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A STUDY OF SUBHARMONIC SOLUTIONS OF
SECOND ORDER EQUATIONS

BY

Whei-Ching Chang Chan

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1985

ABSTRACT

A STUDY OF SUBHARMONIC SOLUTIONS OF
SECOND ORDER EQUATIONS

BY

Whei-Ching Chang Chan

Consider a second order differential equation

R + g(x) = -xi + uf(t)
with small damping and periodic forcing. We will
investigate the condition on the parameters (Lu) to

ensure the existence of subharmonic solutions of order l<
by deriving the bifurcation equation. We find in the
(Lin-plane there are two disjoint regions such that the
equation has at least 2k k-periodic solutions in one
region and none on the other region. The stability of
these solutions is also discussed by computing the
characteristic multipliers. Finally, some numerical
experiments, such as locating those periodic solutions and
increasing parameters to obtain period doubling phenomena,

are performed on the forced pendulum problem.

TABLE OF CONTENTS

PAGE

LIST OF TABLES . . . . . . . - . . . . . . . . . . . ii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . iii
SECTION 1: INTRODUCTION . . . . . . . . . . . . . . l
2: BIFURCATION EQUATION . . . . . . . . . . 3

3: EXISTENCE . . . . . . . . . - . . . . . 12

4: STABILITY - . . . . . . . - . . . . . . l9

5: UNIFORMITY . . . . . . . . . . . . - . . 35

6: NUMERICAL STUDY . . . . . . . . . . . . 49

LIST OF REFERENCES . . . . . . . . . . . . . . . . . 65

 

LIST OF TABLES

PAGE
TABLE 6.1 . . . . . . . . . . . . . . . . . . . . . 55

TABLE 6.2 . . . . . . . . . . . . . . . . . . . . . 56

ii

 

FIGURE

1 .

FIGURE 2

FIGURE

FIGURE

FIGURES 6.2,6.3

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

3

6.1

6.4-A

6.4-B

LIST

OF FIGURES

iii

PAGE
30
30

31,32
57
58
59
60
61
62
63

64

SECTION 1. INTRODUCTION

In [6], D'Humieres, Beasley, Huberman, and Libchaber
presents a series of numerical experiments on the forced
pendulum problem, which includes the period-doubling
cascades which lead to chaotic motions. They also indicate
that there is a non period-doubling case which also leads
to chaotic states. A. Ito [9] also gave some evidence of
successive subharmonic bifurcations which lead to chaos.
Other experiments may be found in [7], [8]. However, in
many cases it is not clear how these motions are created.
Here we attempt to explain some of these phenomena.

Consider the 2nd order differential equation

(1.1) i + g(x) = —xk + uf(t)
where f(t) has least period 1, x,u are parameters and
g(x) is a function such that when (x,u) = (0,0),

equation (1.1) has either a homoclinic orbit or heroclinic
orbit F. Inside the orbit F, it is well—known that
there exist periodic solutions with least periods tending
to infinity as these periodic solutions tend to F.

For each periodic orbit Pk with least period k, we
will investigate the condition on the parameters (x,u) to

ensure the existence of subharmonic solutions of (1.1) of

 

order k. To do this, we first invert a differential
operator, whose inverse is denoted by Gk. Next, we derive
the bifurcation equation for these subharmonic solutions in
order to find the bifurcation diagram. We find that in the
(x,u)-plane there are two disjoint regions such that (1.1)
has at least 2k k—periodic solutions in one region, none
on the other region, and exactly one on the curve. The
above phenomena are called the saddle-node bifurcations,
i.e., two periodic solutions coalesce and then disappear.

To prove that, we need to discuss the stability of the
solutions by' computing their characteristic multipliers.
We also find a neighborhood of (Ln) = (0,0) such that
the stability arguments holds uniformly for each k.

