s 1 ¼ 0; Eq. (5) has a pair of roots with positive real parts when s 2 ðsj ; sj Þ for j ¼ 0; 1; . . . ; m 1; and Eq. (5) has at least a pair of roots with positive real parts when s > sþ , as well as all roots of Eq. (5) with s ¼ s m j (j ¼ 0; 1; . . . ; m for the superindex +, and j ¼ 0; 1; . . . ; m 1 for the super-index ) have negative real parts except the imaginary roots ib , respectively. (3) (i) If either ðH1 Þ or ðH2 Þ is satisfied, then Eq. (5) has at least a pair of roots with positive real parts for all s P 0. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ii) Suppose a2 þ x2 < k1 < jaj is satisfied, and neither of ðH1 Þ and ðH2 Þ holds. Then there exists a integer þ þ þ m P 0 such that 0 < s 0 < s0 < < sm < sm < smþ1 < smþ1 , and Eq. (5) has a pair of roots with positive real þ parts when s 2 ðsþ ; s Þ, for j ¼ 0; 1; . . . ; m with s ¼ 0; and all roots of Eq. (5) have negative real parts when j1 j 1 þ s 2 ðs ; s Þ, for j ¼ 0; 1; . . . ; m; and Eq. (5) has at least a pair of roots with positive real parts when s > sþ j j m ; as well as all roots of Eq. (5) with s ¼ sj , j ¼ 0; 1; . . . ; m have negative real parts except the imaginary roots ib , respectively. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4) If k1 6 a2 þ x2 , then Eq. (5) has at least a pair of roots with positive real parts for all s P 0. Proof. We only verify the case of a P 0. The situation a < 0 is similar to a P 0. (1) By Lemma 2.2 we have sþ 0 ¼
arccosða=k1 Þ arccosða=k1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ s 0 k12 a2 x þ k12 a2
xþ
and a0 ðs j Þ > 0, j ¼ 0; 1; . . . By Lemma 2.1 we know that the couple of roots of Eq. (5) with s ¼ 0 have negative real parts. Applying Corollary 2.4 due to Ruan and Wei [20], the conclusion of (1) follows.
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
821
(2) By Lemma 2.2 we have 1 a p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ; ¼ 2ðj þ 1Þp arccos j k1 x k12 a2 1 a p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sþ ; j ¼ 0; 1; . . . ; ¼ 2jp þ arccos j 2 2 k 1 x þ k1 a which yield sþ 0 ¼
arccosða=k1 Þ 2p arccosða=k1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ s 0 2 2 x þ k1 a x k12 a2
and Msþ j ¼
2p 2p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ms j : 2 2 x þ k1 a x k12 a2
þ þ þ Hence there exists an integer m P 0 such that 0 < sþ 0 < s0 < < sm1 < sm1 < sm < smþ1 < sm . Notice that the 0 þ 0 couple of roots of Eq. (5) with s ¼ 0 have negative real parts, and a ðsj Þ > 0, a ðsj Þ < 0, for j ¼ 0; 1; . . . ; the conclusion of (2) follows. (3) By Lemma 2.2 we have 1 a ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p s ð2j þ 1Þp arccos ; ¼ j k1 x k12 a2 1 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2j þ 1Þp þ arccos sþ ; j ¼ 0; 1; . . . ; j ¼ k1 x þ k12 a2
which imply that 2p 2p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ms j : k12 a2 x k12 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 þ 0 Hence, arccosð ka1 Þ < xp k12 a2 ensures that sþ 0 < s0 . By a ðsj Þ > 0, a ðsj Þ < 0, for j ¼ 0; 1; . . . ; and the couple of roots of Eq. (5) with s ¼ 0phave positive real parts, the conclusion of (i) follows. ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ þ From arccosð ka1 Þ > xp k12 a2 , we have s 0 < s0 . Notice Msj < Msj , similar to the proof of (2), we can get the conclusion of (ii). (4) Notice that the couple of roots of Eq. (5) with s ¼ 0 have positive real parts, and a0 ðs j Þ > 0 for j ¼ 0; 1; . . . ; the conclusion of (iv) follows. The proof is complete. Msþ j ¼
xþ
From Lemmas 2.1–2.4 and applying the Hopf bifurcation theorem for functional differential equations [11, Chapter 11, Theorem 1.1], we have the following results on stability and bifurcation to system (2). Theoremp2.5. Forffi system (2), ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1) if a2 þ x2 6 k1 , then system (2) undergoes a Hopf bifurcation at the origin ð0; 0Þ when s ¼ s j ; j ¼ 0; 1; . . . , as well þ þ as the zero solution p isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi asymptotically stable when s 2 ð0; s Þ, and is unstable when s > s ; 0 0 ffi (2) if jaj < k1 < a2 þ x2 , then system (2) undergoes a Hopf bifurcation at the origin ð0; 0Þ when sm¼ s ; j ¼ 0; 1; . . . S j þ Particularly, there exists an integer m P 0 such that the zero solution is asymptotically stable when s 2 ðs j1 ; sj Þ, and is m1 S þ S þ j¼0 unstable when s 2 ðsj ; sj Þ ðsm ; þ1Þ; j¼0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3) if a2 þ x2 < k1 < jaj, then system (2) undergoes a Hopf bifurcation at the origin ð0; 0Þ when s ¼ s j ; j ¼ 0; 1; . . . , and (i) if either ðH1 Þ or ðH2 Þ is satisfied, then the zero solution is unstable for all s P 0; (ii) if neither of ðH1 Þ and ðH2 Þ is satisfied, then there exists an integer m P 0 such that the zero solution is unstable when m m S S þ S þ s 2 ðsþ ðsm ; þ1Þ, and asymptotically stable when s 2 ðs j1 ; sj Þ j ; sj Þ; j¼0 j¼0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4) if k1 6 a2 þ x2 , then system (2) undergoes a Hopf bifurcation at the origin ð0; 0Þ when s ¼ s j , j ¼ 0; 1; . . ., and the zero solution is unstable for all s P 0;
822
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 1. The curves k1 ¼ a2 þ x2 and k1 ¼ a divide the ða; k1 Þ-plane into six regions, reads D1 ; D2 ; D3 ; D4 ; D5 and D6 . D1 and D2 are conditionally stable regions, and there are stability switches for the parameters being locating in the region D2 ; D3 is a conditionally stable region also under certain conditions; D5 is an absolutely stable region; and D4 and D6 are unstable regions for all s P 0.
(5) if a < k1 < a, then the zero solution is asymptotically stable for all s P 0; (6) if a < k1 < a, then the zero solution is unstable for all s P 0. According to the conclusions of Theorem 2.5, we can draw the bifurcation diagram in the parameter plane as in Fig. 1. 3. Direction and stability of Hopf bifurcation In this section we shall study the direction of the Hopf bifurcations, the stability and the period of the bifurcating periodic solutions. The method we used is based on the normal form method and the center manifold theory introduced by Hassard et al. [14]. We first re-scale the time by t 7! ðt=sÞ to normalize the delay so that system (2) can be written as the form x_ ðtÞ ¼ sða x2 ðtÞ y 2 ðtÞÞxðtÞ sxyðtÞ sk1 xðt 1Þ sk2 ðx2 ðt 1Þ y 2 ðt 1ÞÞ; ð9Þ _ ¼ sxxðtÞ þ sða x2 ðtÞ y 2 ðtÞÞyðtÞ sk1 yðt 1Þ 2sk2 xðt 1Þyðt 1Þ: yðtÞ The linearization around the origin ð0; 0Þ is given by x_ ðtÞ ¼ saxðtÞ sxyðtÞ sk1 xðt 1Þ; _ ¼ sxxðtÞ þ sayðtÞ sk1 yðt 1Þ; yðtÞ and the nonlinear term is sðx2 ðtÞ þ y 2 ðtÞÞxðtÞ sk2 ðx2 ðt 1Þ y 2 ðt 1ÞÞ : f ¼ sðx2 ðtÞ þ y 2 ðtÞÞyðtÞ 2sk2 xðt 1Þyðt 1Þ
ð10Þ
ð11Þ
The characteristic equation associated with Eq. (10) is v sa þ sk1 ev ¼ isx:
ð12Þ
Comparing Eq. (12) with Eq. (5), one can find out that v ¼ sk, and hence, Eq. (12) has a pair of imaginary roots is j b when s ¼ s j for j ¼ 0; 1; . . . ; and the transversal condition holds. Let s ¼ s0 þ l, l 2 R and s0 be taken in fsþ j g [ fsj g. Then l ¼ 0 is the Hopf bifurcation value for Eq. (9). Let is0 b0 be the root of Eq. (12) with s ¼ s0 , where either b0 ¼ bþ or b0 ¼ b . For u 2 Cð½1; 0; R2 Þ, let ðs0 þ lÞk1 ðs0 þ lÞa ðs0 þ lÞx 0 uð0Þ þ uð1Þ; ð13Þ Ll u ¼ x ðs0 þ lÞa 0 ðs0 þ lÞk1
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
823
and f ðl; uÞ ¼
ðs0 þ lÞ½ðu21 ð0Þ þ u22 ð0ÞÞu1 ð0Þ þ k2 ðu21 ð1Þ u22 ð1ÞÞ : ðs0 þ lÞ½ðu21 ð0Þ þ u22 ð0ÞÞu2 ð0Þ 2k2 u1 ð1Þu2 ð1Þ
ð14Þ
By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions gðh; lÞ in h 2 ½1; 0 such that Z 0 Ll u ¼ dgðh; lÞuðhÞ for u 2 C: ð15Þ 1
In fact, we choose
x dðhÞ þ k1 ðs0 þ lÞIdðh þ 1Þ; a 1; h ¼ 0 where I is the identity matrix and dðhÞ ¼ . Then Eq. (15) is satisfied. 0; h 6¼ 0 1 2 For u 2 C ð½1; 0; R Þ, define duðhÞ=dh; h 2 ½1; 0Þ; AðlÞu ¼ R 0 dgðt; lÞuðtÞ; h ¼ 0; 1 gðh; lÞ ¼ ðs0 þ lÞ
a x
ð16Þ
and RðlÞu ¼
0; f ðl; uÞ;
h 2 ½1; 0Þ; h ¼ 0:
ð17Þ
Hence, we can rewrite Eq. (9) in the following form: u_ t ¼ AðlÞut þ RðlÞut ;
ð18Þ
where u ¼ ðx; yÞT ; ut ¼ uðt þ hÞ for h 2 ½1; 0. For w 2 C 1 ð½0; 1; R2 Þ, define dwðsÞ=ds; s 2 ð0; 1; A wðsÞ ¼ R 0 wðtÞ dgðt; 0Þ; s ¼ 0: 1 For u 2 C½1; 0 and w 2 C½0; 1, define the bilinear form Z 0 Z h hÞ dgðhÞ uðnÞ dn; wðn hw; ui ¼ wð0Þuð0Þ 1
ð19Þ
ð20Þ
n¼0
where gðhÞ ¼ gðh; 0Þ. Then A and Að0Þ are adjoint operators, and is0 b0 are eigenvalues of Að0Þ. Thus, they are also eigenvalues of A . By direct computation, we obtain that i is0 b0 h qðhÞ ¼ e 1 is the eigenvector of Að0Þ corresponding to is0 b0 , and q ðsÞ ¼ Dði; 1Þeis0 b0 s is the eigenvector of A corresponding to is0 b0 . Moreover, hq ; qi ¼ 1;
hq ; qi ¼ 0;
where D ¼ ð2 2s0 k1 eis0 b0 Þ1 : Using the same notation as in Hassard et al. [14], we first compute the coordinates to describe the center manifold C0 at l ¼ 0. Let ut be the solution of Eq. (9) when l ¼ 0.
