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On the Stability of Some Exponential Polynomials
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
205, 259]272 Ž1997.
AY965152
On the Stability of Some Exponential Polynomials Margarete Baptistini* Uni¨ ersidade Federal de Sao ˜ Carlos-DM, C. Postal 676, Sao ˜ Carlos, SP 13560-970, Brazil
and † Placido Taboas ´ ´
Uni¨ ersidade de Sao ˜ Paulo-ICMSC, C. Postal 668, Sao ˜ Carlos, SP 13560-970, Brazil Submitted by Jack K. Hale Received January 30, 1996
This paper studies elementary transcendental equations of the type Ž z 2 q pz q . q et z q rz n s 0, where p, q, r g R, t , p ) 0, q G 0, r / 0, n s 0, 1, 2. We are mainly interested in the case n s 0 for which a characterization of stability is accomplished; that is, we state a necessary and sufficient condition for all the roots to lie to the left of the imaginary axis. Also a characterization of stability independent of delays is given. Sufficient conditions for instability are stated for the cases n s 1, 2. The proofs are carried by elementary arguments independent of usual tools. Q 1997 Academic Press
1. INTRODUCTION We are concerned with transcendental equations of the type P Ž z, e z . s 0, where P Ž x, y . is a polynomial in x and y. These equations were systematically studied by Pontryagin w6x, extending an earlier work of ˇebotarev. The analysis on how their roots locate with respect to the C imaginary axis is an old problem that, besides being important by itself, plays a role in the study of asymptotic behavior in the theory of delay differential equations. On the basis of that, there is a natural interest in determining the stability of P Ž z, e z . s 0, that is, when all of its roots have *E-mail address: [email protected]. † E-mail address: [email protected]. 259 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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BAPTISTINI AND TABOAS ´
negative real parts. The ingredients usually involved to handle questions of this nature are classical results on zeros of analytic functions, such as Hurwitz’ Theorem or Rouche’s ´ Theorem and Pontryagin’s monograph. Also Bellman and Cooke’s book w2x is a widely cited reference in this subject. It devotes its thirteenth chapter to applications of Pontryagin’s results in regard to stability theory of delay differential equations. There is a special interest in low degrees of P Ž x, y . motivated by applications. In the setting of these equations this paper treats a specialization discussed in Bellman and Cooke’s book in its Section 13.8, named ‘‘An Important Equation.’’ There the equation Ž z 2 q pz q q . e z q rz n s 0, with p, q, r g R, p ) 0, q G 0, r / 0 and n a positive integer or zero, is treated with respect to its stability. The case n s 0, which appears quite often in the literature of delay differential equations is our main concern here. This case is also the main concern of Section 13.8 of w2x. Our approach can be adopted even in the cases n s 1 and n s 2, but except for particular equations, the analysis becomes considerably more complicated to lead to simple general results like Theorems 2.1 and 2.2. We accomplish sufficient conditions for instability in these cases. We do not focus on the cases n G 3 because in this circumstance the corresponding polynomial P Ž x, y . has no principal term and, in this case, it is known that the equation P Ž z, e z . s 0 has infinitely many roots with arbitrarily large positive real parts. See w2x or w6x, for instance. The lower degree case Ž z q p . e z q q s 0 is studied by Hayes w5x. For this equation, a more complete analysis including the question of stability independent of delays is given by Hale and Lunel in w4x, based on a Liapunov functional for the corresponding delay equation. The main results are stated in the Section 2. Theorems 2.1 and 2.2 characterize the equations Ž z 2 q pz q q . e z q r s 0 having all roots to the left of the imaginary axis. They have the same goal of Theorem 13.9 of w2x, but our statements are simpler and our proofs, based only in elementary arguments, are completely independent of Pontryagin’s results. Indeed, Professor Cooke communicated to us in person that Theorem 13.9 of w2x contains some mistakes, so that Theorems 2.1 and 2.2 are a new characterization of stability for the equations under consideration. Section 2 ends with Theorem 2.4, which gives a necessary and sufficient condition for the equation Ž z 2 q pz q q . et z q r s 0 to be stable and the stability to be independent of the delay. That is, for any t ) 0, all the roots of the equation have negative real part. In Section 3 we investigate the cases n s 1 and n s 2. The same approach of Section 2 is applied to get sufficient conditions for instability. Section 4 gives some applications. We use Theorem 2.1 to simplify a result related to a problem of nonlinear oscillations theory: a damped nonlinear oscillator, with a time delay in the restoring force. See Section
261
EXPONENTIAL POLYNOMIALS
11.6 of w4x, for instance. Also Cooke and Grossman w3x investigate this model in regard to stability switches. Finally, we apply Theorem 2.1 in the investigation of stability of the characteristic equation of the linear part of a system of two delay differential equations near equilibria. 2. AN IMPORTANT EQUATION The complex function H is given by H Ž z . s Ž z 2 q pz q q . e z q r ,
Ž 2.1.
where p, q, r g R, p ) 0, q G 0, r / 0. Thus the case n s 0 of the equation we wish to study is H Ž z . s 0. THEOREM 2.1. Let the 2-¨ ectors ¨ Ž b . s Ž pb, q y b 2 ., w Ž b . s Žcos b, sin b ., b G 0, be gi¨ en. If r ) 0, a necessary and sufficient condition for all roots of the equation H Ž z . s 0 to ha¨ e negati¨ e real part is that the orthogonality condition ¨ Ž b . ? w Ž b . s 0, with b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., implies < ¨ Ž b .< ) r. Proof. Let us denote z s a q bi. There is no loss of generality in restricting the proof to the case b G 0 since H Ž z . s H Ž z ., where z denotes the complex conjugate to z. If H Ž z . s F Ž z . q iGŽ z ., the equation H Ž z . s 0 is equivalent to the system FŽ z. s GŽ z . s
Ž a2 q pa q q y b 2 . cos b y Ž 2 a q p . b sin b Ž a2 q pa q q y b 2 . sin b q Ž 2 a q p . b cos b
e a q r s 0, e a s 0,
Ž 2.2.
or, in a vector form, e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . ? Ž ysin b, cos b . s yr ,
Ž 2 ab, a2 q pa. q Ž pb, q y b 2 .
? Ž cos b, sin b . s 0.
Ž 2.3.
Defining w H Ž b . s Žsin b, ycos b ., uŽ a, b . s Ž2 ab, a2 q pa., the system Ž2.3. reduces to the single vector equation:
n Ž a, b . [ e a u Ž a, b . q ¨ Ž b . s rw H Ž b . . Ž 2.4. Ž . For each fixed a G 0, the vector n a, b describes clockwise an unbounded arc of parabola in the jh-plane, while w H Ž b . describes counterclockwise the unit circle, as b G 0 increases. Thus, in each interval Ž2 kp , Ž2 k q 1.p ., k s 0, 1, . . . , there exists precisely one number, bk Ž a., for which n Ž a, bk Ž a.. is a positive multiple of w H Ž bk Ž a... Moreover, the condition n Ž a, bk Ž a.. ? w Ž b . s 0 holds only if b s bk Ž a., k s 0, 1, . . . . For each nonnegative integer k, bk Ž a. depends continuously on a, and Ž a, bk Ž a.. satisfies Eq. Ž2.4. if, and only if, < n Ž a, bk Ž a..< s r.
