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Local and Global Hopf Bifurcation in a Scalar Delay Differential Equation
Journal of Mathematical Analysis and Applications 237, 425᎐450 Ž1999. Article ID jmaa.1999.6431, available online at http:rrwww.idealibrary.com on
Local and Global Hopf Bifurcation in a Scalar Delay Differential Equation Fotios Giannakopoulos* and Andreas Zapp Mathematical Institute, Uni¨ ersity of Cologne, Weyertal 86-90, D-50931 Cologne, Germany E-mail: [email protected], [email protected] Submitted by U. Kirchgraber Received September 3, 1998
1. INTRODUCTION In this paper we study the bifurcation properties of Hopf branches in the scalar retarded functional differential equation
˙x Ž t . s yx Ž t . q F Ž x Ž t y . .
Ž 1.
with the time delay ) 0 as bifurcation parameter, where F g C k Ž⺢, ⺢., k G 3, and F Ž0. s 0. Equation Ž1. is a prototype for a retarded functional differential equation which has many applications in sciences. Special cases of Ž1. have been proposed as models of both physical and physiological phenomena Žsee, for instance w1, 15, 21, 26x; see also w16, pp. 68᎐81; 24; 28x.. For a wide class of functions F Že.g., monotone F . the dynamics of Ž1. are determined by periodic solutions Žsee w25, 31x.. In the applications periodic dynamics are also of particular interest Žsee w16, pp. 57᎐81x.. There exist many papers which are concerned with the existence of periodic solutions of Ž1. and explore their dependence on the time delay ) 0 Žsee w5, 17, 23, 27, 31, 32x and the references in w23, 32x.. The existence of a sequence of time delays 0 - 0 - 1 - ⭈⭈⭈ , with k ª ⬁ as k ª ⬁, such that a local Hopf bifurcation occurs at s k , provided < F X Ž0.< ) 1, is proved in w32x Žsee also w27x.. In the case F X Ž0. - y1 a condition for supercritical Hopf bifurcation at s 0 is given in w3x. In w33x *Present address: GMD-German Research Center for Information Technology, Schloss Birlinghoven, D-53754 Sankt Augustin, Germany. 425 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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GIANNAKOPOULOS AND ZAPP
conditions for both supercritical and subcritical Hopf bifurcations at s 0 are provided for F X Ž0. - y1, F Y Ž0. s 0, and F Z Ž0. / 0. The global continuation for all ) k of the locally existing Hopf branches at s k is demonstrated in w23x, provided that there exists a compact interval I ; ⺢, 0 g intŽ I ., such that F Ž I . ; I Žinvariance condition., xF Ž x . - 0 for all x g I _ 04 , and F X Ž0. - y1 Žnegative feedback condition.. However, many examples of functions F which do not satisfy the conditions mentioned above arise in applications Žsee w2; 15; 16, pp. 68᎐81; 21; 24x.. Furthermore, from the point of view of qualitative analysis, it is also of interest to know how the properties of Eq. Ž1. change if one of the conditions above fails. The main objective of this work is to extend the results mentioned above to a more general class of functions F. More precisely, we give both sufficient and necessary conditions for the local existence and the bifurcation direction of Hopf branches and for the stability of the periodic solutions bifurcating from the trivial solution x s 0 at s k , k g ⺞ 0 , provided F g C 3. The Hopf bifurcation formula derived here has a readily applicable form and provides all possible bifurcation scenarios. For the derivation of this formula we consider a system of two ordinary differential equations describing the flow of Ž1. on an appropriate center manifold Žsee Sect. 2.. Furthermore, we present a result on the global continuation of the local Hopf branches for functions F g C 2 , where F X Ž x . / 1 for all fixed points x of F and < F X Ž0.< ) 1 Žsee Sect. 3.. Finally, we discuss several examples which illustrate our results and indicate what can happen if the invariance condition or the negative feedback condition is violated Žsee Sect. 4..
2. LOCAL BEHAVIOR Throughout this section we assume ŽLH.
F g C 3, F Ž0. s 0, and < F X Ž0.< ) 1.
2.1. Main Results In this subsection we state the main results on the local Hopf bifurcation. First we need some definitions. DEFINITION 2.1. For k g ⺞ 0 let bk , k: ⺢ _ w y1, 1 x ª ⺢q
LOCAL AND GLOBAL BIFURCATION
427
be positive functions of defined by
¡arccos
bk Ž . [
~
1
ž /
q 2 k
for - y1
Ž 2.
¢2Ž k q 1. y arccos ž / 1
k Ž . [
for ) 1
bk Ž .
Ž 3.
'2 y 1
with arccosŽ x . in Ž0, .. Furthermore we define
¡ 11 q 6 y 2
2
Ž q 1. q2 for - y1 Ž 5 q 4. Ž y 1. Ž 5 q 4 . Ž 1 q 2 . Ck Ž . [~ 11 3 q 35 2 q 24 y 6 Ž q 1 . Ž 2 y 3 . y 2 Ž y 1 . Ž 5 2 q 15 q 12 . Ž 5 2 q 15 q 12 . Ž 1 q 2 . 2
¢
for ) 1
with s k Ž ., k g ⺞ 0 . THEOREM 2.1 ŽLocal Hopf Bifurcation..
Let < < ) 1 with s F X Ž0..
1. At s k Ž ., k g ⺞ 0 , Eq. Ž1. undergoes a Hopf bifurcation; that is, in e¨ ery small neighborhood of Ž x s 0, s k Ž .. there is a unique branch of periodic solutions x k Ž t; . with x k Ž t; . ª 0 as ª k Ž .. For the period pk Ž , . of x k Ž t; ., pk Ž , . ª 2 k Ž .rbk Ž . s 2r '2 y 1 \ pŽ . as ª k Ž .. 2. Assume F Z Ž 0 . F X Ž 0 . - F Y Ž 0 . Ck Ž . . 2
Then the bifurcating branch of periodic solutions exists for ) k Ž . Ž supercritical bifurcation.. Moreo¨ er the arising periodic solutions are stable if
- y1
and k s 0
and they are unstable if
- y1 3.
and k G 1
or
) 1.
Assume F Z Ž 0 . F X Ž 0 . ) F Y Ž 0 . Ck Ž . . 2
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GIANNAKOPOULOS AND ZAPP
Then the bifurcating branch of periodic solutions exists for - k Ž . Ž subcritical bifurcation.. All periodic solutions on this branch are unstable. Proof. The proof is given in Section 2.5. Remark 2.1. 1. For < < F 1 there are no Hopf bifurcation points. 2. s ˜ is a Hopf bifurcation point if and only if ˜ s k Ž . with k g ⺞0 . Using the properties of the function Ck Ž . we get the following criteria for the direction of the bifurcating periodic solutions Žsee Fig. 1.. COROLLARY 2.1. Let F Y Ž0. and F Z Ž0. be not simultaneously equal to zero and < < ) 1 with s F X Ž0.. 1. 2.
If F Z Ž0. F X Ž0. F 0 then all bifurcations are supercritical. If F Y Ž0. s 0 and F Z Ž0. F X Ž0. ) 0 then all bifurcations are subcriti-
cal. If F Y Ž0. / 0 and F Z Ž0. F X Ž0. ) 0 then Ža. For g Žy⬁, y1. j Ž'3 , ⬁. the Hopf bifurcations occurring at s k Ž . are either supercritical for all k g ⺞ 0 or subcritical for all k g ⺞ 0 or there is a unique k c g ⺞ such that the Hopf bifurcations occurring at s k Ž . are subcritical for k - k c and supercritical for k ) k c . Žb. For g Ž1, '3 . the Hopf bifurcations occurring at s k Ž . are either supercritical for all k g ⺞ 0 or subcritical for all k g ⺞ 0 , or there is a unique k c g ⺞ such that the Hopf bifurcations occurring at s k Ž . are supercritical for k - k c and subcritical for k ) k c . 3.
FIG. 1. All possible bifurcation scenarios Žsee Corollary 2.1..
LOCAL AND GLOBAL BIFURCATION
429
Žc. For s '3 and F Z Ž0. F X Ž0. / F Y Ž0. 2 Ck Ž'3 . the Hopf bifurcations occurring at s k Ž . are either supercritical or subcritical for all k g ⺞0 . Proof. The proof can be found in Section 2.4. 2.2. The Characteristic Equation The objective of this section is to study the properties of the characteristic equation associated with the linearization of Ž1.. For the sake of simplicity we rescale the time t by t ¬ t . This provides a scalar delay differential equation with constant time delay 1,
˙x Ž t . s y x Ž t . q F Ž x Ž t y 1 . . .
Ž 4.
After linearization about the equilibrium x s 0 we get
[ F X Ž 0. .
˙x Ž t . s y x Ž t . q x Ž t y 1 . ,
Ž 5.
The characteristic equation of Ž5. is given by 0 s ⌬ Ž z ; , . [ z q y exp Ž yz . .
Ž 6.
The analysis of Ž6. yields conditions for the stability of the equilibrium x s 0 and critical parameter values at which a Hopf bifurcation can occur. Much is known about the characteristic equation ⌬ s 0; for instance, see w4, 9, 20, 23, 27x. Here we present only some relevant results for completeness. LEMMA 2.1. The characteristic equation Ž6. has a pair of purely imaginary, conjugate complex solutions z s "ib, b ) 0, if and only if < < ) 1 and b s bk Ž ., s k Ž ., k g ⺞ 0 . bk , k are defined by Ž2. and Ž3.. The functions bk , k are C 1, and bk satisfy bk Ž . g x Ž 2 k q
. , Ž 2 k q 2. w bk Ž . g x Ž 2 k q 12 . , Ž 2 k q 1 . w 3 2
for k g ⺞ 0 , ) 1, for k g ⺞ 0 , - y1.
Proof. See w9, pp. 305᎐309x, Proposition A.2 in w23x, or Lemma 3.1 in w32x. The estimates for bk follow from Definition 2.1. LEMMA 2.2 ŽProperties of k .. k is a decreasing function of ) 1 and an increasing function of - y1, k g ⺞ 0 . Furthermore, the following hold for k g ⺞ 0
kq 1 Ž . y k Ž . s
2
'
2
y1
᭙ g ⺢ _ w y1, 1 x
Ž monotonicity in k .
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GIANNAKOPOULOS AND ZAPP
and lim k Ž . s 0
ª"⬁
lim k Ž . s ⬁.
ª"1
Proof. The assertions follow from Definition 2.1 and Lemma 2.1. LEMMA 2.3. Let < < ) 1. Then there are an ⑀ ) 0 and a simple characteristic root z Ž . s aŽ . q ibŽ . of Ž6. for < y k Ž .< - ⑀ , k g ⺞ 0 . z is C 1 in and aŽ k Ž .. s 0, bŽ k Ž .. s bk Ž .. Moreo¨ er dd aŽ k Ž .. ) 0. Proof. See Lemma 3.1 in w32x. LEMMA 2.4 ŽD-partition of the Ž , .-Plane.. The characteristic equation Ž6. has 1. only solutions with negati¨ e real part, pro¨ ided G 0 and g wy1, 1. or 0 F - 0 Ž . and - y1; 2. exactly one solution with positi¨ e real part, pro¨ ided 0 F F 0 Ž . with ) 1; 3. exactly 2 k solutions with positi¨ e real part, pro¨ ided ky 1Ž . - F k Ž . with - y1, k g ⺞; 4. exactly 2 k q 1 solutions with positi¨ e real part, pro¨ ided ky 1Ž . F k Ž . with ) 1, k g ⺞. Ž See Fig. 2.. Proof. The proof is an application of XI.2 in w9x. ŽSee also Proposition A.2 in w23x.. 0 Remark 2.2. If Ž , . g wy1, 1. = ⺢q j Žy⬁, y1. = w0, 0 Ž .. the 0 equilibrium x s 0 is asymptotically stable. If Ž , . g Ž1, ⬁. = ⺢q j Žy⬁, y1. = Ž 0 Ž ., ⬁. the equilibrium x s 0 is unstable.