824
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
Define zðtÞ ¼ hq ; ut i;
wðt; hÞ ¼ ut ðhÞ 2RefzðtÞqðhÞg:
On the center manifold C0 we have wðt; hÞ ¼ wðzðtÞ; zðtÞ; hÞ; where wðz; z; hÞ ¼ w20 ðhÞ
z2 z2 þ w11 ðhÞzz þ w02 ðhÞ þ ; 2 2
q . Note that w is real if ut is real. We z and z are local coordinates for center manifold C0 in the direction of q and consider only real solutions. For solution ut in C0 of Eq. (9), since l ¼ 0, z_ ðtÞ ¼ is0 b0 z þ hq ðhÞ; f ðw þ 2RefzðtÞqðhÞgi def
¼ is0 b0 z þ q ð0Þf ðwðz; z; 0Þ þ 2RefzðtÞqð0ÞgÞ¼ is0 b0 z þ q ð0Þf0 ðz; zÞ:
ð21Þ
We rewrite this as z_ ðtÞ ¼ is0 b0 zðtÞ þ gðz; zÞ;
ð22Þ
where f0 ðz; zÞ ¼ fz2
z2 z2 z2z þ ; þ fz2 þ fzz zz þ fz2z 2 2 2
gðz; zÞ ¼ q ð0Þf ðwðz; z; 0Þ þ 2RefzðtÞqð0ÞgÞ ¼ g20
ð23Þ z2 z2 z2z þ þ g11 zz þ g02 þ g21 2 2 2
ð24Þ
Comparing the coefficients of (21) and (22), noticing (24), we have g20 ¼ q ð0Þfz2 ; g11 ¼ q ð0Þfzz ;
ð25Þ
g02 ¼ q ð0Þfz2 ; g21 ¼ q ð0Þfz2z : By (18) and (22), it follows that Aw 2Refq ð0Þf0 qðhÞg; w_ ¼ u_t z_ q z_ q ¼ Aw 2Refq ð0Þf0 qð0Þg þ f0 ;
h 2 ½1; 0Þ def ¼ Aw þ H ðz; z; hÞ; h ¼ 0;
ð26Þ
where H ðz; z; hÞ ¼ H20 ðhÞ
z2 z2 þ H11 ðhÞzz þ H02 ðhÞ þ : 2 2
ð27Þ
Expanding the above series and comparing the coefficients, we obtain ðA 2is0 x0 IÞw20 ðhÞ ¼ H20 ðhÞ;
Aw11 ðhÞ ¼ H11 ðhÞ; . . .
Notice that q ð0Þ ¼ Dði; 1Þ; xðtÞ ¼ iz iz þ wð1Þ ðz; z; 0Þ; yðtÞ ¼ z þ z þ wð2Þ ðz; z; 0Þ; xðt 1Þ ¼ ieis0 b0 z ieis0 b0 z þ wð1Þ ðz; z; 1Þ;
ð28Þ
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
825
and yðt 1Þ ¼ eis0 b0 z þ eis0 b0 z þ wð2Þ ðz; z; 1Þ; where ðjÞ
wðjÞ ðz; z; 0Þ ¼ w20 ð0Þ
z2 z2 ðjÞ ðjÞ þ w11 ð0Þzz þ w02 ð0Þ þ ; 2 2
ðjÞ
wðjÞ ðz; z; 1Þ ¼ w20 ð1Þ
z2 z2 ðjÞ ðjÞ þ w11 ð1Þzz þ w02 ð1Þ þ ; 2 2
j ¼ 1; 2;
and def
f0 ¼ fðs0 ; ut Þ ¼
s0 ½ðx2 ðtÞ þ y 2 ðtÞÞxðtÞ þ k2 ðx2 ðt 1Þ y 2 ðt 1ÞÞ ; s0 ½ðx2 ðtÞ þ y 2 ðtÞÞyðtÞ þ 2k2 xðt 1Þyðt 1Þ
gðz; zÞ ¼ q ð0Þf0 ; we have fzz ¼ 0 and 2is0 b0 ; g20 ¼ 8ik2 s0 De g11 ¼ g02 ¼ 0;
ð29Þ
ð2Þ is0 b0 þ k2 wð1Þ þ ik2 w11 ð1Þeis0 b0 : g21 ¼ 8s0 D½2 11 ð1Þe
From (16), it follows that ( dw11 ðhÞ ; Aw11 ðhÞ ¼ R 0dh dgðtÞw 11 ðtÞ; 1
h 2 ½1; 0Þ; h ¼ 0;
and (26) and (27) yield Refq ð0Þf0 qðhÞg ¼ gqðhÞ gqðhÞ; H ðz; z; hÞ ¼ qð0Þ þ f0 ; Refq ð0Þf0 qð0Þg þ f0 ¼ gqð0Þ g Comparing the coefficients with (27), we get g11 qðhÞ g11 qðhÞ; h 2 ½1; 0Þ; H11 ðhÞ ¼ g11 qð0Þ g11 qð0Þ þ fzz ; h ¼ 0: From Eqs. (28), (30) and (31), we derive w011 ðhÞ ¼ g11 qðhÞ þ g11 qðhÞ;
h 2 ½1; 0Þ:
Notice g11 ¼ 0, we obtain w11 ðhÞ ¼ E2 ; where E2 is constant. From (28), (30) and (31) when h ¼ 0 we have Z 0 dgðhÞE2 ¼ fzz : 1
Hence, E2 ¼ 0 follows from fzz ¼ 0. Thus w11 ðhÞ 0: Consequently, from (29), g21 ¼ 16s0 D: Substituting g11 ; g20 ; g02 and g21 into i 1 g21 2 2 g20 g11 2jg11 j jg02 j þ C1 ð0Þ ¼ 2s0 b0 3 2
ð30Þ
h 2 ½1; 0Þ; h ¼ 0:
ð31Þ
826
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
gives C1 ð0Þ ¼ 8s0 D: Note ¼ D
1 1 ¼ ð1 s0 k1 cos s0 b0 is0 k1 sin s0 b0 Þ 2 2s0 k1 eis0 b0 B
and a ¼ k1 cos s0 b0 , we have Re C1 ð0Þ ¼
8s0 8s0 ð1 s0 k1 cos s0 b0 Þ ¼ ð1 s0 aÞ; B B
where B ¼ 2ð1 þ s20 k12 2s0 k1 cos s0 b0 Þ > 0: Therefore, 16s0 < 0; 1 s0 a > 0; ð1 s0 aÞ b2 ¼ 2RefC1 ð0Þg ¼ > 0; 1 s0 a < 0; B and
8 > > 0; > > > > > < 0; > > > RefC1 ð0Þg < > 0; l2 ¼ a0 ðs0 Þ > < 0; > > > > > < 0; > > > : > 0;
s0 ¼ s j ; 1 s0 a > 0; s0 ¼ s j ; 1 s0 a < 0; s0 ¼ sþ j ; 1 s0 a > 0; s0 ¼ sþ j ; 1 s0 a < 0; s0 ¼ s j ; 1 s0 a > 0; s0 ¼ s j ; 1 s0 a < 0;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jk1 j P a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jk1 j P a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jaj < jk1 j < a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jaj < jk1 j < a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jaj < jk1 j < a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jaj < jk1 j < a2 þ x2 :
ð32Þ
ð33Þ
By the general theory due to Hassard et al. [14], we know that the quantity of b2 determines the stability of the bifurcating periodic solutions on the center manifold, and l2 determines the direction of the bifurcation. So we have the following from (32) and (33). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Theorem 3.1. (i) If jk1 j P a2 þ x2 and s j < a, then the Hopf bifurcation at the origin when s ¼ sj is supercritical and the bifurcating periodic solutions are orbitally asymptotically stable. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 (ii) If jk1 j P a2 þ x2 and s j > a, then the Hopf bifurcation at the origin when s ¼ sj is subcritical and the bifurcating periodic solutions are unstable. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ (iii) If jaj < jk1 j < a2 þ x2 and sþ j < a, then the Hopf bifurcation at the origin when s ¼ sj is supercritical and the bifurcating periodic solutions are stable. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi orbitally asymptotically 1 þ (iv) If jaj < jk1 j < a2 þ x2 and sþ j > a, then the Hopf bifurcation at the origin when s ¼ sj is subcritical and the bifurcating periodic solutions are unstable. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 (v) If jaj < jk1 j < a2 þ x2 and s j < a, then the Hopf bifurcation at the origin when s ¼ sj is subcritical and the bifurcating periodic solutions are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi orbitally 1asymptotically stable. (vi) If jaj < jk1 j < a2 þ x2 and s j > a, then the Hopf bifurcation at the origin when s ¼ sj is supercritical and the bifurcating periodic solutions are unstable. Remark. Theorem 3.1 describes the dynamics of Eq. (2) restricted to the center manifolds. Of course, the existence of the periodic solutions of Eq. (2) is the same as its restriction on the center manifold. Usually, the bifurcating periodic solutions of Eq. (2) from the origin are unstable when the characteristic equation associates with the linearization of Eq. (2) around the origin has at least one root with a positive real part at the critical value. But, when all the roots of the characteristic equation with the bifurcation value have negative real parts except the pair of imaginary roots, the stabilities of the bifurcating periodic solutions are the same for Eq. (2) and its restriction to the center manifolds. So, from Lemma 2.4 and Theorem 3.1, we have the following: For Eq. (2), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1) if a2 þ x2 6 k1 , then 1 i(i) the direction of the Hopf bifurcation at the origin when s ¼ sþ 0 < a is supercritical and the bifurcating periodic solutions are orbitally asymptotically stable; 1 (ii) the direction of the Hopf bifurcation at the origin when s ¼ sþ 0 > a is subcritical and the bifurcating periodic solutions are unstable;