Chow, Hale, and Mallet—Paret [3] indicated that if
there is a neighborhood U of (x,u) = (0,0), such that
for (x,u) e U there exist k-periodic solutions of any k,
then there exist infinitely many periodic solutions which
are derived from successive subharmonic bifurcation. Such
neighborhood would exist if the operator Gk, as mentioned
above, is uniformly bounded in. )9. We prove in Section 5
that Gk is uniformly bounded on the subspace of symmetric
periodic functions.

Finally, we give a continuation method for finding the
periodic orbits numerically. We will use the Runge-Kutta
method to solve the initial value problem and Newton's
method to locate the periodic orbit. Our numerical

experiments are performed on the forced pendulum problem.

SECTION 2- BIFURCATION EQUATION

Consider the equation

(2.1) i + g(x) = —xk + uf(t)
where X,“ are real parameters, g(x) is 3—times
continuously differentiable and f(t) is periodic with

period 1. For x = u = 0, assume the system
(2.2) x + g(x) = 0

has a nontrivial periodic solution p(t) with least period
k, where k is an integer. Let I‘ = {(p(t),p(t);
0 5 t < k}. The problem is to find periodic solutions of
(2.1) with least period k in a sufficiently small
neighborhood of F for small x,u. If such solutions
exist, they are called subharmonic solutions of order k

since their period is k times the period of f(t).

For this discussion, it is convenient to use a
different coordinate system near P. Let G : R4 a R2 be
defined by

6(a.a,x,y) = (p(a) + aha) - Lima) - aim) — y)
Since c(ao,o,p(ao),b(ao)) = (0,0) and

det 532%57 (a0,0,p(ao):b(ao)) = 5(a0)2 + 5(a0)2 y 0 .

3

 

It follows from Implicit function theorem that there exists
5(ao) ) 0, a(ao) ) 0 and two functions a*(x,y), a*(x,y)
with a*(p(ao),i>(ao)) = “0 , a*<p(ao).15(ao)) = 0. such
that G(a*(x,y),a*(x,y),x,y) = 0 for Ix — p(ao)l ( 5(ao),
Iy — p(ao)l < 5(a0) and these solutions are unique for
la — aol < a(ao), Ial < a(ao). This defines a
diffeomorphism from a neighborhood of (p(ao).b(do)) to a

neighborhood of (a0,0). We can apply the above argument

to every point of F, then by compactness of F, there
exists an > 0 and a diffeomorphism F from a
neighborhood of F onto [0,k) x {a : Ial < a0}. In
summary, for any (x,y) near F, there exists unique
(a,a), where a shows the position on I‘ and ap(a)

indicates the distance between the orbit F and the point

(x,y), such that
X = p(a) + aié(a)

y = 5(a) — abta)

If x(t) is a k—periodic solution of (2.l) n1 a
small neighborhood of F, then there exists a unique

(a,a) such that

x(0) = Mon) + aim)

{<(0) = 6(a) — aim)
Therefore we can write x(t) in the form

(2.3) x(t) = p(t + a) + z(t + a)

 

where z(t + a) has small magnitude, (z(a),é(a)) is
orthogonal to (p(a),p(a)) and a is determined by the
initial condition.

Let (2.3) be applied to (2.l), we get

§(t + a) + g'(p(t + a))z(t + a)

= —xi(t + a) — xp(t + a) + uf(t) + G(t + 0:2)

where G(t + a,z) = -g(p(t + a) + z(t + a)) +
g'(p(t + a))z(t + a ) + g(p(t + a)), therefore G(-,z) =
0( I z I 2) . Here we let " , " denote the der ivat ive with
respect to x. Replace t + a by t , we obtain the

following equation
(2.4) i + g'(p)z = —xé — xb + ufa(t) + G(t,z)

where fa(t) = f(t - a). Hence the problem now is to find
k-periodic solutions of (2.4) with (z(a),é(a)) orthogonal
to (p(a).§(a))-

Without loss of generality, suppose p(O) = O and
assume that

(Hl) Every k—periodic solution of the homogeneous equation

(2.5) '2‘ + g'<p)z = o

is a constant multiple of p(t).