2.3. Hopf Bifurcation In the following theorem we show that a Hopf bifurcation occurs at s k Ž .. The proof is an application of the Hopf theorem for retarded functional differential equations Žsee w19, p. 332x.. In w27, 32x the existence of Hopf bifurcations is proved in a similar way. THEOREM 2.2. Let < < ) 1 with s F X Ž0. and k g ⺞ 0 . Then a Hopf bifurcation for Eq. Ž1. occurs at s k Ž .; i.e., in e¨ ery small neighborhood of Ž x s 0, s k Ž .. there is a unique branch of periodic solutions x k Ž t; . with x k Ž t; . ª 0 as ª k Ž .. For the period pk Ž , . of x k Ž t; ., pk Ž , . ª 2 k Ž .rbk Ž . s 2r '2 y 1 \ pŽ . as ª k Ž ..
LOCAL AND GLOBAL BIFURCATION
431
FIG. 2. D-partition of the Ž , .-plane. In every region bounded by s k Ž ., resp. s 1, the number of characteristic solutions with positive real part is constant Žsee Lemma 2.4..
Proof. Lemma 2.1 and Lemma 2.3 provide the assumptions of the Hopf᎐Theorem Žsee w19, p. 332x. and thus the above statements are proved. Remark 2.3. By Lemma 2.1 we get that only at s 0 Ž . with - y1 can the Hopf bifurcation provide slowly oscillating periodic solutions Ži.e., p 0 Ž , . ) 2 .. For - y1 and k G 1, pk Ž , . gxŽrŽ k q 12 ., rŽ k q 14 .w, and for ) 1 and k g ⺞ 0 , pk Ž , . gxrŽ k q 1., rŽ k q 43 .w. For the definition of slowly oscillating periodic solutions see w23x. 2.4. Direction of Hopf Bifurcation In order to be able to analyse the Hopf bifurcation in more detail we compute the reduced system on the center manifold associated with the pair of conjugate complex, purely imaginary solutions ⌳ s ibk Ž ., yibk Ž .4 of the characteristic equation Ž6.. By this reduction we are able to determine the Hopf bifurcation direction, i.e., to answer the question of whether the bifurcating branch of periodic solution exists locally for
432
GIANNAKOPOULOS AND ZAPP
) k Ž . Žsupercritical bifurcation . or - k Ž . Žsubcritical bifurcation .. In the sequel we write bk , k instead of bk Ž . and k Ž .. In general it is difficult to compute the center manifold itself, but in w12x a method is presented providing the reduced system on the center manifold in normal form without computing the manifold itself. A similar method to estimate the direction of the Hopf bifurcation is presented in w30x by using the method of Lyapunov and Schmidt. Since we have two simple, purely imaginary characteristic solutions the center manifold considered is two-dimensional and thus, following w12, 13x, we can compute the reduced system on the center manifold in polar coordinates Ž , .: ˙ s Ž y k . dd a Ž k . q K 3 q O Ž 2 q Ž , .
4
Ž 7.
.
˙s ybk q O Ž < , < . .
Ž 8.
Since dd aŽ k . ) 0 Žsee Lemma 2.3. the sign of the factor K determines the direction of the Hopf bifurcation. A positive K implies a subcritical bifurcation; a negative K implies a supercritical bifurcation. Our aim is now to calculate the factor K in terms of the derivatives of F at zero. To do this let us first give the Taylor expansion of the right hand side of Ž4.. Setting ␣ [ y k , Eq. Ž4. becomes
˙x Ž t . s L␣ x t q G Ž x Ž t y 1 . ; ␣ . with L␣ s y Ž k q ␣ . Ž 0 . q Ž k q ␣ . Ž y1 . s Žk q ␣ .
0
Hy1 d Ž . Ž .
and G Ž u; ␣ . s Ž k q ␣ . F Ž u . y Ž k q ␣ . u. The function G satisfies G Ž 0; ␣ . s 0, G Ž u; ␣ . s k
DG Ž 0; ␣ . s 0, F Y Ž 0. 2
u2 q
F Y Ž 0. 2
␣ u2 q k
F Z Ž 0. 3!
u 3 q O Ž Ž u, ␣ .
This implies G Ž u; 0 . s k
F Y Ž 0. 2
u q k 2
F Z Ž 0. 3!
u3 q O Ž u4 . .
4
..
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LOCAL AND GLOBAL BIFURCATION
Proceeding as in w12, 13x, in the next step we consider the decomposition of the phase space C s C Žwy1, 0x, ⺢. s P [ Q, where P is the two-dimensional eigenspace associated with the set of simple eigenvalues ⌳ s ibk , yibk 4 . Let ⌽ s Ž 1 , 2 . s Ž e i b k , eyi b k . be a basis of P and ⌿ s colŽ 1 , 2 . s Ž 1Ž0. eyi b k , 2 Ž0. e i b k . a basis of the dual space P U in C U with the property
Ž ⌿, ⌽ . [ Ž Ž i , j . , i , j s 1, 2 . s I2 . The above bilinear form is given by Žsee w19, p. 211x.
0
Ž , . s Ž 0. Ž 0. y H
H y1 0
Ž y . d Ž . Ž . d .
Ž ⌿, ⌽ . s I2 yields Žsee w13x.
1Ž 0 . s 1 y L0 Ž e i b k .
y1
2 Ž 0. s 1Ž 0. .
,
Ž 9.
Thus the factor K in Ž7. is given by Žsee w13x. F Z Ž 0. k
Ks
2
½
= y
Re Ž eyi b k 1 Ž 0 . . q F Y Ž 0 . k2 2
Re Ž eyi b k 1 Ž 0 . . L0 Ž 1.
q
1 2
Re
ey3 i b k 1 Ž 0 . 2 ibk y L0 Ž e 2 i b k .
5
.
Ž 10 .
Here we have L0 Ž 1 . s k Ž y 1 . ,
L0 Ž e i b k . s y k eyi b k ,
L0 Ž e 2 i b k . s k Ž y1 q ey2 i b k . ,
1Ž 0. s
1 1 q k eyi b k
.
Substituting the above terms into Eq. Ž10. we obtain K as function of . Now we are in a position to prove the following useful lemma. LEMMA 2.5. Ks
Let s k Ž ., k g ⺞ 0 . Then
F Z Ž 0.
1 q 2
2
Ž 1 q 2 q 2 2 .
F Y Ž 0 . 2 Ž 11 2 q 6 y 2 . q Ž 2 3 q 13 2 q 4 y 4 . 2
y
2
2 Ž 5 q 4 . Ž y 1 . Ž 1 q 2 q 2 2 .
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GIANNAKOPOULOS AND ZAPP
for - y1 and Ks
F Z Ž0.
1 q 2
2
Ž 1 q 2 q 2 2 .
F Y Ž 0 . 2 Ž 11 3 q 35 2 q 24 y 6 . q Ž y2 4 q 11 3 q 43 2 q 24 y 12 . 2
y
2 Ž y 1 .Ž 5 2 q 15 q 12 .Ž 1 q 2 q 2 2 .
2
for ) 1. Proof. The proof will be given in the Appendix. To determine the direction of the Hopf bifurcation one can evaluate the term K or one can use the following conditions which guarantee the sign of K: LEMMA 2.6.
Let < < ) 1 and s k Ž ., k g ⺞ 0 . Then K - 0 m F Z Ž 0 . F X Ž 0 . - F Y Ž 0 . Ck Ž . 2
K ) 0 m F Z Ž 0 . F X Ž 0 . ) F Y Ž 0 . Ck Ž . 2
K s 0 m F Z Ž 0 . F X Ž 0 . s F Y Ž 0 . Ck Ž . , 2
where
¡ 11 q 6 y 2
2
Ž q 1. q2 for - y1 Ž 5 q 4. Ž y 1. Ž 5 q 4 . Ž 1 q 2 . Ck Ž . [~ 11 3 q 35 2 q 24 y 6 Ž q 1 . Ž 2 y 3 . y 2 Ž y 1 . Ž 5 2 q 15 q 12 . Ž 5 2 q 15 q 12 . Ž 1 q 2 . 2
¢
for ) 1.
Proof. Multiplying the term K given in Lemma 2.5 by the positive factor 2 Ž2r .Ž1 q 2 q 2 2 .rŽ1 q 2 . provides the results stated in the lemma. COROLLARY 2.2. Let < < ) 1 with s F X Ž0. and s k Ž ., k g ⺞ 0 . Then the periodic solutions bifurcating at s k Ž . exist 1. for ) k Ž . pro¨ ided F Z Ž0. F X Ž0. - F Y Ž0. 2 Ck Ž . Ž supercritical bifurcation.; 2. for - k Ž . pro¨ ided F Z Ž0. F X Ž0. ) F Y Ž0. 2 Ck Ž . Ž subcritical bifurcation.. Proof. Equations Ž7. and Ž8. and Lemma 2.6 provide the assertions.
435
LOCAL AND GLOBAL BIFURCATION
The next lemma deals with the properties of the sequence Ž Ck Ž .. k g ⺞ 0 . LEMMA 2.7. The following hold: 1. For - y1 or ) '3 the sequence Ž Ck Ž .. k g ⺞ 0 is positi¨ e, bounded, and strictly increasing. 2. For 1 - - '3 the sequence Ž Ck Ž .. k g ⺞ 0 is positi¨ e, bounded, and strictly decreasing. 3. For s '3 Ž Ck Ž .. k g ⺞ is a constant sequence. 0
Proof. The statements are a consequence of the definition of Ck Ž .. Remark 2.4. One can prove that Ck Ž ., k g ⺞ 0 , is bounded by 1.4887 ( 2
126 q 125'7
Ž 2 q 5'7 .Ž 35 q 2'7 .
- Ck Ž . -
11 5
for - y1
and Ck Ž . )
21 10
for ) 1.
Remark 2.5. The monotonicity and boundness of the sequence Ž Ck Ž .. k g ⺞ provide readily applicable criteria for the direction of the 0 Hopf bifurcation at s k Ž .. For instance: The monotonicity implies the assertions of Corollary 2.1.3. From the boundness we get 1. If - y1 and F Z Ž0. F X Ž0. - 2ŽŽ126 q 125'7 .rŽ2 q 5'7 .Ž35 q 2'7 .. F Y Ž0. 2 then all bifurcations are supercritical Ž K - 0.. 2. If - y1 and F Z Ž0. F X Ž0. ) 115 F Y Ž0. 2 then all bifurcations are subcritical Ž K ) 0.. Y Ž . 2 then all bifurcations are 3. If ) 1 and F Z Ž0. F X Ž0. - 21 10 F 0 supercritical Ž K - 0.. Remark 2.6. If K s 0, i.e., C s Ck Ž . Žsee proof of Corollary 2.1. or F Ž0. s F Z Ž0. s 0, one has to calculate higher derivatives of F and to compute higher order terms of the normal form of the reduced system on the center manifold. Y
2.5. Proofs of the Main Results Proof of Theorem 2.1. Theorem 2.2 provides the existence of Hopf bifurcation at s k Ž .. By applying Corollary 2.2 one gets the conditions for the direction of the Hopf bifurcation. The stability properties of the bifurcating periodic solutions follow from Lemma 2.4 and w7x.
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GIANNAKOPOULOS AND ZAPP
Proof of Corollary 2.1. If F Y Ž0. F X Ž0. F 0 then K - 0 and therefore, in this case, all bifurcations are supercritical. If F Y Ž0. s 0 and F Z Ž0. / 0 then sign K s sign F Z Ž0. F X Ž0., and thus all bifurcations are either supercritical or subcritical. Otherwise there is a unique C ) 0 with F Z Ž0. F X Ž0. s F Y Ž0. 2 C. If C - Ck Ž . then the bifurcation is supercritical; if C ) Ck Ž . the bifurcation is subcritical. By applying Lemma 2.7 it is clear that there is at most one k g ⺞ 0 where the direction of the Hopf bifurcations can change, i.e., Ž C y Cky 1Ž ..Ž C y Ckq1Ž .. - 0, provided / '3 . 3. GLOBAL CONTINUATION In this section we study the global continuation of the local Hopf branches bifurcating from the points Ž x, . s Ž0, k Ž ..; k g ⺞ 0 . We prove that each of these branches lies on an unbounded continuum Sk of periodic solutions with pairwise disjoint S k, k g ⺞ 0 . First we need some notation. We set S [ Ž , , p . g C = ⺢q= ⺢q: x Ž t . is a non-constant p-periodic solution of Ž1. with sF X Ž0. and x
p ) 2 pro¨ ided - y1 and k s 0; p gxrŽ k q 12 ., rŽ k q 14 .w pro¨ ided - y1 and k G 1; p gxrŽ k q 1., rŽ k q 34 .w pro¨ ided ) 1 and k g ⺞ 0 .
