1L 6“ 51. .. .«
L. a .
'1‘ _ 7 - w
. ‘1 f, ‘ -
x w . w L ‘ ~
‘ i 0 o .
. '_ h . V
‘3 " V E: U 'V- L a. 1
I
This is to certify that the

dissertation entitled

0n the monotonicity and critical
points of the period function of some second
order equations

presented by

Duo Wang

has been accepted towards fulfillment
of the requirements for

Ph.D. degreein Mathematics

{Le/N)- ow

Major professor

Date October 30, 1986

0-12771

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ON THE MONOTONICITY AND
CRITICAL POINTS OF THE PERIOD FUNCTION
OF SOME SECOND ORDER EQUATIONS

By

Duo Wang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1986

ABSTRACT
ON THE MONO’I‘ONICITY AND
CRITICAL POINTS OF THE PERIOD FUNCTION
OF SOME SECOND ORDER EQUATIONS
By

Duo Wang

The period function p(c) of the equation 5% + g(x) = 0 is studied.
We give sufficient conditions for the monotonicity of p(c) in the cases
where the closed orbits surround only one critical point and also more
than one critical point. The boundedness of the number of critical
points of the period function in an example where g(x) =

x’(x - a)(x - l) (0 i a < l) is a certain polynomial of degree 4 is
established.

TO
MY PARENTS
AND
MY WIFE

ii

ACKNOWLEDGMENTS

I wish to express my deep gratitude to Professor Shui Nee Chow, my
thesis advisor, for all his patience, expert guidance and encouragement
during the course of my research. I also wish to Offer thanks to
Professors David Yen, Lee M. Sonneborn, B. Drachman and J. Cleo Kurtz

for their patient reading of my thesis and attending my defense.

I am also grateful to Chairman Kyung Whan Kwun for his support

and encouragement.

A special thanks goes to my wife, Baozhu, for her unwavering

support and infinite patience.

I wish to thank Mrs. Berle Reiter, the Librarian of the I
Mathematics-Statistics Library and Ms. Barbara S. Miller, the secretary
of the graduate office of the Department of Mathematics for all of their
help during my graduate work at Michigan State University. Finally, I
would like to thank Cindy Lou Smith for typing this dissertation and for

her patience and understanding.

iii

TABLE OF CONTENTS

Chapter

INTRODUCTION . . . . . .....

I

1.1
1.2

II

2.1
2.2

MONOTONICITY OF THE PERIOD
FUNCTION OF I: + g(x) = 0

Main Results
Applications . . . ................

CRITICAL POINTS OF THE PERIOD FUNCTION OF
x-x’(x-a)(x-1)=O(0‘a<l) .......

Analyticity of the Period Function

Boundedness of Number Of Critical

Points of the Period Function . . . ..... . .

LIST OF REFERENCES

iv

Page

17

22

22

28

54

Figure 1.1.1

Figure 1.1.2

Figure 2.1.1

Figure 2.1.2

LIST OF FIGURES

Page

13

23

27

INTRODUCTION

Consider the scalar equation

u n d2)!
(0.1) x + 80‘) = 0 . (x = -2-)
(it
where g(x) is smooth for all x e R. Let
(0.2) coo = [X acme + co .
0

where 0., is an arbitrary real number. If there exist a < O < b
such that G(a) = G(b) = c, G(x) < c for all a < x < b and g(a) °
g(b) 1 0, then there exists a periodic orbit Of (0.1) in the phase plane
with energy c, intersecting the x-axis at (a,0) and (b,O). Let the
least period of this periodic orbit be denoted by p(c), which will be
referred to as the period function in this note. It is well known that
p(c) is a smooth function of c (see [2]). In fact, if g is 07, 7 i 1,

then p is C7. Furthermore, p(c) is given by the following formula

_ b
(0.3) p(c) = «2 I A .
a Jc-G(x)

Since the monotonicity Of- p(c) plays a very important role in the
study of subharmonic bifurcations from a planar Hamiltonian system (see
[4]), there have been many authors who have studied the monotonicity
of p(c). See, for example, Loud [6], Obi [7], Opial [8], and Schaaf [9].

In this note, we will also discuss some of the properties of p(c).

In Chapter I, we study the monotonicity Of the period function of
(0.1) for general g’s. In 91.1, we derive some formulae for p'(c) and
p' (c), and then give some useful theorems and corollaries for the
monotonicity. In 51.2, we apply the results of 91.1 to some specific
equations. We will prove the monotonicity of periodic function of

equation
(0.4) 52+e"-1=o,

that cannot be derived from previous results. This will complement the

results of Wang [11] and will be useful for bifurcation problems [4].

Another important problem about the period function of (0.1) is: If
g(x) is a polynomial in x of degree n, is there a bound (depending
only on n), denoted by C(n), of the number of the critical points of
period functions of (0.1)? This problem is raised by Chow and Sanders
[5], and also by Smoller and Carr independently. It is related to the
"weakened Hilbert's 16th problem" (Arnold [1], p. 303). In [5], Chow
and Sanders proved that 0(2) = 0 and 0(3) 2 3. For n i 4, this
problem is still open. Even for n = 4 we do not know whether this

bound exists.

If g(x) is a quartic polynomial, then by scaling, all the cases
under which the equation (0.1) has periodic solutions can be normalized
to the following two cases:

(1) sh) = w: - 1)(X’ + «X + R). (a’ < 45);

(ii) s(x) = w: - a)(xi- b)(x - 1).

(0‘a6b51,a3+(b-1)‘#0).

In Chapter II, we study the critical points of the period function of

the equation
(0.5) x' - x’(x - a)(x - 1) = 0 , (o c a < 1) ,

which is a special case of the case (ii) (b = 1). Our main result is that
the number of critical points of the period functions p¢(c) of (0.5) for
any a e [0.1) is bounded (Theorem 2.2.24). The proof is mainly based
on the analyticity of pa(c) in a and C, which is proved by using

the Picard-Fuchs equation.

In 92.1. we prove the analyticity of p¢(c) of (0.5) when (a,c) is
in the domain D = D. U D, (see Figure 2.1.2), which is a bounded but

not compact set in a - 0 plane.

In order to prove that there exists a bound for the number of
critical points of p¢(c), we try to find a compact subdomain D, such
that when (a.c) e D - Do. p¢(c) has at most one critical point for any
fixed 0:. SO in 92.2. we first prove that p¢(c) is monotone for each
a e (0,1) and when c varies such that (a.c) is in D, (Lemma
2.2.1). Then we prove that for any so e (0.0.6) and (cameo) e JD.
there is a neighborhood of (some) in which p¢(c) has no critical
points (Lemma 2.2.6 and 2.2.10). The main difficulty in finding 0,. is
to prove that there is a 6 > 0. such that for each a e [0.6) and
a e (0.6 - 6, 0.6), p¢(c) has exactly one critical point (Corollaries

2.2.19 and 2.2.23).

It does not seem easy to obtain the bound for the number of critical
points of the period function p¢(c) Of (0.5) for all a e [0.1). The
computer results suggest that p¢(c) may have 7 critical points for

some a c (0.0.6).

Chapter I

MONOTONICITY 01: THE PERIOD FUNCTION
OF x + g(x) = 0

91.1. MAIN RESULTS

Let g(x), G(x) and p(c) be as in the introduction.

Since we are interested mainly in either the monotonicity or the
number of critical points of p(c) (a critical point of p(c) is the point
c at which p'(c) = 0). we may assume that g(x) has been scaled by

g(x) 9 k g(ax + 3). where k - a > 0. Hence. we will assume g(O) '2 0.

