Periodic solutions of a quartic nonautonomous equation

NonlinearAnalysis, Theory, Methods & Applications, Vol. 11, No. 7, pp. 809-820, 1987. Printed in Great Britain.

PERIODIC

SOLUTIONS

OF A QUARTIC EQUATION

0362-546X/87 $3.00 + .00 PergamonJournals Ltd.

NONAUTONOMOUS

M. A. M. ALWASH* and N. G. LLOYD Pure Mathematics Department, The University College of Wales, Aberystwyth, Dyfed, Wales, U.K.

(Received 17 January 1986; received for publication 4 June 1986) Key words and phrases: Periodic solutions, nonautonomous equations, polynomial systems.

1. I N T R O D U C T I O N

IN [1] we investigated equations of the form = cr(t)z 3 + fl(t)z 2 + y(t)z

(z e C)

(1.1)

in some detail. We explained the connection between such equations and certain polynomial systems in the plane (see also [5]); the whole discussion was motivated by the desire to obtain estimates for the possible number of limit cycles of certain classes of such systems--part of Hilbert's sixteenth problem. Given co E ~, we sought information on the number of "periodic" solutions, that is, solutions satisfying (1.2)

z(O) = z ( w )

- - a n d were especially interested in the multiplicity of z = 0 as a periodic solution. The coefficients a, /3 and y were supposed to be real-valued continuous functions. We explained why it is useful to work with a complex independent variable; briefly, the reason is that periodic solutions cannot then be destroyed by small perturbations of the right-hand side of the equation. We refer to [1] for full details, but here recall that the multiplicity of a solution q~(t) of (1.1) satisfying (1.2) is the multiplicity of q~(0) as a zero of the holomorphic function q : c ~ z(w; O, c) - c; z(t; to, c) is the solution satisfying z(t0; to, c) = c. Note that q is defined and holomorphic in an open set containing the origin. Suppose that q~(t) is a periodic solution of multiplicity k. By applying RouchCs theorem to the function q, we see that, for any sufficiently small perturbation of the equation, there are precisely k periodic solutions in a neighbourhood of q~ (counting multiplicity): see [1, theorem 2.4]. Hence, if e is sufficiently small, the equation = EZ 4 "}- 0l(t)z 3 + t ~ ( t ) z 2 + y ( t ) z

has at least as many isolated periodic solutions as (1.1). Under the scaling z ~ et/3z, this equation becomes ~--- Z a -'b e - 2 / 3 ( t ) z 3 -[- •-1/3(0Z2

-Jr- y(t)z.

* Present address: Department of Mathematics, College of Science, University of Salahuddin, Erbil, Iraq. 809

810

M.A.M. ALWASHand N. G. LLOYD

Thus, we are led to consider equations of the form = Z 4 + a'(/)Z 3 + fl(t)Z 2 + y(t)Z.

(1.3)

It is with such equations that we shall be concerned in this paper, which is partly motivated by the paper of Shahshahani [7]. H e investigated the multiplicity of the origin when or, fl and y are polynomial functions of t; in Section 4 we shall show that a conjecture made in [7] is false. We shall also discuss this question when the coefficients are polynomials in cos t and sin t; this relates to the paper of Lins Neto [2], in which the connection between (1.1) and (1.3) is noted. In [1] we were mainly concerned with (1.1) when the multiplicity of z = 0 was greater than one; we showed that we could then suppose that y(t) = 0 without loss of generality. In this paper we shall investigate the corresponding quartic equation 2 = -74 + a~(t)Z3 + ] ~ ( 0 Z 2.

(1.4)

There is another reason for our interest in (1.3), namely the work described in [3], where equations of the more general form = z N + p l ( t ) z N-1 + . . .