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BAPTISTINI AND TABOAS ´
Let us prove the necessity. Suppose R Ž z . - 0 for all roots of H Ž z . s 0, and assume temporarily the condition stated in the theorem is false. Therefore, for some integer k G 0, the number bk s bk Ž0. g Ž2 kp , Ž2 k q 1.p ., such that ¨ Ž bk . is a multiple Žpositive. of w H Ž bk ., must satisfy < ¨ Ž bk .< F r. If < ¨ Ž bk .< s r, bk must satisfy Ž2.4. with a s 0, that is, ¨ Ž bk . s rw H Ž bk . and, therefore, z s ibk is a pure imaginary root of H Ž z . s 0, a contradiction. It remains to consider the case < ¨ Ž bk .< - r. Under this hypothesis the circle of radius r determines a bounded arc, to which ¨ Ž bk . belongs, on the branch of parabola ¨ Ž b ., b G 0. Since lim aª` < n Ž a, bk Ž a..< s `, the continuity of n Ž a, bk Ž a.. implies there exists a positive number a0 in such a way that < n Ž a0 , bk Ž a0 ..< s r. This implies Ž a0 , bk Ž a0 .. satisfies Eq. Ž2.4. and, therefore, z s a0 q ibk Ž a0 . is a root of H Ž z . s 0 to the right of the imaginary axis, a contradiction. Thus, the condition is necessary. Let us prove now the sufficiency by showing that if there exists a root z of H Ž z . s 0 such that R Ž z . G 0, then for some b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., with ¨ Ž b . ? w Ž b . s 0, we have 0 - < ¨ Ž b .< F r. Suppose ˜ zsa ˜ q ib˜ is a root of H Ž z . s 0, with a˜ G 0. Notice that for any root z s a q ib of H Ž z . s 0, with a G 0, according to Eq. Ž2.4. we have < uŽ a, b . q ¨ Ž b .< F r. Therefore, if z s a q ib varies in the set of roots of H Ž z . s 0 with nonnegative real part, its imaginary part b is bounded. The boundedness of b implies that ˜ z can be choosen in such a way that a ˜ is minimum keeping the property of ˜z. Let k G 0 such that ˜ b g Ž2 kp , Ž2 k q 1.p .. For each a, 0 - a F a, ˜ let Ž . bk a as defined before, that is, a number b g Ž2 kp , Ž2 k q 1.p . such that n Ž a, bk Ž a.. is a multiple of w H Ž bk Ž a... For each a, 0 - a - a, ˜ consider the number r s r Ž a. ) 0 defined by
n Ž a, bk Ž a . . s r w H Ž bk Ž a . . .
Ž 2.5.
From the minimality of a, ˜ it follows that r / r. We claim that r - r. In fact, for any a ) 0 an elementary calculation gives
a
< n Ž a, b . < 2 s 2 e 2 a Ž 2 a q p . 2 b 2 q Ž a2 q pa q q y b 2 .
2
q e 2 a 4 Ž 2 a q p . b 2 q 2 Ž 2 a q p . Ž a2 q pa q q y b 2 . s 2 < n Ž a, b . < 2 q e 2 a Ž 2 a q p . b 2 q Ž 2 a q p . Ž a2 q pa q q .
) 0.
EXPONENTIAL POLYNOMIALS
263
Therefore, if 0 - a - a ˜ we can assure that < n Ž a, ˜b .< - < n Ž a, ˜ ˜b .< s r, that ˜ is, n Ž a, b . lies inside the circle of radius r with center in the origin, according to Fig. 1. If a point Q s Ž j , h . / Ž0, yr . belongs to the circle of radius r with center in the origin, we denote by l Q the length of the arc of this circle from Ž0, yr . to Q in the counterclockwise sense. Suppose for a while the alignment of n Ž a, bk Ž a.. and w H Ž bk Ž a.. occurs with n Ž a, bk Ž a.. outside the circle of radius r with center in the origin, that is, r s < n Ž a, bk Ž a..< G r. Therefore, either bk Ž a. - ˜ b or bk Ž a. ) ˜ b and each of these alternatives leads respectively to l w r w H Ž b k Ž a..x - l w r w H Ž ˜b.x s ln Ž a, ˜ ˜b. - l w rn Ž a, b k Ž a..r r x or l w rn Ž a, b k Ž a..r r x - ln Ž a, ˜ ˜b. s l w r w H Ž ˜b.x - l w r w H Ž b k Ž a..x and this implies, in both cases, that n Ž a, bk Ž a.. and w H Ž bk Ž a.. cannot be aligned, which is a contradiction. Therefore, Eq. Ž2.5. holds with r - r, that is, n Ž a, bk Ž a.. remains inside the circle of radius r with center in the origin for any a, 0 - a - a. ˜ This gives < ¨ Ž bk Ž0..< F r and, therefore, the sufficiency. We have an analogous result for the case r - 0: THEOREM 2.2. If r - 0, keeping the notation of Theorem 2.1, a necessary and sufficient condition for all roots of H Ž z . s 0 to ha¨ e negati¨ e real part is that the orthogonality condition ¨ Ž b . ? w Ž b . s 0, with b g w 0, `. _ D `ks 0 Ž2 kp , Ž2 k q 1.p ., implies < ¨ Ž b .< ) yr.
FIG. 1. Location of n Ž a, ˜ b ..
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BAPTISTINI AND TABOAS ´
Proof. In this case we might repeat the steps of the proof of Theorem 2.1, replacing r by < r < and introducing H w Ž b . s Žysin b, cos b . to play the role of the vector w H Ž b . in Eq. Ž2.4. which becomes
n Ž a, b . [ e a u Ž a, b . q ¨ Ž b . s yr H w Ž b . .
Ž 2.6.
Since real roots are admissible in this case, some minor additional care is needed. For the sufficiency, let us assume the existence of a root ˜ z with RŽ ˜ z . G 0. We need to prove that there exists b g w 0, `. _ D `ks 0 Ž2 kp , Ž2 k q 1.p . such that < ¨ Ž b .< F yr. If ˜ zsa ˜ G 0 is a real root, Eq. Ž2.6. implies n Ž a, ˜ 0. s yr Ž0, 1. which gives < ¨ Ž0.< s q F yr. If J Ž ˜z . cannot vanish, the proof of sufficiency is analogous to the case r ) 0. Doing the proof of necessity by contradiction as in the case r ) 0, in assuming the failure of the condition stated in the theorem, one needs to consider the possibility < ¨ Ž0.< F yr or, equivalently, q F yr. This leads to a real root a G 0 of H Ž z . s 0, where a is the unique nonnegative solution of e a Ž a2 q pa q q . s yr. From this point on the proof follows by the same arguments of the proof of necessity in the case r ) 0. Remark 2.3. If r ) 0, the arguments of the proof of Theorem 2.1 allow us to estimate the number of roots of H Ž z . s 0 with positive real part, by extending the definition of ¨ Ž b . s Ž pb, q y b 2 . to all values of b g R. The number of roots with positive real part coincide with the number of values of b g R, with "b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., such that ¨ Ž b . ? w Ž b . s 0 and < ¨ Ž b .< - r. This fact follows directly from the geometric arguments given in the proof of the necessity of Theorem 2.1. An analogous remark holds for the case r - 0, that is, the number of roots with positive real part coincide with the number of values of b g R, with "b g w 0, `. _ D `ks 0 Ž2 kp , Ž2 k q 1.p . such that ¨ Ž b . ? w Ž b . s 0 and 0 < ¨ Ž b .< - yr. In order to understand the equation H Ž z . s 0 as a characteristic equation of a linear delay differential equation in its extent generality, a normalization is presumed to make the delay equal to 1. By ignoring this fact, the equation should be
Ž z 2 q pz q q . et z q r s 0,
Ž 2.7.
with the same assumptions on p, q, and r, where t ) 0 is the delay. In this setting, it is important to investigate the stability of Eq. Ž2.7. independently of the delay. The next theorem gives a complete answer to this question. THEOREM 2.4. Consider the 2-¨ ector ¨ Ž b . s Ž pb, q y b 2 ., b G 0. A necessary and sufficient condition for Eq. Ž2.7. to be stable for any t ) 0 is that < ¨ Ž b .< ) r, for any b ) 0, if r ) 0 or < ¨ Ž b .< ) yr, for any b G 0, if r - 0.
EXPONENTIAL POLYNOMIALS
265
Proof. Let us suppose r ) 0. After the change of variable z l t z, Eq. Ž2.7. is written in the form
Ž z 2 q t pz q t 2 q . e z q t 2 r s 0
Ž 2.8.
so that, defining the 2-vector ¨t Ž b . s Žt pb, t 2 q y b 2 ., b G 0, and recalling that w Ž b . s Žcos b, sin b ., it follows from Theorem 2.1 that all of the roots of Ž2.8. and, therefore, the roots of Ž2.7. lie to the left of the imaginary axis if and only if, for any b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., ¨t Ž b . ? w Ž b . s 0
«
< ¨t Ž b . < ) t 2 r .
Ž 2.9.
Noticing that ¨t Ž b . is parallel to ¨ Ž brt ., since ¨t Ž b . s t 2 ¨ Ž brt ., the condition Ž2.9. can be reformulated as ¨ Ž brt . ? w Ž b . s 0
«
< ¨ Ž brt . < ) r .