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
827
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2) if jaj < k1 < a2 þ x2 , then þ 1 ii(i) the directions of the Hopf bifurcations at the origin when s ¼ sþ j ðj ¼ 0; 1; . . . ; mÞ and sj < a are supercritical and the bifurcating periodic solutions are orbitally asymptotically stable; þ 1 i(ii) the directions of the Hopf bifurcations at the origin when s ¼ sþ j ðj ¼ 0; 1; . . . ; mÞ and sj > a are subcritical and the bifurcating periodic solutions are unstable; 1 (iii) the directions of the Hopf bifurcations at the origin when s ¼ s j ðj ¼ 0; 1; . . . ; m 1Þ and sj < a are subcritical and the bifurcating periodic solutions are orbitally asymptotically stable; 1 (iv) the directions of the Hopf bifurcations at the origin when s ¼ s j ðj ¼ 0; 1; . . . ; m 1Þ and sj > a are supercritical and ffithe bifurcating periodic solutions are unstable; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p (3) if a2 þ x2 < k1 < jaj, and neither of ðH1 Þ and ðH2 Þ holds, then þ 1 iii(i) the directions of the Hopf bifurcations at the origin when s ¼ sþ j ðj ¼ 0; 1; . . . ; mÞ and sj < a are supercritical and the bifurcating periodic solutions are orbitally asymptotically stable; þ 1 ii (ii) the directions of the Hopf bifurcations at the origin when s ¼ sþ j ðj ¼ 0; 1; . . . ; mÞ and sj > a are subcritical and the bifurcating periodic solutions are unstable; 1 (iii) the directions of the Hopf bifurcations at the origin when s ¼ s j ðj ¼ 0; 1; . . . ; mÞ and sj < a are subcritical and the bifurcating periodic solutions are orbitally asymptotically stable; 1 i(iv) the directions of the Hopf bifurcations at the origin when s ¼ s j ðj ¼ 0; 1; . . . ; mÞ and sj > a are supercritical and the bifurcating periodic solutions are unstable.
4. Numerical simulations In this section, we present some numerical results to system (2) at different values of a, x, ki (i ¼ 1; 2) and s. We choose k2 ¼ 1; x ¼ 1, then system (2) is given by x_ ðtÞ ¼ axðtÞ yðtÞ k1 xðt sÞ x3 ðtÞ xðtÞy 2 ðtÞ x2 ðt sÞ þ y 2 ðt sÞ; ð34Þ y_ ðtÞ ¼ xðtÞ þ ayðtÞ k1 yðt sÞ x2 ðtÞyðtÞ y 3 ðtÞ 2xðt sÞyðt sÞ: By Eqs. (7), (8), (32) and (33) we can calculate l2 and b2 so that the dynamical behavior of system (34) will be determined. The results are given in Table 1. We choose k1 ¼ 1, a ¼ 2 ðða; k1 Þ 2 D5 Þ and a ¼ 2 ðða; k1 Þ 2 D6 Þ. By Theorem 2.5 the zero solution of Eq. (34) is asymptotically stable and unstable, respectively. The corresponding waveform and phase plots are shown in Figs. 2 and 3, respectively, where s ¼ 1. We choose a ¼ 2, k1 ¼ 2:5 ðða; k1 Þ 2 D4 Þ. By Theorem 2.5 the zero solution of Eq. (34) is unstable. One finds out that l2 > 0 and b2 < 0 from Table 1. Hence, by Theorem 3.1 we know that the direction of the bifurcation at the critical value s0 ¼ 2:26 is supercritical, and the bifurcating periodic solutions are stable, as shown in Fig. 4, where s ¼ 2:4. We choose a ¼ 2, k1 ¼ 2:5 ðða; k1 Þ 2 D1 Þ. With these parameters, sþ 0 ¼ 1:0033, l2 > 0; and b2 < 0 are found from Table 1. Hence, by Theorem 2.5 the zero solution of Eq. (34) is asymptotically stable when s 2 ð0; 1:0033Þ, as shown on the first line in Fig. 5, where s ¼ 0:5, and unstable when s 2 ð1:0033; 1Þ. By Theorem 3.1 we know that the direction of
Table 1 Parameter values at different quantities of a and k1 when x ¼ 1 and k2 ¼ 1 a
k1
s0
)2 )2 )1
)2.5 2.5 1.09
)1
)1.09
)0.5
)1
sþ 0 sþ 0 sþ 0 s 0 sþ 1 sþ 2 s 1 s 0 sþ 0 sþ 1 s 1 sþ 0 s 0
¼ 2:26 ¼ 1:0033 ¼ 2:1010 ¼ 5:0741 ¼ 6:9342 ¼ 11:7674 ¼ 14:0501 ¼ 0:5861 ¼ 4:5176 ¼ 9:3508 ¼ 9:5621 ¼ 2:806 ¼ 7:8167
sign(l2 )
sign(b2 )
) ) ) + )
+ + + ) +
) )
) +
)
+
828
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
Fig. 2. Waveform plot and phase for system (34) with a ¼ 2; k1 ¼ 1 and s ¼ 1. The equilibrium ð0; 0Þ is asymptotically stability.
Fig. 3. Waveform plot and phase for system (34) with a ¼ 2; k1 ¼ 1 and s ¼ 1. The equilibrium ð0; 0Þ is unstable.
the bifurcation at the critical value s0 ¼ 1:0033 is supercritical, and the bifurcating periodic solutions are stable, as shown on the last line in Fig. 5, where s ¼ 1:1. We choose a ¼ 1, k1 ¼ 1:09 ðða; k1 Þ 2 D2 Þ. With these parameters, from Table 1 and by Theorem 2.5 one can get that the zero solution of Eq. (34) is asymptotically stable when s 2 ð0; 2:1010Þ [ ð5:0741; 6:9342Þ, and unstable when s 2 ð2:1010; 5:0741Þ [ ð6:9342; 1Þ. This shows that the equilibrium ð0; 0Þ switches 2 times from stability to instability and unstable for any s > 6:9342. By Theorem 3.1 we know that the directions of the Hopf bifurcations at the critical values s0 ¼ 2:1010; 5:0741 and 6:9342 are supercritical, subcritical and supercritical, respectively. And the bifurcating
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
829
Fig. 4. Waveform plot and phase for system (34) with a ¼ 2; k1 ¼ 2:5 and s ¼ 2:4. s ¼ 2:4 being near to sþ 0 ¼ 2:26, a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable.
Fig. 5. Waveform plot and phase for system (34) with a ¼ 2; k1 ¼ 2:5 and s ¼ 0:5 or s ¼ 1, respectively. When s ¼ 0:5 < 1:0033 ¼ sþ 0 þ the equilibrium ð0; 0Þ is asymptotically stability. When s ¼ 1:1 > sþ 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable.
periodic solutions are stable, unstable and stable, respectively, as shown on each line from the top to the bottom in Fig. 6, where s ¼ 1; 2:3 and 6, respectively. We choose a ¼ 1, k1 ¼ 1:09 ðða; k1 Þ 2 D3 Þ. With these parameters from Table 1 and by Theorem 2.5, we conclude that the zero solution of Eq. (34) is unstable when s 2 ð0; 0:5861Þ [ ð4:5176; 1Þ, and asymptotically stable when s 2 ð0:5861; 4:5176Þ. This shows that the equilibrium ð0; 0Þ switches 1 times from stability to instability and unstable for any s > 4:5176. By Theorem 3.1 we know that the directions of the bifurcation at the critical values s0 ¼ 0:5861 and
830
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
Fig. 6. Waveform plot and phase for system (34) with a ¼ 1; k1 ¼ 1:09; and s ¼ 1; 2:3 or 6, respectively. When s ¼ 1 < 2:1010 ¼ sþ 0 þ the equilibrium ð0; 0Þ is asymptotically stability. When s ¼ 2:3 > sþ 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable. When s ¼ 5 < 5:0741 ¼ s 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is unstable. þ When s 0 < s ¼ 6 < 6:9342 ¼ s1 . The equilibrium ð0; 0Þ is asymptotically stable.