+pN(t)

(1.5)

were considered; the coefficients were supposed to be periodic functions. It was shown that, when N - - 3, (1.5) has exactly three periodic solutions (counting multiplicity). It had already been shown by Pliss [6] that, when N -- 4, there may be more than four periodic solutions. In Section 4 we give an example of an equation of the form (1.3) with at least eight periodic solutions; the coefficients are quadratic polynomials in cos t and sin t, as were the coefficients in Pliss's example. It is clear that there is no upper bound on the number of periodic solutions unless some restrictions are placed on the coefficients and, in particular, on the number of their zeros. In Section 2, we recall some results from [3]; we adapt and apply them to the present context. In Sections 3 and 4 we shall be concerned with the multiplicity of z -- 0 as a periodic solution; we shall consider various classes of coefficients, and the discussion will be closely related to that of Sections 4 and 5 of [1]. In the final section we shall present some "global" results on the number of periodic solutions of (1.3). 2. THE PHASE-PORTRAIT Let 9Obe the set of all equations of the form (1.3), where or,/3, • are continuous real-valued functions on [0, w]. We identify the equation with the triplet P = (a~, fl, y), and define II/'11 = max I max Io~(t)l, max Ifl(/)l, max IY(/)I~. J L0~ 0 if Ixl is sufficiently large. We refer to [3] for full details. Let q ( P , c ) = Z e ( O g ; O , c ) - c , where ze(t;to, C) is the solution of P Ego satisfying Zp(to; to, C) = C. Suppose that (Pn) and (cn) are sequences in 9o and C, respectively, such that

Periodic solutions of a quartic nonautonomous equation

811

o

Fig. 1.

q(Pn, cn) = 0; that is, Pn has a periodic solution with initial point cn. If P~---~ P in 50 and c~---~ c in C as n---> ~, then either q(P, c ) = 0, in which case Zp(to; O, c) is a periodic solution, or ze(t; O, c) is not defined for the whole interval 0 ~< t ~ to. In the latter case, we say that Zp(t; O, c) is a singular periodic solution (see Section 5 of [3]); thus, ze(t; O, c) becomes unbounded as t increases and, moreover, ze(t; to, c) becomes unbounded as t decreases. We also say that P has a singular periodic solution if c~---~ ~; in this case there are ~ and c such that ze(t; z,c) becomes unbounded at finite time both as t increases and as t decreases. Let M be the subset of 5° consisting of all equations which have no singular periodic solutions. Then [3, Proposition 6.3] all equations in the same component of M have the same number of periodic solutions; those in the component of the equation ~ = z 4 have exactly four (Proposition 6.4). We showed that M is open in 50 (Proposition 5.3) and in a well-defined sense "nearly" dense (Proposition 6.5). The other result that we quote from [3] is Proposition 6.7, which when applied to (1.3) states that the number of periodic solutions of (1.3) is even. We now prove two results about equation (1.3) using the ideas which we have just summarized. THEOREM 2.1. Suppose that

fl(t) <~0 for all t. Then (1.3) has exactly four periodic solutions.

Proof. Let ~ = {(tr, r , 7) E 50; fl(t) ~< 0 for 0 ~< t ~< 09}. With z = re i°, we have /~ = r 4 sin 30 + r3o~(t) sin 20 + r21~(t) sin 0; thus, 0 < 0 when 0 = ½:r. This means that there are no singular periodic solutions, for otherwise there would be a solution crossing 0 = ½~rwith 0 > 0. Now )~P E ~ if P E ~t and consequently is contained in the component of ~ containing the origin of 50. We deduce that all equations in ~ have exactly four periodic solutions. The next result highlights a significant difference between equations of the forms (1.3) and (1.1): the origin is necessarily an isolated periodic solution of (1.3). In the parlance of [1] this means that z = 0 is never a centre, in contrast to (1.1), for which criteria for the existence of a centre were there established. THEOREM 2.2. The solution z = 0 is isolated as a periodic solution of (1.3).

812

M.A.M. ALWASHand N. G. LLOYD

Proof. Suppose, if possible, that there is an open set U C C containing the origin such that all solutions starting in U are periodic. Then q -= 0 in the component of its domain of definition containing the origin. But, as is clear from the description of Fig. 1, the real zeros of q are contained in the disc D. Thus inf{c E R; c > 0, z(t; O, c) is not defined for 0 ~< t ~< to} < o0. It follows that there is a real singular periodic solution; but a positive real solution which tends to infinity can do so only as t increases. This is a contraction, and the result follows. 3. CALCULATION OF MULTIPLICITY TO calculate the multiplicity of the zero solution of (1.3) we note that in a neighbourhood of z = 0, we can write o0

z(t; O, c) = ~ an(t)c n

(3.1)

n=l

for 0 ~< t ~< to; the an(t) are continuous and al(0) = 1,

an(0) = 0

for

n > 1.