Thus, by the change of parameters b l brt the necessary and sufficient condition for the stability of Ž2.7. becomes that for any b g D `ks 0 Ž2 kprt , Ž2 k q 1.prt ., ¨ Ž b. ? w Žt b. s 0
«
<¨ Ž b. < ) r .
Ž 2.10.
Let us prove now the assertion of the theorem. The sufficiency is immediate since if < ¨ Ž b .< ) < r < for any b G 0, the inequality of the right side of Ž2.10. holds for any b g D `ks 0 Ž2 kprt , Ž2 k q 1.prt ., for any t ) 0. To prove the necessity suppose Eq. Ž2.7. is stable independently of the delay. Therefore, the condition Ž2.10. holds for any t ) 0. Assuming temporarily that there exists ˆ b ) 0 such that < ¨ Ž ˆ b .< F r, one can adjust ` ˆ Ž Ž . . t ) 0 in order that t b g D ks 0 2 kp , 2 k q 1 p and ¨ Ž ˆ brt . ? w Ž ˆ b . s 0. Ž . This contradicts 2.10 and so the condition is necessary. In the case r - 0 the proof is carried similarly by using Theorem 2.2 instead of Theorem 2.1. 3. THE CASES n s 1 AND n s 2 The same approach is adequate to examine the cases n s 1 and n s 2. For the equation
Ž z 2 q pz q q . e z q rz s 0
Ž 3.1.
BAPTISTINI AND TABOAS ´
266 we have the following
THEOREM 3.1. A sufficient condition for the existence of a root z of Ž3.1. with nonnegati¨ e real part is that, for some integer k G 0, the unique solution bk in ŽŽ2 k q 1r2.p , Ž2 k q 3r2.p . of tan b s satisfies rbk G
q y b2 pb
Ž 3.2.
'Ž pb . q Ž q y b . . 2 2 k
2
k
Proof. Replacing z s a q bi in Ž3.1. we are lead to the equivalent system e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . Ž sin b, ycos b . s ra, e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . Ž cos b, sin b . s yrb.
Ž 3.3.
Since if z is a root of Ž3.1., z also is a root, we again confine ourselves to the case b G 0. For any a G 0 the system Ž3.3. means that z s a q ˜ bi is a solution of Ž3.1. if and only if the point of the plane Ž j , h . in the parabola e awŽ2 ab, a2 q pa. q Ž pb, q y b 2 .x, corresponding to b s ˜ b, has coordinates ˜. in the moving coordinates system wŽsin b, ycos b ., Žcos b, sin b .x. Ž ra, yrb When b G 0 varies, the points Ž ra, yrb . describe the spiral
ga Ž b . :
½
j h
s r Ž a sin b y b cos b . s yr Ž b sin b q a cos b . .
Ž 3.4.
So, in other terms, z s a q ˜ bi is a solution of Ž3.1. if and only if the points ˜ corresponding to b in the parabola ¨ aŽ b . s e awŽ2 ab, a2 q pa. q Ž pb, q y b 2 .x and in the spiral Ž3.4. coincide. When b varies positively in the interval w 0, `. , at each complete lap of Ž ga b . on the spiral in the counterclockwise sense, the point ¨ aŽ b . describes an arc of parabola in the clockwise sense in the semi-plane j G 0. Therefore, at the kth lap there exists a unique value bk Ž a. ) 0 such that ¨ aŽ bk Ž a.. is a positive multiple of gaŽ bk Ž a.. and bk Ž a. depends continuously on a. For a s 0, the is spiral g 0 Ž b . Ž j , h . s yrbŽcos b, sin b . and the parabola Ž ¨ 0 b . is given by Ž j , h . s Ž pb, q y b 2 .. Let G be the intersection of the spiral g 0 Ž b . and the semi-plane j G 0. In this case, for any integer k G 0, the point of alignment bk s bk Ž0. is reached when b varies positively in the interval ŽŽ2 k q 1r2.p , Ž2 k q 3r2.p .. This is because ¨ 0 Ž b . describes in the clockwise sense an arc of the parabola in the semi-plane j G 0 while and an entire connected component of G is described by g 0 Ž b . in the counterclockwise sense.
EXPONENTIAL POLYNOMIALS
267
The condition stated in the theorem says that bk s bk Ž0., that is, the points in the parabola ¨ 0 Ž bk . and in the spiral g 0 Ž bk . are aligned and, furthermore, < ¨ 0 Ž bk Ž0..< F < g 0 Ž bk Ž0..<. See Fig. 2. Thus, for small a G 0, the point ¨ aŽ bk Ž a.. belongs to an arc V a of the parabola ¨ aŽ b . determined by the spiral gaŽ b .. Since the parabola ¨ aŽ b . dilates with the order O Ž e aa2 . while the spiral gaŽ b . dilates with the order O Ž a., as a ª `, one sees that the arc V a collapses as a increases approaching `. Before that, we must reach a value a ˜ G 0 such that ¨ a˜Ž bk Ž a˜.. s ga˜Ž bk Ž a˜... For the equation
Ž z 2 q pz q q . e z q rz 2 s 0
Ž 3.5.
we have the following result THEOREM 3.2. A sufficient condition for the existence of a root z of Ž3.5. with nonnegati¨ e real part is that, for some integer k G 0, the unique solution bk in ŽŽ2 k q 1.p , 2Ž k q 1.p . of
ycot b s
satisfies rbk2 G
q y b2
'Ž pb . q Ž q y b . . 2
k
2 2 k
FIGURE 2
pb
Ž 3.6.
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BAPTISTINI AND TABOAS ´
A proof can be accomplished by noticing that if z s a q bi, b G 0, then Eq. Ž3.5. is equivalent to the system e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . Ž sin b, ycos b . s r Ž a2 y b 2 . , e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . Ž cos b, sin b . s yrab.
Ž 3.7.
For any a G 0, let us consider the parabola ¨ nŽ b . s e awŽ2 ab, a2 q pa. q Ž pb, q y b 2 .x and the spiral gaŽ b ., described by the points having coordinates Ž r Ž a2 y b 2 ., y2 rab. in the moving coordinate system wŽsin b, ycos b ., Žcos b, sin b .x. In this setting, the system Ž3.7. means that z s a q ˜ bi is a solution of Ž3.5. if and only if the points corresponding to b s ˜ b in the parabola ¨ aŽ b . and in the spiral gaŽ b . coincide. The proof now follows the same steps of the proof of Theorem 3.2 by noticing that the parabola ¨ aŽ b . dilates with the order O Ž e aa2 ., while the spiral dilates with the order O Ž a2 ., as a ª `.
4. APPLICATIONS In Section 11.6 of w4x, Hale and Lunel discuss the existence of nonconstant periodic solutions of the equation
¨x Ž t . q f Ž x Ž t . . ˙x Ž t . q g Ž x Ž t y 1 . . s 0,
Ž 4.1.
where r G 0, f is continuous, f Ž0. s yk ) 0, and g is continuous together with its first derivative, g 9Ž0. s 1. The authors state a Hopf bifurcation for Ž4.1., considering k as a parameter. Besides, they apply an ejective fixed point principle to achieve an existence theorem for nonconstant periodic solutions, for k greater than a certain critical value. For that, it is necessary to find out the first value of k for which the characteristic equation associated to the linear part of Eq. Ž4.1.,
l2 y k l q eyl r s 0,
Ž 4.2.
has a pair of roots on the imaginary axis. Their discussion depends in a crucial way on Theorem A.6 of the Appendix of w4x. We state below theorem A.6 of w4x and, since Ž4.2. is a special case of the equation H Ž z . s 0, we may give a simpler proof based on Theorem 2.1. THEOREM 4.1 w4, Theorem A.6x. All roots of the equation Ž z 2 q az . e z q 1 s 0 ha¨ e negati¨ e real parts if and only if a ) Žsin z .rz , where z is the unique root of the equation z 2 s cos z , 0 - z - pr2.