Fig. 7. Waveform plot and phase for system (34) with a ¼ 1; k1 ¼ 1:09, and s ¼ 0:5; 3 or 4.6, respectively. When s ¼ 0:5 < 0:5861 ¼ s 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable. When s ¼ 3 the þ equilibrium ð0; 0Þ is asymptotically stability. When s ¼ 4:6 > 4:5176 ¼ sþ 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable.
4.5176 are subcritical and supercritical, respectively. And the bifurcating periodic solutions are stable, as shown on each line from the top to the bottom in Fig. 7, where s ¼ 0:5; 3 and 4.6, respectively.
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
831
Fig. 8. Waveform plot and phase for system (34) with a ¼ 0:5; k1 ¼ 1; s ¼ 3. The equilibrium ð0; 0Þ is unstable.
We choose a ¼ 0:5, k1 ¼ 1 ðða; k1 Þ 2 D3 Þ. By Theorem 2.5 the zero solution of Eq. (34) is unstable for all s P 0, as shown in Fig. 8, where s ¼ 3. References [1] Reddy D, Sen A, Johnston G. Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks. Physica D 2000;144:335–57. [2] Zou S, Liao X, Yu J, Wong K. Chaos and its synchronization in two-neuron systems with discrete delays. Chaos, Solitons & Fractals 2004;21:133–42. [3] Aronson D, Ermentrout G, Koppel N. Amplitude response of coupled oscillators. Physica D 1990;41:403. [4] Schuster H, Wagnar P. Mutual entrainment of two limit cycle oscillators with time delayed coupling. Prog Theoret Phys 1989;81:939–45. [5] Niebur E, Schuster H, Kammen D. Colective frequencies and metastability in networks of limit-cycle oscillators with time delay. Phys Rev Lett 1991;67:2753–6. [6] Nakamura Y, Tominaga F, Munakata T. Clustering behavior of time-delayed nearest-neighbor coupled oscillators. Phys Rev E 1994;49:4849–56. [7] Kim S, Park SH, Ryu CS. Multistability in coupled oscillator systems with time delay. Phys Rev Lett 1997;79:2911–4. [8] Reddy D, Sen A, Jonston G. Time delay induced death in coupled limit cycle oscillators. Phys Rev Lett 1998;80:5109–12. [9] Reddy D, Sen A, Jonston G. Time delay effects on coupled limit cycle oscillators at Hopf bifurcation. Physica D 1999;129:15–34. [10] Yeung M, Strogatz S. Time delay in the Kuramoto model of coupled oscillators. Phys Rev Lett 1999;82:648–51. [11] Hale J, Verduyn Lunel S. Introduction to functional differential equations. New York: Springer Verlag; 1993. [12] Goppalsamy K. Stability and oscillations in delay differential equations of population dynamics. Dordrecht: Kluwer Academic Publishers; 1992. [13] Wu J. Introduction to neural dynamics and signal transmission delay. Berlin, New York: Walter de Gruyter; 2001. [14] Hassard BD, Kazarinoff ND, Wan YH. Theory and application of hopf bifurcation. Cambridge: Cambridge University Press; 1981. [15] Meng X, Wei J. Stability and bifurcation of mutual system with time delay. Chaos, Solitons & Fractals 2004;21:729–40. [16] Song Y, Wei J. Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos. Chaos, Solitons & Fractals 2004;22:75–91. [17] Yuan S, Song Y, Han M. Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays. Chaos, Solitons & Fractals 2004;21:1109–23. [18] Liao X, Wong K, Leung C, Wu Z. Hopf bifurcation and chaos in a single delayed neural equation with non-monotonic activation function. Chaos, Solitons & Fractals 2001;12:1352–547. [19] Li C, Liao X, Yu J. Hopf bifurcation in a prototype delayed system. Chaos, Solitons & Fractals 2004;19:779–87. [20] Ruan S, Wei J. On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn Continuous, Discrete Impuls Syst A: Math Anal 2003;10:863–74.
Bifurcation analysis in a limit cycle oscillator with delayed feedback q Weihua Jiang, Junjie Wei
*
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China Accepted 5 May 2004
Abstract A kind of limit cycle oscillator with delayed feedback is considered. Firstly, the linear stability is investigated. According to the analysis results, the bifurcation diagram is drawn in the parameter plane. It is found that there are stability switches for time delay, and Hopf bifurcations when time delay crosses through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results found. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Recently, there has been extensive interest in studying the effect of time delay on the collective dynamics of coupled models [1–10]. It is well know that time delay is ubiquitous in most physical, chemical, biological, neural, and other natural system due to finite propagation speeds of signals, finite processing times in synapses, and finite reaction times. We refer the reader to [11–13]. Time delayed coupling introduces interesting new features in the collective dynamics, for example, simultaneous existence of several different synchronized states [4–7], regions of amplitude death even among identical oscillators [8,9], and bi-stability between synchronized and incoherent states [9,10]. One of the markable aspects of this cooperative dynamics is that many of its salient features can be observed even in a system consisting of just two coupled oscillators [1–3,9,10]. The temporal behavior of either of the two oscillators in such a case reveals a great deal about the collective aspects of larger systems. In fact, a useful point of view to adopt is to regard each oscillator as being driven autonomously by a source term that represents the collective feedback of the rest of the system. Based on the consideration, Reddy et al. [1] proposed the following model system of an autonomously driven single limit cycle oscillator: z_ ðtÞ ða þ ix jzðtÞj2 ÞzðtÞ ¼ k1 zðt sÞ k2 z2 ðt sÞ;
ð1Þ
where z ¼ x þ iy is a complex quantity, x, the frequency of oscillation, a, real constant, s P 0 is the time delay of the autonomous feedback term, and k1 and k2 represent the strengths of the linear and nonlinear contributions of the feedback. Using a combination of analytical methods and numerical analysis, Reddy et al. investigated the temporal dynamics of system (1) in various regimes characterized by the natural parameters of the oscillator (e.g. its frequency x, linear growth rate a), strengths of the feedback components ðk1 ; k2 Þ and the time delay parameter s. Their principal
q
Supported by the National Natural Science Foundation of China. Corresponding author. E-mail address: [email protected] (J. Wei).
*
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.05.028
818
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
results are presented in the form of bifurcation diagrams in these parameter spaces, where the bifurcation diagrams are given in numerical simulation. The goal of the present paper is to study system (1) also. We shall regard a, k1 and the delay s as parameters to study the linear stability to system (1). According to the analysis results, the bifurcation diagram in ða; k1 Þ-plane is drawn. We get four curves which divide the ða; k1 Þ-plane into six regions, where one is an absolutely stable region, two regions are unstable for all s P 0, another two regions are conditionally stable, and the other is a conditionally stable region in which it is found that there exist stability switches. Hopf bifurcation occurs when stability is lost. The nature questions are what is the direction of the bifurcation, and how about the stability of the bifurcating periodic solutions? It is important to answer these questions for studying the dynamics of differential systems. The normal form theory and center manifold theorem, introduced by Hassard et al. [14], are very useful for answering these questions. Recently, there are several papers, for example [15–19], to investigate the properties of Hopf bifurcation in delayed differential equations employing the method. Here, we also use the normal form method and center manifold theorem due to Hassard et al. [14] to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. We would like to mention that, for system (1), the directions of the Hopf bifurcations and the stability of the bifurcating periodic solutions on the center manifolds are given precisely. The remainder of the paper is organized as follows. In Section 2, the linear stability is studied, and the bifurcation set is drawn on ða; k1 Þ-plane. In Section 3, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions on the center manifold are determined, using normal form method and center manifold introduced by Hassard et al. [14]. Finally, some numerical simulations are performed to illustrate the analytical results found.
2. Local stability and Hopf bifurcation In this section, we shall employ the result due to Ruan and Wei [20] to study the distribution of the eigenvalues of the linearization of system (1), and hence, get the stability of the fixed point and the existence of Hopf bifurcations. Let z ¼ x þ iy. Then Eq. (1) becomes x_ ðtÞ ¼ ða x2 ðtÞ y 2 ðtÞÞxðtÞ xyðtÞ k1 xðt sÞ k2 ðx2 ðt sÞ y 2 ðt sÞÞ; ð2Þ _ ¼ xxðtÞ þ ða x2 ðtÞ y 2 ðtÞÞyðtÞ k1 yðt sÞ 2k2 xðt sÞyðt sÞ: yðtÞ The linearization of Eq. (2) around the origin ð0; 0Þ is given by x_ ðtÞ ¼ axðtÞ xyðtÞ k1 xðt sÞ; _ ¼ xxðtÞ þ ayðtÞ k1 yðt sÞ: yðtÞ
ð3Þ
Its characteristic equation is ðk a þ k1 eks Þ2 þ x2 ¼ 0;
ð4Þ
that is k a þ k1 eks ¼ ix:
ð5Þ
Clearly, we have the following for Eq. (5). Lemma 2.1. Assume s ¼ 0. Then the couple of roots of Eq. (5) have negative real parts when k1 > a, and have positive real parts when k1 < a, as well as are purely imaginary when k1 ¼ a. Let ib ðb > 0Þ be a root of Eq. (5), then b solves a ¼ k1 cosðbsÞ; b x ¼ k1 sinðbsÞ: It follows that a2 þ ðb xÞ2 ¼ k12 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and hence b ¼ x k12 a2 .