(3.2)

The multiplicity is k if al(to) = 1, a2(to) = . • • = ak-l(to) = O, ak(to) --k O. TO calculate the functions an(t), we substitute the expansion (3.1) into (1.3); we obtain a set o f linear differential equations which are solved recursively. Clearly

al(t) = exp

(I0

),(s) ds

Hence, z = 0 is a multiple solution if and only if fS' y(s) ds = 0. Since we are here interested equations in which the multiplicity of the origin is greater than one, we suppose that fS'Y = 0. We then make the transformation z ~ zp -1, where p(t) = exp(f~ y), and obtain z" = (p(t))3z 4 + (p(t))za,(t)z 3 + p(t)fl(t)z 2.

(3.3)

Since p is periodic, the function q is unchanged by the transformation, and hence the periodic solutions of (1.3) and (3.3) have the same initial points and the same multiplicities. A further transformation of the independent variable leads to equations of the form (1.4)'which we introduced in Section 1; it is with such equations that we now work. For (1.4), al(t) - 1 and the equations satisfied by the an(t) (for n > 1) are

(In =

~

i+]+k+l=n i,],k,l>~ l

aiajaka I + tr(t)

~

i+]+k=n i,j,k >-I

aia]a k + fl(t) ~

i+j=n i,j>~ l

aia #

As in the proof of theorem 2.7 in [1], we integrate by parts repeatedly. The calculations soon become extremely heavy and time-consuming, as may be inferred from the expressions given below for the a n up to n -- 8; this is as far as it seems reasonable to proceed. We omit the details of the computations, of course, and merely state the results. A bar above a function denotes its indefinite integral: ~(t) =

fo

~(s) ds.

Periodic solutions of a quartic nonautonomous equation THEOREM 3.1. For equation (1.4), the functions a l , . . . , follows.

813

a8 in the expansion (3.1) are as

al=l. a 2 = ft. a3 = f12 -b ft. a4 =/~3 + 2flti + fla~ + t. a5 = f14 + 3f12& + fl2ac + 3flfl-~ + 2/~ + ]ff2 + 2tfl. a6 = f15 q_ aft3& +/~3a~ +

3f12fl-~ + 3/~-~ + 4flfl + 2/~/~2o~ - ½flt~ ----~ + ~flf2 + 3fltrff

+ t~ + 3/(/~ 2 + &). a7 = f16 _1_5f14t~ _[_f14~ _1_4fl3/~-~ + 2fl3trfl + 4 ~ + 3/~2fl2--~ + 6 ~ f l + 6fl2/~ + ~ ~2 &2 + 3f12=-"~ff_ f l f f ~ _ 2flflf2 + 6(~-~)2 + 2flt~ + 4fl-ff - 4(ff~) 2 + 6/~f + ~ ff3 + 4/(fl3 + 3/~--~+ 2ffff) + 2/2. a8 = ~7 -t- ~5"-~ q_ 6flsf

+ 2]~4----~fl_l_5f14fl'~q_ 3fl2fl3cr q- 4fl3fl2tl, q- 8fl'-3fl -t- 9fl2~

_ 25fi2ff~ff _ ]fiEfl~----~ _ 4fi"~ff~ _ 25fiEfffl-~ + 5fi(ff~)2 _ 4flflflff2

+ 9fi2&_ f i 2 ~ _ 2fifiO¢ + 9fiiff + 3fi2~ + 16tiff& + 8fi'~fi + 10flfitr- 2]~ --½flff---3-- ]fltr-~-2C~+ 10tiff 3 + ~ fia~ff,2 + ~a72 + 3ff07 + t[5/~ 4 + 15fi 2& + 5flEer + 10fl-flo: + 10~, + ~ t~21 + 5t2fl.