We consider now periodic orbits which contain only one critical point
in their interior. In this case. we define G(x) by (0.2) with Co = 0

and then consider the hypothesis:

(HI) There exist - l at < 0 < b! i +0, an integer n i 0 and a

smooth function h(x) such that

h(x) > 0 . a* < x < b* ,
(1.1.1)

211+

g(x) = x 1 h(x) , a* < x < b* ,

and

0 < G(a*) = c(b*) = c* . +. .

Note that under the above hypothesis, the graph of y = G(x) and
the corresponding phase portrait of (0.1) are shown in Figure 1.1.1.

Furthermore. p(c) is defined for every 0 < c < c‘.

Y'9 (x) x //"\ x

 

 

 

FIGURE 1.1.1

For simplicity, let

(1.1.2) 7(x,c) = 2(c - G(x)) .
Note that
a - _
(1.1.3) 3-; - 2m) .
(1.1.4) p(c) = 2f 9-“.- .
a J1

where at < a < 0 < b < b3. 7(a.c) = 7(b.c) = 0. 7(x.c) > 0 if
a < x < b.

6

Theorem 1.1.1. Assume that (H1) holds. Then for any 0 < c < c*.

1)
(1.1.5) cp'(c) =I 7'1?— dx . (p'(c) = 35>
a 41 s (X)

where a* < a < 0 < b < b‘, G(a) = G(b) = C. and

(1.1.6) R(x) = g2(x) — 2G(x)g'(x) .

Proof: Let

I = Ib J? dx .
8

and
(1.1.7) J = [b (1 - 2c) J; dx .
a
Then
b
(1.1.8) 1' = I -§ dx ,
3 J7
b - 2c
J':I Lr—dX-T-I‘ZCI'
8 J1
Hence
(1.1.9) J' = -I' - 2cI' .

On the other hand. integration by parts in (1.1.7) yields

3 a (91)
3x
. _ 2 lb ,3/2 d w
3 a (3%)

 

: _ g Ib 3/2 20‘) - G X ' x'll (1X
3 2 °
8 s (X)

Differentiating the above equality with respect to c twice, we have

 

2 .
(1.1.10) J" = -2 [b KPQX) ' g(x)5-$¥l dx .
a fish)

Then from (1.1.8). (1.1.9) and (1.1.10). we have

b 2 .
(1.1.11) 2c1' = 2 I 5"“) ’ “(‘15-151 dx - I'

a «1; 320:)

 

b

__(_)__Rx dx
.— z '
a Jr a (X)

Note that p(c) = 21'. Therefore (1.1.11) gives the desired result.

Remark 1.1.2. The hypothesis (H1) guarantees that all the
integrations in the proof of Theorem 1.1.1 make sense.
Corollary 1.1.3. If (H1) holds and
x g'(x) < 0 (or > 0) , x i 0 , a* < x < b* ,
then

p'(c) > 0 (or < 0) , 0 < c < c* .

gm: Since R'(x) = -26(x)g'(x), R(0) = 0, then R(x) > 0

(or<0), x10.a‘(x(b‘.

Corollary 1.1.4. If (H1) holds and

_l31(£)_ _ 431.15.15.21 (0 (or>0). a*<x<0.
s (X) s (A(X))

where R(x) = g'(x) - 26(x)g'(x) and A(x) is defined by

(1.1.12) G(A(x)) = G(x) , a* < x < 0 , 0 < A(x) < b*,

then

p'(c) > 0 (or < 0) , 0 < c < c* .

Proof: By the implicit function theorem. A(x) e C‘ (a‘,0) and

. = __£1§l_ x
(1.1.13) A (x) g(A(x)) . a < x < 0 .
In the integral
Ib Rgx) dx
_ 2 .
0 47 s (X)

related to (1.1.5) we change variables by x = My) to Obtain

 

b a
(1.1.14) I 1312‘de = I jug“) A'(x)dx .
0 47 8 (X) 0 “‘7 8 (M10)

From Theorem 1.1.1 and (1.1.13), (1.1.14). we have

cp'(c):Io Mlfll-Mde.
a 3(X) 33(A(X))

 

Note that g(x) < 0. a’ < x < 0. The conclusion now follows.
Corollarz 1.1. . Suppose (H1) holds. If g‘(0) > 0 and

(1.1.15) H(x) = g2(x) +.-411191-§ g3(x) - 26(x)g‘(x) > 0 (or < 0) ,
3(8’(0))
x750. a*<x<b*.

then

p'(c) > 0 (or < 0) . 0 < c < c* .

Proof: By L’Hopital’s rule.

1., .1200 = -IJIJQL.
M .30.) 3 (won2

Thus H(x) > 0 (or < 0) implies

_§(& (-3; .1112)? < w ’ a¥<x<o’
g(x) (8'(0)) 9(A(X))

(or lgfl ) _%_flgl_2 ) €122.11, 8*<X(0),
8 (X) (8'(0)) 8 (4(8))

since A(x) > 0, x s (a‘,0) and x - g(x) > 0 for x 1 0, x e (a*,b*).

By Corollary 1.1.4. we thus have

p'(c) > 0 (or < o) , 0 < c < c* .

Corollary 1.1.6. Suppose (H1) holds. If g'(0) > 0 and

(1.1.16) v = 5(g'(0))2 - 3g'(0)g‘(0) > 0 (or < 0) ,

10

then there exists 6 > 0 such that

p'(c) > 0 (or < o) , 0 < c < a .

Proof: By using Taylor’s series we Obtain.

x4 - v + 0(IXI5) . as le . 0

SI...

H(x)=
The conclusion now follows from Corollary 1.1.5.

Theorem 1.1.7. Suppose (H1) holds. Then for any 0 < c < c‘,

b 2
2c2p'(c) = I ‘--1-1—- dx. (p'(c) = 9-5)
8 J7 s4 (X) do

where a‘ < a < 0 < b < b", G(a) = G(b) = c. and

(1.1.17) 5(x) = —g4(x) - 40(x)gz<x)g'<x) — 402(x)g(x)z'(x) +
12 cz<x><g’<x))2

Proof: Let
(1.1.18) K = [b -151i1 dx,
8 s 2(X)
and
(1.1.19, .:I M d.
82 (X)

Differentiating (1.1.18) and (1.1.19) with respect to c, we Obtain

11

Rgx) dx ’

(1.1.20) 11' _
4? 62(X)

ebb

L' =Ib 211(x) G(x) dx .
a J; 82(X)

By Theorem 1.1.1 it follows that K' :: cp'(c). Since 7 - 2c = -ZG(x).

-L' = K- ZCK' , we have

p'(c) + cp'(¢) = K" .
(1.1.21)

K' + ZCK' = L"
0n the other hand, integration by parts in (1.1.8) yields

_ g Ib Rgx) G(x) (1 73/2
3

(1.1.22) 1. = 3
a s (X)
= g_ ib 13/24 dla (x) 36(x)]
3 a 9 3(3)
2 Ib_13/ 28 W(X)
= -3- ,
a s (X)
where

(1.1.23) 51‘“) = ¢4(x> — sc<x)¢2<x)c'(x)
- 262(x)s(x)s'(x) + 602<x)(¢'(x))2

Differentiating (1.1.22) with respect to c twice, we then have

12
81(X)
(1.1.24) 1." = 2r 7-?— dx .
a J7 g (x)
From (1.1.20), (1.1.21) and (1.1.24) and by Theorem 1.1.1.
2 2 U _. U l
c p (C) - L ‘ 3C? (C)

S(x)
2r+.-.r-__w..
a «”00 . 41:20:)

2810:) - amazon
a «'7' 340:)

 

The desired result now follows from (1.1.6) and (1.1.23).
Ila-r1: 1.1.8. The hypothesis (H1) guarantees that all the
integrations in the above proof make sense.

We now extend the previous results to periodic orbits whose interior

may contain more than one critical points.