ð6Þ
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
819
From the above discussion and Lemma 2.1, we can easily obtain the following results. Lemma 2.2. (1) Suppose k12 < a2 is satisfied. Then all the roots of Eq. (5) have negative real parts when a < k1 < a, and Eq. (5) has exactly a couple of roots with positive real parts when a < k1 < a. (2) Suppose k12 > a2 is satisfied. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ii(i) If x < k12 a2 , then Eq. (5) has a pair of imaginary roots ib when s ¼ s j , respectively, where b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þ k12 a2 , and i 8 h 1 a > 2jp þ arccos ; k1 > a P 0; > k1 > i > b h > > 1 a < ð2j þ 1Þp þ arccos k1 ; k1 < a 6 0; b h i s ð7Þ j ¼ 1 > ð2j þ 1Þp arccos ka1 ; k1 > a > 0; > > b > h i > > : 1 2ðj þ 1Þp arccos a ; k < a < 0; j ¼ 0; 1; . . . b
k1
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i(ii) If xp >ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 ffi a2 , then Eq. (5) has a pair 2 x k1 a2 , sþ j is defined by (7), and i 8 h 1 a > ; k1 2ðj þ 1Þp arccos > b k1 > > > h i > > > < b1 ð2j þ 1Þp arccos ka1 ; k1 ¼ s h i j > 1 > ð2j þ 1Þp þ arccos ka1 ; k1 > b > > > h i > > : 1 2jp þ arccos a ; k1 b k1
of imaginary roots ib when s ¼ s j , respectively, where b ¼ > a P 0; < a 6 0; ð8Þ > a > 0; < a < 0; j ¼ 0; 1; . . .
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 and (iii) If x ¼ k12 a2 , then Eq. (5) has a pair of imaginary roots ibþ when s ¼ sþ j , where bþ ¼ 2x ¼ 2 þ sj is defined by (7). (3) If k12 ¼ a2 , then Eq. (5) has a pair of imaginary roots ib (b ¼ x) when s ¼ sj , where sj ¼ 2jp ðj ¼ 0; 1; . . .Þ when x k1 ¼ a, and sj ¼ ð2jþ1Þp ðj ¼ 0; 1; . . .Þ when k1 ¼ a, respectively. x Let k ¼ aðsÞ þ ibðsÞ be the root of Eq. (5) satisfying aðs j Þ ¼ 0, and bðsj Þ ¼ b .
Lemma 2.3. If k12 > a2 , then a0 ðsþ j Þ > 0; and a0 ðs j Þ > 0; a0 ðs j Þ < 0;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi when x > k12 a2 ; j ¼ 0; 1; . . . when x <
Proof. Substituting kðsÞ into Eq. (5) and taking the derivative with respect to s, we get dkðsÞ k1 keks : ¼ 1 k1 seks ds Then " # da dkðsÞ ik1 b cosðbsÞ þ k1 b sinðbsÞ k1 b sinðbsÞ þ ik1 b sinðbsÞ ¼ Re ¼ Re ¼ Re ds a¼0 ds a¼0 1 k1 s cosðbsÞ þ ik1 s sinðbsÞ ð1 k1 s cosðbsÞÞ2 þ ðk1 s sinðbsÞÞ2 ¼
k1 b sinðbsÞ ; D
where D ¼ ð1 k1 s cosðbsÞÞ2 þ ðk1 s sinðbsÞÞ2 .
820
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
From (6), we have sinðbþ sþ j Þ and
b x ¼ þ ¼ k1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 k1
8 pffiffiffiffiffiffiffiffiffi < b þx ¼ k12 a2 ; k1 pk1ffiffiffiffiffiffiffiffiffi sinðb s j Þ ¼ : b x k12 a2 ¼ ; k1 k1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x > k12 a2 ; j ¼ 0; 1; . . .
x<
Hence, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi da bþ k12 a2 ¼ >0 ds s¼sþ D j
and
8 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < b k12 a2 da > 0; when x < k12 a2 ; Dpffiffiffiffiffiffiffiffiffi ¼ ffi ds s¼s : b k12 a2 < 0; when x > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k12 a2 ; j ¼ 0; 1; . . . j D
This completes the proof.
For the convenience, we make the following assumptions. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a p k12 a2 ; < ðH1 Þ a2 þ x2 < k1 < a 6 0; and arccos k1 x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a ðH2 Þ a2 þ x2 < k1 < a < 0; and x k12 a2 < arccos : p k1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lemma 2.4. (1) If a2 þ x2 6 k1 , then all roots of Eq. (5) have negative real parts when s 2 ð0; sþ 0 Þ, and Eq. (5) has at least þ a pair of roots with positive real parts when s > sþ , as well as all roots of Eq. (5) with s have negative real parts except the 0 0 pair of imaginary roots ibþ . pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ (2) If jaj < k1 < a2 þ x2 , then there exists a integer m P 0 such that 0 < sþ 0 < s0 < < sm1 < þ þ þ sm1 < sm < smþ1 < sm , and all roots of Eq. (5) have negative real parts when s 2 ðsj1 ; sj Þ, for j ¼ 0; 1; . . . ; m with þ s 1 ¼ 0; Eq. (5) has a pair of roots with positive real parts when s 2 ðsj ; sj Þ for j ¼ 0; 1; . . . ; m 1; and Eq. (5) has at least a pair of roots with positive real parts when s > sþ , as well as all roots of Eq. (5) with s ¼ s m j (j ¼ 0; 1; . . . ; m for the superindex +, and j ¼ 0; 1; . . . ; m 1 for the super-index ) have negative real parts except the imaginary roots ib , respectively. (3) (i) If either ðH1 Þ or ðH2 Þ is satisfied, then Eq. (5) has at least a pair of roots with positive real parts for all s P 0. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ii) Suppose a2 þ x2 < k1 < jaj is satisfied, and neither of ðH1 Þ and ðH2 Þ holds. Then there exists a integer þ þ þ m P 0 such that 0 < s 0 < s0 < < sm < sm < smþ1 < smþ1 , and Eq. (5) has a pair of roots with positive real þ parts when s 2 ðsþ ; s Þ, for j ¼ 0; 1; . . . ; m with s ¼ 0; and all roots of Eq. (5) have negative real parts when j1 j 1 þ s 2 ðs ; s Þ, for j ¼ 0; 1; . . . ; m; and Eq. (5) has at least a pair of roots with positive real parts when s > sþ j j m ; as well as all roots of Eq. (5) with s ¼ sj , j ¼ 0; 1; . . . ; m have negative real parts except the imaginary roots ib , respectively. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4) If k1 6 a2 þ x2 , then Eq. (5) has at least a pair of roots with positive real parts for all s P 0. Proof. We only verify the case of a P 0. The situation a < 0 is similar to a P 0. (1) By Lemma 2.2 we have sþ 0 ¼
arccosða=k1 Þ arccosða=k1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ s 0 k12 a2 x þ k12 a2
xþ
and a0 ðs j Þ > 0, j ¼ 0; 1; . . . By Lemma 2.1 we know that the couple of roots of Eq. (5) with s ¼ 0 have negative real parts. Applying Corollary 2.4 due to Ruan and Wei [20], the conclusion of (1) follows.
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
821
(2) By Lemma 2.2 we have 1 a p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ; ¼ 2ðj þ 1Þp arccos j k1 x k12 a2 1 a p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sþ ; j ¼ 0; 1; . . . ; ¼ 2jp þ arccos j 2 2 k 1 x þ k1 a which yield sþ 0 ¼
arccosða=k1 Þ 2p arccosða=k1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ s 0 2 2 x þ k1 a x k12 a2
and Msþ j ¼
2p 2p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ms j : 2 2 x þ k1 a x k12 a2
þ þ þ Hence there exists an integer m P 0 such that 0 < sþ 0 < s0 < < sm1 < sm1 < sm < smþ1 < sm . Notice that the 0 þ 0 couple of roots of Eq. (5) with s ¼ 0 have negative real parts, and a ðsj Þ > 0, a ðsj Þ < 0, for j ¼ 0; 1; . . . ; the conclusion of (2) follows. (3) By Lemma 2.2 we have 1 a ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p s ð2j þ 1Þp arccos ; ¼ j k1 x k12 a2 1 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2j þ 1Þp þ arccos sþ ; j ¼ 0; 1; . . . ; j ¼ k1 x þ k12 a2
which imply that 2p 2p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ms j : k12 a2 x k12 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 þ 0 Hence, arccosð ka1 Þ < xp k12 a2 ensures that sþ 0 < s0 . By a ðsj Þ > 0, a ðsj Þ < 0, for j ¼ 0; 1; . . . ; and the couple of roots of Eq. (5) with s ¼ 0phave positive real parts, the conclusion of (i) follows. ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ þ From arccosð ka1 Þ > xp k12 a2 , we have s 0 < s0 . Notice Msj < Msj , similar to the proof of (2), we can get the conclusion of (ii). (4) Notice that the couple of roots of Eq. (5) with s ¼ 0 have positive real parts, and a0 ðs j Þ > 0 for j ¼ 0; 1; . . . ; the conclusion of (iv) follows. The proof is complete. Msþ j ¼
xþ
From Lemmas 2.1–2.4 and applying the Hopf bifurcation theorem for functional differential equations [11, Chapter 11, Theorem 1.1], we have the following results on stability and bifurcation to system (2). Theoremp2.5. Forffi system (2), ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1) if a2 þ x2 6 k1 , then system (2) undergoes a Hopf bifurcation at the origin ð0; 0Þ when s ¼ s j ; j ¼ 0; 1; . . . , as well þ þ as the zero solution p isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi asymptotically stable when s 2 ð0; s Þ, and is unstable when s > s ; 0 0 ffi (2) if jaj < k1 < a2 þ x2 , then system (2) undergoes a Hopf bifurcation at the origin ð0; 0Þ when sm¼ s ; j ¼ 0; 1; . . . S j þ Particularly, there exists an integer m P 0 such that the zero solution is asymptotically stable when s 2 ðs j1 ; sj Þ, and is m1 S þ S þ j¼0 unstable when s 2 ðsj ; sj Þ ðsm ; þ1Þ; j¼0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3) if a2 þ x2 < k1 < jaj, then system (2) undergoes a Hopf bifurcation at the origin ð0; 0Þ when s ¼ s j ; j ¼ 0; 1; . . . , and (i) if either ðH1 Þ or ðH2 Þ is satisfied, then the zero solution is unstable for all s P 0; (ii) if neither of ðH1 Þ and ðH2 Þ is satisfied, then there exists an integer m P 0 such that the zero solution is unstable when m m S S þ S þ s 2 ðsþ ðsm ; þ1Þ, and asymptotically stable when s 2 ðs j1 ; sj Þ j ; sj Þ; j¼0 j¼0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4) if k1 6 a2 þ x2 , then system (2) undergoes a Hopf bifurcation at the origin ð0; 0Þ when s ¼ s j , j ¼ 0; 1; . . ., and the zero solution is unstable for all s P 0;
822
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 1. The curves k1 ¼ a2 þ x2 and k1 ¼ a divide the ða; k1 Þ-plane into six regions, reads D1 ; D2 ; D3 ; D4 ; D5 and D6 . D1 and D2 are conditionally stable regions, and there are stability switches for the parameters being locating in the region D2 ; D3 is a conditionally stable region also under certain conditions; D5 is an absolutely stable region; and D4 and D6 are unstable regions for all s P 0.