Remark. The formulae given in theorem 3.1 contain t explicitly; this did not happen in the corresponding result in [1], and reflects the fact proved in Section 2 that the origin is never a centre for equation (1.3). We can now give formulae for the calculation of the multiplicity of the origin; these will be used in Section 4. THEOREM 3.2. The solution z = 0 of (1.4) has multiplicity k (2 ~< k ~< 8) if and only if 772 "~- 773 = " ' " = ~ k - 1 = 0 a n d 7]k ~ 0, where r/2 =

n4 =

foo 3

+ 1)

814

M.A.M. ALWASHand N. G. LLOYD

?]5 =

?]6 =

?]v --

?]8=

f0t° (~2 a~ + 2fl)

f; I; f;

( 2 ~ 3 ~ + 6~ 2 - / ~ s 2 + 2o7)

(fi4a~ + 4/~ 3 - 2fl/~& 2 + 4fl&)

(a'~+5B4-6B2es2-B2~-B2~+9B2~-2B/~

_ ½/~3 _ 2 ~ + ~ 2 - 2o9B). P r o o f . L e t the multiplicity of z = 0 be/~. W e recall that # = k if a2(o9) = • • • = a~_ 1(o9) = 0 and ak(og) 4: 0. T h e idea o f the p r o o f is to use the relations a2(o9) = . . . = ak- 1(o9) = 0 in the calculation o f ak(og). It is i m m e d i a t e f r o m t h e o r e m 3.1 that a2(o9) = f~o fl and that if a2(o9) = 0, then a3(o9 ) = f~o o: and a4(o9) = fS' (/~a~ + 1). H e n c e /~ = 2 if ?]2 :j/=0, ~ = 3 if ?]2 = 0 and ?]3 ~ 0, w h i l e / , = 4 if ?]2 = ?]3 = 0 and ?]4 :j/=0. N o w we suppose that ?]2 = ?]3 = 0; then as(to) =

I0

(fi2~ + 2fi)

and

ado,) =

fo

(B3o~+

3/~2 _ ½fl&2 + S). Thus, # = 5 if ?]2 = ?]3 = ?]4 = 0 and

?]5 :/= 0, while/~ = 6 if ?]j = 0 for 2 ~< j ~< 5 and ?]6 =~ 0. Finally, if ?]2 = ?]3 = ?]4 = 0, we find that a7(o9) = ?]v and a8(o9) = ?]8. T h e p r o o f is thus complete. R e m a r k . In the p r o o f of t h e o r e m 3.2, we did not m a k e use of all the available relations in o u r calculations. F o r example, we only used the relations ?]2 = ?]3 = 0 to obtain o u r expressions for ?]5 and ?]6, and only the relations ?]2 = ?]3 = ?]4 = 0 to c o m p u t e ?]v and ?]8. W h e n we consider particular classes o f coefficients in the next section, further reductions will be possible. N o t a t i o n . In the next section, we shall write ~max(~) for the m a x i m u m possible multiplicity of z = 0 for e q u a t i o n s in a class %.

4. SOME P A R T I C U L A R CASES In [7] S h a h s h a h a n i c o n s i d e r e d equations of the f o r m (1.4) in which o~ and fl are p o l y n o m i a l functions o f t. H e c o n j e c t u r e d that if tx and fl are of d e g r e e k, then the multiplicity of the origin is at m o s t k + 3. W e shall show that this is not so, at least for k -- 2, 3 or 4. W e use the f o r m u l a e given in t h e o r e m 3.2 and again omit the routine details of o u r calculations. W e take o9=1. Let fl(t)=a+bt+ct 2 and t r ( t ) = d + e t + f i E . Write u = a + ½ b + ½ c and v = d + ½ e + ~ f ; then ?]2 ~---U and ?]3 = v. If u = v = 0, it m a y be verified that ?]4 = 1 + ( b f - c e ) / 3 6 0 and ?]s = - ( b + c ) [ b f - ce + 420]/2520.