Note that we can also define G(x) as follows:
G(X) = Ix s(£)dt + co .
0
where co can be any real number.
We need the following hypothesis:

(H2) Thereexist «ia*<a6066<b*£+o, integers mbo.

n h 0 and a smooth function h(x) such that

l3

h(x)50, a’<x<b‘,
xg(x)>0. a*<x<a, p<x<b*,
0<G(s*)=0(b’)=c*‘+0,
and

2l+

(1.1.25) G(x) = (x - a) 1 (x — ”2”“ h(x). a* < x < 5*.

The graph of y = G(x) and the corresponding phase portrait of
(0.1) are shown in Figure 1.1.2.

 

 

FIGURE 1.1.2

14

Thggr_em 1.1. . Suppose (H2) holds. Then for 0 < c < c‘ ,

a b
(1.1.26) cp'(c) = [ + TEEL— dX - 2C In '3!- ,
I. In «7 320:) a 73/2

where a* < a < a, p < b < b‘. G(a) = G(b) = c. R(x). 7(x,c) are the

same as those in Theorem 1.1.1.

Proof: Define

J=Ib(1-2c)~/;dx.
a

Note that G(a) = 6(5) = 0. Hence

(.1
ll

zl‘”+ b]z-2cd3/2 J" -
- 7 + (v-ZC)~/1dx
3 Ia I, (3:5) 0

 

-% l J.“ + Ib] 3/2: i2(x)2___ G(X) :0 {x22 dx
a F 8 (X)

+Ip(1-2c)~l;dx.
0

The rest Of the proof is similar to that of Theorem 1.1.1.

Remark 1.1.10. Hypothesis (H2) guarantees that all the integrations

in the above proof make sense.

M It (112) holds and

(1.1.27) -§-§’-‘1-—§$M¥11>0, a*<x(aa
8 (X) 8 (4(3))

where A(x) is defined by

15

C(A(x)) = G(x) , 8* < x < a , p < A(x) < b‘ ,
and R(X) = s'(X) - 26(X)¢'(X). then

p'(c) < o , o < c < c* .

Proof. It is similar to that of Corollary 1.1.4.

W. Suppose (32) holds. If
(1) sun in odd.

(ii) s(«l = 0.

(iii) g'(x)60.a*<x<a,

then

p'(c) < 0 , o < c < c‘ .

Proof: By the Oddness of g(x), (1.1.26) becomes

a ‘C

ZR 1: dx
(1.1.28) cp'(c) = _—-(—L dx - 2c ——
I. J, g2(x) I“ 73/2

Because g'(x) ‘ 0. so R‘(x) I 0, a‘ < x < 0:. Therefore
20:) . 8020 = :20») - more») = o. a* < x < .. .
From (1.1.28), the conclusion of the corollary is obvious.

Theorem 1.1.13. Suppose (H2) holds. Then for any 0 < c < c‘,

2 - - “ _£(212_ 2 dX
2cp(c)—[ +Ib _ dx+12cr—’
In I J7 140:) a 15/2

where s(x), 1(x.c), a, b, c are the same as those in Theorem 1.1.7.

16

1.1.14. Suppose (H2) holds and

_(_)_Sx__L_(_)_)_§AX (0, a*<x<a.
850:) s 0100)

where A(x) is the same as that in Corollary 1.1.11, S(x) is the same

as that in Theorem 1.1.7. then

p'(c) > 0 , 0 < c < c* .

Remark 1.1.15. Theorems 1.1.1 and 1.1.7 are special cases of

Theorems 1.1.9 and 1.1.13 respectively.

91.2. APPLICATIONS

In this section, the results of 91.1 will be applied to several
examples to show the monotonicity of the period function p(c). The

following theorem is useful in applications.

Theorem 1.2.1. Suppose (H1) holds. If g‘(0) > 0, g'(0) i 0. then
each of the following conditions implies H(x) > 0 for x 7- 0, x e (a.,b.)

(see (1.1.15)).

(i) 8"(x) > 0 and

s(x) = x (s'(0)s'(x) - g'(0)s'<x)) . 0. x e (81.51) .

where at is. 6 0 6 b. ‘ b3.

(ii) i s'(X) > 0. 3"(X) ‘ 0. x e (anbt). where

at ‘ 81 ‘ O ‘ b; ‘ b‘o
(iii) 8"(1) < 0, “'(X) fi 0. 0 ‘ a} < x < b! ‘ bt, and H‘fiy) . 0.
(iv) g'(x)£0, 0<a.<x<b.6b¥.

(v) g'(x) < 0. g"(x) i 0. at i a. < x < b. < 0 and H(a.) i 0.
H‘bg) i 0.

Example 1. Let

g(x) = ex - 1 , -o < x < +0 .

17

18

Since g'(x) : g'(x) 2 ex > 0 , -. < X ( +0 , and s(x) =

x(g'(0)g'(x) -g'(0)g"(x)) I 0. By Theorem 1.2.1 (i) and Corollary 1.1.5,

p'(c))O, 0<c<+o.

From the results Of Opial [8],

lim p(c) = 2n . li- p(c) = +6 .
c+0+ c-H-O

Remark 1.2.2. The above result does not follow from the

monotonicity results in [6]. [7]. [8] and [9].

Exam le . Let g(x) be a quadratic polynomial. We may consider
the normal form [5]:

g(x) = x(x + l) , -l < x < +6 .
Since g'(x) = 2, g'(x) I 0. by Theorem 1.2.1 (ii) and Corollary 1.1.5,
o * _ 1
p(c)>0, 0<c<c-§.

Since c* corresponds to a homoclinic orbit, so 11mm; - p(c) : +6.

By the result of Opial [8]. limcgo... p(c) = 21!.

Example . Let g(x) be a cubic polynomial. For periodic orbits
with only one critical point in the interior, we may consider the

following normal forms:

(3.a) g(x) -(x + a)x(x - 1), 0 < a 6 1, -a < x < 1 .

(3.b) g(x) x(x+a)(x+1). 0<a61. -a<x<+0.

l9

(3.c) g(x) x(x'+bx+1). 06b<2,-6<x<+o.
(3.d) g(x)=x’. 4<x<+6.

For (3.a). g(x) = -x’ + (1 - a)x' 4- ax. Then

g'(x) -6x + 2(1 - a) ,

g'(x) -6 < 0 .

By Theorem 1.2.1 (ii). (iii), (iv) and Corollary 1.1.5.

p'(c) ) 0 , 0 < c < ct = G(-a) .

For (3.b), g(x) = x3 + (1 + a)xa 4- ax. Then

s'(X) 6X + 2(1 + a) .

6>0.

s'(X)
Hence g'(x) ) 0 if and only if x > -% (1 + a).

x2[6(1 + a)x + 4(1 + a)2 - 6a]

s(x)

I'

xztsu + tax-3% (1 + a» + 4(1 + a)2 - 6a]

x2[2(1 + a)2 - 6a]

5 0 , x > - § (1 + a) .

By Theorem 1.2.1 (i). (v) and Corollary 1.1.5. we conclude that

p‘(c) > 0 , 0 < c < c* = G(-a) .

For (3.0), if b = 0. then g(x) = x3 + x. g"(x) = 6x. Thus by
Corollary 1.1.3.

20

p'(c)<0. 0<c<+o.
If b > J9/10. then

urwnz-uwmrm>
2

4
ll

20b -18)0.

By Corollary 1.1.6. there exists 6 > 0 such that p'(c) > 0, 0 < c < 6.
On the other hand. by a result of Opial [8]. p(c) 4 0 as c 9 +.. This
implies that p(c) is not monotone.

For (3.d), g“(x) = 6x. Then by Corollary 1.1.3,

p'(c) < 0 , 0 < c < +6 .

Remark 1.2.3. In [5]. Chow and Sanders proved that there are at
most three critical points of period function when g(x) is a polynomial

of degree three.