LOCAL AND GLOBAL BIFURCATION
437
Proof. To prove that Sk is unbounded we apply Theorem 3.3 in w11x Žsee also Theorem 2.5 in w32x.. The theorem mentioned above provides that, under certain assumptions, either Sk is unbounded or Sk is bounded and additionally S k l Ž x, , p . g C = ⺢q= ⺢q : x g ⺢, x s F Ž x . 4 s Ž x l , l , pl . g ⺢ = ⺢q= ⺢q : l s 0, 1, 2, . . . , n4
and
n
Ý ␥ Ž x l , l , pl . s 0,
Ž 11 .
ls0
where ␥ Ž x l , l , pl . is the so-called crossing number of Ž x l , l , pl . whose definition will be given below. Consequently, in order to prove that S k is unbounded, it suffices to show that Eq. Ž11. cannot be fulfilled. First we need to give the definition of ␥ Ž x l , l , pl . Žsee w11, 32x.. Let Ž x, , p . be an isolated center of Ž1.; i.e., x s F Ž x ., Ž ⌬ iŽ2rp .; , F X Ž x .. s 0, and there exists a neighborhood UŽ x, , p . of Ž x, , p . such that Ž x, , p . is the only center in UŽ x, , p .. Using the definition of ⌬ Žsee Eq. Ž6.. and Lemma 2.3 one can show that ⌬Ž z; , . is analytic in z g ⺓ and continuous in gx y ⑀ 0 , q ⑀ 0 w, where ⑀ 0 ) 0 is an appropriate positive constant and s F X Ž x . Žsee Condition Ž A3. in w32x.. Moreover, it follows from Lemma 2.1 and 2.3 that there exist small positive constants ␦ , ⑀ gx0, ⑀ 0 w such that on ⭸ ⍀ ⑀ , p = w y ␦, q ␦ x ⌬ aqi
ž
2 p
, , s 0 if and only if s , a s 0, and p s p,
/
where ⍀ ⑀ , p [ Ž a, p . : 0 - a - ⑀ , p g x p y ⑀ , p q ⑀ w 4
Ž see Condition Ž A4 . in w 32x . . Notice that z s "iŽ2rp . are the only zeroes of ⌬Ž z; , . on the imaginary axis Žsee Lemma 2.1.. We define H Ž a, p . [ ⌬ a q i
ž
2 p
,, .
/
H : ⍀ ⑀ , p ª ⺢ 2 ( ⺓ is a differentiable function of Ž a, p . g ⍀ ⑀ , p . Now the crossing number ␥ Ž x, , p . can be given by
␥ Ž x, , p . [ deg Ž Hy ␦ , ⍀ ⑀ , p . y deg Ž Hq ␦ , ⍀ ⑀ , p . ,
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GIANNAKOPOULOS AND ZAPP
where deg denotes the Brouwer degree with respect to Ž0, 0. g ⺢ 2 ( ⺓ and ⍀ ⑀ , p Žsee w11, 32x.. This implies that the crossing number ␥ Ž x, , p . is well defined. A calculation shows det Ž DH Ž a, p . <Ž a, p .gHy1 Ž0, 0. . s y
2 p2
ž
2 Ž1 q a q . q
4 2 2 p2
/
- 0.
Using the definition of the degree Žsee w6, p. 65x. we obtain deg Ž H , ⍀ ⑀ , p . s
Ý
Ž a,
p .gHy1 Ž0, 0 .
Ž y1. s y噛Hy1 Ž 0, 0 . ,
where 噛Hy1 Ž0, 0. denotes the number of elements in the set Hy1 Ž0, 0., provided Hy1 Ž0, 0. consists of only a finite number of points in ⍀ ⑀ , p . Ž . Lemma 2.4 provides that, for sufficiently small ␦ and ⑀ ) 0, Hy1 y ␦ 0, 0 s ⭋ Ž . and Hy1 0, 0 consists of only one point in ⍀ . This implies q␦ ⑀, p
␥ Ž x, , p . s y1, provided Ž x, , p . is an isolated center. If Ž x, , p . is not a center, we can easily prove that
␥ Ž x, , p . s 0. Now, in order to be able to apply Theorem 3.3 in w11x Žsee also Theorem 2.5 in w32x., we still have to prove that all centers of Ž1. are isolated. Lemma 2.2 provides that if x g ⺢ is a stationary solution, all centers Ž x, k , pk ., k g ⺞ 0 , corresponding to x, are isolated. By assumption ŽGH2. this yields that all the centers are isolated, and thus we have proved that Sk is unbounded and Ž x, , p . g S k if and only if x s 0, s k Ž ., and p s pŽ . with s F X Ž0.. By applying Lemma 4.1 in w8x to Ž4., we get that Eq. Ž1. has no non-constant periodic solutions of period 2m for any m g ⺞. From Theorem 2.1 and Lemma 2.1 we know p Ž . s
2 bk Ž .
k Ž . , k g ⺞0 ,
where bk Ž . g x Ž 2 k q
3 2
. , Ž 2 k q 2. w
for k g ⺞ 0 , ) 1
and bk Ž . g x Ž 2 k q
1 2
. , Ž 2 k q 1. w
for k g ⺞ 0 , - y1.
LOCAL AND GLOBAL BIFURCATION
439
This provides p Ž . ) 2 0 Ž . p Ž . g
k Ž . kq
1 2
,
for k s 0, if - y1,
k Ž . kq
1 4
for any k g ⺞, if - y1
and p Ž . g
k Ž . kq1
,
k Ž . kq
3 4
for any k g ⺞ 0 , if ) 1,
and thus we have shown that if Ž , , p . g Sk and / 0, the minimal period p of the periodic solution x Ž t . of Ž1. with x
Assume ŽGH1., ŽGH2., and ŽGH3. are satisfied. Then
1. For s F X Ž0. - y1 there exists ˜ k Ž . gx0, k Ž .x such that for any G˜ k Ž . there are g C and p gxrŽ k q 12 ., rŽ k q 14 .w such that Ž , , p . g Sk, k g ⺞. Furthermore, if Ž , , p . g S k, k g ⺞, and x Ž t . is the unique existing p-periodic solution of Ž1. with x
440
GIANNAKOPOULOS AND ZAPP
From Corollary 1.1 in w23x we know, because of the invariance condition ŽGH3., that if x Ž t . is a non-constant periodic solution of Ž1. satisfying x Ž t . g I for all t G y then x Ž t . g intŽ I . for all t G y . On the other hand, using the local Hopf bifurcation theorem 2.1, we can prove that for ) 0 close to k Ž . there exists a non-constant p-periodic solution x Ž t . with x
xF Ž x . - 0 for x g I _ 04 .
Notice that in our proof we do not need the assumption ŽGH4.. On the other hand, ŽGH1. is stronger than ŽGH1X .. 2. Results similar to Corollary 3.1 are presented in w32, Theorem 4.1, x 4.2 under the additional assumption that F is a bounded and monotone function with maximum slope at x s 0. Notice that the proof provided in w32, Remark 4.8x is only valid for F X Ž0. - 1 Žsee Eq. Ž4.4. in w32x.. Remark 3.3. In order to be able to prove that the -component of S0, where - y1, is unbounded, we need the additional assumption that F satisfies the negative feedback condition ŽGH4. Žsee Remark 3.2.1.. More precisely, Theorem 1.1 in w23x provides: Let ŽGH1X ., ŽGH3., and ŽGH4. be satisfied. Then for - y1 there exists ˜ 0 Ž . gx0, 0 Ž .w such that for any )˜ 0 Ž . there are a g C Žwy , 0x, I . and a p ) 2 with Ž , , p . g S0. Furthermore, if Ž , , p . g S0 and x Ž t . is the unique solution of Ž1. with x
441
LOCAL AND GLOBAL BIFURCATION
4. EXAMPLES In this last section we present and discuss five examples which illustrate our local ŽTheorem 2.1 and Corollary 2.1. and global results ŽTheorem 3.1 and Corollary 3.1.. Furthermore we provide bifurcation diagrams by using a numerical continuation method for delay differential equations Žsee w10x. where the periodic solutions are approximated by a Fourier polynomial. The last two examples indicate what can happen if the invariance condition ŽGH3. or feedback condition ŽGH4. is violated. 1. In our first example we use the sigmoid nonlinearity
F Ž x . s F1 Ž x . [
3
1
2 1 q exp Ž y4 x .
y
3 4
.
It follows that F is bounded by y 34 - F Ž x . - 34 for all x g ⺢, is strictly increasing, and has exactly one turning point at 0 Ž F Y Ž0. s 0, F Z Ž0. - 0.. Moreover it holds F Ž0. s 0 and s F X Ž0. s 32 ) 1. F has three fixed points: x s 0, ˜ x ( 0.644, and yx. ˜ By Theorem 2.1 we get that a supercritical Hopf bifurcation occurs at Ž x, . s Ž0, k Ž .., k g ⺞ 0 . Corollary 3.1 provides a ˜ k Ž . gx0, k Ž .x, k g ⺞ 0 , such that for all ) ˜ k Ž . there is a periodic solution x Ž t . belonging to Sk. Furthermore we know that the period p of x Ž t . satisfies p gxrŽ k q 1., rŽ k q 34 .w. Figures 3 and 4 show the numerical results. One can observe that the amplitude of the solution is strictly increasing in . 2. Now we use the nonlinearity F Ž x . s yF1Ž x . with F1 as in the first example. x s 0 is the only fixed point of F and it holds s F X Ž0. s y Y Z 3 Ž . Ž0. ) 0. Theorem 2.1 yields that a supercritical 2 - y1, F 0 s 0, F Hopf bifurcation occurs at s k Ž ., k g ⺞ 0 . Since F satisfies ŽGH1., ŽGH2., ŽGH3., and ŽGH4. with I s wyx, ˜ ˜x x with ˜x as in Example 1, from Corollary 3.1 and Remark 3.3, we obtain that the branches of periodic solutions Sk are unbounded in . For the period p of the solution x Ž t . belonging to Sk, p ) 2 for k s 0 and p gxrŽ k q 12 ., rŽ k q 14 .w for k G 1. Numerical simulations indicate that the amplitude of the solution is strictly increasing in and for every ) 0 Ž . there is a unique slowly oscillating periodic solution. These properties for the slowly oscillating periodic solutions can be proved analytically. Applying Corollary 2.1 in w5x one can prove the uniqueness of the slowly oscillating periodic solution for ) 0 Ž .. For F 0 Ž . there are no slowly oscillating periodic solutions. Moreover Corollary 2.1 in w5x provides the monotonicity property of the branch of slowly oscillating periodic solutions S0 as function of ) 0 Ž .: The orbit ⌫ Ž . of the unique existing slowly oscillating periodic solution in
442
GIANNAKOPOULOS AND ZAPP
FIG. 3. Branches of periodic solutions in the Ž , < x <.-plane. All bifurcations are supercritical and the corresponding periodic solutions are bounded by ˜ x Žsee Example 1..
p
FIG. 4. Branches of periodic solutions in the Ž , .-plane Žsee Example 1..