H e n c e # = 4 if and only if ~ = ?] = O, b f - ce 4= - 360

(4.1)

ce + 360 = 0,

(4.2)

and # = 5 if and only if = ?] = b f -

b + c 4: 0.

Periodic solutions of a quartic nonautonomous equation

815

If T/k = 0 for 2 ~< k ~< 5, we have that r/6 = c2/2310 - 60/7c and ~7 = -cf/2970. Hence # = 6 if a = ~c, b = - c , o = 0, c ( f + e) = 360,)

(4.3)

/

c :/= c o = (19800) 1/3.

If (4.3) is satisfied and c = c o, then /l = 7 if f ¢ : 0 and/~ = 8 if f = 0. We have proved the following result. THEOREM 4.1. Let %, be the class of equations of the form (1.4) in which tr and fl are polynomials of degree k in t. Then #max(~l) -- 4 and gmax(%2) -- 8. There is a unique equation in (~2 with # = 8. COROLLARY 4.2. Shahshahani's conjecture is false if a~ and fl are of degree 2, 3 or 4. Having determined #max(~2) we use the technique described in Section 2 of [1] to construct equations with this maximum number of periodic solutions, all of them real. The idea is to make a sequence of perturbations in cr and r , each of which reduces the multiplicity of the oriein bv one: a oeriodic solution thus bifurcates out of 0, and it must be real simply because non-real solutions occur in conjugate pairs. THEOREM 4.3. Consider the equation (1.4) with tr(t)

--

e 0 - 3e0 6(1-el)

e3 4 +e4+

(eo - eo)t 1-el

( d-

eo 1-el

+

e3

) t2

and

Co(1 - el) fl(t) =

6

+

- (1 - e l ) ( C o

e2(1 - el)(3e0 - eo) 6(eo - e0) + e5 - e 2 ) t + [(c0(1 - / 7 1 ) q-

\

g2y..y_.x_ _~.(1 -~_E1)/\ t2, eo - eo

/

where c o = (19800) 1/3 and e 0 = 360Co 1. If [ekl are sufficiently small (for 0 ~< k ~< 5), then there are six distinct nontrivial real periodic solutions.

Proof. Referring to the proof of theorem 4.1, it may be verified that ~t = 8 if ek = 0 for 0 ~< k ~< 5 and that # = 7 if e k = 0 for 1 ~< k ~< 5 and e0 :/= 0. By choosing e0 ~ 0, we therefore ensure that a real periodic solution (9, say) bifurcates out of the origin. If e I =/=0 and Ek = 0 for 2 ~ k ~< 5, then (4.3) holds, and # = 6; then another real periodic solution bifurcates out of the origin, and moreover, ~ is not destroyed if el is small enough. We continue in this way, and using (4.1) and (4.2), for example, eventually find that six real periodic solutions have bifurcated out of the origin. We can take the above procedure further to deduce the following corollary. COROLLARY 4.4. With cr and fl as in theorem 4.3, the equation = z 4 + ~ ( O z 3 + 3 ( t ) z 2 + o l z + 02

has eight distinct real periodic solutions if ol and 02 are sufficiently small.

816

M . A . M . ALWASHand N. G. LLOYD

W e n o w consider e q u a t i o n s o f the f o r m (1.4) in which te and fl are polynomials in cos t and sin t. W e shall give examples o f equations in which te and fl are quadratic having at least eight periodic solutions, so improving on Pliss's example. W e take to = 2Jr. LEMMA 4.5. S u p p o s e that a~ and fl are polynomials in cos t and sin t all o f w h o s e terms are o f o d d degree. T h e n / ~ = 4 o r / t > 6. P r o o f . L e t Po be the set o f all polynomials in cos t and sin t all of w h o s e terms are o f o d d d e g r e e ; we use the fact that e v e r y m e m b e r o f P0 has m e a n value zero. S u p p o s e that a~, fl E P0; then certainly ?]2 = ?]3 = 0. N o w write fl = fl0 + k, where fl0 E P0 and k ~ R. If ?]4 = 0, then

fl0 tr = - 2 ~ . Since fl2a~ E P0, we have that ?]5 = 2k

fl0 tr + 4~tk = 0. T h e result follows.