Example . Let g(x) = -x‘ + x3. -6 < x < +1. A direct calculation

shows that

16 15 14 13

S(x) = 5% (156 x - 624 x + 896 x - 550 x + 125 x12) .

Hence S(x) > 0 for x < 0. Furthermore.

4

S(x) = 55 x12[156(x - 0.65)4 + 218.4(x - 0.65)2(1-x)

+ 4.879025(1 — x) + 2.439025(1 - x)x + 0.659025 x2]
> 0, 0 ( x < 1 .
By Theorem 1.1.7.
1

p'(c) > o , 0 < c < 6* = 6(1) = 55 .

21

Since

11. p(c) = 11- P(C) = +° .
c-DO"' c-Dcr

p(c) has exactly one critical point.
Exam 1e . Let g(x) = x(x’ - 1)’ and
_ 1
G(x) - J: 8(£)d€ - g

Then

G(x) =%(x + 1)3 (x - 1)3.

Since g(x) is odd, g(-1) = G(-1) = 0, and g'(x) = 20 x3 - 12 x < 0.
x < -1. by Corollary 1.1.12, the period function of the periodic orbits
with three critical points in their interior is decreasing for c e (0,+6).

If we let

G(x) = I: 8(€)d€ a

then by Theorem 1.2.1 (i). (iii), (iv), (v). we have H(x) > 0 for x 1 0,

x e (-l.1). Therefore

p'(c) > 0, 0 < c < C! =

Oflhi

We may thus conclude that there are no critical points of the period

function Of equation

k + x(x2- 1)2 = 0 .

Chapter II

CRITICAL POINTS OF THE PERIOD FUNCTION
OF x-x’(x-¢)(x-1)=0 (06¢<1)

92.1. ANALYTICITY OF THE PERIOD FUNCTION.

Let
_ 2
8(X) - -X (X - a)(X - 1).
G(x)=rg(£)d£=—lx5+l(a+1)x4-lax3
o 5 4 3 '

The curves y = g(x). y = G(x) and the corresponding phase
portraits are shown in Figure 2.1.1.
Let
(2.1.1) I = Ib xny dx. n = 0.1.2.3,
n a

where a < a < b < 1, y = (2c - ZG(x))‘/’. y(a) = y(b) = 0.

Lemma 2.1.1.
b n
I' = I 5- dx ,
n a y
where "'" denotes the differentiation with respect to c.
Proof. Since
2
y = Zc - 2G(x) ,
then

22

23

 

 

 

 

 

 

Y Y
y-s (x) y-g (x)
x x
7 A / A
V .3 V
y (X) y-Gix)
x _\_ __/\ x

 

 

 

Figure 2.1.1

Lem 2.1.3. Let 1 : (I.,I.,I,,I,)T. Then

(2.1.2) ’I' = Q1 ’

where

24

 

 

 

' 42 0 o 0 ‘
. ___ -(3¢ + 3) 54 0 0
-(3a2 - 24 + 3) -(6a + 6) 66 0
. -(3a3 - 242 - 25 + 3) —(6¢2 - 4a + 6) -(9s + 9) 78 .
' 60c 0 3a2+3a -3a2+2a—3
_ 0 60c 3G3-2¢2+3¢ -3a3+2a2+2a-3
’ ‘ 4 3 2 4 3 2
0 0 60¢+3a -2a -2a +3a -3a +2¢ +2a +2a-3
0 0 3a5-2a4-2a3-2a2+3a 60c-3¢5+2a4+263+2a2+2a-3
Proof. Since
yyx = 10:) = x4 — (. + m" + axz .

x = yyx + (a + 1)):3 - axz

It follows then

 

 

b
I0 = a y dx
b3
:J‘de
air
= Ib 20 + 2/5x5 - l/2(a + 1):.4 + 2/3ax3 dx
8 y
b x(yy* + (a + 1)..3 - axz)
30 I = 60 cI' + 12 dx
0 o a y

4
- 15 + 1 Ib 5- dx + 20 I'
(«083, «3

 

25

60 C10 + 12 I: xyxdx - 12 «13

2

yyx+(a+l)x3—ax
-3(a+1) dx+20¢I'
a y 3

 

60 cIé - 12 I0 + 8a 15 - 3(a + 1)215 + 3a(a + 1)Ié .

Hence

(2.1.3) 42 I = 60 CI'

2 . _ 2 _ .
o+(3a +mu2 ma za+m§.

0

Similarly. we have
(2.1.4) —(3a 4- 3)Io + 54 I1 =

= 60 cIi + (3c3 - 2a2

+ 3a)Ié - (3a3 - 262 - 2s + 3)Ié

(2.1.5) -(3az-2a+3)Io - (6a+6)I1 + 6512 =

432

= (60c+3¢ -2a -2a +3¢)Ié - (3.4-2.3-2az

-2¢+3)I3

(2.1.6) -(3a3-2..2-2¢+3)1o - (6a2—4a+6)1l - (9.49):2 + 7813 =

4 2 5432

= (ass-Zn -2a3-2¢ +3¢)Ié + (60c-3a +2a +2a +2¢ +2¢-3)Ié.

Combining (2.1.3) ~ (2.1.6), we have the desired result.

The equation

2

(2.1.7) § - x (x - a)(x - 1) = 0

has three critical points: (0.0). (01.0) and (1.0), which have energy
constants C, = 6(0), c. = c.(a) : G(a) : 1/20 s' - 1/12 s‘ and C. =

c,(¢) = 6(1) = 1/20 - 1/12 a respectively. Then c.(¢) and c,(s) are

26

both strictly decreasing. c.(a) < c,(a). 0 6 a < 1. Furthermore,
Cg(0) : 0, 03(006) : 00

Although g(x), G(x) and the period function Of (2.1.7) depend on
a, for simplicity, we usually suppress the subscript a in g“(x). G¢(x)

and p¢(x).

Theorem 2.1.3. p(c) is analytic in a and C. provided (a.c) is

in the domain D = D. 0 D3, where

D {(a.c) e R2 | o 5 a < 0.6, o < c < 02(G)}

1

D

2 {(a.c) e R2 l 0 < a < 1, c1(a) < c ( nin(0,cz(a))}

(see Figure 2.1.2).

Proof. A direct calculation shows that

det ! = 604 c2(c - c1(a))(c - c2(a)) .

Therefore (2.1.2) is equivalent to

(2.1.8) I' = 3;%-; +91, if c ¢ 0, c y 61(6), 6 y cz(a),

where v is the adjoint matrix of !. So the right hand side Of (2.1.8)
is analytic in I, a. c, when (3.0) e D. -o < Ij ( a, j = 0.1.2.3.
Therefore any solution of (2.1.8) I = (10.1.,I,,I,)T is analytic in a. c.
when (a.c) e D (see. e.g., [10]). Since P(c) = 2I5. the conclusion now

follows.

Remark 2.1.4. (2.1.8) is generally called the Picard-Fuchs equation

to the algebraic curve i— y3 + G(x) = C.

27

0.05

 

 

 

 

Figure 2.1.2

92.2. BOUNDEDNESS OF NUMBER OF CRITICAL POINTS OF THE
PERIOD FUNCTION.

Consider the period function p(c) of the equation
(2.2.1) 1 — x2(x - a)(x - 1) = o, 0 4 a < 1 .

We have

Lemma 2.2.1. For any a e (0.1). p(c) is strictly increasing when
c varies from c.(a) to min(0,c,(a)).
Proof. By scaling g(x). we have

1
(1‘9)

 

Rn 4n41-nx+o

X(X + 1)(X - a)2 .

where a = all-a. Let

&O=C&mu.

Thus it suffices to prove p'(c) > 0. 0 < c < c* = min(G(-1).G(a)).