LOCAL AND GLOBAL BIFURCATION
443
the Ž x, ˙ x .-plane possesses the following property. ⌫ Ž 1 . is in the closure of the exterior of the orbit ⌫ Ž 2 . whenever 2 ) 1 ) 0 Ž .. 3. This example demonstrates the change of the Hopf bifurcation direction Žsee Corollary 2.1.. We consider Ž1. with F Ž x . s y575 arctan Ž x q 10'5 . q 575 arctan Ž 10'5 . . It holds F Ž0. s 0, [ F X Ž0. s y575rŽ1 q 500. - y1, F Y Ž0. s Ž11500r251,001.'5 , F Z Ž0. s y1,723,850r125,751,501. So we get C s Ž F Z Ž0. F X Ž0.rF Y Ž0. 2 . s 1499r1000 s 1.4990 Žsee proof of Corollary 2.1.. It follows that 1.49716 ( C0 Ž . - C - C1 Ž . ( 1.49952. Applying Theorem 2.1 we can show that a Hopf bifurcation occurs at s k Ž ., k g ⺞ 0 . The first Hopf bifurcation at s 0 Ž . is subcritical; all other Hopf bifurcations are supercritical Žsee Figs. 5 and 6.. 4. Here we discuss an example where the function F does not satisfy condition ŽGH3.. Consider F Ž x . s y15 sinh Ž x . . It holds that F Ž0. s 0, [ F X Ž0. s y15, F Y Ž0. s 0, and F Z Ž0. s y15 F is strictly decreasing and satisfies conditions ŽGH1. and ŽGH2. but does not satisfy condition ŽGH3.. x s 0 is the only fixed point of F. Using Theorem 2.1 we get that a subcritical Hopf bifurcation occurs at s k Ž ..
FIG. 5. Subcritical Hopf bifurcation at s 0 Ž . s 4.66712 Žsee Example 3..
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GIANNAKOPOULOS AND ZAPP
FIG. 6. Supercritical Hopf bifurcation at s 1Ž . s 15.82280 Žsee Example 3..
From Theorem 3.1 we know that the branches of periodic solutions S k bifurcating from Ž x s 0, s k Ž .. are unbounded but we do not know which component of S k is unbounded. Actually the numerically computed bifurcation diagram Žsee Fig. 7. shows that the branches are bounded in . The numerically computed p-periodic solutions x k Ž t; . belonging to Sk exist only for 0 - - k Ž . and < x k < ª ⬁ as ª 0. Furthermore for the p period p of x 0 Ž t; . we get ª 4 as ª 0.
FIG. 7. Branches of periodic solutions for F Ž x . s y15 sinhŽ x . Žsee Example 4..
445
LOCAL AND GLOBAL BIFURCATION
5. The last example deals with the case in which F does not satisfy the negative feedback condition ŽGH4.. We consider Eq. Ž1. with FŽ x. s 2
xq1 1 q Ž x q 1.
10
y 2;
see w22x. This equation was introduced by Mackey and Glass as a mathematical model for the generation of white blood cells. F is a bounded function with F Ž0. s 0, s F X Ž0. s y4 - y1, F Y Ž0. s y5, and F Z Ž0. s 255. Using Theorem 2.1 we get that a supercritical Hopf bifurcation takes place at Ž x s 0, s k Ž .., k g ⺞ 0 . From Corollary 3.1 we know that for k G 1 the -component of Sk is unbounded. Since ŽGH4. is not satisfied we cannot investigate S0 by applying Theorem 1.1 in w23x Žsee Remark 3.3.. In w14, 18, 22x the Mackey᎐Glass equation is studied numerically. Numerical simulations show that the branch S0 of slowly oscillating periodic solutions undergoes a cascade of period doubling bifurcations. Moreover numerical calculations indicate the existence of chaotic attractors for sufficiently large delays .
APPENDIX: PROOF OF LEMMA 2.5 Let s k Ž . and b s bk Ž ., k g ⺞ 0 . We compute the factor K Žsee Eq. Ž10.. in several steps. 1. First, notice that the following equalities hold: cos Ž b . s sin Ž b . s
1
cos Ž 2 b . s
'2 y 1
2 y 2
sin Ž 2 b . s 2
< <
2
'2 y 1 < <
cos Ž 3b . s sin Ž 3b . s
b s '2 y 1 . 2. We get 1Ž0. s
1 
with  [ 1 q eyi b. This provides
<  < 2 s  s Ž 1 q eyi b . Ž 1 q e i b . s 1 q 2 cos Ž b . q 22 s 1 q 2 q 22 .
4 y 3 2
3
'2 y 1 2 < <
446
GIANNAKOPOULOS AND ZAPP
3. We set C1 [ ReŽ eyi b 1Ž0... It holds that C1 s Re s Re s
eyi b <  <2 eyi b q 1 q 2 q 22 1 q 2
.
Ž 1 q 2 q 22 .
4. y
Re Ž eyi b 1 Ž 0 . . L Ž 1.
sy
1 q 2
Ž 1 q 2 q 22 . Ž y 1 .
\ T2 .
5.
␣ [ 2 ib y Ž y1 q ey2 i b . s y cos Ž 2 b . q i Ž 2 b q sin Ž 2 b . . s 1y
ž
s
2 y 2
2
q i 2 b q 2
/ ž
Ž q 2. Ž y 1. q i
'2 y 1
2'2 y 1 < <
< <
/
Ž < < q 1. .
Therefore
␣s
Ž q 2. Ž y 1. y i
2'2 y 1 < <
Ž < < q 1.
and it follows that
2
< ␣ < 2 s ␣␣ s s
2
2 2 Ž q 2. Ž y 1. q
2 Ž y 1.
4 2 Ž 2 y 1 . < < 2
Ž < < q 1.
2
Ž Ž y 1. Ž q 2. 2 q 4 Ž q 1. Ž < < q 1. 2 . .
We distinguish between ) 1 and - y1. Ža. The case ) 1: <␣ <2 s s
2
2 Ž y 1. 2 2 Ž y 1.
Ž Ž y 1 . Ž 2 q 4 q 4 . q 4 Ž 3 q 3 2 q 3 q 1 . . Ž 5 2 q 15 q 12 . .
447
LOCAL AND GLOBAL BIFURCATION
Žb. The case - y1:
<␣ <2 s s
2 Ž y 1.
2
Ž 2 q 4 q 4 q 4 Ž 2 y 1 . .
2 2 Ž y 1.
2
Ž 5 q 4. .
6. We compute two auxiliary terms A and B.
A [ cos Ž 3b . q cos Ž 2 b . s s
3
q
2 y 2
2
4 y 3 2 q 2 Ž 2 y 2 .
3
B [ sin Ž 3b . q sin Ž 2 b . s s
4 y 3 2
'2 y 1 2 < <
'2 y 1 Ž 4 y 2 . 2 < <
q 2
'2 y 1 < <
Ž 4 y 2 q 22 . .
7. It holds that Re Ž ey3 i b␣ .
ž ž
s Re ey3 i b Ž 1 q e i b .
s Re Ž A y iB .
s s
A
ž
ž
Ž q 2. Ž y 1. y i
Ž y 1. Ž q 2. y
Ž y 1. 4
Ž q 2. Ž y 1. y i
2'2 y 1 < <
2'2 y 1
2'2 y 1 < <
< <
Ž < < q 1.
Ž < < q 1.
//
//
B Ž < < q 1.
Ž Ž q 2 . Ž 4 y 3 2 . y 2 Ž q 1 . Ž < < q 1 . Ž 4 y 2 . q2 Ž Ž 2 y 2 . Ž q 2 . y 4 Ž q 1 . Ž < < q 1 . . . .
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GIANNAKOPOULOS AND ZAPP
8. We set T3 [ Re
ey3 i b 1 Ž 0 . L Ž e 2 i b .
s
Re Ž ey3 i b␣ . <␣ <2 <  <2
.
Ža. The case ) 1:
Ž q2 . Ž 4y3 2 y2 Ž q1 . 2 Ž 2y . . q 2 Ž Ž 2y 2 .Ž q2 . y4 Ž q1 . 2 . T3 s . 3 Ž 5 2 q15 q12 .Ž 1q2 q 22 . Žb. The case - y1: T3 s
Ž q2 . Ž 4y3 2 y2 Ž q1 . 2 Ž 2y . . q 2 Ž Ž 2y 2 .Ž q2 . q4 Ž 2 y1 . . . 3 Ž y1 .Ž 5 q4 .Ž 1q2 q 22 .
9. Now we calculate C2 [ T2 q 12 T3 : Ža. The case ) 1: C2 s
y2 Ž 1 q 2 . 2 Ž 5 2 q 15 q 12 . 2 Ž y 1 .Ž 5 2 q 15 q 12 .Ž 1 q 2 q 22 . 3
Ž y 1 . Ž Ž q 2 . Ž 4 y 3 2 y 2 Ž q 1 . 2 Ž 2 y . . 2 q2 Ž Ž 2 y 2 .Ž q 2 . y 4 Ž q 1 . . . q 23 Ž y 1 .Ž 5 2 q 15 q 12 .Ž 1 q 2 q 22 . s
2 Ž y11 3 y 35 2 y 24 q 6 . q Ž 2 4 y 11 3 y 43 2 y 24 q 12 . 22 Ž y 1 .Ž 5 2 q 15 q 12 .Ž 1 q 2 q 22 .
.
Žb. The case - y1: 2
C2 s
y2 Ž 1 q 2 . 2 Ž 5 q 4 . q Ž q 2 . Ž 4 y 3 2 y 2 Ž q 1 . Ž 2 y . . 23 Ž y 1 . Ž 5 q 4 . Ž 1 q 2 q 22 . q
s
2 Ž Ž 2 y 2 . Ž q 2 . q 4 Ž 2 y 1 . . 23 Ž y 1 . Ž 5 q 4 . Ž 1 q 2 q 22 .
2 Ž y11 2 y 6 q 2 . q Ž y2 3 y 13 2 y 4 q 4 . 22 Ž y 1 . Ž 5 q 4 . Ž 1 q 2 q 22 .
.
10. Finally we compute K s Ž F Z Ž0.r2.C1 q F Y Ž0. 2 2 C2 and the proof is finished.
LOCAL AND GLOBAL BIFURCATION
449
REFERENCES 1. U. an der Heiden, Delay in physiological systems, J. Math. Biol. 8 Ž1979., 345᎐364. 2. U. an der Heiden and M. C. Mackey, The dynamics of production and destruction: Analytic insight into complex behaviour, J. Math. Biol. 16 Ž1982., 75᎐101. 3. J. Belair, Stability in a model of a delayed neural network, J. Dynam. Differential ´ Equations 5 Ž1993., 607᎐623. 4. F. G. Bose, The stability chart for the linearized Cushing equation with a discrete delay and with gamma-distributed delays, J. Math. Anal. Appl. 140 Ž1989., 510᎐536. 5. Y. Cao, Uniqueness of the periodic solution for differential delay equations, J. Differential Equations 128 Ž1996., 46᎐57. 6. S.-N. Chow and J. K. Hale, ‘‘Methods of Bifurcation Theory,’’ Springer-Verlag, New York, 1982. 7. S.-N. Chow and J. Mallet-Paret, Integral averaging and bifurcation, J. Differential Equations 26 Ž1977., 112᎐159. 8. S.-N. Chow and J. Mallet-Paret, The Fuller index and global Hopf bifurcation, J. Differential Equations 29 Ž1978., 66᎐85. 9. O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. O. Walther, ‘‘Delay Equations, Functional-, Complex- and Nonlinear Analysis,’’ Springer-Verlag, New York, 1995. 10. E. J. Doedel and P. C. Leung, A numerical technique for bifurcation problems in delay differential equations, Congr. Numer. 34 Ž1982., 225᎐237. 11. L. H. Erbe, W. Krawcewicz, K. Geba, and J. Wu, S 1-degree and global Hopf bifurcation theory of functional differential equations, J. Differential Equations 98 Ž1992., 227᎐298. 12. T. Faria and L. T. Magalhaes, ˜ Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations 122 Ž1995., 181᎐200. 13. T. Faria and L. T. Magalhaes, ˜ Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities, J. Dynam. Differential Equations 8 Ž1995., 35᎐70. 14. J. Doyne Farmer, Chaotic attractors of an infinite-dimensional dynamical system, Phys. D 4 Ž1982., 366᎐393. 15. H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Observation of chaos in optical bistability, Phys. Re¨ . Lett. 46 Ž1981., 474᎐477. 16. L. Glass and M. C. Mackey, ‘‘From Clocks to Chaos,’’ Princeton Univ. Press, Princeton, NJ, 1988. 17. K. P. Hadeler and J. Tomiuk, Periodic solutions of difference᎐differential equations, Arch. Rational Mech. Anal. 65 Ž1977., 87᎐95. 18. J. K. Hale and N. Sternberg, Onset of chaos in differential delay equations, J. Comput. Phys. 77 Ž1988., 271᎐287. 19. J. K. Hale and S. M. Verduyn Lunel, ‘‘Introduction to Functional Differential Equations,’’ Springer-Verlag, New York, 1993. 20. N. D. Hayes, Roots of the transcendential equation associated with a certain difference᎐differential equation, J. London Math. Soc. 25 Ž1950., 226᎐232. 21. K. Ikeda, K. Kondo, and O. Akimoto, Successive higher-harmonic bifurcations in systems with delayed feedback, Phys. Re¨ . Lett. 49 Ž1982., 1467᎐1470. 22. M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science 197 Ž1977., 287᎐289. 23. J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential᎐delay equation, Ann. Mat. Pura. Appl. Ž4. 145 Ž1986., 33᎐128.