E x a m p l e 4.6. L e t fl(t) = cos t sin 2 t and or(t) = a cos t + b cos 2 t sin t + c sin 3 t.

W e first c o m p u t e that ?]4 = ~r(5c + b + 48)/24. It is easily verified that r/5 = 0. By further calculation, we find that ?]6

=

~'~(640c2 "1- 448bc + 6221c + 64b 2 + 3079b + 960)/2304

and ?]7 = ~ta(9c + b + 96)/96.

T h e r e f o r e , if ?]4 = 0, we have ?]6 = ~r(7c + 104)/383. W e then conclude that # = 4 if 5c+b+48:/:0, #=6 if 5 c + b + 4 8 = 0 and c ~ - 1 0 4 / 7 , while # = 7 if b = 1 8 4 / 7 , c = - 1 0 4 / 7 and a ~: 0. THEOREM 4.7. L e t $'k be the class of equations in which cr and fl are h o m o g e n e o u s polynomials in cos t and sin t of d e g r e e k. Then/~ma~(B-1) = ~ma~(~-2) ----6 and ~max($-3)/> 8. P r o o f . F o r k = 1, write fl(t) = a cos t + b sin t and re(t) = c cos t + d sin t. Clearly 772 = ?]3 = 0. W e calculate that ?]4 = Jt(ad - bc + 2); if ?]4 = 0, then by l e m m a 4.5, ?]5 = 0 also. F u r t h e r calculation gives ?]6 = 3~r( a2 + b2) • But if a = b = 0, then ?]4 :~ 0. H e n c e / ~ = 6 or 4, according to w h e t h e r a d - b c + 2 = 0 o r not. For k=2, we write fl(t)=acos 2t+bcostsint+csin 2t and a~(t)=dcos 2t+ e cos t sin t + f s i n 2 t. T h e n ?]2 = (a + c)~r and ?]3 = (d + f ) : t . If ?]z = ? ] 3 = 0, then fl(t) = a cos 2t + ½b sin 2t and o~(t) = d cos 2t + ½e sin 2t. W e can n o w use the calculations o f the p r e v i o u s p a r a g r a p h ; we d e d u c e that again # ~< 6. ( W e n o w have ?]4 = ¼~r(ae - b d + 8) and, if ?]4 = 0, ?]5 = ~ ( 4a2 + b2) -) T h e result for k = 3 follows f r o m e x a m p l e 4.6, noting that cos t -- cos 3 t + cos t sin 2 t. T h e m e t h o d used in t h e o r e m 4.3 to separate a solution o f multiplicity k into k distinct real solutions does n o t w o r k for the classes ~'k, for we c a n n o t arrange for ?]E, ?]3 and ?]5 to be n o n z e r o w h e n k = 1 and k = 3, n o r for ?]2 = ? ] 3 = 0 but ?]5 :/: 0 w h e n k = 2. H o w e v e r , the technique is successful if cr and fl are i n h o m o g e n e o u s polynomials in cos t and sin t, as we n o w show.

Periodic solutions of a quartic nonautonomous equation

817

C o n s i d e r the following forms for a and fl:

fl(t) = cos t sin t + a cos 2 t + b sin 2 t + c sin t, ] c~(t)

d cos t sin t - 2 cos t + f s i n t + g cos z t.

A f t e r s o m e comi, utation, 7-z = (a + b) and at3 = g a r . Suppose that 72 = 73 = 0; t h e n we calculate that 174 = ¼ar(ad + 8c + 8) and 75 = ~zr(-4afc + 4acd + ad + 32c z + 36c + 8). If 74 = 0, we substitute ad = - 8 ( c + 1) in the expression for 75 and find that 75 = -¼arc(1 + aj0. W e n o w s u p p o s e that c = 0 and ad = - 8 . Certainly 74 = 75 = 0 and, m o r e o v e r , 76

=

zr( 12a2 -- 8 a f 2 - 1 6 f + 32a + 3)/16

and 77 = a r ( - 8 a f 2 + 12aZ + 32a - 1 6 f + 4d + 3)/32. S u p p o s e that 76

=- 0;

t h e n 77 = ~zrd. C o n s e q u e n t l y , # = 7 if ad = - 1 , c = 0 and 12a 2 -- 8 a f 2 + 32a -- 16f + 3 = 0.