EXPONENTIAL POLYNOMIALS
269
Proof. In this case we have H Ž z . s Ž z 2 q az . e z q 1 and ¨ Ž b . s Ž ab, yb 2 .. Let b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., then < ¨ Ž b .< F 1 holds only if b g Ž0, 1. ; Ž0, pr2.. There is a unique ˆ b g Ž0, pr2., with ¨ Ž ˆ b. ? wŽ ˆ b . s 0, so that, by Theorem 2.1, all roots of the H Ž z . s 0 have negative real parts if and only if < ¨ Ž ˆ b .< ) 1. Let z g Ž0, pr2. be the unique root of the equation z 2 s cos z . Since ¨ Ž z . s Ž az , yz 2 . and w H Ž z . s Žsin z , cos z . lie in the same horizontal line, a necessary and sufficient condition for < ¨ Ž ˆ b .< ) 1 is that ¨ Ž z . s Ž az , ycos z . lies outside the unit circle. But this is equivalent to az ) sin z . Remark 4.2. By taking into consideration the delay in the equation of Theorem 4.1, it becomes Ž z 2 q az . et z q 1 s 0. Since the parabola ¨ Ž b . s Ž ab, yb 2 . passes through the origin, in this case, according to the Theorem 2.4, there is no hope to obtain stability independent of the delays. In w7x the second author studies the planar delay differential equation
˙x Ž t . s yx Ž t . q a F Ž x Ž t y 1 . . ,
Ž 4.3.
where x s Ž x 1 , x 2 . g R 2 , F s Ž F1 , F2 . is a bounded continuous map from R 2 into itself, and a ) 0 is a real parameter. Besides some smoothness condition on F near the origin, with Ž F1r x 2 .Ž0., Ž F2r x 1 .Ž0. / 0, the following hypothesis is assumed, x 2 F1 Ž x . ) 0, x 2 / 0;
x 1 F2 Ž x . - 0, x 1 / 0,
Ž H1.
and F is normalized in such a way that Ž F1r x 2 .Ž0. s yŽ F2r x 1 .Ž0. s 1. The analysis in w7x depends on a certain return map on a closed convex subset of the phase space. In order to get an ejectivity condition for the origin, with respect to this map, it is needed to study the location of the zeros of the characteristic equation of the linear part of Ž4.3., 2 Ž l q 1 . e 2 l s ya 2
Ž 4.4.
which, replacing l by zr2, becomes
Ž z 2 q 4 z q 4 . e z q 4a 2 s 0.
Ž 4.5.
Although another approach is adopted, the results in w7x contain the following theorem, which is proved here applying Theorem 2.1: THEOREM 4.3. All roots of Eq. Ž4.4. ha¨ e negati¨ e real parts if and only if a - z 2 q 1 , where z g Ž0, pr2. is uniquely defined by z tan z s 1.
'
BAPTISTINI AND TABOAS ´
270
Proof. If H Ž z . s Ž z 2 q 4 z q 4. e z q 4a 2 , we have ¨ Ž b . s Ž4 b, 4 y b 2 .. Thus, when b g w0, 2x increases, ¨ Ž b . describes an arc of parabola, from Ž4, 0. to Ž0, 8., while w H Ž b . varies counterclockwise on the unit circle, from Ž0, y1. to Žsin 2, ycos 2.. There is a unique ˆ b g Žpr2, 2. such that ¨ Žˆ b . is a positive multiple of w H Ž ˆ b . and < ¨ Ž ˆ b .< is minimum among the ¨ Ž b .’s, b G 0, with this alignment property. Therefore, all roots of Ž4.4. lie to the left of the imaginary axis if and only if 4a 2 - < ¨ Ž ˆ b .<. The alignment condition of ¨ Ž ˆ b . and w H Ž ˆ b . is cot ˆ bs
ˆb 4
y
1
Ž 4.6.
ˆb
and the condition 4a 2 - < ¨ Ž ˆ b .< is 4a 2 - 4 q ˆ b2 . Replacing ˆ b by 2 z , Eq. Ž4.7. becomes a uniqueness of ˆ b, Ž4.6. leads to z tan z s 1.
Ž 4.7.
'z
2
q 1 and, by the
Considering the equation corresponding to Ž4.3. with an arbitrary delay t ) 0, that is,
˙x Ž t . s yx Ž t . q a F Ž x Ž t y t . . ,
t ) 0,
the characteristic equation Ž4.5. becomes
Ž z 2 q 4 z q 4 . et z q 4a 2 s 0,
a ) 0.
Ž 4.8.
We can state the following result on stability independent of the delay: THEOREM 4.4. only if a - 1.
Equation Ž4.8. is stable independently of the delay if and
Proof. This is a consequence of Theorem 2.4, by noticing that in this case we have < ¨ Ž b . < s < Ž 4 b, 4 y b 2 . < ) 4a 2 for any b ) 0
m
a - 1.
In w1x the authors consider Eq. Ž4.3. with a slight change in the position of the parameter a ,
˙x Ž t . s ya x Ž t . q a F Ž x Ž t y 1 . . assuming one condition further on F, y
F1 x2
Ž 0.
F2 x1
Ž 0 . s d ) 1.
Ž 4.9.
EXPONENTIAL POLYNOMIALS
271
Now, the characteristic equation of the linear part is 2 Ž l q a . e 2 l s yd 2a 2
Ž 4.10.
and we can state THEOREM 4.5.
All roots of Eq. Ž4.10. ha¨ e negati¨ e real parts if and only
if
a'd 2 y 1 - z ,
where z s arccot'd 2 y 1 .
Ž 4.11.
Proof. With the introduction of a new variable z, by 2 l s z, Eq. Ž4.10. becomes
Ž z 2 q 4a z q 4a 2 . e z s y4d 2a 2
Ž 4.12.
and, in this case, ¨ Ž b . s Ž4a b, 4a 2 y b 2 .. There is a unique ˆ b g Ž0, p . such that ¨ Ž ˆ b . is a positive multiple of H Ž ˆ. ˆ ˆ Ž . w b s sin b, ycos b . This condition of alignment is cot ˆ bs
ˆb 4a
y
a
ˆb
.
Ž 4.13.
Introducing z by ˆ b s 2 z one has 2 cot 2 z s zra y arz and, considering the identity 2 cot 2 x s cot x y tan x, the uniqueness of ˆ b implies that Eq. Ž4.13. is equivalent to
z tan z s a .
Ž 4.14.
Since < ¨ Ž ˆ b .< is the lower value of < ¨ Ž b .<, b G 0, such that ¨ Ž b . is a positive multiple of w H Ž b ., Theorem 2.1 implies that a necessary and sufficient condition for the roots of Eq. Ž4.10. to have negative real parts is that 4d 2a 2 - < ¨ Ž ˆ b. <.
Ž 4.15.
If we have the equality 4d 2a 2 s < ¨ Ž ˆ b .< instead of Ž4.15. we could write it equivalently
a'd 2 y 1 s z ,
Ž 4.16.
'd 2 y 1 s cot z .
Ž 4.17.
which, according to Ž4.14., is
272
BAPTISTINI AND TABOAS ´
Therefore, a necessary and sufficient condition for the roots of Eq. Ž4.10. to have negative real parts is that, instead of Ž4.16., we have
a'd 2 y 1 - z ,
where z s arccot'd 2 y 1 .
Ž 4.18.
Remark 4.6. In w1x the value stated for z is z s arcsinŽ1rd ., but if we define u , 0 - u - pr2, by 1rd s sin u , Eq. Ž4.17. gives cot u s cot z , that is, z s arcsinŽ1rd ..
REFERENCES 1. M. Baptistini and P. Taboas, On the existence and global bifurcation of periodic solutions ´ to planar differential delay equations, J. Differential Equations 127, Ž1996., 391]425. 2. R. Bellman and K. Cooke, ‘‘Differential-Difference Equations,’’ Academic Press, New York, 1963. 3. K. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 Ž1982., 592]627. 4. J. Hale and S. Lunel, ‘‘Introduction to Functional Differential Equations,’’ SpringerVerlag, New York, 1993. 5. N. Hayes, Roots of the transcendental equation associated to a certain differencedifferential equation, J. London Math. Soc. 25 Ž1950., 226]232. 6. L. S. Pontryagin, On the zeros of some elementary transcendental functions, in ‘‘Amer. Math. Soc. Transl. Ser. 2,’’ Vol. 1, pp. 95]110, Amer. Math. Soc., Providence, 1955. 7. P. Taboas, Periodic solutions of a planar delay equation, Proc. Roy. Soc. Edinburgh Sect. A ´ 116 Ž1990., 85]101.
205, 259]272 Ž1997.