(1) Suppose a 6 2. Then

§'(x) = 4x3 + 3(1 - 28)X2 + 2(a2 - 2a)x + 82 ,
~. _ 2 2

g (x) — 12x + 6(1 - 2a)x + 2(a - 2a) ,

§"(x) = 24x + 6(1 - 2a).

28

29

It is easy to Check

(i) g'(x) > 0. g'(x) < 0 for -1 < x < x,. where

 

x1 = -% (1 - 28) - 3% «9(1—26)’ - 24(a’-2a) u o .

(ii) there exists x. s (x..a) such that
g'(x) < 0, g'(x) > 0, x e (x1,xo) and
g'(x) < o, x 5 0.0.6) .
Then by Theorem 1.2.1 (ii). (iii), (iv) and Corollary 1.1.5,
p'(c) > 0, 0 < c < ct .
(2) Suppose 0 < a < 2. We consider

31(X) = -§(-X) = -X(X - 1)(X + a)2 .

Then
Eio.) = -4x3 4» 3(1 - 28))(2 + (4a - 282))! + 82 ,
310.) = ~12x2 + 6(1 - 28))! + (4a - 2&2) ,
§;(X) = -24x + 6(1 — 28) .

It is easy to check

(i) g.'(x) > 0, if x. < x < x,, where

 

 

x1 = 71!- (1 - 28) - T11: «9(1-2a)’ - 24(a'-2a) ,
x2 = fi- (1 - 28) + 11% «9(1-28)’ - 24(a’—2a) ,

30

and -a<x.<0<x,<1;
(ii) There exists xo 6 (x,,1) such that
Ei<x) < 0. Eicx) > 0. x . (x2.xo> .

Ei(x) < 0. x e (xo.1> ;

(111) Ei(x) < 0, gi(x) > o, x e (-a,x1).

(iv) {g(x) < 0, if x > l (1 - 2a) .

.h

We rewrite s(x) = x’Q(x). where

G(x) = 8(a2 - 2a)x2 - [6(a2 - 2a)(1 - 2a) - 12a2]x

2

+ 4(a2 - 28)2 - 6a (1 - 28) .

Note that Q(x) is quadratic and 8(aa - 2a) < 0. SO G(x) > 0 if and
only if x c (11.3,), where i. < 0 < II, are the two real roots Of
Q(x). Therefore s(x) 6 0 if and only if x e [52.51,]. Thus to prove
s(x) 6 0. x. 6 x 6 x,. it suffices to show s(x.) 6 0 and A(x,) 6 0.
By definition. s(x) = x(g'.'(0)g;(x) - E;(0)E:(::)). Since x. and x,
are the roots of Em), s(xi) = Em) - xi gun). i = 1.2, in this case.
Note that 2:10) > 0. The sign of s(xi) is the same as that of
xi ° Em.) (i = 1.2). It is easy to see gflx.) < 0. an.) > 0 and

x. < 0 < x,. Then s(x) 6 0 (x. < x < x.) follows.

Then by Theorem 1.2.1 (i). (iii), (iv). (v) and Corollary 1.1.5, we

conclude

Pi(c) > o, o < c < ct .

31

Remark 2.2.2. We can prove in a similar manner that the period

function p(c) of periodic orbits which contain only one critical point
in their interiors Of the equation

i + X(x - 1)(x - a)(x — b) = 0, 0 4 a 4 b 4 1,

82 + (1 - b)2

is strictly increasing.

Remark 2.2.3. Lemma 2.2.1 implies that p(c) has no critical points
when (a,c) is in D3. Thus. in order to study the critical points of
p(c). we can turn our attention on the case (a,c) e D.. which
corresponds to the periodic orbits of (2.2.1) that contain more than one
critical points in their interior. We will henceforth always assume

(“10) 5 01°

Lemma 2.2.4. When (me) e D.,

(2.2... c...) [1° 1217:4744): 137,.

 

 

0
2 ' _ SI I 2 dx
(2.2.3) 20 P (C) - [ Ia + I: )5 g‘(x ) dx + 12 c I: 73/3 ’

where a < 0 < b < 1 such that G(a) = G(b) = c; R(x). S(x). 7 = 1(x.c)
are the same as those in Theorem 1.1.9 and 1.1.13; 0: 6 p < 1 such that

6(9) =

Proof: (2.2.2) and (2.2.3) follow from Theorem 1.1.9 and 1.1.13

respectively.

32

Remark 2.2.5. By the implicit function theorem. the functions F =

5(a). s ' a(a.c) and b = b(a.c) implicitly defined by G(fl) = 0

(a 6 p < 1). G(a) = G(b) = c (a < 0 < b < 1) are continuous.
M For any a. s (0. 0.6), there exists 6 > 0 such that

P'(C) < 0. la - col < 6, 0 < c < 6 .

Proof: Let A(x) be the function implicitly defined by

(2.2.4) G(A(x)) = G(x). for x < 0, A(x) > 6.

Then A(x) e C'(a’.0). where a" < 0 such that G(a’) = (3(1). and

. _ 8‘32 1
A (x) - g(A(x)) , a < x < 0.
Since
11- 40104)) = «mo» 7‘ o .
M‘
“*“o
11- ((8) = 0 a
230’
“2‘0
we have

11- % A'(x) = o .
N90- 8 (4(8))

On the other hand, it is easy to show that

33

(2.2.5) —’}Q‘l<-%, x<0.
s(X)

Then there exists 6. > 0 such that

—(—l§"- M‘” A(x)<---(- %)<0.
9(8) 8(A(X))'

if la - sol < 6., -61 < x < 0 .
Hence

-§£§l- -§£AL§11 ) 0, |¢ - sol < 61, -61 < x < 0 .
9 (X) 8(A(X))

Take

6 = min(6 . min G(-61)) .
1 l“‘¢ol‘5i

Now by Corollary 1.1.11. we have

P'(c) < 0, la - «0| < 6, 0 < c < 6 .

Lemma 2.2.7. For any “0 e [0,0.6). there exist 6 > 0, M ) 0

such that

Io—EL’Q—ax>—M, Ia-ao|<6, lc-c2(ao)|<6.
“4180!)

Proof: Let G = 1/2 a(¢..c.(¢.)). Then G < 0. By continuity of

 

a(¢.c). there exists 6. > 0 such that

34

a(a,c) ( 6, la - «0| ( 61. Ic - c2(¢o)| ( 61 .

It follows that
C = G(8(¢,C)) > 6(6): I“ - “0' < 619 IC - c2(“o)l < 61 °

Therefore

 

 

IOQ=JIO (,3 (514m d,
t J; J2 f JC-G(x) ‘ 40(£)-G(X)

P(G(G))‘* Pa (Ga (6)). as «‘9 a.
e o

where A(x) is defined by (2.2.4). On the other hand, by the mean

value theorem,

 

 

 

 

I‘g=_1I‘_g_=_1I‘ dx
4 J; «2 *1 «(z—(area) «5 a «ti—(nun)
G —_
<-}_- 2 (x-a)‘/3I =M,a<n<£.
J2 J>s(e) 3 4‘8(6)
Note that a(c,a) > -1 always. Then
6 ——
I 9.1:: < 42 +1 _, 42 +1 ’ as a _, a0 .
a 4? 4>s(:) 4‘Eé (6)

Therefore there exist M > 0. 6 > 0 (6 6.) such that

I

0 0

a: ‘ 3x. ._.
; (Ia + Ir) J; < M. | on < a,

k

lc - c2(¢o)| < 6 .

It is easy to show that

35

—121L’£l>-1, x<0
80:)

Hence the conclusion now follows.

Lemma 2.2.8. For any so 5 [0. 0.6) and 1., e (p(ao).1). there
exist 6 > 0, M > 0 such that if la - sol < 6, lo - c.(ao)| < 6, then

(i) 9(a) < 7o. New!) > 76.

(11) Iyo—iflL-dx > —M.
3 4; f(x)

Proof: (i) is obvious. Now suppose (1) holds if la -ao| < 6..