Note that equation (2.2) is Hamiltonian. Hence p(t) must
be embedded on a one—parameter family of periodic solutions
of (2.2). Let the parameter be denoted by b and the

corresponding period be T(b). Assume that T(bo) = k.

 

6

Proposition 2.1. Let T(b) be the period function as

above. Then hypothesis (H1) is equivalent to T'(bo) # 0.

Egggf. For simplicity, let's assume that k) is the
place where the periodic orbit crosses the positive
x-axis, therefore p(O) = 0, p(0) = b0.

Let 'p(t,b) be the periodic solution of (2.2) with

period T(b), then
(2.6) p(t + T(b),b) = p(t,b)

and p(t,b0) = p(t). Let

Q1

(2.7) q(t) = 53 p(t.b)|b=bo

then q satisfies (2.5) with initial condition q(O) = l,
q(O) = 0. Take the derivative with respect to b of both

sides of (2.6) and set b = b0. We obtain

(2 8) é<t>T'(bo) + q(t + T(bo)) = q(t)

Set t:== 0, we get q(k) = 1. Take the derivative with
respect to t of both sides of (2.8) and set t = 0, and
we get em.) = -p(0)T'(bo). If T'(bo) = 0, then
q(k) = 0 which implies q(t) is a k-periodic solution of

(2.5), therefore (Hl) does not hold. If T'(bo) # 0, then

q(t) is not a k—periodic solution . Let
'(t)

(2.9) r(t) = .‘—?——
10(0)

then

[ q(t) r(t) J
(2.10) X(t) =

é(t) f(t)
is a fundamental matrix solution of (2.5), that is the
solution of (2.5) is a linear combination of q and r.
This shows that the only k—periodic solution of (2.5) is a

constant multiple of p(t). Q.E.D.

We now apply the method of Liapunov-Schmidt to
equation (2.4). Let Pkr be the space of r—times
continuously differentiable periodic function with period
l< with lflr = sup{lf(i)(t)l : i = 0,1,...,r,t e [o,k)}.

For any y e sz let

(2.11) Ay § + g'(p)y

—x§ — xfi + ufa(t) + G(t,y)

NY

where fa and G(t,y) are the same as in (2.4). Then A
is a continuous linear operator from sz » Pk° and N is
a continuous operator from sz » Pko. (H1) implies that

the null space of A is one dimensional. Define

P : Pko a Pko by

0 k c
(2.12) Py = npfo pydt
where

k
(2.13) n = (I p2 dt) 1

0

Then P is a continuous projection.

Lemma 2.2. Assume (H1) holds. Let X(t) be the

 

fundamental matr ix of (2 . 5) . For any d: 6 Pk" , define

5 : (I - P)Pk° » Pko by

 

 

 

 

 

G¢(t) w0 t _1 0
(2.14) ~. = X(t) + X(t) f x (5) ds
G¢(t) 0 0 ¢(S)
where
k
l
wo = j' q(s)¢(s)ds
-Q(k) 0
and q(t) is given by (2.7). Then G is a continuous
linear operator, and G¢(t) is a solution of
Hi + g'(p)z = NH
(2.15) 1 z is k—periodic
3 2(0) = 0
Proof: It follows from the variation of constants

formula that the solution of (2.l5) can be written as

wo t _1 0
J + X(t) I x (s)[ ] ds
0 0

= X(t)[
¢(S)

 

[ z(t)
é(t)

Then z(t) is a k-periodic solution if and only if

-r(s)¢(s)ds

 

 

+ X(k) I:

 

 

Q(s)¢(s)ds

 

 

-r(S)¢(s)ds
(1 - x(k))

 

"° 1 IN

0 q(S)¢(S)ds

 

IE —r(s)¢<s)ds

‘é(k) IE r(S)¢(S)ds + f5 q(S)¢(S)ds

 

3 -If r(s)¢(s)ds

If q(s)¢(s)ds

Therefore I? r(s)¢(s)ds = 0, and

k
w0 = _q%k) f0 q(s)¢(s)ds

 

Q.E.D.

Note that since both G and P are continuous linear
operators, therefore the operator G(I — P) has continuous
second Frechet derivative.