AY965152
On the Stability of Some Exponential Polynomials Margarete Baptistini* Uni¨ ersidade Federal de Sao ˜ Carlos-DM, C. Postal 676, Sao ˜ Carlos, SP 13560-970, Brazil
and † Placido Taboas ´ ´
Uni¨ ersidade de Sao ˜ Paulo-ICMSC, C. Postal 668, Sao ˜ Carlos, SP 13560-970, Brazil Submitted by Jack K. Hale Received January 30, 1996
This paper studies elementary transcendental equations of the type Ž z 2 q pz q . q et z q rz n s 0, where p, q, r g R, t , p ) 0, q G 0, r / 0, n s 0, 1, 2. We are mainly interested in the case n s 0 for which a characterization of stability is accomplished; that is, we state a necessary and sufficient condition for all the roots to lie to the left of the imaginary axis. Also a characterization of stability independent of delays is given. Sufficient conditions for instability are stated for the cases n s 1, 2. The proofs are carried by elementary arguments independent of usual tools. Q 1997 Academic Press
1. INTRODUCTION We are concerned with transcendental equations of the type P Ž z, e z . s 0, where P Ž x, y . is a polynomial in x and y. These equations were systematically studied by Pontryagin w6x, extending an earlier work of ˇebotarev. The analysis on how their roots locate with respect to the C imaginary axis is an old problem that, besides being important by itself, plays a role in the study of asymptotic behavior in the theory of delay differential equations. On the basis of that, there is a natural interest in determining the stability of P Ž z, e z . s 0, that is, when all of its roots have *E-mail address: [email protected]. † E-mail address: [email protected]. 259 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
260
BAPTISTINI AND TABOAS ´
negative real parts. The ingredients usually involved to handle questions of this nature are classical results on zeros of analytic functions, such as Hurwitz’ Theorem or Rouche’s ´ Theorem and Pontryagin’s monograph. Also Bellman and Cooke’s book w2x is a widely cited reference in this subject. It devotes its thirteenth chapter to applications of Pontryagin’s results in regard to stability theory of delay differential equations. There is a special interest in low degrees of P Ž x, y . motivated by applications. In the setting of these equations this paper treats a specialization discussed in Bellman and Cooke’s book in its Section 13.8, named ‘‘An Important Equation.’’ There the equation Ž z 2 q pz q q . e z q rz n s 0, with p, q, r g R, p ) 0, q G 0, r / 0 and n a positive integer or zero, is treated with respect to its stability. The case n s 0, which appears quite often in the literature of delay differential equations is our main concern here. This case is also the main concern of Section 13.8 of w2x. Our approach can be adopted even in the cases n s 1 and n s 2, but except for particular equations, the analysis becomes considerably more complicated to lead to simple general results like Theorems 2.1 and 2.2. We accomplish sufficient conditions for instability in these cases. We do not focus on the cases n G 3 because in this circumstance the corresponding polynomial P Ž x, y . has no principal term and, in this case, it is known that the equation P Ž z, e z . s 0 has infinitely many roots with arbitrarily large positive real parts. See w2x or w6x, for instance. The lower degree case Ž z q p . e z q q s 0 is studied by Hayes w5x. For this equation, a more complete analysis including the question of stability independent of delays is given by Hale and Lunel in w4x, based on a Liapunov functional for the corresponding delay equation. The main results are stated in the Section 2. Theorems 2.1 and 2.2 characterize the equations Ž z 2 q pz q q . e z q r s 0 having all roots to the left of the imaginary axis. They have the same goal of Theorem 13.9 of w2x, but our statements are simpler and our proofs, based only in elementary arguments, are completely independent of Pontryagin’s results. Indeed, Professor Cooke communicated to us in person that Theorem 13.9 of w2x contains some mistakes, so that Theorems 2.1 and 2.2 are a new characterization of stability for the equations under consideration. Section 2 ends with Theorem 2.4, which gives a necessary and sufficient condition for the equation Ž z 2 q pz q q . et z q r s 0 to be stable and the stability to be independent of the delay. That is, for any t ) 0, all the roots of the equation have negative real part. In Section 3 we investigate the cases n s 1 and n s 2. The same approach of Section 2 is applied to get sufficient conditions for instability. Section 4 gives some applications. We use Theorem 2.1 to simplify a result related to a problem of nonlinear oscillations theory: a damped nonlinear oscillator, with a time delay in the restoring force. See Section
261
EXPONENTIAL POLYNOMIALS
11.6 of w4x, for instance. Also Cooke and Grossman w3x investigate this model in regard to stability switches. Finally, we apply Theorem 2.1 in the investigation of stability of the characteristic equation of the linear part of a system of two delay differential equations near equilibria. 2. AN IMPORTANT EQUATION The complex function H is given by H Ž z . s Ž z 2 q pz q q . e z q r ,
Ž 2.1.
where p, q, r g R, p ) 0, q G 0, r / 0. Thus the case n s 0 of the equation we wish to study is H Ž z . s 0. THEOREM 2.1. Let the 2-¨ ectors ¨ Ž b . s Ž pb, q y b 2 ., w Ž b . s Žcos b, sin b ., b G 0, be gi¨ en. If r ) 0, a necessary and sufficient condition for all roots of the equation H Ž z . s 0 to ha¨ e negati¨ e real part is that the orthogonality condition ¨ Ž b . ? w Ž b . s 0, with b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., implies < ¨ Ž b .< ) r. Proof. Let us denote z s a q bi. There is no loss of generality in restricting the proof to the case b G 0 since H Ž z . s H Ž z ., where z denotes the complex conjugate to z. If H Ž z . s F Ž z . q iGŽ z ., the equation H Ž z . s 0 is equivalent to the system FŽ z. s GŽ z . s
Ž a2 q pa q q y b 2 . cos b y Ž 2 a q p . b sin b Ž a2 q pa q q y b 2 . sin b q Ž 2 a q p . b cos b
e a q r s 0, e a s 0,
Ž 2.2.
or, in a vector form, e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . ? Ž ysin b, cos b . s yr ,
Ž 2 ab, a2 q pa. q Ž pb, q y b 2 .
? Ž cos b, sin b . s 0.
Ž 2.3.
Defining w H Ž b . s Žsin b, ycos b ., uŽ a, b . s Ž2 ab, a2 q pa., the system Ž2.3. reduces to the single vector equation:
n Ž a, b . [ e a u Ž a, b . q ¨ Ž b . s rw H Ž b . . Ž 2.4. Ž . For each fixed a G 0, the vector n a, b describes clockwise an unbounded arc of parabola in the jh-plane, while w H Ž b . describes counterclockwise the unit circle, as b G 0 increases. Thus, in each interval Ž2 kp , Ž2 k q 1.p ., k s 0, 1, . . . , there exists precisely one number, bk Ž a., for which n Ž a, bk Ž a.. is a positive multiple of w H Ž bk Ž a... Moreover, the condition n Ž a, bk Ž a.. ? w Ž b . s 0 holds only if b s bk Ž a., k s 0, 1, . . . . For each nonnegative integer k, bk Ž a. depends continuously on a, and Ž a, bk Ž a.. satisfies Eq. Ž2.4. if, and only if, < n Ž a, bk Ž a..< s r.