 

|c - c.(a.)| < 6. for some 6. > 0.
Claim: There exist 6, > 0 (6 6.), M. > 0 such that

(2.2.6) gig > -Ml, Ia — «0| ( 62, 6(a) < x < 70 .

In fact. if a. 1 0. then p(ao) > a... and (2.2.6) holds by

continuity. When “0 = 0, we let

 

4
s (X) 68 (X)

where f(x) = 4 a3 + 3 x2 - 6 ex 4- 0 (”II + |a|)°) as |x| 4 0. a -6 0.
Since 4 a3 - 6 ex 4- 3 x’ is positive definite, there exists 6. > 0

(6 6;) such that

f(x))O, 06¢<62, o<x<32.

Therefore

36

(2.2.7) -§$¥l > -1, '

0 4 a < 52, o = 3(0) < x < E
s (X)

2 .

Take 63 = 1/2 :30 Then

40040. 04444 ‘

2. 626x670.

Therefore there exists M. > 1 such that

(2.2.8) -§1¥1 >-4ul, 0 4 a 4 a ‘
s (X)

2, 626x610.

Combining (2.27) and (2.28). we Obtain (2.26).

Proof of (ii). Since

660) -> 9.0”.) < 6%”) = c260). as a -> «o .
there exists 6, > 0 (6 6.) such that

C (a ) + 26 (7 )
2 0 an 0 _
Take

G(a)-G (7)
6=min[63. 2 0 31%]:

M 1
M = —_ ° ‘/2.
43 [g(ao) - Ga (70)
3

fl

Then

 

 

 

 

 

 

 

I’°T!2131_.__1I° 11-3
4130:) «'2' 2c<a>+c(1)
p p 2 o 3 an o _ G(To)
) _ 52’ (1o - 3)
J2 J/'2°a(“o)+Ga (1°) - C,(¢o)+ZGa (7°)
3 3
> "Mo l“‘°‘ol < 6’ IC‘C2(°‘O)| < 6 °

Lemma 2.2.9. Suppose so 6 [0,0.6). If there exist 70 e (p(ao).1)

and 6.. > 0 such that

g'(x) ( 0. 70 < x 6 1, la - «0| < 60.

then for any M > 0. there exists 6 > 0 such that

b
4051...... ..-.0..., ..

c (a )| < 6 .
70 J; 8 2(X) 2 0

Proof: Since

11- F(a) = 5(ao) < 70 .
mo

11- % +6 ,

aT¢o 8 (x)

x-Dl‘

11- £99. = 1‘“o>°’

mo x

1191’

and

lim b(a,c) = 1 ,
mo

CTCa(¢o)

38

there exist 6 > 0. 7. c (70.1) such that
fl(¢) < 70 0

-£K¥l' > 42'“ 41 - a

 

 

 

s'(X) °
8(71) ’/' _
1
1 + 11 .
b = b(a.c) > 2 . 1f la — «0| < 6,

IC - c2(so)| < 6. and 11 6 x < l .

Note that R(x) > 0, 7.. 6 x 6 b. In: - 6.] < 60 under the condition of

Lemma 2.2.9. Therefore

RIx) dx > J'b RIx) dx

7. J; g'(X) 1. J; 83(3)

1)
> M 41 - so I dx
1; JB(b)-G(x)

 

=Mn-ao d" 0.444!»
7! 48(6)(b-X)
—/‘”"
)mJl-ao W
> M, if [C — «0| < 6, IC - c2(ao)| < a .

Lemma 2.2.10. For any “0 s [0,0.6). there exists 6 > 0 such that

P'(C) > 0. |a - aol ( 6, IC - c2(ao)| < 6 .

39

gm: Sin“ 813123.13 (1) =¢e" 1 <09 88 X*1-:“"¢ev
there exist 6. > 0 and 1. c (l(¢.).1) such that g'(x) < 0, la - so]

(6;,7°<X<lo

It is not difficult to show that there exists M. > 0 such that

2c

1
o 73/: dx < M1, if c )'5 02(60), '3 - col ( 6

1 .

By Lemma 2.2.7 and 2.2.8, there exist M. > 0. M, > 0 and 6. > 0

(6 min(6., 1/2 C.(%))) such that

I0 -3$51- dx >
a 4'7- 3’04)

-”2

I7°-§£¥1- dx > -M

F 47 g'(X) 3 o

2 9
By Lemma 2.2.9. there exists 6 > 0 (6 6.) such that

__£1§1__ dx > H

1. J? g’(x)

1+5+%'
Ia - col < 6 , Ic - 62(¢0)| < 6 .
It then follows from (2.2.2) that.

p'(C) > 0. la - «0| < 6, IC - c2(¢o)| < a .

M. There exists 6 > 0 such that

(i) 4/3 a 6 6(6) 6 1.3333401. if 0 6 a < 6 ,

(ii) g'(x))0. g'(x))O. 06a(6. ¢<x<1/3.

40

(iii) R(X) = s‘h) - 26(X)s'(X) < 0. 0 6 a < 6.

1.5794 4 x < 1/3.
Prod: Since
G(ks) = - 5} (ka)3 . a . (314(4). — 5) + 5(4 — 31.)] .

So (i) is Obvious from the definition of 6(a).

(ii) is trivial. Therefore. there exists 6 . > 0 such that

031-0

R‘(x) = -26(x)g"(x) < 0, 0 6 a < 61, 6(a) < x < - .

Note that

R(1.579a) = 1.579446[—0.000954 + 0(1)], as a .. 0.

Then (iii) holds.

Lemma 2.2.12. If k., k. > 0 and

4 4

_ 3_ _ 3
Hl(kl,kz) _ (151‘ + 2°“. ) (151.2 201.2 ) > o ,

then there exists 6 > 0 such that

G(-kla) 6 G(kza), 0 6 a < 6 .

Proof: It follows from the fact that

G(-k1¢) - G(kza)

4
= 30' (8101.22) + 0(1)], as a -> o .

41
Lemma 2.2.13. If 11.. k. > 0, and
3
) 0 ,

Hz(k1.k2) = -l.40475(k13 + 112) + (123 - k

then there exists 6 > 0 such that

482(fl)s(-kla) + 63(kza> 4 0. o 4 a < a .

Proof: By Lemma 2.2.11 there exists 6. > 0 such that

p(«) 4 1.333344 < l , 0 4 a < 6‘.

0)

8'(X)>0 . 06a<6‘. 6(a)6x<%.

Therefore. since g(-k.a) < 0, we have

4gz<n>s(-k,a) + 23(kza)
4 4g2(1.33334a)g(-k1a) + g3(k2a)

tahgmpg)+muh a «+0.

Lemma 2.2.14. If

(0 1., > k.. . 0. 1.. > 1 .

(ii) Haflhukn) > 0 .
(iii) H.<k...k..) > 0 .

where H.. H. are as defined in the previous lemmas, then there exists

6 > 0 such that

_4¥wmu>+?umD40. 04a<a.

'k12“ ‘ x ‘ ‘k11“ '

42

where A(x) is, the function implicitly defined by (2.2.4).
PM: By lemma 2.2.12 and (ii), there exists 6. > 0 such that

0 6 a < 6 .

G(-k11¢) 6 G(k21a), 1

Then

c<4<x>) = G(x) t G""11“) . “(“21“) .

0 ‘ a < 6“ “kg,“ ‘ x ‘ ‘kggae SO

-k a6x6-k a.

0 ‘ “ < ‘1' 12 11

A0!) 6 16210:,

Since

Mar) -’ 0, as a 9 0*, x -’ 0“,

there exists 6, > 0 (6 6.) such that

H

A(x)(-. 06a<6 —ka6x6-ka.