(4.4)

T o bifurcate six distinct real periodic solutions out of the origin, we use the same technique as in t h e o r e m 4.3 and corollary 4.4. W e start with an e q u a t i o n in which ad = - 1 , ¢ = 0 and (4.4) is satisfied; t h e n # = 7. W e first p e r t u r b f so that (4.4) is violated; t h e n # = 6. Next, we p e r t u r b c, at the same time adjusting a and b so that 74 and 72 remain zero; at this stage, # = 5. W e n o w p e r t u r b a so that 74 :/: 0, retaining b = - a ; then # = 4. By p e r t u r b i n g g to be n o n z e r o and t h e n b so that a + b :/: 0, we have # = 2. Finally we introduce a sufficiently small linear t e r m into the equation. W e have the following result. THEOREM 4.8. In the e q u a t i o n = Z 4 --1-Clg(0Z 3 -[- ~ ( 0 Z 2 -[- orz,

(4.5)

take

fl(t) = cos t s i n t + a l ( 1 + 62)COS2 t - - a1(1 + e2 + es) sin 2 t + e2 s i n t and or(t) = d cos t sin t - 2 cos t + ( f + el )sin t + E4 C O S 2 t, w h e r e d 4 : 0 and a I = - 8 d -l. C h o o s e e~ . . . .

Iol

levi

, e5 and o so that

te,I

1.

T h e n (4.5) has at least eight distinct real periodic solutions.

Proof. W e follow the p r o c e d u r e described in the p a r a g r a p h preceding the s t a t e m e n t of the t h e o r e m ; the origin bifurcates into seven distinct real periodic solutions. But we n o t e d in Section 2 that the n u m b e r o f real periodic solutions is even; there are, t h e r e f o r e , at least eight such solutions. Remarks. (1) T h e coefficients in Pliss's e x a m p l e [6] of an e q u a t i o n with five periodic solutions were also i n h o m o g e n e o u s quadratic polynomials in cos t and sin t.

818

M . A . M . ALWASHand N. G. LLOYD

(2) The calculations described in example 4.6, theorem 4.7 and theorem 4.8 have been checked on a computer using the R E D U C E symbolic manipulation package. This allows further calculation to be done, for example, we can calculate r/6 and r/7 for (4.4) without setting C=0:

?76

=

: r [ - 6 4 a f ~ - 384afc 2 - 96arc + 64adf+ 512cf + 3 8 4 f + 12a3d + 192a2c + 192a 2 + 288acEd + 96acd + 16adz + 15ad + 256a + 1920c 3 + 2688c z + 192cd + 816c + 128d + 144]/128

r17 = :r[-512afZ c - 64af 2 - 48a3 cf - 896fac 3 - 384ac2f + 320acdf - 60acf + 6 4 d f + 2560fc 2 + 2816cf + 3 8 4 f + 48a3cd + 12a3d + 768a2c 2 + 912a2c + 192a 2 + 640ac3d + 264ac2d + 64acd 2 + 60acd + 1024ac + 20ad 2 + 7ad + 256a + 3584c 4 + 6016c 3 + 768c2d + 2112c 2 + 704cd + 596c + 192d + 80]/256. 5. D E R I V A T I V E S

OF THE POINCARI~ MAP

The functions ak(t) introduced in Section 3 are closely related to the derivatives of the function q : c ~ z(w; 0, c) - c. Referring to the expansion (3.1), we see that

ak(W) --

q(k)(o) k!

Thus, information about the real periodic solutions of the equation (1.3) can be gleaned from the signs of the derivatives of q over a range of values of c. In a different context, formulae for the first three derivatives of q were obtained in [4]. To use these consider an equation

.~ = f(x, 0

(x E

~1),

(5.1)

where f i s as smooth as is required in the argument; let x(t; c) be the solution satisfying x(0; c) = c. Write fk for okf/Ox k and define

E(t, c) = exp I ~ o f l ( x ( r ; c), r) d r ] , O(t, c) = E(t, c)f2 (x(t; c), t),

G(t, c) --

D(r, c) dr.