262
BAPTISTINI AND TABOAS ´
Let us prove the necessity. Suppose R Ž z . - 0 for all roots of H Ž z . s 0, and assume temporarily the condition stated in the theorem is false. Therefore, for some integer k G 0, the number bk s bk Ž0. g Ž2 kp , Ž2 k q 1.p ., such that ¨ Ž bk . is a multiple Žpositive. of w H Ž bk ., must satisfy < ¨ Ž bk .< F r. If < ¨ Ž bk .< s r, bk must satisfy Ž2.4. with a s 0, that is, ¨ Ž bk . s rw H Ž bk . and, therefore, z s ibk is a pure imaginary root of H Ž z . s 0, a contradiction. It remains to consider the case < ¨ Ž bk .< - r. Under this hypothesis the circle of radius r determines a bounded arc, to which ¨ Ž bk . belongs, on the branch of parabola ¨ Ž b ., b G 0. Since lim aª` < n Ž a, bk Ž a..< s `, the continuity of n Ž a, bk Ž a.. implies there exists a positive number a0 in such a way that < n Ž a0 , bk Ž a0 ..< s r. This implies Ž a0 , bk Ž a0 .. satisfies Eq. Ž2.4. and, therefore, z s a0 q ibk Ž a0 . is a root of H Ž z . s 0 to the right of the imaginary axis, a contradiction. Thus, the condition is necessary. Let us prove now the sufficiency by showing that if there exists a root z of H Ž z . s 0 such that R Ž z . G 0, then for some b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., with ¨ Ž b . ? w Ž b . s 0, we have 0 - < ¨ Ž b .< F r. Suppose ˜ zsa ˜ q ib˜ is a root of H Ž z . s 0, with a˜ G 0. Notice that for any root z s a q ib of H Ž z . s 0, with a G 0, according to Eq. Ž2.4. we have < uŽ a, b . q ¨ Ž b .< F r. Therefore, if z s a q ib varies in the set of roots of H Ž z . s 0 with nonnegative real part, its imaginary part b is bounded. The boundedness of b implies that ˜ z can be choosen in such a way that a ˜ is minimum keeping the property of ˜z. Let k G 0 such that ˜ b g Ž2 kp , Ž2 k q 1.p .. For each a, 0 - a F a, ˜ let Ž . bk a as defined before, that is, a number b g Ž2 kp , Ž2 k q 1.p . such that n Ž a, bk Ž a.. is a multiple of w H Ž bk Ž a... For each a, 0 - a - a, ˜ consider the number r s r Ž a. ) 0 defined by
n Ž a, bk Ž a . . s r w H Ž bk Ž a . . .
Ž 2.5.
From the minimality of a, ˜ it follows that r / r. We claim that r - r. In fact, for any a ) 0 an elementary calculation gives
a
< n Ž a, b . < 2 s 2 e 2 a Ž 2 a q p . 2 b 2 q Ž a2 q pa q q y b 2 .
2
q e 2 a 4 Ž 2 a q p . b 2 q 2 Ž 2 a q p . Ž a2 q pa q q y b 2 . s 2 < n Ž a, b . < 2 q e 2 a Ž 2 a q p . b 2 q Ž 2 a q p . Ž a2 q pa q q .
) 0.
EXPONENTIAL POLYNOMIALS
263
Therefore, if 0 - a - a ˜ we can assure that < n Ž a, ˜b .< - < n Ž a, ˜ ˜b .< s r, that ˜ is, n Ž a, b . lies inside the circle of radius r with center in the origin, according to Fig. 1. If a point Q s Ž j , h . / Ž0, yr . belongs to the circle of radius r with center in the origin, we denote by l Q the length of the arc of this circle from Ž0, yr . to Q in the counterclockwise sense. Suppose for a while the alignment of n Ž a, bk Ž a.. and w H Ž bk Ž a.. occurs with n Ž a, bk Ž a.. outside the circle of radius r with center in the origin, that is, r s < n Ž a, bk Ž a..< G r. Therefore, either bk Ž a. - ˜ b or bk Ž a. ) ˜ b and each of these alternatives leads respectively to l w r w H Ž b k Ž a..x - l w r w H Ž ˜b.x s ln Ž a, ˜ ˜b. - l w rn Ž a, b k Ž a..r r x or l w rn Ž a, b k Ž a..r r x - ln Ž a, ˜ ˜b. s l w r w H Ž ˜b.x - l w r w H Ž b k Ž a..x and this implies, in both cases, that n Ž a, bk Ž a.. and w H Ž bk Ž a.. cannot be aligned, which is a contradiction. Therefore, Eq. Ž2.5. holds with r - r, that is, n Ž a, bk Ž a.. remains inside the circle of radius r with center in the origin for any a, 0 - a - a. ˜ This gives < ¨ Ž bk Ž0..< F r and, therefore, the sufficiency. We have an analogous result for the case r - 0: THEOREM 2.2. If r - 0, keeping the notation of Theorem 2.1, a necessary and sufficient condition for all roots of H Ž z . s 0 to ha¨ e negati¨ e real part is that the orthogonality condition ¨ Ž b . ? w Ž b . s 0, with b g w 0, `. _ D `ks 0 Ž2 kp , Ž2 k q 1.p ., implies < ¨ Ž b .< ) yr.
FIG. 1. Location of n Ž a, ˜ b ..
264
BAPTISTINI AND TABOAS ´
Proof. In this case we might repeat the steps of the proof of Theorem 2.1, replacing r by < r < and introducing H w Ž b . s Žysin b, cos b . to play the role of the vector w H Ž b . in Eq. Ž2.4. which becomes
n Ž a, b . [ e a u Ž a, b . q ¨ Ž b . s yr H w Ž b . .
Ž 2.6.
Since real roots are admissible in this case, some minor additional care is needed. For the sufficiency, let us assume the existence of a root ˜ z with RŽ ˜ z . G 0. We need to prove that there exists b g w 0, `. _ D `ks 0 Ž2 kp , Ž2 k q 1.p . such that < ¨ Ž b .< F yr. If ˜ zsa ˜ G 0 is a real root, Eq. Ž2.6. implies n Ž a, ˜ 0. s yr Ž0, 1. which gives < ¨ Ž0.< s q F yr. If J Ž ˜z . cannot vanish, the proof of sufficiency is analogous to the case r ) 0. Doing the proof of necessity by contradiction as in the case r ) 0, in assuming the failure of the condition stated in the theorem, one needs to consider the possibility < ¨ Ž0.< F yr or, equivalently, q F yr. This leads to a real root a G 0 of H Ž z . s 0, where a is the unique nonnegative solution of e a Ž a2 q pa q q . s yr. From this point on the proof follows by the same arguments of the proof of necessity in the case r ) 0. Remark 2.3. If r ) 0, the arguments of the proof of Theorem 2.1 allow us to estimate the number of roots of H Ž z . s 0 with positive real part, by extending the definition of ¨ Ž b . s Ž pb, q y b 2 . to all values of b g R. The number of roots with positive real part coincide with the number of values of b g R, with "b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., such that ¨ Ž b . ? w Ž b . s 0 and < ¨ Ž b .< - r. This fact follows directly from the geometric arguments given in the proof of the necessity of Theorem 2.1. An analogous remark holds for the case r - 0, that is, the number of roots with positive real part coincide with the number of values of b g R, with "b g w 0, `. _ D `ks 0 Ž2 kp , Ž2 k q 1.p . such that ¨ Ž b . ? w Ž b . s 0 and 0 < ¨ Ž b .< - yr. In order to understand the equation H Ž z . s 0 as a characteristic equation of a linear delay differential equation in its extent generality, a normalization is presumed to make the delay equal to 1. By ignoring this fact, the equation should be
Ž z 2 q pz q q . et z q r s 0,
Ž 2.7.
with the same assumptions on p, q, and r, where t ) 0 is the delay. In this setting, it is important to investigate the stability of Eq. Ž2.7. independently of the delay. The next theorem gives a complete answer to this question. THEOREM 2.4. Consider the 2-¨ ector ¨ Ž b . s Ž pb, q y b 2 ., b G 0. A necessary and sufficient condition for Eq. Ž2.7. to be stable for any t ) 0 is that < ¨ Ž b .< ) r, for any b ) 0, if r ) 0 or < ¨ Ž b .< ) yr, for any b G 0, if r - 0.
EXPONENTIAL POLYNOMIALS
265
Proof. Let us suppose r ) 0. After the change of variable z l t z, Eq. Ž2.7. is written in the form
Ž z 2 q t pz q t 2 q . e z q t 2 r s 0
Ž 2.8.
so that, defining the 2-vector ¨t Ž b . s Žt pb, t 2 q y b 2 ., b G 0, and recalling that w Ž b . s Žcos b, sin b ., it follows from Theorem 2.1 that all of the roots of Ž2.8. and, therefore, the roots of Ž2.7. lie to the left of the imaginary axis if and only if, for any b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., ¨t Ž b . ? w Ž b . s 0
«
< ¨t Ž b . < ) t 2 r .
Ž 2.9.
Noticing that ¨t Ž b . is parallel to ¨ Ž brt ., since ¨t Ž b . s t 2 ¨ Ž brt ., the condition Ž2.9. can be reformulated as ¨ Ž brt . ? w Ž b . s 0
«
< ¨ Ž brt . < ) r .