We obtain the following Lemma by adding an appropriate

constant multiple of p(t).

Lemma 2.3. Let ¢ 6 Pk0 . Then

 

E + g'(p)z = ¢<t) — P¢(t)

z is k—periodic

10

has a unique solution G(I —- P)¢ which has continuous
second Frechet derivative with respect to ¢ and
P(G(I - P)¢) = 0.

It follows from Lemma 2.3 that AG = I on (I - P)Pk°

and GA = I — U on sz where

U = Plpkz

For this definition of CL (2.1) will have a solution of
the form given in (2.3) if and only if the following two

equations are satisfied.

(2.16) w = G(I - P)N(z)
= G(I — P)[—xé — x5 + ufa + c( ,z)]
(2.17) PN(z) = p[—xé — xb + ufa + G( ,z)] = 0

where z=w+ap and Pw=0. Let B:Pk2xRx[0,k)x

R x R » Pko be defined by
B(w,a,a,x,u) = w — G(I — P)[-xé — xp + ufa + G( ,z)]

Since B(0,0,a,0,0) = 0 and

6B

5W (0,0,a,0,0) = I

on mg; by the Implicit function theorem there is a
neighborhood U _C_ sz of zero and a neighborhood V E R3
of (a,).,u) = (0,0,0), such that (2.16) has a solution
w*(a,a,x,u) for (a,x,u) e V, 0 s a ( k. This solution

is unique and has continuous second derivative with respect

11

to a,a,x,u and w*(a,a,0,0) = 0 for all a. Therefore
(2.1) will have k—periodic solutions if and only if
(a,a,k,p.) satisfies equation (2.17) with 2 replaced by

w*(a,a,x,u) + ap.

Proposition 2.4. Let

(2.18) C(a.a,x,u)

n Ik pI-XW*(a,a,x,u) - in + ufa(t)
0

+ G(t.w*(a,a,x,u))]dt

—x + h(a)n + h.o.t.

where
k 0
h(a) = n f p(t)f(t — a)dt and
o

h.o.t. = 0(le2 + lulz + Ial) as x,u,a a 0

Then (2.1) has a k-periodic solution for some small 1, E,

a if and only if there exists an & such that
(2.19) C(é,&,i,i) = 0

Remark, 2.5. Equation (2.19) is often called the

bifurcation equation.

 

Remark 2.6. Since w*(a,a,X,u) is a solution of
(2.16) and f(t) is l—periodic, it follows that
C(a,a + 1,1,u) == C(a,a,x,u), that is (3 is l-periodic

in a with x A fixed.

 

 

 

 

SECTION 3. EXISTENCE

To obtain the complete picture of the existence of
k—periodic subharmonic solutions of (2.1), it remains to
analyse the bifurcation equation (2.19). The procedure is
similar to Chow and Hale [3]. Since a in the definition
of the bifurcation equation is not important, we drop its
dependence in our analysis.

Let h(a) be defined by (2.18) which has period 1.

Assume that

(H2) There are only finitely many numbers ai 6 [0,1] 1 =

l,...,N, such that h'(ai) = 0 and h"(ai) f 0.

For any u f 0, (2.19) is equivalent to

h(a) _ [AL + G(arfirxru) = 0

where

x
u - G(a,;,x.u) = x h(a)u - C(a,x,u)

and

G(a,B,0,0)

I
O
E

II

Hence to finding all possible solutions of (2.19) for small

x, u, is equivalent to finding all possible solutions of

12

13

(3.1) F(a,B,u) = h(a) - B + G(a,B,Bu,u) = 0

for all 0 s a < k, B e R and ,u. near 0. It follows
from Remark 2.6 that F(a,B,u) is l—periodic in a with
x,u fixed. For 0 s a < 1 and u = 0, the only solution
of (3.1) is (ao,Bo), which satisfies h(ao) = 80. If

h'(ao) f 0, then

6F
6—“; ((10,30'0) 7‘ 0 °

The Implicit function theorem implies that there exists
6010,80) > 0 and a unique solution a*(B,/1.) such that
F(a*(B,u),B,u) = 0 for m — 30: < Magma). lul <
6(ao,Bo) and a*(Bo,0) = a0.
If h'(ao) = 0 then (H2) implies
2

a F

aa2

 