From [4] we have the following formulae:

q'(c) = E(co, c) - 1, q"(c) = E(co, c)

q'(c) = E(,o, c)

foo D(t, c) dt, (G(,o, c)) z +

(E(t, c))Zf3(x(t; c), t) dt.

The results proved in [4] depend on the partial derivatives of f with respect to x being of one

Periodic solutions of a quartic nonautonomous equation

819

sign for c > 0. For equation (1.3), f2 = 12x2 + 6o~x + 213 and f3 = 36x + 6or (x being x(t; c)). Thus, f3 > 0 for c > 0 if o~I> 0 for all t, and f2 >/0 if 3o~z - 8fl ~< 0 for all t or if c~ I> 0 and fl~> 0. Using theorems 6 and 9 of [4], we have the following result. THEOREM 5.1. (1) Suppose that o~(t)/> 0 for all t. Then equation (1.3) has at most two real positive periodic solutions. (2) Suppose that or(t)/> 0 and fl(t) >I 0 for all t or 3(o~(t)) 2 ~< 8fl(t) for all t. Then (1.3) has at most one real positive periodic solution. Formulae for q~iv) and q~°) for equation (5.1) can also be obtained, and we record them here. Their exploitation is considerably more subtle than that of q" and q'", and we defer this to another paper. We calculate that 0

-~CE(t, c) = E(t, c)G(t, c), 0 G(t, c) = Oc

fo•

D ( r , c)G(r, c)dr +

= ½(G(t, c)) 2 +

fo•

(E(r, c))2f3(x(~; c), c) d~

( E ( r , c))2f3(x(r; c), c) dr,

and 0

~ccfk (x(t; c), c) = E(t, C)fk+ 1(x(t; C), C). Using these expressions, the proof of the following result is straightforward. THEOREM 5.2. For equation (5.1),

E~olqq°)(c) = 3G 3 + 4G,o

I/

E2f3 + 2

fo

E2Gtf3 +

f;

E3f4

and

E~olq(°)(c) = ~ G 4 + 15G 2

E2f3 + 10Go

E2Gtf3 + 5

f0

Et2Gtf32

+ 5 (fo ° E 2 f 3 ) 2 + 5G~o fo ° E3f4 + 5 fo ° E?Gtf 4 + fo ° E4fs, where we have written Et and G t for E(t, c) and G(t, c), respectively, and the partial derivatives of f are evaluated at (x(t; c), c).

Acknowledgment--We

are very grateful to Dr T. P. McDonough for his assistance with the computing.

REFERENCES 1. ALWASH M. A. M. & LLOYD N. G., Non-autonomous equations related to polynomial two-dimensional systems, Proc. R. Soc. Edinb., Section A (to appear). 2. LINS NETO A., On the number of solutions of the equation dx/dt = YT=oaj(t)#, 0 ~< t ~< 1, for which x(O) = x(1), Invent. Math. 59, 67-76 (1980).

820

M. A. M. ALWASHand N. G. LLOYD

3. LLOYD N. G., The number of periodic solutions of the equation i = 9 +pr(f)zN-’ + . . . +pN(t), Proc. Lond. marh. Sot. 27, 667-700 (1973). 4. LLOYDN. G., A note on the number of limit cycles in certain two-dimensional systems, /. Lond. math. Sot. 20, 277-286 (1979). 5. LLOYD N. G., Small-amplitude limit cycles of polynomial differential equations, in Ordinary Differential Equations and Operators (Edited by W. N. EVERITT and R. T. LEWIS),Lecture Notes in Mathematics 1032,346-357, Springer, Berlin (1982). 6. PLISSV. A., Nonlocal Problems in the Theory of Nonlinear Oscillations, Academic Press, New York (1966). 7. %AHSHAHANI S., Periodic solutions of polynomial first order differential equations, Nonlinear Analysis 5, 157165 (1981).