Thus, by the change of parameters b l brt the necessary and sufficient condition for the stability of Ž2.7. becomes that for any b g D `ks 0 Ž2 kprt , Ž2 k q 1.prt ., ¨ Ž b. ? w Žt b. s 0
«
<¨ Ž b. < ) r .
Ž 2.10.
Let us prove now the assertion of the theorem. The sufficiency is immediate since if < ¨ Ž b .< ) < r < for any b G 0, the inequality of the right side of Ž2.10. holds for any b g D `ks 0 Ž2 kprt , Ž2 k q 1.prt ., for any t ) 0. To prove the necessity suppose Eq. Ž2.7. is stable independently of the delay. Therefore, the condition Ž2.10. holds for any t ) 0. Assuming temporarily that there exists ˆ b ) 0 such that < ¨ Ž ˆ b .< F r, one can adjust ` ˆ Ž Ž . . t ) 0 in order that t b g D ks 0 2 kp , 2 k q 1 p and ¨ Ž ˆ brt . ? w Ž ˆ b . s 0. Ž . This contradicts 2.10 and so the condition is necessary. In the case r - 0 the proof is carried similarly by using Theorem 2.2 instead of Theorem 2.1. 3. THE CASES n s 1 AND n s 2 The same approach is adequate to examine the cases n s 1 and n s 2. For the equation
Ž z 2 q pz q q . e z q rz s 0
Ž 3.1.
BAPTISTINI AND TABOAS ´
266 we have the following
THEOREM 3.1. A sufficient condition for the existence of a root z of Ž3.1. with nonnegati¨ e real part is that, for some integer k G 0, the unique solution bk in ŽŽ2 k q 1r2.p , Ž2 k q 3r2.p . of tan b s satisfies rbk G
q y b2 pb
Ž 3.2.
'Ž pb . q Ž q y b . . 2 2 k
2
k
Proof. Replacing z s a q bi in Ž3.1. we are lead to the equivalent system e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . Ž sin b, ycos b . s ra, e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . Ž cos b, sin b . s yrb.
Ž 3.3.
Since if z is a root of Ž3.1., z also is a root, we again confine ourselves to the case b G 0. For any a G 0 the system Ž3.3. means that z s a q ˜ bi is a solution of Ž3.1. if and only if the point of the plane Ž j , h . in the parabola e awŽ2 ab, a2 q pa. q Ž pb, q y b 2 .x, corresponding to b s ˜ b, has coordinates ˜. in the moving coordinates system wŽsin b, ycos b ., Žcos b, sin b .x. Ž ra, yrb When b G 0 varies, the points Ž ra, yrb . describe the spiral
ga Ž b . :
½
j h
s r Ž a sin b y b cos b . s yr Ž b sin b q a cos b . .
Ž 3.4.
So, in other terms, z s a q ˜ bi is a solution of Ž3.1. if and only if the points ˜ corresponding to b in the parabola ¨ aŽ b . s e awŽ2 ab, a2 q pa. q Ž pb, q y b 2 .x and in the spiral Ž3.4. coincide. When b varies positively in the interval w 0, `. , at each complete lap of Ž ga b . on the spiral in the counterclockwise sense, the point ¨ aŽ b . describes an arc of parabola in the clockwise sense in the semi-plane j G 0. Therefore, at the kth lap there exists a unique value bk Ž a. ) 0 such that ¨ aŽ bk Ž a.. is a positive multiple of gaŽ bk Ž a.. and bk Ž a. depends continuously on a. For a s 0, the is spiral g 0 Ž b . Ž j , h . s yrbŽcos b, sin b . and the parabola Ž ¨ 0 b . is given by Ž j , h . s Ž pb, q y b 2 .. Let G be the intersection of the spiral g 0 Ž b . and the semi-plane j G 0. In this case, for any integer k G 0, the point of alignment bk s bk Ž0. is reached when b varies positively in the interval ŽŽ2 k q 1r2.p , Ž2 k q 3r2.p .. This is because ¨ 0 Ž b . describes in the clockwise sense an arc of the parabola in the semi-plane j G 0 while and an entire connected component of G is described by g 0 Ž b . in the counterclockwise sense.
EXPONENTIAL POLYNOMIALS
267
The condition stated in the theorem says that bk s bk Ž0., that is, the points in the parabola ¨ 0 Ž bk . and in the spiral g 0 Ž bk . are aligned and, furthermore, < ¨ 0 Ž bk Ž0..< F < g 0 Ž bk Ž0..<. See Fig. 2. Thus, for small a G 0, the point ¨ aŽ bk Ž a.. belongs to an arc V a of the parabola ¨ aŽ b . determined by the spiral gaŽ b .. Since the parabola ¨ aŽ b . dilates with the order O Ž e aa2 . while the spiral gaŽ b . dilates with the order O Ž a., as a ª `, one sees that the arc V a collapses as a increases approaching `. Before that, we must reach a value a ˜ G 0 such that ¨ a˜Ž bk Ž a˜.. s ga˜Ž bk Ž a˜... For the equation
Ž z 2 q pz q q . e z q rz 2 s 0
Ž 3.5.
we have the following result THEOREM 3.2. A sufficient condition for the existence of a root z of Ž3.5. with nonnegati¨ e real part is that, for some integer k G 0, the unique solution bk in ŽŽ2 k q 1.p , 2Ž k q 1.p . of
ycot b s
satisfies rbk2 G
q y b2
'Ž pb . q Ž q y b . . 2
k
2 2 k
FIGURE 2
pb
Ž 3.6.
268
BAPTISTINI AND TABOAS ´
A proof can be accomplished by noticing that if z s a q bi, b G 0, then Eq. Ž3.5. is equivalent to the system e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . Ž sin b, ycos b . s r Ž a2 y b 2 . , e a Ž 2 ab, a2 q pa. q Ž pb, q y b 2 . Ž cos b, sin b . s yrab.
Ž 3.7.
For any a G 0, let us consider the parabola ¨ nŽ b . s e awŽ2 ab, a2 q pa. q Ž pb, q y b 2 .x and the spiral gaŽ b ., described by the points having coordinates Ž r Ž a2 y b 2 ., y2 rab. in the moving coordinate system wŽsin b, ycos b ., Žcos b, sin b .x. In this setting, the system Ž3.7. means that z s a q ˜ bi is a solution of Ž3.5. if and only if the points corresponding to b s ˜ b in the parabola ¨ aŽ b . and in the spiral gaŽ b . coincide. The proof now follows the same steps of the proof of Theorem 3.2 by noticing that the parabola ¨ aŽ b . dilates with the order O Ž e aa2 ., while the spiral dilates with the order O Ž a2 ., as a ª `.
4. APPLICATIONS In Section 11.6 of w4x, Hale and Lunel discuss the existence of nonconstant periodic solutions of the equation
¨x Ž t . q f Ž x Ž t . . ˙x Ž t . q g Ž x Ž t y 1 . . s 0,
Ž 4.1.
where r G 0, f is continuous, f Ž0. s yk ) 0, and g is continuous together with its first derivative, g 9Ž0. s 1. The authors state a Hopf bifurcation for Ž4.1., considering k as a parameter. Besides, they apply an ejective fixed point principle to achieve an existence theorem for nonconstant periodic solutions, for k greater than a certain critical value. For that, it is necessary to find out the first value of k for which the characteristic equation associated to the linear part of Eq. Ž4.1.,
l2 y k l q eyl r s 0,
Ž 4.2.
has a pair of roots on the imaginary axis. Their discussion depends in a crucial way on Theorem A.6 of the Appendix of w4x. We state below theorem A.6 of w4x and, since Ž4.2. is a special case of the equation H Ž z . s 0, we may give a simpler proof based on Theorem 2.1. THEOREM 4.1 w4, Theorem A.6x. All roots of the equation Ž z 2 q az . e z q 1 s 0 ha¨ e negati¨ e real parts if and only if a ) Žsin z .rz , where z is the unique root of the equation z 2 s cos z , 0 - z - pr2.