(00,80,0) = h"(ao) # 0

By the Implicit function theorem there exists 5(ao,Bo) > 0

and a unique solution a*(8,u) such that

6F 2 _
‘6‘; (a (B:#)'B'#) - 0

for IB — Bol < 5(ao,£o),lu| < 6(a0,Bo). Hence
F(a*(B,u),B,u) is a maximum or minimum of F(a,B,u) with
respect to a for B, 11. fixed. For fixed 8,11,, let
g(a) = F(a,Bru) then

82F

(a,B,u)
aa2

8"(a) =

In particular

l4

82F
s”(a*(Bo,0)) = ——; (a0,30,0) = h"(ao) e o .
6a
If h"(ao) > 0 then for (3,u) near (30,0),

g"(a*(B,u.)) > 0 therefore F(a*(B,u),3,u) is a minimum.
If h"(ao) ( 0 then by the same argument F(a*(£,u),3,u)
is a maximum. The number of solutions of (3.1) will depend

on the sign of F(a*(3,u),3,u). Let
(3.2) 7(B,u) = sign h"(ao) - F(a*(e,u),e,u)

Then the following holds
(1) y(3,u) > 0 => there are no solutions of (3.1)
(2) 7(B,u) = 0 => there is only one solution of
(3.1)
(3) 7(B,u) < 0 => there are exactly two solutions

of (3.1)
Let H(B,u) = F(a*(B,u),B,u) = 0. Since

6H _ aF 6a

aF _ _
5? (30:0) ‘ (5&- ‘6—5 + fi)(aorfior0) - l

and H(30,0) = 0, it follows from the Implicit function
theorem that there exists 5(60) > 0 and a unique solution
3*(u). such that H(3*(u),u) = 0 for nu < 5(30).

Therefore F(a*(e*(u).u),a*(u),u) = 0 or 7‘1(0) = {(B,u)

3 = 3*(u),lul < 5(Bo)}. We conclude that there are two

solutions of (3.1) near do on one side of the curve

,8 = 3*(11) and none on the other side. In terms of the

 

15

coordinates (1,11,), the curve becomes 1 = 3*(u)u. which
is tangent to x = h(ao)u at x = u = 0.

The above argument can be applied to each a0 + j,
j = l,...,k - 1. Since F(a + l,>.,u.) = F(a,).,u.), hence
the curve we obtain will be the same for each a0 + j.
This shows that altogether there are 2k solutions on one

side of x = 3*(u)u and none on the other side. Let

h(a*) = max h(a) and h(a*) = min h(a)
ae[0,l) ae[0,l)
and two curves 1 = C*(u) x = Cx(u) which are
respectively tangent to x = h(a*),u., x = h(a*)u at

x = u = 0. We obtain the following theorem.l

Theorem 3.1. If hypotheses (H1) and (H2) are

satisfied, then there are neighborhoods U of P, V of x

= u = O and a finite number of curves Cj e V defined by
x = Cj(u.) which is tangent to the straight line
x = h(aj)u at x = u. = 0, j = l,...,N. The number of

k-periodic subharmonic solutions of (2.1) in U changes by

2k as each curve Cj is crossed. Moreover if
s = {(Mu) e v : c*(u) < x < c.(u.)}

then there are no solutions of (2.1) in U for (x,u) t S

and at least 2k in S

 

16

Proof. It remains to find the neighborhood V of

x = u = 0. For each 013 in Hypothesis (H2), by the same

argument as before, there exists 6(Bj),e(aj) and
bifur cat ion curve ). = .83 (11,)”. such that there are two
solutions of (3.1) on one side of x = 83(u)u and none on
the other side for IXI, lul ( 6(83) and
la - 013! < €(aj). Let B be the complement of the union
of {a;la - ajl < €(dj)}, j 1= 142,‘-°,N then B is
compact in [0,1] and h'(a) 74 0 on B. Therefore no

further bifurcation will take place. By the same argument
as before, for each a0 6 B, there exists 6(ao,Bo) > 0,