(5) if a < k1 < a, then the zero solution is asymptotically stable for all s P 0; (6) if a < k1 < a, then the zero solution is unstable for all s P 0. According to the conclusions of Theorem 2.5, we can draw the bifurcation diagram in the parameter plane as in Fig. 1. 3. Direction and stability of Hopf bifurcation In this section we shall study the direction of the Hopf bifurcations, the stability and the period of the bifurcating periodic solutions. The method we used is based on the normal form method and the center manifold theory introduced by Hassard et al. [14]. We first re-scale the time by t 7! ðt=sÞ to normalize the delay so that system (2) can be written as the form x_ ðtÞ ¼ sða x2 ðtÞ y 2 ðtÞÞxðtÞ sxyðtÞ sk1 xðt 1Þ sk2 ðx2 ðt 1Þ y 2 ðt 1ÞÞ; ð9Þ _ ¼ sxxðtÞ þ sða x2 ðtÞ y 2 ðtÞÞyðtÞ sk1 yðt 1Þ 2sk2 xðt 1Þyðt 1Þ: yðtÞ The linearization around the origin ð0; 0Þ is given by x_ ðtÞ ¼ saxðtÞ sxyðtÞ sk1 xðt 1Þ; _ ¼ sxxðtÞ þ sayðtÞ sk1 yðt 1Þ; yðtÞ and the nonlinear term is sðx2 ðtÞ þ y 2 ðtÞÞxðtÞ sk2 ðx2 ðt 1Þ y 2 ðt 1ÞÞ : f ¼ sðx2 ðtÞ þ y 2 ðtÞÞyðtÞ 2sk2 xðt 1Þyðt 1Þ
ð10Þ
ð11Þ
The characteristic equation associated with Eq. (10) is v sa þ sk1 ev ¼ isx:
ð12Þ
Comparing Eq. (12) with Eq. (5), one can find out that v ¼ sk, and hence, Eq. (12) has a pair of imaginary roots is j b when s ¼ s j for j ¼ 0; 1; . . . ; and the transversal condition holds. Let s ¼ s0 þ l, l 2 R and s0 be taken in fsþ j g [ fsj g. Then l ¼ 0 is the Hopf bifurcation value for Eq. (9). Let is0 b0 be the root of Eq. (12) with s ¼ s0 , where either b0 ¼ bþ or b0 ¼ b . For u 2 Cð½1; 0; R2 Þ, let ðs0 þ lÞk1 ðs0 þ lÞa ðs0 þ lÞx 0 uð0Þ þ uð1Þ; ð13Þ Ll u ¼ x ðs0 þ lÞa 0 ðs0 þ lÞk1
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
823
and f ðl; uÞ ¼
ðs0 þ lÞ½ðu21 ð0Þ þ u22 ð0ÞÞu1 ð0Þ þ k2 ðu21 ð1Þ u22 ð1ÞÞ : ðs0 þ lÞ½ðu21 ð0Þ þ u22 ð0ÞÞu2 ð0Þ 2k2 u1 ð1Þu2 ð1Þ
ð14Þ
By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions gðh; lÞ in h 2 ½1; 0 such that Z 0 Ll u ¼ dgðh; lÞuðhÞ for u 2 C: ð15Þ 1
In fact, we choose
x dðhÞ þ k1 ðs0 þ lÞIdðh þ 1Þ; a 1; h ¼ 0 where I is the identity matrix and dðhÞ ¼ . Then Eq. (15) is satisfied. 0; h 6¼ 0 1 2 For u 2 C ð½1; 0; R Þ, define duðhÞ=dh; h 2 ½1; 0Þ; AðlÞu ¼ R 0 dgðt; lÞuðtÞ; h ¼ 0; 1 gðh; lÞ ¼ ðs0 þ lÞ
a x
ð16Þ
and RðlÞu ¼
0; f ðl; uÞ;
h 2 ½1; 0Þ; h ¼ 0:
ð17Þ
Hence, we can rewrite Eq. (9) in the following form: u_ t ¼ AðlÞut þ RðlÞut ;
ð18Þ
where u ¼ ðx; yÞT ; ut ¼ uðt þ hÞ for h 2 ½1; 0. For w 2 C 1 ð½0; 1; R2 Þ, define dwðsÞ=ds; s 2 ð0; 1; A wðsÞ ¼ R 0 wðtÞ dgðt; 0Þ; s ¼ 0: 1 For u 2 C½1; 0 and w 2 C½0; 1, define the bilinear form Z 0 Z h hÞ dgðhÞ uðnÞ dn; wðn hw; ui ¼ wð0Þuð0Þ 1
ð19Þ
ð20Þ
n¼0
where gðhÞ ¼ gðh; 0Þ. Then A and Að0Þ are adjoint operators, and is0 b0 are eigenvalues of Að0Þ. Thus, they are also eigenvalues of A . By direct computation, we obtain that i is0 b0 h qðhÞ ¼ e 1 is the eigenvector of Að0Þ corresponding to is0 b0 , and q ðsÞ ¼ Dði; 1Þeis0 b0 s is the eigenvector of A corresponding to is0 b0 . Moreover, hq ; qi ¼ 1;
hq ; qi ¼ 0;
where D ¼ ð2 2s0 k1 eis0 b0 Þ1 : Using the same notation as in Hassard et al. [14], we first compute the coordinates to describe the center manifold C0 at l ¼ 0. Let ut be the solution of Eq. (9) when l ¼ 0.
824
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
Define zðtÞ ¼ hq ; ut i;
wðt; hÞ ¼ ut ðhÞ 2RefzðtÞqðhÞg:
On the center manifold C0 we have wðt; hÞ ¼ wðzðtÞ; zðtÞ; hÞ; where wðz; z; hÞ ¼ w20 ðhÞ
z2 z2 þ w11 ðhÞzz þ w02 ðhÞ þ ; 2 2
q . Note that w is real if ut is real. We z and z are local coordinates for center manifold C0 in the direction of q and consider only real solutions. For solution ut in C0 of Eq. (9), since l ¼ 0, z_ ðtÞ ¼ is0 b0 z þ hq ðhÞ; f ðw þ 2RefzðtÞqðhÞgi def
¼ is0 b0 z þ q ð0Þf ðwðz; z; 0Þ þ 2RefzðtÞqð0ÞgÞ¼ is0 b0 z þ q ð0Þf0 ðz; zÞ:
ð21Þ
We rewrite this as z_ ðtÞ ¼ is0 b0 zðtÞ þ gðz; zÞ;
ð22Þ
where f0 ðz; zÞ ¼ fz2
z2 z2 z2z þ ; þ fz2 þ fzz zz þ fz2z 2 2 2
gðz; zÞ ¼ q ð0Þf ðwðz; z; 0Þ þ 2RefzðtÞqð0ÞgÞ ¼ g20
ð23Þ z2 z2 z2z þ þ g11 zz þ g02 þ g21 2 2 2
ð24Þ
Comparing the coefficients of (21) and (22), noticing (24), we have g20 ¼ q ð0Þfz2 ; g11 ¼ q ð0Þfzz ;
ð25Þ
g02 ¼ q ð0Þfz2 ; g21 ¼ q ð0Þfz2z : By (18) and (22), it follows that Aw 2Refq ð0Þf0 qðhÞg; w_ ¼ u_t z_ q z_ q ¼ Aw 2Refq ð0Þf0 qð0Þg þ f0 ;
h 2 ½1; 0Þ def ¼ Aw þ H ðz; z; hÞ; h ¼ 0;
ð26Þ
where H ðz; z; hÞ ¼ H20 ðhÞ
z2 z2 þ H11 ðhÞzz þ H02 ðhÞ þ : 2 2
ð27Þ
Expanding the above series and comparing the coefficients, we obtain ðA 2is0 x0 IÞw20 ðhÞ ¼ H20 ðhÞ;
Aw11 ðhÞ ¼ H11 ðhÞ; . . .