EXPONENTIAL POLYNOMIALS
269
Proof. In this case we have H Ž z . s Ž z 2 q az . e z q 1 and ¨ Ž b . s Ž ab, yb 2 .. Let b g D `ks 0 Ž2 kp , Ž2 k q 1.p ., then < ¨ Ž b .< F 1 holds only if b g Ž0, 1. ; Ž0, pr2.. There is a unique ˆ b g Ž0, pr2., with ¨ Ž ˆ b. ? wŽ ˆ b . s 0, so that, by Theorem 2.1, all roots of the H Ž z . s 0 have negative real parts if and only if < ¨ Ž ˆ b .< ) 1. Let z g Ž0, pr2. be the unique root of the equation z 2 s cos z . Since ¨ Ž z . s Ž az , yz 2 . and w H Ž z . s Žsin z , cos z . lie in the same horizontal line, a necessary and sufficient condition for < ¨ Ž ˆ b .< ) 1 is that ¨ Ž z . s Ž az , ycos z . lies outside the unit circle. But this is equivalent to az ) sin z . Remark 4.2. By taking into consideration the delay in the equation of Theorem 4.1, it becomes Ž z 2 q az . et z q 1 s 0. Since the parabola ¨ Ž b . s Ž ab, yb 2 . passes through the origin, in this case, according to the Theorem 2.4, there is no hope to obtain stability independent of the delays. In w7x the second author studies the planar delay differential equation
˙x Ž t . s yx Ž t . q a F Ž x Ž t y 1 . . ,
Ž 4.3.
where x s Ž x 1 , x 2 . g R 2 , F s Ž F1 , F2 . is a bounded continuous map from R 2 into itself, and a ) 0 is a real parameter. Besides some smoothness condition on F near the origin, with Ž F1r x 2 .Ž0., Ž F2r x 1 .Ž0. / 0, the following hypothesis is assumed, x 2 F1 Ž x . ) 0, x 2 / 0;
x 1 F2 Ž x . - 0, x 1 / 0,
Ž H1.
and F is normalized in such a way that Ž F1r x 2 .Ž0. s yŽ F2r x 1 .Ž0. s 1. The analysis in w7x depends on a certain return map on a closed convex subset of the phase space. In order to get an ejectivity condition for the origin, with respect to this map, it is needed to study the location of the zeros of the characteristic equation of the linear part of Ž4.3., 2 Ž l q 1 . e 2 l s ya 2
Ž 4.4.
which, replacing l by zr2, becomes
Ž z 2 q 4 z q 4 . e z q 4a 2 s 0.
Ž 4.5.
Although another approach is adopted, the results in w7x contain the following theorem, which is proved here applying Theorem 2.1: THEOREM 4.3. All roots of Eq. Ž4.4. ha¨ e negati¨ e real parts if and only if a - z 2 q 1 , where z g Ž0, pr2. is uniquely defined by z tan z s 1.
'
BAPTISTINI AND TABOAS ´
270
Proof. If H Ž z . s Ž z 2 q 4 z q 4. e z q 4a 2 , we have ¨ Ž b . s Ž4 b, 4 y b 2 .. Thus, when b g w0, 2x increases, ¨ Ž b . describes an arc of parabola, from Ž4, 0. to Ž0, 8., while w H Ž b . varies counterclockwise on the unit circle, from Ž0, y1. to Žsin 2, ycos 2.. There is a unique ˆ b g Žpr2, 2. such that ¨ Žˆ b . is a positive multiple of w H Ž ˆ b . and < ¨ Ž ˆ b .< is minimum among the ¨ Ž b .’s, b G 0, with this alignment property. Therefore, all roots of Ž4.4. lie to the left of the imaginary axis if and only if 4a 2 - < ¨ Ž ˆ b .<. The alignment condition of ¨ Ž ˆ b . and w H Ž ˆ b . is cot ˆ bs
ˆb 4
y
1
Ž 4.6.
ˆb
and the condition 4a 2 - < ¨ Ž ˆ b .< is 4a 2 - 4 q ˆ b2 . Replacing ˆ b by 2 z , Eq. Ž4.7. becomes a uniqueness of ˆ b, Ž4.6. leads to z tan z s 1.
Ž 4.7.
'z
2
q 1 and, by the
Considering the equation corresponding to Ž4.3. with an arbitrary delay t ) 0, that is,
˙x Ž t . s yx Ž t . q a F Ž x Ž t y t . . ,
t ) 0,
the characteristic equation Ž4.5. becomes
Ž z 2 q 4 z q 4 . et z q 4a 2 s 0,
a ) 0.
Ž 4.8.
We can state the following result on stability independent of the delay: THEOREM 4.4. only if a - 1.
Equation Ž4.8. is stable independently of the delay if and
Proof. This is a consequence of Theorem 2.4, by noticing that in this case we have < ¨ Ž b . < s < Ž 4 b, 4 y b 2 . < ) 4a 2 for any b ) 0
m
a - 1.
In w1x the authors consider Eq. Ž4.3. with a slight change in the position of the parameter a ,
˙x Ž t . s ya x Ž t . q a F Ž x Ž t y 1 . . assuming one condition further on F, y
F1 x2
Ž 0.
F2 x1
Ž 0 . s d ) 1.
Ž 4.9.
EXPONENTIAL POLYNOMIALS
271
Now, the characteristic equation of the linear part is 2 Ž l q a . e 2 l s yd 2a 2
Ž 4.10.
and we can state THEOREM 4.5.
All roots of Eq. Ž4.10. ha¨ e negati¨ e real parts if and only
if
a'd 2 y 1 - z ,
where z s arccot'd 2 y 1 .
Ž 4.11.
Proof. With the introduction of a new variable z, by 2 l s z, Eq. Ž4.10. becomes
Ž z 2 q 4a z q 4a 2 . e z s y4d 2a 2
Ž 4.12.
and, in this case, ¨ Ž b . s Ž4a b, 4a 2 y b 2 .. There is a unique ˆ b g Ž0, p . such that ¨ Ž ˆ b . is a positive multiple of H Ž ˆ. ˆ ˆ Ž . w b s sin b, ycos b . This condition of alignment is cot ˆ bs
ˆb 4a
y
a
ˆb
.
Ž 4.13.
Introducing z by ˆ b s 2 z one has 2 cot 2 z s zra y arz and, considering the identity 2 cot 2 x s cot x y tan x, the uniqueness of ˆ b implies that Eq. Ž4.13. is equivalent to
z tan z s a .
Ž 4.14.
Since < ¨ Ž ˆ b .< is the lower value of < ¨ Ž b .<, b G 0, such that ¨ Ž b . is a positive multiple of w H Ž b ., Theorem 2.1 implies that a necessary and sufficient condition for the roots of Eq. Ž4.10. to have negative real parts is that 4d 2a 2 - < ¨ Ž ˆ b. <.
Ž 4.15.
If we have the equality 4d 2a 2 s < ¨ Ž ˆ b .< instead of Ž4.15. we could write it equivalently
a'd 2 y 1 s z ,
Ž 4.16.
'd 2 y 1 s cot z .
Ž 4.17.
which, according to Ž4.14., is
272
BAPTISTINI AND TABOAS ´
Therefore, a necessary and sufficient condition for the roots of Eq. Ž4.10. to have negative real parts is that, instead of Ž4.16., we have
a'd 2 y 1 - z ,
where z s arccot'd 2 y 1 .
Ž 4.18.
Remark 4.6. In w1x the value stated for z is z s arcsinŽ1rd ., but if we define u , 0 - u - pr2, by 1rd s sin u , Eq. Ž4.17. gives cot u s cot z , that is, z s arcsinŽ1rd ..
REFERENCES 1. M. Baptistini and P. Taboas, On the existence and global bifurcation of periodic solutions ´ to planar differential delay equations, J. Differential Equations 127, Ž1996., 391]425. 2. R. Bellman and K. Cooke, ‘‘Differential-Difference Equations,’’ Academic Press, New York, 1963. 3. K. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 Ž1982., 592]627. 4. J. Hale and S. Lunel, ‘‘Introduction to Functional Differential Equations,’’ SpringerVerlag, New York, 1993. 5. N. Hayes, Roots of the transcendental equation associated to a certain differencedifferential equation, J. London Math. Soc. 25 Ž1950., 226]232. 6. L. S. Pontryagin, On the zeros of some elementary transcendental functions, in ‘‘Amer. Math. Soc. Transl. Ser. 2,’’ Vol. 1, pp. 95]110, Amer. Math. Soc., Providence, 1955. 7. P. Taboas, Periodic solutions of a planar delay equation, Proc. Roy. Soc. Edinburgh Sect. A ´ 116 Ž1990., 85]101.