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24. J. Mallet-Paret and R. D. Nussbaum, A differential᎐delay equation arising in optics and physiology, SIAM J. Math. Anal. 20 Ž1989., 249᎐292. 25. J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations 125 Ž1996., 385᎐440. 26. C. M. Marcus and R. M. Westervelt, Stability of analog neural networks with delay, Phys. Re¨ . A 39 Ž1989., 347᎐359. 27. M. Martelli, K. Schmitt, and H. L. Smith, Periodic solutions of some nonlinear delay-differential equations, J. Math. Anal. Appl. 74 Ž1980., 494᎐503. 28. J. Milton, ‘‘Dynamics of Small Neural Populations,’’ CRM Monograph Series, No. 7, Amer. Math. Soc., Providence, RI, 1996. 29. R. D. Nussbaum, A Hopf global bifurcation theorem for retarded functional differential equations, Trans. Amer. Math. Soc. 238 Ž1978., 139᎐164. 30. H. W. Stech, Hopf bifurcation calculations for functional differential equations, J. Math. Anal. Appl. 109 Ž1985., 472᎐491. X 31. H.-O. Walther, The 2-dimensional attractor of x Ž t . s y x Ž t . q f Ž x Ž t y 1.., Mem. Amer. Math. Soc. 544 Ž1995.. 32. J. Wu, Delay-induced discrete waves of large amplitudes in neural networks with circulant connection matrices, preprint Ž1997.. 33. J. Wu and X. Zou, Patterns of sustained oscillations in neural networks with delayed interactions, Appl. Math. Math. Comput. 73 Ž1995., 55᎐75.
Local and Global Hopf Bifurcation in a Scalar Delay Differential Equation Fotios Giannakopoulos* and Andreas Zapp Mathematical Institute, Uni¨ ersity of Cologne, Weyertal 86-90, D-50931 Cologne, Germany E-mail: [email protected], [email protected] Submitted by U. Kirchgraber Received September 3, 1998
1. INTRODUCTION In this paper we study the bifurcation properties of Hopf branches in the scalar retarded functional differential equation
˙x Ž t . s yx Ž t . q F Ž x Ž t y . .
Ž 1.
with the time delay ) 0 as bifurcation parameter, where F g C k Ž⺢, ⺢., k G 3, and F Ž0. s 0. Equation Ž1. is a prototype for a retarded functional differential equation which has many applications in sciences. Special cases of Ž1. have been proposed as models of both physical and physiological phenomena Žsee, for instance w1, 15, 21, 26x; see also w16, pp. 68᎐81; 24; 28x.. For a wide class of functions F Že.g., monotone F . the dynamics of Ž1. are determined by periodic solutions Žsee w25, 31x.. In the applications periodic dynamics are also of particular interest Žsee w16, pp. 57᎐81x.. There exist many papers which are concerned with the existence of periodic solutions of Ž1. and explore their dependence on the time delay ) 0 Žsee w5, 17, 23, 27, 31, 32x and the references in w23, 32x.. The existence of a sequence of time delays 0 - 0 - 1 - ⭈⭈⭈ , with k ª ⬁ as k ª ⬁, such that a local Hopf bifurcation occurs at s k , provided < F X Ž0.< ) 1, is proved in w32x Žsee also w27x.. In the case F X Ž0. - y1 a condition for supercritical Hopf bifurcation at s 0 is given in w3x. In w33x *Present address: GMD-German Research Center for Information Technology, Schloss Birlinghoven, D-53754 Sankt Augustin, Germany. 425 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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GIANNAKOPOULOS AND ZAPP
conditions for both supercritical and subcritical Hopf bifurcations at s 0 are provided for F X Ž0. - y1, F Y Ž0. s 0, and F Z Ž0. / 0. The global continuation for all ) k of the locally existing Hopf branches at s k is demonstrated in w23x, provided that there exists a compact interval I ; ⺢, 0 g intŽ I ., such that F Ž I . ; I Žinvariance condition., xF Ž x . - 0 for all x g I _ 04 , and F X Ž0. - y1 Žnegative feedback condition.. However, many examples of functions F which do not satisfy the conditions mentioned above arise in applications Žsee w2; 15; 16, pp. 68᎐81; 21; 24x.. Furthermore, from the point of view of qualitative analysis, it is also of interest to know how the properties of Eq. Ž1. change if one of the conditions above fails. The main objective of this work is to extend the results mentioned above to a more general class of functions F. More precisely, we give both sufficient and necessary conditions for the local existence and the bifurcation direction of Hopf branches and for the stability of the periodic solutions bifurcating from the trivial solution x s 0 at s k , k g ⺞ 0 , provided F g C 3. The Hopf bifurcation formula derived here has a readily applicable form and provides all possible bifurcation scenarios. For the derivation of this formula we consider a system of two ordinary differential equations describing the flow of Ž1. on an appropriate center manifold Žsee Sect. 2.. Furthermore, we present a result on the global continuation of the local Hopf branches for functions F g C 2 , where F X Ž x . / 1 for all fixed points x of F and < F X Ž0.< ) 1 Žsee Sect. 3.. Finally, we discuss several examples which illustrate our results and indicate what can happen if the invariance condition or the negative feedback condition is violated Žsee Sect. 4..
2. LOCAL BEHAVIOR Throughout this section we assume ŽLH.
F g C 3, F Ž0. s 0, and < F X Ž0.< ) 1.
2.1. Main Results In this subsection we state the main results on the local Hopf bifurcation. First we need some definitions. DEFINITION 2.1. For k g ⺞ 0 let bk , k: ⺢ _ w y1, 1 x ª ⺢q
LOCAL AND GLOBAL BIFURCATION
427
be positive functions of defined by
¡arccos
bk Ž . [
~
1
ž /
q 2 k
for - y1
Ž 2.
¢2Ž k q 1. y arccos ž / 1
k Ž . [
for ) 1
bk Ž .
Ž 3.
'2 y 1
with arccosŽ x . in Ž0, .. Furthermore we define
¡ 11 q 6 y 2
2
Ž q 1. q2 for - y1 Ž 5 q 4. Ž y 1. Ž 5 q 4 . Ž 1 q 2 . Ck Ž . [~ 11 3 q 35 2 q 24 y 6 Ž q 1 . Ž 2 y 3 . y 2 Ž y 1 . Ž 5 2 q 15 q 12 . Ž 5 2 q 15 q 12 . Ž 1 q 2 . 2
¢
for ) 1
with s k Ž ., k g ⺞ 0 . THEOREM 2.1 ŽLocal Hopf Bifurcation..
Let < < ) 1 with s F X Ž0..
1. At s k Ž ., k g ⺞ 0 , Eq. Ž1. undergoes a Hopf bifurcation; that is, in e¨ ery small neighborhood of Ž x s 0, s k Ž .. there is a unique branch of periodic solutions x k Ž t; . with x k Ž t; . ª 0 as ª k Ž .. For the period pk Ž , . of x k Ž t; ., pk Ž , . ª 2 k Ž .rbk Ž . s 2r '2 y 1 \ pŽ . as ª k Ž .. 2. Assume F Z Ž 0 . F X Ž 0 . - F Y Ž 0 . Ck Ž . . 2
Then the bifurcating branch of periodic solutions exists for ) k Ž . Ž supercritical bifurcation.. Moreo¨ er the arising periodic solutions are stable if
- y1
and k s 0
and they are unstable if
- y1 3.
and k G 1
or
) 1.
Assume F Z Ž 0 . F X Ž 0 . ) F Y Ž 0 . Ck Ž . . 2
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GIANNAKOPOULOS AND ZAPP
Then the bifurcating branch of periodic solutions exists for - k Ž . Ž subcritical bifurcation.. All periodic solutions on this branch are unstable. Proof. The proof is given in Section 2.5. Remark 2.1. 1. For < < F 1 there are no Hopf bifurcation points. 2. s ˜ is a Hopf bifurcation point if and only if ˜ s k Ž . with k g ⺞0 . Using the properties of the function Ck Ž . we get the following criteria for the direction of the bifurcating periodic solutions Žsee Fig. 1.. COROLLARY 2.1. Let F Y Ž0. and F Z Ž0. be not simultaneously equal to zero and < < ) 1 with s F X Ž0.. 1. 2.
If F Z Ž0. F X Ž0. F 0 then all bifurcations are supercritical. If F Y Ž0. s 0 and F Z Ž0. F X Ž0. ) 0 then all bifurcations are subcriti-
cal. If F Y Ž0. / 0 and F Z Ž0. F X Ž0. ) 0 then Ža. For g Žy⬁, y1. j Ž'3 , ⬁. the Hopf bifurcations occurring at s k Ž . are either supercritical for all k g ⺞ 0 or subcritical for all k g ⺞ 0 or there is a unique k c g ⺞ such that the Hopf bifurcations occurring at s k Ž . are subcritical for k - k c and supercritical for k ) k c . Žb. For g Ž1, '3 . the Hopf bifurcations occurring at s k Ž . are either supercritical for all k g ⺞ 0 or subcritical for all k g ⺞ 0 , or there is a unique k c g ⺞ such that the Hopf bifurcations occurring at s k Ž . are supercritical for k - k c and subcritical for k ) k c . 3.
FIG. 1. All possible bifurcation scenarios Žsee Corollary 2.1..
LOCAL AND GLOBAL BIFURCATION
429
Žc. For s '3 and F Z Ž0. F X Ž0. / F Y Ž0. 2 Ck Ž'3 . the Hopf bifurcations occurring at s k Ž . are either supercritical or subcritical for all k g ⺞0 . Proof. The proof can be found in Section 2.4. 2.2. The Characteristic Equation The objective of this section is to study the properties of the characteristic equation associated with the linearization of Ž1.. For the sake of simplicity we rescale the time t by t ¬ t . This provides a scalar delay differential equation with constant time delay 1,
˙x Ž t . s y x Ž t . q F Ž x Ž t y 1 . . .
Ž 4.
After linearization about the equilibrium x s 0 we get
[ F X Ž 0. .
˙x Ž t . s y x Ž t . q x Ž t y 1 . ,
Ž 5.
The characteristic equation of Ž5. is given by 0 s ⌬ Ž z ; , . [ z q y exp Ž yz . .
Ž 6.
The analysis of Ž6. yields conditions for the stability of the equilibrium x s 0 and critical parameter values at which a Hopf bifurcation can occur. Much is known about the characteristic equation ⌬ s 0; for instance, see w4, 9, 20, 23, 27x. Here we present only some relevant results for completeness. LEMMA 2.1. The characteristic equation Ž6. has a pair of purely imaginary, conjugate complex solutions z s "ib, b ) 0, if and only if < < ) 1 and b s bk Ž ., s k Ž ., k g ⺞ 0 . bk , k are defined by Ž2. and Ž3.. The functions bk , k are C 1, and bk satisfy bk Ž . g x Ž 2 k q
. , Ž 2 k q 2. w bk Ž . g x Ž 2 k q 12 . , Ž 2 k q 1 . w 3 2
for k g ⺞ 0 , ) 1, for k g ⺞ 0 , - y1.