Notice that q ð0Þ ¼ Dði; 1Þ; xðtÞ ¼ iz iz þ wð1Þ ðz; z; 0Þ; yðtÞ ¼ z þ z þ wð2Þ ðz; z; 0Þ; xðt 1Þ ¼ ieis0 b0 z ieis0 b0 z þ wð1Þ ðz; z; 1Þ;
ð28Þ
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
825
and yðt 1Þ ¼ eis0 b0 z þ eis0 b0 z þ wð2Þ ðz; z; 1Þ; where ðjÞ
wðjÞ ðz; z; 0Þ ¼ w20 ð0Þ
z2 z2 ðjÞ ðjÞ þ w11 ð0Þzz þ w02 ð0Þ þ ; 2 2
ðjÞ
wðjÞ ðz; z; 1Þ ¼ w20 ð1Þ
z2 z2 ðjÞ ðjÞ þ w11 ð1Þzz þ w02 ð1Þ þ ; 2 2
j ¼ 1; 2;
and def
f0 ¼ fðs0 ; ut Þ ¼
s0 ½ðx2 ðtÞ þ y 2 ðtÞÞxðtÞ þ k2 ðx2 ðt 1Þ y 2 ðt 1ÞÞ ; s0 ½ðx2 ðtÞ þ y 2 ðtÞÞyðtÞ þ 2k2 xðt 1Þyðt 1Þ
gðz; zÞ ¼ q ð0Þf0 ; we have fzz ¼ 0 and 2is0 b0 ; g20 ¼ 8ik2 s0 De g11 ¼ g02 ¼ 0;
ð29Þ
ð2Þ is0 b0 þ k2 wð1Þ þ ik2 w11 ð1Þeis0 b0 : g21 ¼ 8s0 D½2 11 ð1Þe
From (16), it follows that ( dw11 ðhÞ ; Aw11 ðhÞ ¼ R 0dh dgðtÞw 11 ðtÞ; 1
h 2 ½1; 0Þ; h ¼ 0;
and (26) and (27) yield Refq ð0Þf0 qðhÞg ¼ gqðhÞ gqðhÞ; H ðz; z; hÞ ¼ qð0Þ þ f0 ; Refq ð0Þf0 qð0Þg þ f0 ¼ gqð0Þ g Comparing the coefficients with (27), we get g11 qðhÞ g11 qðhÞ; h 2 ½1; 0Þ; H11 ðhÞ ¼ g11 qð0Þ g11 qð0Þ þ fzz ; h ¼ 0: From Eqs. (28), (30) and (31), we derive w011 ðhÞ ¼ g11 qðhÞ þ g11 qðhÞ;
h 2 ½1; 0Þ:
Notice g11 ¼ 0, we obtain w11 ðhÞ ¼ E2 ; where E2 is constant. From (28), (30) and (31) when h ¼ 0 we have Z 0 dgðhÞE2 ¼ fzz : 1
Hence, E2 ¼ 0 follows from fzz ¼ 0. Thus w11 ðhÞ 0: Consequently, from (29), g21 ¼ 16s0 D: Substituting g11 ; g20 ; g02 and g21 into i 1 g21 2 2 g20 g11 2jg11 j jg02 j þ C1 ð0Þ ¼ 2s0 b0 3 2
ð30Þ
h 2 ½1; 0Þ; h ¼ 0:
ð31Þ
826
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
gives C1 ð0Þ ¼ 8s0 D: Note ¼ D
1 1 ¼ ð1 s0 k1 cos s0 b0 is0 k1 sin s0 b0 Þ 2 2s0 k1 eis0 b0 B
and a ¼ k1 cos s0 b0 , we have Re C1 ð0Þ ¼
8s0 8s0 ð1 s0 k1 cos s0 b0 Þ ¼ ð1 s0 aÞ; B B
where B ¼ 2ð1 þ s20 k12 2s0 k1 cos s0 b0 Þ > 0: Therefore, 16s0 < 0; 1 s0 a > 0; ð1 s0 aÞ b2 ¼ 2RefC1 ð0Þg ¼ > 0; 1 s0 a < 0; B and
8 > > 0; > > > > > < 0; > > > RefC1 ð0Þg < > 0; l2 ¼ a0 ðs0 Þ > < 0; > > > > > < 0; > > > : > 0;
s0 ¼ s j ; 1 s0 a > 0; s0 ¼ s j ; 1 s0 a < 0; s0 ¼ sþ j ; 1 s0 a > 0; s0 ¼ sþ j ; 1 s0 a < 0; s0 ¼ s j ; 1 s0 a > 0; s0 ¼ s j ; 1 s0 a < 0;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jk1 j P a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jk1 j P a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jaj < jk1 j < a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jaj < jk1 j < a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jaj < jk1 j < a2 þ x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jaj < jk1 j < a2 þ x2 :
ð32Þ
ð33Þ
By the general theory due to Hassard et al. [14], we know that the quantity of b2 determines the stability of the bifurcating periodic solutions on the center manifold, and l2 determines the direction of the bifurcation. So we have the following from (32) and (33). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Theorem 3.1. (i) If jk1 j P a2 þ x2 and s j < a, then the Hopf bifurcation at the origin when s ¼ sj is supercritical and the bifurcating periodic solutions are orbitally asymptotically stable. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 (ii) If jk1 j P a2 þ x2 and s j > a, then the Hopf bifurcation at the origin when s ¼ sj is subcritical and the bifurcating periodic solutions are unstable. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ (iii) If jaj < jk1 j < a2 þ x2 and sþ j < a, then the Hopf bifurcation at the origin when s ¼ sj is supercritical and the bifurcating periodic solutions are stable. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi orbitally asymptotically 1 þ (iv) If jaj < jk1 j < a2 þ x2 and sþ j > a, then the Hopf bifurcation at the origin when s ¼ sj is subcritical and the bifurcating periodic solutions are unstable. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 (v) If jaj < jk1 j < a2 þ x2 and s j < a, then the Hopf bifurcation at the origin when s ¼ sj is subcritical and the bifurcating periodic solutions are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi orbitally 1asymptotically stable. (vi) If jaj < jk1 j < a2 þ x2 and s j > a, then the Hopf bifurcation at the origin when s ¼ sj is supercritical and the bifurcating periodic solutions are unstable. Remark. Theorem 3.1 describes the dynamics of Eq. (2) restricted to the center manifolds. Of course, the existence of the periodic solutions of Eq. (2) is the same as its restriction on the center manifold. Usually, the bifurcating periodic solutions of Eq. (2) from the origin are unstable when the characteristic equation associates with the linearization of Eq. (2) around the origin has at least one root with a positive real part at the critical value. But, when all the roots of the characteristic equation with the bifurcation value have negative real parts except the pair of imaginary roots, the stabilities of the bifurcating periodic solutions are the same for Eq. (2) and its restriction to the center manifolds. So, from Lemma 2.4 and Theorem 3.1, we have the following: For Eq. (2), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1) if a2 þ x2 6 k1 , then 1 i(i) the direction of the Hopf bifurcation at the origin when s ¼ sþ 0 < a is supercritical and the bifurcating periodic solutions are orbitally asymptotically stable; 1 (ii) the direction of the Hopf bifurcation at the origin when s ¼ sþ 0 > a is subcritical and the bifurcating periodic solutions are unstable;
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
827
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2) if jaj < k1 < a2 þ x2 , then þ 1 ii(i) the directions of the Hopf bifurcations at the origin when s ¼ sþ j ðj ¼ 0; 1; . . . ; mÞ and sj < a are supercritical and the bifurcating periodic solutions are orbitally asymptotically stable; þ 1 i(ii) the directions of the Hopf bifurcations at the origin when s ¼ sþ j ðj ¼ 0; 1; . . . ; mÞ and sj > a are subcritical and the bifurcating periodic solutions are unstable; 1 (iii) the directions of the Hopf bifurcations at the origin when s ¼ s j ðj ¼ 0; 1; . . . ; m 1Þ and sj < a are subcritical and the bifurcating periodic solutions are orbitally asymptotically stable; 1 (iv) the directions of the Hopf bifurcations at the origin when s ¼ s j ðj ¼ 0; 1; . . . ; m 1Þ and sj > a are supercritical and ffithe bifurcating periodic solutions are unstable; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p (3) if a2 þ x2 < k1 < jaj, and neither of ðH1 Þ and ðH2 Þ holds, then þ 1 iii(i) the directions of the Hopf bifurcations at the origin when s ¼ sþ j ðj ¼ 0; 1; . . . ; mÞ and sj < a are supercritical and the bifurcating periodic solutions are orbitally asymptotically stable; þ 1 ii (ii) the directions of the Hopf bifurcations at the origin when s ¼ sþ j ðj ¼ 0; 1; . . . ; mÞ and sj > a are subcritical and the bifurcating periodic solutions are unstable; 1 (iii) the directions of the Hopf bifurcations at the origin when s ¼ s j ðj ¼ 0; 1; . . . ; mÞ and sj < a are subcritical and the bifurcating periodic solutions are orbitally asymptotically stable; 1 i(iv) the directions of the Hopf bifurcations at the origin when s ¼ s j ðj ¼ 0; 1; . . . ; mÞ and sj > a are supercritical and the bifurcating periodic solutions are unstable.