2’ 12 11

0.1

Since

g'(x) > 0, x < 0,
and by Lemma 2.2.11. there exists 6, > 0 (6 6.) such that

g'(x))O. 0602(63, a<x<%,

we have

482(fl)s(x) + 23(A<x)) . 442(n>s(—k124) + 83(k219)

06a<6 -ka6x6-ka.

3’ 12 11

By Lemma 2.2.13 and (iii), the conclusion now follows.

43

Lem. 2.2.15. There exist 6 > 0. xo > 0 such that

-§L§1§211A’(x) a _ % ’ 0 6 a < 6, -x0 ( x < 0.
s (A(X))

Proof: It suffices to show that there exist 6 > 0 and x. > 0

 

such that
f(x) = 4R(A(x))g(x) + 83(A(X)) :- 0, 0 4 a < 8, -x0 < x < 0.
From Lemma 2.2.11. there exists 6. > 0 such that
1
fl(¢) ‘ 1.579“ ( '3- ,
g'(x) > 0 , 0 6 a < 6‘, a < x < %',
and
R(A(x)) < 0, 0 4 a < a, 1.5794 4 A(x) < % .
Therefore
(2.2.9) f(x) 4 0, 0 4 a < 3‘, 1,5794 4 A(x) < % .

On the other hand,

1

R'(x) = -26(x)g'(x) < 0, 0 6 a < 6‘, 6(a) < x < '5 .

Thus

80:) 4 801(4)) = 4204(4)). o 4 a < .1, 4(4) < x < § .

Hence

(2.2.10) f(x) 4 482(P(¢))8(X) + 30100).
0‘¢<6‘, fl(¢)<A(x)<%-.

Let

44

0 = k‘(0) < k‘(1) = 0.33256 ( k‘(2) = 0.38391

II
0

< k‘(3) 0.41305 < k:(4) 0.43379 < k:(5) .45058

It
O

.49426

< k‘(6)

0.46549 < k’(7) 0.47973 < k1(8)

0.55094

< k‘(9) 0.51004 < k‘(10) = 0.52830 < k‘(11)

< k‘(12) 0.58129 < kl(l3) = 0.62617 < k‘(14) = 0.70162

< k‘(15) 0.77200 .

Correspondingly. let

1.333 = k2(0) < k2(1) = 1.35771 < ka(2) = 1.37094

< ka(3) 1.39353

1.38005 < k3(4) 1.38725 < k2(5)

1.41166

< k2(6) 1.39942 < ka(7) 1.40534 < k,(8)

< ka(9) 1.41884 < ka(10) = 1.42756 < ka(11) = 1.43896

< k2(12) 1.45524 < ka(13) = 1.48123 < ka(14) = 1.52958

< k3(15) 1.57900 .

Computer results show that

Hl(k‘(I),ka(I)) > 0, I O.1,2.....14,15.

0.1,2.....l4.

nogr+nagn>>m I
By Lemma 2.2.14, there exist 6(1) > 0, (I = 0.....14). such that

4Ewonnmfia<uw)4m u 04a<un ma

-k‘(I + 1)a 6 x 6 -k‘(I)a, I = 0,1,...,14 .

Taking 6. = min(6..6(1),...,6(14)). we have

waan 4fwonnw+gkum)4m u 04a<%.

-0.772a ( x ( 0 .

45

By Lemma 2.2.12, there exists 6, > 0 (6 6.) such that
G(A(-0.772¢)) = G(—0.772a) 6 G(1.579a), 0 6 a < 63 ,
i.e.

A(-0.772¢) 6 1.57901, 0 6 a < 63 ,

(2.2.11) then implies that

(2.2.12) 422(4)::(x) + 330104)) 4 0. o 4 .. < 43 .

p(«) 6 MR) < 1.5790: .
Take 6 > 0 (6 6,) and x0 > 0 such that
G(-xo) 4 0(319, o 4 a. < a .
The conclusion follows from (2.2.9), (2.2.10)) and (2.2.12).
Lemma 2.2.16. There exists 6 > 0 such that
p'(c)<0, 06a<6, 0<C<6.

Proof: From (2.2.5) and Lemma 2.2.15, there exist 6. > 0 and x. >

0 such that

 

M__m21)(-%+1). 1 ,0

, 06a66 ,-x <x<0.
430:) 330100) 4 3‘”) ‘ °

Take

6 = min(6 , sin G(—x )) .
1 Odada‘ 0

By Corollary 1.1.11. we have

46

p'(c)<0. 06a<6, 0<C<6.

Lemma 2.2.17. There exists 6 > 0 such that

-§L§l > -4, 0 6 a < 6.

4 flaz<1~c<6,
s(X)

3
where S(x) is as defined in Theorem 1.1.7.

Proof: It is equivalent to show that

%S(x)+g4(x)>0. 06a<6, §a(x<6.
A direct calculation shows that
% S(x) + g“(x)
-_1_84 2.3.4. 2-3.3 12-32
— 602 x [x (4300(x 4) 4900(x 4) + 2212 2(x 4)

— 56 21“; - I) + 10 g) + R(x.a)] .

where R(x,a) is a polynomial containing higher order terms only. The

conclusion now follows.
Lemma 2.2.18. There exist a > 0, E > 0 (< 1/20) such that

p‘(c)<0. 04a<5, 0<c4E

P'(C) > 0. 0 6 a < 6, C < c < c2(a).

Proof.

(i) By Lemma 2.2.16, there exist 6. > 0. C ) 0 (< 1/20) such that

47

p'(C) < 0. 0 6 a < 61, o < c 6 E.

(ii) a(c-:,a) -6 a(C,0) < 0. c.(¢) 9 1/20, as a -4 0. Therefore, there

exist a. < 0, 6. > 0(6 6.) such that

a(;.a) < a1, c203) > C, 0 6 a 6 62 .

It is not difficult to show that S(x) > 0, if x < 0. Then

1
feel—Mr“ Tam—d... 0......
a «y g‘(x) a. «1 g‘(x) '

C<c<ca(a).

Since
1
a 8‘ S(x) dx
- O
8. ¢7 s (X)

is continuous in a and C when ~0 6 a 6 6., C 6 c 6 C,(a), there

exists In > 0, such that

I
I' --§151- dx 6 m > 0, 0 6 a < 6’,

._ ‘ -c-<C<ca(a).
3: 478(3)

Therefore
0 SIX)
dx>m>0, 06a<6a,

_ ‘ C<c<ca(a).
a 47 s (X)

(iii) 55...) 4 56.0) > 0, as a 4 0

G(x)-’0. as 6+0, x40.

Therefore there exist 6. > 0 (6 6.). b. > 0 such that

48
_ 3 _
b(c,a) > b‘, G(x) < 4 c. 0 6 a < 63, 0 < x < 63.
From Lemma 2.2.17. there exists 6. ) 0 (6 6,) and

70 > 0 (4 min(63,b1. 15%))

such that
6(a) < 70 O 6 a < 6‘ ,
and
-§$¥l > —4, o 4 a < 6‘, 8(a) < x < 70
s (X)
Then

I70 SEX) dx > _4 170
9 J; 8 (X) 0 Jb‘G(X)

To 7
> —4 I -——9¥-- = -8 5% > -m
042-33 «E
4
06a<6. C<C<C(a)
Therefore.

II:+ I I—J-L-dx>m-m'0 06a<6‘.
47 8 (X)

E < c < 62(e) .

(iv) S(x) is continuous in a: and x. and from Example 4 of 91.2,

when a = 0.

S(x) > 0, 0 ( 1° 6 x 6 1 .

Then there exists 6 ) 0 (6 6.) such that

49

S(x) > 0, 0 6 a < 6, 1° 6 x 6 1 .

Therefore

26%.“)- [1° I:°]_LL¢HJ‘ _J_2_d,‘
4? a (X) 1o 4; 8 (X)
+ 12c2 I,

0

  
 

2 dx > o, o ‘ a < a, E < c < c2(¢).