Proof. See w9, pp. 305᎐309x, Proposition A.2 in w23x, or Lemma 3.1 in w32x. The estimates for bk follow from Definition 2.1. LEMMA 2.2 ŽProperties of k .. k is a decreasing function of ) 1 and an increasing function of - y1, k g ⺞ 0 . Furthermore, the following hold for k g ⺞ 0
kq 1 Ž . y k Ž . s
2
'
2
y1
᭙ g ⺢ _ w y1, 1 x
Ž monotonicity in k .
430
GIANNAKOPOULOS AND ZAPP
and lim k Ž . s 0
ª"⬁
lim k Ž . s ⬁.
ª"1
Proof. The assertions follow from Definition 2.1 and Lemma 2.1. LEMMA 2.3. Let < < ) 1. Then there are an ⑀ ) 0 and a simple characteristic root z Ž . s aŽ . q ibŽ . of Ž6. for < y k Ž .< - ⑀ , k g ⺞ 0 . z is C 1 in and aŽ k Ž .. s 0, bŽ k Ž .. s bk Ž .. Moreo¨ er dd aŽ k Ž .. ) 0. Proof. See Lemma 3.1 in w32x. LEMMA 2.4 ŽD-partition of the Ž , .-Plane.. The characteristic equation Ž6. has 1. only solutions with negati¨ e real part, pro¨ ided G 0 and g wy1, 1. or 0 F - 0 Ž . and - y1; 2. exactly one solution with positi¨ e real part, pro¨ ided 0 F F 0 Ž . with ) 1; 3. exactly 2 k solutions with positi¨ e real part, pro¨ ided ky 1Ž . - F k Ž . with - y1, k g ⺞; 4. exactly 2 k q 1 solutions with positi¨ e real part, pro¨ ided ky 1Ž . F k Ž . with ) 1, k g ⺞. Ž See Fig. 2.. Proof. The proof is an application of XI.2 in w9x. ŽSee also Proposition A.2 in w23x.. 0 Remark 2.2. If Ž , . g wy1, 1. = ⺢q j Žy⬁, y1. = w0, 0 Ž .. the 0 equilibrium x s 0 is asymptotically stable. If Ž , . g Ž1, ⬁. = ⺢q j Žy⬁, y1. = Ž 0 Ž ., ⬁. the equilibrium x s 0 is unstable.
2.3. Hopf Bifurcation In the following theorem we show that a Hopf bifurcation occurs at s k Ž .. The proof is an application of the Hopf theorem for retarded functional differential equations Žsee w19, p. 332x.. In w27, 32x the existence of Hopf bifurcations is proved in a similar way. THEOREM 2.2. Let < < ) 1 with s F X Ž0. and k g ⺞ 0 . Then a Hopf bifurcation for Eq. Ž1. occurs at s k Ž .; i.e., in e¨ ery small neighborhood of Ž x s 0, s k Ž .. there is a unique branch of periodic solutions x k Ž t; . with x k Ž t; . ª 0 as ª k Ž .. For the period pk Ž , . of x k Ž t; ., pk Ž , . ª 2 k Ž .rbk Ž . s 2r '2 y 1 \ pŽ . as ª k Ž ..
LOCAL AND GLOBAL BIFURCATION
431
FIG. 2. D-partition of the Ž , .-plane. In every region bounded by s k Ž ., resp. s 1, the number of characteristic solutions with positive real part is constant Žsee Lemma 2.4..
Proof. Lemma 2.1 and Lemma 2.3 provide the assumptions of the Hopf᎐Theorem Žsee w19, p. 332x. and thus the above statements are proved. Remark 2.3. By Lemma 2.1 we get that only at s 0 Ž . with - y1 can the Hopf bifurcation provide slowly oscillating periodic solutions Ži.e., p 0 Ž , . ) 2 .. For - y1 and k G 1, pk Ž , . gxŽrŽ k q 12 ., rŽ k q 14 .w, and for ) 1 and k g ⺞ 0 , pk Ž , . gxrŽ k q 1., rŽ k q 43 .w. For the definition of slowly oscillating periodic solutions see w23x. 2.4. Direction of Hopf Bifurcation In order to be able to analyse the Hopf bifurcation in more detail we compute the reduced system on the center manifold associated with the pair of conjugate complex, purely imaginary solutions ⌳ s ibk Ž ., yibk Ž .4 of the characteristic equation Ž6.. By this reduction we are able to determine the Hopf bifurcation direction, i.e., to answer the question of whether the bifurcating branch of periodic solution exists locally for
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GIANNAKOPOULOS AND ZAPP
) k Ž . Žsupercritical bifurcation . or - k Ž . Žsubcritical bifurcation .. In the sequel we write bk , k instead of bk Ž . and k Ž .. In general it is difficult to compute the center manifold itself, but in w12x a method is presented providing the reduced system on the center manifold in normal form without computing the manifold itself. A similar method to estimate the direction of the Hopf bifurcation is presented in w30x by using the method of Lyapunov and Schmidt. Since we have two simple, purely imaginary characteristic solutions the center manifold considered is two-dimensional and thus, following w12, 13x, we can compute the reduced system on the center manifold in polar coordinates Ž , .: ˙ s Ž y k . dd a Ž k . q K 3 q O Ž 2 q Ž , .
4
Ž 7.
.
˙s ybk q O Ž < , < . .
Ž 8.
Since dd aŽ k . ) 0 Žsee Lemma 2.3. the sign of the factor K determines the direction of the Hopf bifurcation. A positive K implies a subcritical bifurcation; a negative K implies a supercritical bifurcation. Our aim is now to calculate the factor K in terms of the derivatives of F at zero. To do this let us first give the Taylor expansion of the right hand side of Ž4.. Setting ␣ [ y k , Eq. Ž4. becomes
˙x Ž t . s L␣ x t q G Ž x Ž t y 1 . ; ␣ . with L␣ s y Ž k q ␣ . Ž 0 . q Ž k q ␣ . Ž y1 . s Žk q ␣ .
0
Hy1 d Ž . Ž .
and G Ž u; ␣ . s Ž k q ␣ . F Ž u . y Ž k q ␣ . u. The function G satisfies G Ž 0; ␣ . s 0, G Ž u; ␣ . s k
DG Ž 0; ␣ . s 0, F Y Ž 0. 2
u2 q
F Y Ž 0. 2
␣ u2 q k
F Z Ž 0. 3!
u 3 q O Ž Ž u, ␣ .
This implies G Ž u; 0 . s k
F Y Ž 0. 2
u q k 2
F Z Ž 0. 3!
u3 q O Ž u4 . .
4
..
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LOCAL AND GLOBAL BIFURCATION
Proceeding as in w12, 13x, in the next step we consider the decomposition of the phase space C s C Žwy1, 0x, ⺢. s P [ Q, where P is the two-dimensional eigenspace associated with the set of simple eigenvalues ⌳ s ibk , yibk 4 . Let ⌽ s Ž 1 , 2 . s Ž e i b k , eyi b k . be a basis of P and ⌿ s colŽ 1 , 2 . s Ž 1Ž0. eyi b k , 2 Ž0. e i b k . a basis of the dual space P U in C U with the property
Ž ⌿, ⌽ . [ Ž Ž i , j . , i , j s 1, 2 . s I2 . The above bilinear form is given by Žsee w19, p. 211x.
0
Ž , . s Ž 0. Ž 0. y H
H y1 0
Ž y . d Ž . Ž . d .
Ž ⌿, ⌽ . s I2 yields Žsee w13x.
1Ž 0 . s 1 y L0 Ž e i b k .
y1
2 Ž 0. s 1Ž 0. .
,
Ž 9.
Thus the factor K in Ž7. is given by Žsee w13x. F Z Ž 0. k
Ks
2
½
= y
Re Ž eyi b k 1 Ž 0 . . q F Y Ž 0 . k2 2
Re Ž eyi b k 1 Ž 0 . . L0 Ž 1.
q
1 2
Re
ey3 i b k 1 Ž 0 . 2 ibk y L0 Ž e 2 i b k .
5
.
Ž 10 .
Here we have L0 Ž 1 . s k Ž y 1 . ,
L0 Ž e i b k . s y k eyi b k ,
L0 Ž e 2 i b k . s k Ž y1 q ey2 i b k . ,
1Ž 0. s
1 1 q k eyi b k
.
Substituting the above terms into Eq. Ž10. we obtain K as function of . Now we are in a position to prove the following useful lemma. LEMMA 2.5. Ks
Let s k Ž ., k g ⺞ 0 . Then
F Z Ž 0.
1 q 2
2
Ž 1 q 2 q 2 2 .
F Y Ž 0 . 2 Ž 11 2 q 6 y 2 . q Ž 2 3 q 13 2 q 4 y 4 . 2
y
2
2 Ž 5 q 4 . Ž y 1 . Ž 1 q 2 q 2 2 .
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GIANNAKOPOULOS AND ZAPP
for - y1 and Ks
F Z Ž0.
1 q 2
2
Ž 1 q 2 q 2 2 .
F Y Ž 0 . 2 Ž 11 3 q 35 2 q 24 y 6 . q Ž y2 4 q 11 3 q 43 2 q 24 y 12 . 2
y
2 Ž y 1 .Ž 5 2 q 15 q 12 .Ž 1 q 2 q 2 2 .
2
for ) 1. Proof. The proof will be given in the Appendix. To determine the direction of the Hopf bifurcation one can evaluate the term K or one can use the following conditions which guarantee the sign of K: LEMMA 2.6.
Let < < ) 1 and s k Ž ., k g ⺞ 0 . Then K - 0 m F Z Ž 0 . F X Ž 0 . - F Y Ž 0 . Ck Ž . 2
K ) 0 m F Z Ž 0 . F X Ž 0 . ) F Y Ž 0 . Ck Ž . 2
K s 0 m F Z Ž 0 . F X Ž 0 . s F Y Ž 0 . Ck Ž . , 2
where
¡ 11 q 6 y 2
2
Ž q 1. q2 for - y1 Ž 5 q 4. Ž y 1. Ž 5 q 4 . Ž 1 q 2 . Ck Ž . [~ 11 3 q 35 2 q 24 y 6 Ž q 1 . Ž 2 y 3 . y 2 Ž y 1 . Ž 5 2 q 15 q 12 . Ž 5 2 q 15 q 12 . Ž 1 q 2 . 2
¢
for ) 1.
Proof. Multiplying the term K given in Lemma 2.5 by the positive factor 2 Ž2r .Ž1 q 2 q 2 2 .rŽ1 q 2 . provides the results stated in the lemma. COROLLARY 2.2. Let < < ) 1 with s F X Ž0. and s k Ž ., k g ⺞ 0 . Then the periodic solutions bifurcating at s k Ž . exist 1. for ) k Ž . pro¨ ided F Z Ž0. F X Ž0. - F Y Ž0. 2 Ck Ž . Ž supercritical bifurcation.; 2. for - k Ž . pro¨ ided F Z Ž0. F X Ž0. ) F Y Ž0. 2 Ck Ž . Ž subcritical bifurcation.. Proof. Equations Ž7. and Ž8. and Lemma 2.6 provide the assertions.
435
LOCAL AND GLOBAL BIFURCATION
The next lemma deals with the properties of the sequence Ž Ck Ž .. k g ⺞ 0 . LEMMA 2.7. The following hold: 1. For - y1 or ) '3 the sequence Ž Ck Ž .. k g ⺞ 0 is positi¨ e, bounded, and strictly increasing. 2. For 1 - - '3 the sequence Ž Ck Ž .. k g ⺞ 0 is positi¨ e, bounded, and strictly decreasing. 3. For s '3 Ž Ck Ž .. k g ⺞ is a constant sequence. 0
Proof. The statements are a consequence of the definition of Ck Ž .. Remark 2.4. One can prove that Ck Ž ., k g ⺞ 0 , is bounded by 1.4887 ( 2
126 q 125'7
Ž 2 q 5'7 .Ž 35 q 2'7 .
- Ck Ž . -
11 5
for - y1
and Ck Ž . )
21 10
for ) 1.