4. Numerical simulations In this section, we present some numerical results to system (2) at different values of a, x, ki (i ¼ 1; 2) and s. We choose k2 ¼ 1; x ¼ 1, then system (2) is given by x_ ðtÞ ¼ axðtÞ yðtÞ k1 xðt sÞ x3 ðtÞ xðtÞy 2 ðtÞ x2 ðt sÞ þ y 2 ðt sÞ; ð34Þ y_ ðtÞ ¼ xðtÞ þ ayðtÞ k1 yðt sÞ x2 ðtÞyðtÞ y 3 ðtÞ 2xðt sÞyðt sÞ: By Eqs. (7), (8), (32) and (33) we can calculate l2 and b2 so that the dynamical behavior of system (34) will be determined. The results are given in Table 1. We choose k1 ¼ 1, a ¼ 2 ðða; k1 Þ 2 D5 Þ and a ¼ 2 ðða; k1 Þ 2 D6 Þ. By Theorem 2.5 the zero solution of Eq. (34) is asymptotically stable and unstable, respectively. The corresponding waveform and phase plots are shown in Figs. 2 and 3, respectively, where s ¼ 1. We choose a ¼ 2, k1 ¼ 2:5 ðða; k1 Þ 2 D4 Þ. By Theorem 2.5 the zero solution of Eq. (34) is unstable. One finds out that l2 > 0 and b2 < 0 from Table 1. Hence, by Theorem 3.1 we know that the direction of the bifurcation at the critical value s0 ¼ 2:26 is supercritical, and the bifurcating periodic solutions are stable, as shown in Fig. 4, where s ¼ 2:4. We choose a ¼ 2, k1 ¼ 2:5 ðða; k1 Þ 2 D1 Þ. With these parameters, sþ 0 ¼ 1:0033, l2 > 0; and b2 < 0 are found from Table 1. Hence, by Theorem 2.5 the zero solution of Eq. (34) is asymptotically stable when s 2 ð0; 1:0033Þ, as shown on the first line in Fig. 5, where s ¼ 0:5, and unstable when s 2 ð1:0033; 1Þ. By Theorem 3.1 we know that the direction of
Table 1 Parameter values at different quantities of a and k1 when x ¼ 1 and k2 ¼ 1 a
k1
s0
)2 )2 )1
)2.5 2.5 1.09
)1
)1.09
)0.5
)1
sþ 0 sþ 0 sþ 0 s 0 sþ 1 sþ 2 s 1 s 0 sþ 0 sþ 1 s 1 sþ 0 s 0
¼ 2:26 ¼ 1:0033 ¼ 2:1010 ¼ 5:0741 ¼ 6:9342 ¼ 11:7674 ¼ 14:0501 ¼ 0:5861 ¼ 4:5176 ¼ 9:3508 ¼ 9:5621 ¼ 2:806 ¼ 7:8167
sign(l2 )
sign(b2 )
) ) ) + )
+ + + ) +
) )
) +
)
+
828
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
Fig. 2. Waveform plot and phase for system (34) with a ¼ 2; k1 ¼ 1 and s ¼ 1. The equilibrium ð0; 0Þ is asymptotically stability.
Fig. 3. Waveform plot and phase for system (34) with a ¼ 2; k1 ¼ 1 and s ¼ 1. The equilibrium ð0; 0Þ is unstable.
the bifurcation at the critical value s0 ¼ 1:0033 is supercritical, and the bifurcating periodic solutions are stable, as shown on the last line in Fig. 5, where s ¼ 1:1. We choose a ¼ 1, k1 ¼ 1:09 ðða; k1 Þ 2 D2 Þ. With these parameters, from Table 1 and by Theorem 2.5 one can get that the zero solution of Eq. (34) is asymptotically stable when s 2 ð0; 2:1010Þ [ ð5:0741; 6:9342Þ, and unstable when s 2 ð2:1010; 5:0741Þ [ ð6:9342; 1Þ. This shows that the equilibrium ð0; 0Þ switches 2 times from stability to instability and unstable for any s > 6:9342. By Theorem 3.1 we know that the directions of the Hopf bifurcations at the critical values s0 ¼ 2:1010; 5:0741 and 6:9342 are supercritical, subcritical and supercritical, respectively. And the bifurcating
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
829
Fig. 4. Waveform plot and phase for system (34) with a ¼ 2; k1 ¼ 2:5 and s ¼ 2:4. s ¼ 2:4 being near to sþ 0 ¼ 2:26, a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable.
Fig. 5. Waveform plot and phase for system (34) with a ¼ 2; k1 ¼ 2:5 and s ¼ 0:5 or s ¼ 1, respectively. When s ¼ 0:5 < 1:0033 ¼ sþ 0 þ the equilibrium ð0; 0Þ is asymptotically stability. When s ¼ 1:1 > sþ 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable.
periodic solutions are stable, unstable and stable, respectively, as shown on each line from the top to the bottom in Fig. 6, where s ¼ 1; 2:3 and 6, respectively. We choose a ¼ 1, k1 ¼ 1:09 ðða; k1 Þ 2 D3 Þ. With these parameters from Table 1 and by Theorem 2.5, we conclude that the zero solution of Eq. (34) is unstable when s 2 ð0; 0:5861Þ [ ð4:5176; 1Þ, and asymptotically stable when s 2 ð0:5861; 4:5176Þ. This shows that the equilibrium ð0; 0Þ switches 1 times from stability to instability and unstable for any s > 4:5176. By Theorem 3.1 we know that the directions of the bifurcation at the critical values s0 ¼ 0:5861 and
830
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
Fig. 6. Waveform plot and phase for system (34) with a ¼ 1; k1 ¼ 1:09; and s ¼ 1; 2:3 or 6, respectively. When s ¼ 1 < 2:1010 ¼ sþ 0 þ the equilibrium ð0; 0Þ is asymptotically stability. When s ¼ 2:3 > sþ 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable. When s ¼ 5 < 5:0741 ¼ s 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is unstable. þ When s 0 < s ¼ 6 < 6:9342 ¼ s1 . The equilibrium ð0; 0Þ is asymptotically stable.
Fig. 7. Waveform plot and phase for system (34) with a ¼ 1; k1 ¼ 1:09, and s ¼ 0:5; 3 or 4.6, respectively. When s ¼ 0:5 < 0:5861 ¼ s 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable. When s ¼ 3 the þ equilibrium ð0; 0Þ is asymptotically stability. When s ¼ 4:6 > 4:5176 ¼ sþ 0 and being near to s0 , a periodic solution bifurcate from ð0; 0Þ and is asymptotically stable.
4.5176 are subcritical and supercritical, respectively. And the bifurcating periodic solutions are stable, as shown on each line from the top to the bottom in Fig. 7, where s ¼ 0:5; 3 and 4.6, respectively.
W. Jiang, J. Wei / Chaos, Solitons and Fractals 23 (2005) 817–831
831
Fig. 8. Waveform plot and phase for system (34) with a ¼ 0:5; k1 ¼ 1; s ¼ 3. The equilibrium ð0; 0Þ is unstable.
We choose a ¼ 0:5, k1 ¼ 1 ðða; k1 Þ 2 D3 Þ. By Theorem 2.5 the zero solution of Eq. (34) is unstable for all s P 0, as shown in Fig. 8, where s ¼ 3. References [1] Reddy D, Sen A, Johnston G. Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks. Physica D 2000;144:335–57. [2] Zou S, Liao X, Yu J, Wong K. Chaos and its synchronization in two-neuron systems with discrete delays. Chaos, Solitons & Fractals 2004;21:133–42. [3] Aronson D, Ermentrout G, Koppel N. Amplitude response of coupled oscillators. Physica D 1990;41:403. [4] Schuster H, Wagnar P. Mutual entrainment of two limit cycle oscillators with time delayed coupling. Prog Theoret Phys 1989;81:939–45. [5] Niebur E, Schuster H, Kammen D. Colective frequencies and metastability in networks of limit-cycle oscillators with time delay. Phys Rev Lett 1991;67:2753–6. [6] Nakamura Y, Tominaga F, Munakata T. Clustering behavior of time-delayed nearest-neighbor coupled oscillators. Phys Rev E 1994;49:4849–56. [7] Kim S, Park SH, Ryu CS. Multistability in coupled oscillator systems with time delay. Phys Rev Lett 1997;79:2911–4. [8] Reddy D, Sen A, Jonston G. Time delay induced death in coupled limit cycle oscillators. Phys Rev Lett 1998;80:5109–12. [9] Reddy D, Sen A, Jonston G. Time delay effects on coupled limit cycle oscillators at Hopf bifurcation. Physica D 1999;129:15–34. [10] Yeung M, Strogatz S. Time delay in the Kuramoto model of coupled oscillators. Phys Rev Lett 1999;82:648–51. [11] Hale J, Verduyn Lunel S. Introduction to functional differential equations. New York: Springer Verlag; 1993. [12] Goppalsamy K. Stability and oscillations in delay differential equations of population dynamics. Dordrecht: Kluwer Academic Publishers; 1992. [13] Wu J. Introduction to neural dynamics and signal transmission delay. Berlin, New York: Walter de Gruyter; 2001. [14] Hassard BD, Kazarinoff ND, Wan YH. Theory and application of hopf bifurcation. Cambridge: Cambridge University Press; 1981. [15] Meng X, Wei J. Stability and bifurcation of mutual system with time delay. Chaos, Solitons & Fractals 2004;21:729–40. [16] Song Y, Wei J. Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos. Chaos, Solitons & Fractals 2004;22:75–91. [17] Yuan S, Song Y, Han M. Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays. Chaos, Solitons & Fractals 2004;21:1109–23. [18] Liao X, Wong K, Leung C, Wu Z. Hopf bifurcation and chaos in a single delayed neural equation with non-monotonic activation function. Chaos, Solitons & Fractals 2001;12:1352–547. [19] Li C, Liao X, Yu J. Hopf bifurcation in a prototype delayed system. Chaos, Solitons & Fractals 2004;19:779–87. [20] Ruan S, Wei J. On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn Continuous, Discrete Impuls Syst A: Math Anal 2003;10:863–74.