Gorollarz 2.2.19. There exists 6 > 0 such that for any a: e [0.6),
p(c) has exactly one critical point.
Proof. The proof follows from Lemma 2.2.10 and Lemma 2.2.18.

Lemma 2.2.20. There exists 6 > 0 such that

(i) g'(x) < o, g'(x) < o, g'(x) < o, 1 - a < x < 1,
(ii) 3(1 - 0.54JBTE'Z'3) > 0,
(iii) G(-0.51(0.6 — g)‘/’) . G(l - 0.54J576'3'2),

(iv) 1 - 0.54JBTE'I'; > p(«) > 1 — 0.65J6761313,

provided 0.6 - 6 < a < 0.6, where S(x) is as defined in Theorem

1.1.3.

Proof: (i) is trivial. (ii), (iii) and (iv) are proved by using the

 

Taylor’s series.

Lemma 2.2.21. There exists 6 > 0 such that

M A'(x) ( - , if p(c) ( A(x) < 1,
S(A(X))

0.6-6<a<0.6,

50

where MI!) is as defined by (2.2.4).

Proof: From Lemma 2.2.20(i) it is easy to show that there exists

6, > 0 such that
S'(x) > 0, if 0.6 - 61 < a < 0.6, 5(a) < x < 1.

Therefore it suffices to show that there exists 6 > 0 such that

334(fl(a))g(x) + 2g5(A(x)) > o, 0.6 — a < a < 0.6 ,

fi(a) < A(x) < 1 - 0.5440.6 — a ,

since S(1 - 0.5445???) > 0 when a is close to 0.6 by Lemma
2.2.20(ii).

Let
_ 4 5
f(X.y) - 38 (fi(«))s(X) + 28 (y) .
Then

'35 (st) :: 334(fi(a))g'(x) > 09 if X < 0.

From Lemma 2.2.20(i), there exists 6, > 0 (6 6.) such that

g; (x,y) = 1034(y)g'(y) < o, p(«) < y < 1, 0.6 - 62 < a < 0.6

Therefore, if 11., < x < O, 3(a) < y < A0 ( 1, then

(2.2.13) f(x.y> > 3:4(n<a>)g(xo) + 285(A.>»

0.6-62<a<0.6.

Let Ao(a) = 1 - 0.54JO.6 —'3. Take 1(a) < 0 such that

G(7(a)) = G(Ao(a)). Then by Lemma 2.2.20(iii), there exists 6;, > 0

51

(6 6,) such that

(2.2.14) 7(a) . -o.51 (0.6 - «)1/3, if 0.6 - 53 < a < 0.6 .

By (2.2.13), (2.2.14) and Lemma 2.2.20(i), (iv), there exists 6. > 0 (6
a.) such that if p(a) < A(x) < 1 - o.54¢o.s :52, 0.6 - a. < a < 0.6,

then

x > -0.51(0.6 - «)1/3 ,

and
4 5 -
38 “(0)800 + 28 (M8)) - f(X.A(X))
> 3g4(1 - 0.65Jb.6 - a)g(-0.51(0.6 - «)1/3)

+ 235(1 - 0.54Jb.6 2'3)
= 2 - 0.2165(o.6 - «)5/2 + o(0.6 — «)5/2), as a a 0.6.
Therefore there exists 6 > 0 (6 6.) such that

3g4(p(a))g(x) + 2g5(A(x)) > o, p(a) < h(x) < 1 - 0.54Jb.6'=':

0.6-6<a<0.6.

Lemma 2.2.22. There exists 6 > 0 such that

p'(c) > 0, 0.6 - 6 < a < 0.6, 0 < c < c2(a) .

Proof: Since

, asx+0-, «+0.6-,

s(X)

ODIN

there exists 6, > 0 such that

52

-§$¥l > 3», 0.6 - a < a < 0.6, -a < x < o .
e 3 I 1
s (X)

By Lemma 2.2.21, there exists 6, > 0 (6 6,) such that

hymn “x, < .3, 0.6 - a, < a < 0.6. 5(a) < am < 1.
s (A(X))

Note A(x) + 1" as x 9 0" and a + 0.6“. There exists 6 ) 0 (6 6,)

such that

S(x) _ smxn A'(x) > g _ g 0 ’

g‘(x) (mm 3

if 0.6 - 6 < a < 0.6, -6 < x < 0. Then by Corollary 1.1.14 and note

that c,(a) a O as a + 0.6, the conclusion now follows.

Corollary 2.2.23. There exists 6 > 0 such that p(c) has exactly

one critical point if 0.6 - 6 < a < 0.6.
Proof. The proof follows from Lemmas 2.2.6, 2.2.10 and 2.2.22.

Theorem 2.2.24. There exists a uniform bound for the number of
critical points of the period function p(c) of equation (2.2.1) for

06a<l.

Proof: From Lemmas 2.2.1, 2.2.6, 2.2.10 and Corollaries 2.2.19 and
2.2.23, there exists 6 > 0 such that p(c) has at most one critical

point when (a,c) e D5 U D,, where
D6 = {(a,c) 5 D1 l dist((a,c),JDl) < 6} .

Theorem 2.1.3 shows that p(c) is analytic in a and c when

53

(a,c) e D ,\D5. We claim that there exists a uniform bound for the
number of zeros of p'(c) for any a e [6.0.6 - 6] and c varying

from 6 to c,(a) -6.

Suppose not. Then there is a sequence {an} C [6,0.6 - 6] such

that

(2.2.15) a 4 a , N 9 o, as n -) +a,
where Nn denotes the number of zeros of p&n(c) in [6,c,(an) - 6].

Since p&o(c) is analytic in c, it has a finite nudaer N
(counting by multiplicity) of zeros in [6,c,(a.) - 6]. From the
analyticity of p&(c) and the compactness of [6,c,(a.) - 6], it is clear
that there is a 6, > 0 such that for any a e (a, - 6,, a, + 6,),
p&(c) has at most N distinct zeros when c varies from 6 to

c,(a) - 6. This contradicts (2.2.15).

LIST 01" REFERENCES

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

LIST OF REFERENCES

v.1. Arnold, Geometric methods in the theory of ordinary
differential equations, Springer-Verlag, N.Y., 1983.

P. Brunovsky and S.N. Chow, Generic properties of
stationary state solutions of Reaction-Diffusion
Equations, J. Diff. Eqn. Vol. 53, No. 1, 1-23.

J. Carr, S.N. Chow and J.K. Hale, Abelian integrals and
bifurcation theory, J. Diff. Eqn., to appear.

S.N. Chow and J.K. Hale, Methods of Bifurcation Theory,
Springer-Verlag, N.Y., 1982.

S.N. Chow and J.A. Sanders, On the number of critical
points of the period, to appear.

W.S. Loud, Periodic solution of x' + cx' + g(x) = c f(t).
Mem. Amer. Math. Soc., No. 31 (1959), 1-57.

C. Obi, Analytical theory of nonlinear oscillation, VII,
The periods of the periodic solutions of the equation
x + g(x) = 0, J. Math. Anal. Appl. 55 (1976), 295-301.

2. Opial, Sur les periodes des solutions de le’ equation
differentialle x + g(x) = 0, Ann. Pol. Math. 10 (1961),
49-720

R. Schaaf, Global behavior of solution branches for
some Neumann problems depending on one or several
parameters, J. Reine Angew. Math. 346 (1984), 1-31.

R.A. Struble, Nonlinear Differential Equations,
McGraw-Hill Book Company, Inc., New York, Toronto,
London, 1972.

D. Wang, On the existence of Zn—periodic solutions of
differential equation x" + g(x): p(t), Chin. Ann. Math.,
5A(1) (1984), 61-72.

54