Remark 2.5. The monotonicity and boundness of the sequence Ž Ck Ž .. k g ⺞ provide readily applicable criteria for the direction of the 0 Hopf bifurcation at s k Ž .. For instance: The monotonicity implies the assertions of Corollary 2.1.3. From the boundness we get 1. If - y1 and F Z Ž0. F X Ž0. - 2ŽŽ126 q 125'7 .rŽ2 q 5'7 .Ž35 q 2'7 .. F Y Ž0. 2 then all bifurcations are supercritical Ž K - 0.. 2. If - y1 and F Z Ž0. F X Ž0. ) 115 F Y Ž0. 2 then all bifurcations are subcritical Ž K ) 0.. Y Ž . 2 then all bifurcations are 3. If ) 1 and F Z Ž0. F X Ž0. - 21 10 F 0 supercritical Ž K - 0.. Remark 2.6. If K s 0, i.e., C s Ck Ž . Žsee proof of Corollary 2.1. or F Ž0. s F Z Ž0. s 0, one has to calculate higher derivatives of F and to compute higher order terms of the normal form of the reduced system on the center manifold. Y
2.5. Proofs of the Main Results Proof of Theorem 2.1. Theorem 2.2 provides the existence of Hopf bifurcation at s k Ž .. By applying Corollary 2.2 one gets the conditions for the direction of the Hopf bifurcation. The stability properties of the bifurcating periodic solutions follow from Lemma 2.4 and w7x.
436
GIANNAKOPOULOS AND ZAPP
Proof of Corollary 2.1. If F Y Ž0. F X Ž0. F 0 then K - 0 and therefore, in this case, all bifurcations are supercritical. If F Y Ž0. s 0 and F Z Ž0. / 0 then sign K s sign F Z Ž0. F X Ž0., and thus all bifurcations are either supercritical or subcritical. Otherwise there is a unique C ) 0 with F Z Ž0. F X Ž0. s F Y Ž0. 2 C. If C - Ck Ž . then the bifurcation is supercritical; if C ) Ck Ž . the bifurcation is subcritical. By applying Lemma 2.7 it is clear that there is at most one k g ⺞ 0 where the direction of the Hopf bifurcations can change, i.e., Ž C y Cky 1Ž ..Ž C y Ckq1Ž .. - 0, provided / '3 . 3. GLOBAL CONTINUATION In this section we study the global continuation of the local Hopf branches bifurcating from the points Ž x, . s Ž0, k Ž ..; k g ⺞ 0 . We prove that each of these branches lies on an unbounded continuum Sk of periodic solutions with pairwise disjoint S k, k g ⺞ 0 . First we need some notation. We set S [ Ž , , p . g C = ⺢q= ⺢q: x Ž t . is a non-constant p-periodic solution of Ž1. with sF X Ž0. and x
p ) 2 pro¨ ided - y1 and k s 0; p gxrŽ k q 12 ., rŽ k q 14 .w pro¨ ided - y1 and k G 1; p gxrŽ k q 1., rŽ k q 34 .w pro¨ ided ) 1 and k g ⺞ 0 .
LOCAL AND GLOBAL BIFURCATION
437
Proof. To prove that Sk is unbounded we apply Theorem 3.3 in w11x Žsee also Theorem 2.5 in w32x.. The theorem mentioned above provides that, under certain assumptions, either Sk is unbounded or Sk is bounded and additionally S k l Ž x, , p . g C = ⺢q= ⺢q : x g ⺢, x s F Ž x . 4 s Ž x l , l , pl . g ⺢ = ⺢q= ⺢q : l s 0, 1, 2, . . . , n4
and
n
Ý ␥ Ž x l , l , pl . s 0,
Ž 11 .
ls0
where ␥ Ž x l , l , pl . is the so-called crossing number of Ž x l , l , pl . whose definition will be given below. Consequently, in order to prove that S k is unbounded, it suffices to show that Eq. Ž11. cannot be fulfilled. First we need to give the definition of ␥ Ž x l , l , pl . Žsee w11, 32x.. Let Ž x, , p . be an isolated center of Ž1.; i.e., x s F Ž x ., Ž ⌬ iŽ2rp .; , F X Ž x .. s 0, and there exists a neighborhood UŽ x, , p . of Ž x, , p . such that Ž x, , p . is the only center in UŽ x, , p .. Using the definition of ⌬ Žsee Eq. Ž6.. and Lemma 2.3 one can show that ⌬Ž z; , . is analytic in z g ⺓ and continuous in gx y ⑀ 0 , q ⑀ 0 w, where ⑀ 0 ) 0 is an appropriate positive constant and s F X Ž x . Žsee Condition Ž A3. in w32x.. Moreover, it follows from Lemma 2.1 and 2.3 that there exist small positive constants ␦ , ⑀ gx0, ⑀ 0 w such that on ⭸ ⍀ ⑀ , p = w y ␦, q ␦ x ⌬ aqi
ž
2 p
, , s 0 if and only if s , a s 0, and p s p,
/
where ⍀ ⑀ , p [ Ž a, p . : 0 - a - ⑀ , p g x p y ⑀ , p q ⑀ w 4
Ž see Condition Ž A4 . in w 32x . . Notice that z s "iŽ2rp . are the only zeroes of ⌬Ž z; , . on the imaginary axis Žsee Lemma 2.1.. We define H Ž a, p . [ ⌬ a q i
ž
2 p
,, .
/
H : ⍀ ⑀ , p ª ⺢ 2 ( ⺓ is a differentiable function of Ž a, p . g ⍀ ⑀ , p . Now the crossing number ␥ Ž x, , p . can be given by
␥ Ž x, , p . [ deg Ž Hy ␦ , ⍀ ⑀ , p . y deg Ž Hq ␦ , ⍀ ⑀ , p . ,
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GIANNAKOPOULOS AND ZAPP
where deg denotes the Brouwer degree with respect to Ž0, 0. g ⺢ 2 ( ⺓ and ⍀ ⑀ , p Žsee w11, 32x.. This implies that the crossing number ␥ Ž x, , p . is well defined. A calculation shows det Ž DH Ž a, p . <Ž a, p .gHy1 Ž0, 0. . s y
2 p2
ž
2 Ž1 q a q . q
4 2 2 p2
/
- 0.
Using the definition of the degree Žsee w6, p. 65x. we obtain deg Ž H , ⍀ ⑀ , p . s
Ý
Ž a,
p .gHy1 Ž0, 0 .
Ž y1. s y噛Hy1 Ž 0, 0 . ,
where 噛Hy1 Ž0, 0. denotes the number of elements in the set Hy1 Ž0, 0., provided Hy1 Ž0, 0. consists of only a finite number of points in ⍀ ⑀ , p . Ž . Lemma 2.4 provides that, for sufficiently small ␦ and ⑀ ) 0, Hy1 y ␦ 0, 0 s ⭋ Ž . and Hy1 0, 0 consists of only one point in ⍀ . This implies q␦ ⑀, p
␥ Ž x, , p . s y1, provided Ž x, , p . is an isolated center. If Ž x, , p . is not a center, we can easily prove that
␥ Ž x, , p . s 0. Now, in order to be able to apply Theorem 3.3 in w11x Žsee also Theorem 2.5 in w32x., we still have to prove that all centers of Ž1. are isolated. Lemma 2.2 provides that if x g ⺢ is a stationary solution, all centers Ž x, k , pk ., k g ⺞ 0 , corresponding to x, are isolated. By assumption ŽGH2. this yields that all the centers are isolated, and thus we have proved that Sk is unbounded and Ž x, , p . g S k if and only if x s 0, s k Ž ., and p s pŽ . with s F X Ž0.. By applying Lemma 4.1 in w8x to Ž4., we get that Eq. Ž1. has no non-constant periodic solutions of period 2m for any m g ⺞. From Theorem 2.1 and Lemma 2.1 we know p Ž . s
2 bk Ž .
k Ž . , k g ⺞0 ,
where bk Ž . g x Ž 2 k q
3 2
. , Ž 2 k q 2. w
for k g ⺞ 0 , ) 1
and bk Ž . g x Ž 2 k q
1 2
. , Ž 2 k q 1. w
for k g ⺞ 0 , - y1.
LOCAL AND GLOBAL BIFURCATION
439
This provides p Ž . ) 2 0 Ž . p Ž . g
k Ž . kq
1 2
,
for k s 0, if - y1,
k Ž . kq
1 4
for any k g ⺞, if - y1
and p Ž . g
k Ž . kq1
,
k Ž . kq
3 4
for any k g ⺞ 0 , if ) 1,
and thus we have shown that if Ž , , p . g Sk and / 0, the minimal period p of the periodic solution x Ž t . of Ž1. with x
Assume ŽGH1., ŽGH2., and ŽGH3. are satisfied. Then
1. For s F X Ž0. - y1 there exists ˜ k Ž . gx0, k Ž .x such that for any G˜ k Ž . there are g C and p gxrŽ k q 12 ., rŽ k q 14 .w such that Ž , , p . g Sk, k g ⺞. Furthermore, if Ž , , p . g S k, k g ⺞, and x Ž t . is the unique existing p-periodic solution of Ž1. with x
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GIANNAKOPOULOS AND ZAPP
From Corollary 1.1 in w23x we know, because of the invariance condition ŽGH3., that if x Ž t . is a non-constant periodic solution of Ž1. satisfying x Ž t . g I for all t G y then x Ž t . g intŽ I . for all t G y . On the other hand, using the local Hopf bifurcation theorem 2.1, we can prove that for ) 0 close to k Ž . there exists a non-constant p-periodic solution x Ž t . with x
xF Ž x . - 0 for x g I _ 04 .
Notice that in our proof we do not need the assumption ŽGH4.. On the other hand, ŽGH1. is stronger than ŽGH1X .. 2. Results similar to Corollary 3.1 are presented in w32, Theorem 4.1, x 4.2 under the additional assumption that F is a bounded and monotone function with maximum slope at x s 0. Notice that the proof provided in w32, Remark 4.8x is only valid for F X Ž0. - 1 Žsee Eq. Ž4.4. in w32x.. Remark 3.3. In order to be able to prove that the -component of S0, where - y1, is unbounded, we need the additional assumption that F satisfies the negative feedback condition ŽGH4. Žsee Remark 3.2.1.. More precisely, Theorem 1.1 in w23x provides: Let ŽGH1X ., ŽGH3., and ŽGH4. be satisfied. Then for - y1 there exists ˜ 0 Ž . gx0, 0 Ž .w such that for any )˜ 0 Ž . there are a g C Žwy , 0x, I . and a p ) 2 with Ž , , p . g S0. Furthermore, if Ž , , p . g S0 and x Ž t . is the unique solution of Ž1. with x
441
LOCAL AND GLOBAL BIFURCATION
4. EXAMPLES In this last section we present and discuss five examples which illustrate our local ŽTheorem 2.1 and Corollary 2.1. and global results ŽTheorem 3.1 and Corollary 3.1.. Furthermore we provide bifurcation diagrams by using a numerical continuation method for delay differential equations Žsee w10x. where the periodic solutions are approximated by a Fourier polynomial. The last two examples indicate what can happen if the invariance condition ŽGH3. or feedback condition ŽGH4. is violated. 1. In our first example we use the sigmoid nonlinearity
F Ž x . s F1 Ž x . [
3
1
2 1 q exp Ž y4 x .
y
3 4
.
It follows that F is bounded by y 34 - F Ž x . - 34 for all x g ⺢, is strictly increasing, and has exactly one turning point at 0 Ž F Y Ž0. s 0, F Z Ž0. - 0.. Moreover it holds F Ž0. s 0 and s F X Ž0. s 32 ) 1. F has three fixed points: x s 0, ˜ x ( 0.644, and yx. ˜ By Theorem 2.1 we get that a supercritical Hopf bifurcation occurs at Ž x, . s Ž0, k Ž .., k g ⺞ 0 . Corollary 3.1 provides a ˜ k Ž . gx0, k Ž .x, k g ⺞ 0 , such that for all ) ˜ k Ž . there is a periodic solution x Ž t . belonging to Sk. Furthermore we know that the period p of x Ž t . satisfies p gxrŽ k q 1., rŽ k q 34 .w. Figures 3 and 4 show the numerical results. One can observe that the amplitude of the solution is strictly increasing in . 2. Now we use the nonlinearity F Ž x . s yF1Ž x . with F1 as in the first example. x s 0 is the only fixed point of F and it holds s F X Ž0. s y Y Z 3 Ž . Ž0. ) 0. Theorem 2.1 yields that a supercritical 2 - y1, F 0 s 0, F Hopf bifurcation occurs at s k Ž ., k g ⺞ 0 . Since F satisfies ŽGH1., ŽGH2., ŽGH3., and ŽGH4. with I s wyx, ˜ ˜x x with ˜x as in Example 1, from Corollary 3.1 and Remark 3.3, we obtain that the branches of periodic solutions Sk are unbounded in . For the period p of the solution x Ž t . belonging to Sk, p ) 2 for k s 0 and p gxrŽ k q 12 ., rŽ k q 14 .w for k G 1. Numerical simulations indicate that the amplitude of the solution is strictly increasing in and for every ) 0 Ž . there is a unique slowly oscillating periodic solution. These properties for the slowly oscillating periodic solutions can be proved analytically. Applying Corollary 2.1 in w5x one can prove the uniqueness of the slowly oscillating periodic solution for ) 0 Ž .. For F 0 Ž . there are no slowly oscillating periodic solutions. Moreover Corollary 2.1 in w5x provides the monotonicity property of the branch of slowly oscillating periodic solutions S0 as function of ) 0 Ž .: The orbit ⌫ Ž . of the unique existing slowly oscillating periodic solution in
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GIANNAKOPOULOS AND ZAPP
FIG. 3. Branches of periodic solutions in the Ž , < x <.-plane. All bifurcations are supercritical and the corresponding periodic solutions are bounded by ˜ x Žsee Example 1..
p
FIG. 4. Branches of periodic solutions in the Ž , .-plane Žsee Example 1..
LOCAL AND GLOBAL BIFURCATION
443
the Ž x, ˙ x .-plane possesses the following property. ⌫ Ž 1 . is in the closure of the exterior of the orbit ⌫ Ž 2 . whenever 2 ) 1 ) 0 Ž .. 3. This example demonstrates the change of the Hopf bifurcation direction Žsee Corollary 2.1.. We consider Ž1. with F Ž x . s y575 arctan Ž x q 10'5 . q 575 arctan Ž 10'5 . . It holds F Ž0. s 0, [ F X Ž0. s y575rŽ1 q 500. - y1, F Y Ž0. s Ž11500r251,001.'5 , F Z Ž0. s y1,723,850r125,751,501. So we get C s Ž F Z Ž0. F X Ž0.rF Y Ž0. 2 . s 1499r1000 s 1.4990 Žsee proof of Corollary 2.1.. It follows that 1.49716 ( C0 Ž . - C - C1 Ž . ( 1.49952. Applying Theorem 2.1 we can show that a Hopf bifurcation occurs at s k Ž ., k g ⺞ 0 . The first Hopf bifurcation at s 0 Ž . is subcritical; all other Hopf bifurcations are supercritical Žsee Figs. 5 and 6.. 4. Here we discuss an example where the function F does not satisfy condition ŽGH3.. Consider F Ž x . s y15 sinh Ž x . . It holds that F Ž0. s 0, [ F X Ž0. s y15, F Y Ž0. s 0, and F Z Ž0. s y15 F is strictly decreasing and satisfies conditions ŽGH1. and ŽGH2. but does not satisfy condition ŽGH3.. x s 0 is the only fixed point of F. Using Theorem 2.1 we get that a subcritical Hopf bifurcation occurs at s k Ž ..
FIG. 5. Subcritical Hopf bifurcation at s 0 Ž . s 4.66712 Žsee Example 3..
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GIANNAKOPOULOS AND ZAPP
FIG. 6. Supercritical Hopf bifurcation at s 1Ž . s 15.82280 Žsee Example 3..
From Theorem 3.1 we know that the branches of periodic solutions S k bifurcating from Ž x s 0, s k Ž .. are unbounded but we do not know which component of S k is unbounded. Actually the numerically computed bifurcation diagram Žsee Fig. 7. shows that the branches are bounded in . The numerically computed p-periodic solutions x k Ž t; . belonging to Sk exist only for 0 - - k Ž . and < x k < ª ⬁ as ª 0. Furthermore for the p period p of x 0 Ž t; . we get ª 4 as ª 0.
FIG. 7. Branches of periodic solutions for F Ž x . s y15 sinhŽ x . Žsee Example 4..
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LOCAL AND GLOBAL BIFURCATION
5. The last example deals with the case in which F does not satisfy the negative feedback condition ŽGH4.. We consider Eq. Ž1. with FŽ x. s 2
xq1 1 q Ž x q 1.
10
y 2;
see w22x. This equation was introduced by Mackey and Glass as a mathematical model for the generation of white blood cells. F is a bounded function with F Ž0. s 0, s F X Ž0. s y4 - y1, F Y Ž0. s y5, and F Z Ž0. s 255. Using Theorem 2.1 we get that a supercritical Hopf bifurcation takes place at Ž x s 0, s k Ž .., k g ⺞ 0 . From Corollary 3.1 we know that for k G 1 the -component of Sk is unbounded. Since ŽGH4. is not satisfied we cannot investigate S0 by applying Theorem 1.1 in w23x Žsee Remark 3.3.. In w14, 18, 22x the Mackey᎐Glass equation is studied numerically. Numerical simulations show that the branch S0 of slowly oscillating periodic solutions undergoes a cascade of period doubling bifurcations. Moreover numerical calculations indicate the existence of chaotic attractors for sufficiently large delays .
APPENDIX: PROOF OF LEMMA 2.5 Let s k Ž . and b s bk Ž ., k g ⺞ 0 . We compute the factor K Žsee Eq. Ž10.. in several steps. 1. First, notice that the following equalities hold: cos Ž b . s sin Ž b . s
1
cos Ž 2 b . s
'2 y 1
2 y 2
sin Ž 2 b . s 2
< <
2
'2 y 1 < <
cos Ž 3b . s sin Ž 3b . s
b s '2 y 1 . 2. We get 1Ž0. s
1 
with  [ 1 q eyi b. This provides
<  < 2 s  s Ž 1 q eyi b . Ž 1 q e i b . s 1 q 2 cos Ž b . q 22 s 1 q 2 q 22 .
4 y 3 2
3
'2 y 1 2 < <
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GIANNAKOPOULOS AND ZAPP
3. We set C1 [ ReŽ eyi b 1Ž0... It holds that C1 s Re s Re s
eyi b <  <2 eyi b q 1 q 2 q 22 1 q 2
.
Ž 1 q 2 q 22 .
4. y
Re Ž eyi b 1 Ž 0 . . L Ž 1.
sy
1 q 2
Ž 1 q 2 q 22 . Ž y 1 .
\ T2 .
5.
␣ [ 2 ib y Ž y1 q ey2 i b . s y cos Ž 2 b . q i Ž 2 b q sin Ž 2 b . . s 1y
ž
s
2 y 2
2
q i 2 b q 2
/ ž
Ž q 2. Ž y 1. q i
'2 y 1
2'2 y 1 < <
< <
/
Ž < < q 1. .
Therefore
␣s
Ž q 2. Ž y 1. y i
2'2 y 1 < <
Ž < < q 1.
and it follows that
2
< ␣ < 2 s ␣␣ s s
2
2 2 Ž q 2. Ž y 1. q
2 Ž y 1.
4 2 Ž 2 y 1 . < < 2
Ž < < q 1.
2
Ž Ž y 1. Ž q 2. 2 q 4 Ž q 1. Ž < < q 1. 2 . .
We distinguish between ) 1 and - y1. Ža. The case ) 1: <␣ <2 s s
2
2 Ž y 1. 2 2 Ž y 1.
Ž Ž y 1 . Ž 2 q 4 q 4 . q 4 Ž 3 q 3 2 q 3 q 1 . . Ž 5 2 q 15 q 12 . .
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LOCAL AND GLOBAL BIFURCATION
Žb. The case - y1:
<␣ <2 s s
2 Ž y 1.
2
Ž 2 q 4 q 4 q 4 Ž 2 y 1 . .
2 2 Ž y 1.
2
Ž 5 q 4. .
6. We compute two auxiliary terms A and B.
A [ cos Ž 3b . q cos Ž 2 b . s s
3
q
2 y 2
2
4 y 3 2 q 2 Ž 2 y 2 .
3
B [ sin Ž 3b . q sin Ž 2 b . s s
4 y 3 2
'2 y 1 2 < <
'2 y 1 Ž 4 y 2 . 2 < <
q 2
'2 y 1 < <
Ž 4 y 2 q 22 . .
7. It holds that Re Ž ey3 i b␣ .
ž ž
s Re ey3 i b Ž 1 q e i b .
s Re Ž A y iB .
s s
A
ž
ž
Ž q 2. Ž y 1. y i
Ž y 1. Ž q 2. y
Ž y 1. 4
Ž q 2. Ž y 1. y i
2'2 y 1 < <
2'2 y 1
2'2 y 1 < <
< <
Ž < < q 1.
Ž < < q 1.
//
//
B Ž < < q 1.
Ž Ž q 2 . Ž 4 y 3 2 . y 2 Ž q 1 . Ž < < q 1 . Ž 4 y 2 . q2 Ž Ž 2 y 2 . Ž q 2 . y 4 Ž q 1 . Ž < < q 1 . . . .
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GIANNAKOPOULOS AND ZAPP
8. We set T3 [ Re
ey3 i b 1 Ž 0 . L Ž e 2 i b .
s
Re Ž ey3 i b␣ . <␣ <2 <  <2
.
Ža. The case ) 1:
Ž q2 . Ž 4y3 2 y2 Ž q1 . 2 Ž 2y . . q 2 Ž Ž 2y 2 .Ž q2 . y4 Ž q1 . 2 . T3 s . 3 Ž 5 2 q15 q12 .Ž 1q2 q 22 . Žb. The case - y1: T3 s
Ž q2 . Ž 4y3 2 y2 Ž q1 . 2 Ž 2y . . q 2 Ž Ž 2y 2 .Ž q2 . q4 Ž 2 y1 . . . 3 Ž y1 .Ž 5 q4 .Ž 1q2 q 22 .
9. Now we calculate C2 [ T2 q 12 T3 : Ža. The case ) 1: C2 s
y2 Ž 1 q 2 . 2 Ž 5 2 q 15 q 12 . 2 Ž y 1 .Ž 5 2 q 15 q 12 .Ž 1 q 2 q 22 . 3
Ž y 1 . Ž Ž q 2 . Ž 4 y 3 2 y 2 Ž q 1 . 2 Ž 2 y . . 2 q2 Ž Ž 2 y 2 .Ž q 2 . y 4 Ž q 1 . . . q 23 Ž y 1 .Ž 5 2 q 15 q 12 .Ž 1 q 2 q 22 . s
2 Ž y11 3 y 35 2 y 24 q 6 . q Ž 2 4 y 11 3 y 43 2 y 24 q 12 . 22 Ž y 1 .Ž 5 2 q 15 q 12 .Ž 1 q 2 q 22 .
.
Žb. The case - y1: 2
C2 s
y2 Ž 1 q 2 . 2 Ž 5 q 4 . q Ž q 2 . Ž 4 y 3 2 y 2 Ž q 1 . Ž 2 y . . 23 Ž y 1 . Ž 5 q 4 . Ž 1 q 2 q 22 . q
s
2 Ž Ž 2 y 2 . Ž q 2 . q 4 Ž 2 y 1 . . 23 Ž y 1 . Ž 5 q 4 . Ž 1 q 2 q 22 .
2 Ž y11 2 y 6 q 2 . q Ž y2 3 y 13 2 y 4 q 4 . 22 Ž y 1 . Ž 5 q 4 . Ž 1 q 2 q 22 .
.
10. Finally we compute K s Ž F Z Ž0.r2.C1 q F Y Ž0. 2 2 C2 and the proof is finished.
LOCAL AND GLOBAL BIFURCATION
449
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