6(ao,Bo) > 0 such that equation (3.1) has exactly one

solution for Ik|,lul < 6(ao,Bo) and la - aol < 6(ao,£o).
The sets {azla — aol < e(ao,Bo)} as a0 varies over B,
serves as an open covering of B. By the compactness of
B, there exists a finite covering, [a;|a: — aoil (
€(aoi,Boi)} i = 1,2,~'~,M, of B. Let
5 = min {5(aoi,Boi).5(Bj)} ,

i=l,~--,M

j=lr°"rN
then. ‘V = {(x,u);|1|,lul < a} will be the required
neighborhood.

Remark 3.2. The above result can be generalized to a

 

two dimensional systems

i = g(X) + f(t,x,u)

 

17

where g : R2 -+ R2, f(t,x,fl-) : R x R2 x R2 —9 R2 are
r-times continuously differentiable and f(t + 1,x,u) =

f(t,x,u). Assume

i = g(x)
has a periodic solution p(t) of least period k. Let

q(t) be a nontrivial k—periodic solution of the equation

& = -y.A(t)
where

A(t) = 52 g(p(t))

Then the bifurcation equation becomes

k
(3 2) C(a.u) = [0 q(t) - F(t,z*(a,u)(t),u.a)dt = 0

where F(t,z,u,a) = f(p(t) + z) — f(p(t)) - A(t)z + 8(t -
a,p(t) + z,u) and "°" is the inner product. Finding the

solutions of (3.2) is equivalent to finding the solutions

of
(3.3) B(a,B,C) = 8 ° h(a) + Bo(a,B,C)
where u = BC, 3 6 R2, IBI = 1, C e R and
k
h(a) = )0 q(t) - [af(t — a,p(t).0)/au1dt
Apply the proof of Theorem 3.1 to (3.3). We obtain a

result similar to that of Theorem 3.1.

18

Remark 3.3. For those (a,x,u) such that

 

C(a,l,u) = x - h(a)u + h.o.t = 0 ,
since

ac _
“6‘7 (auroro) " l f 0 I

the Implicit function theorem implies there exists (5 > 0
and a unique function x*(a,u) such that if Iul,

Ia - a0: < a then C(a,x*(a,u),u) = o.

SECTION 4. STABILITY

Assume (H1) holds, we will discuss the stability of
the subharmonic solution of (2.1) by computing the
characteristic multipliers of the linearized equation.

It follows from Lemma 2.3 that for small x,u and 0 <

a < k there exists a unique solution 2 of

E+g'(p)z = —x2 - xp + uf(t—a) + G(t,z) — C(a,x,u)p
(4.1) 2 is k-periodic
z(anbw) + 2(a)i$(a) = 0

where G is given by (2.4) and C(a,x,u) is the
expression given by (2.18). Note that the solution
z(t,a,x,u) has continuous second derivatives with respect
to a,x,u. Let ¢(t,a,x,u) == p(t) -+ z(t,a,x,u). Then
¢(t,a,x,u) is a k-periodic subharmonic solution of

Q + g(x) = -xi + uf(t-a) - C(a,x,u)b

Note that ¢(t,a,0,0) = p(t). We will find the information
needed to decide the stability of ¢(t,a,x,u).

Consider the linearized equation around ¢(t,a,x,u)

(4.2) E + g'(¢)x + xx = 0

which can be rewritten as

19

20

[:J=[-g‘3(.) -illii]

(4.3) = A(t) [x]
2
Let
Y1(tvarxrfl) Y2(trapxrfl)
(4.4) Y(t.a,x'u) y1(t,a,x,u) y2(t,a,x,u)

 

Y(0,a,x,u)

[2 i}

be the fundamental matrix of (4.3). Note that

 

Q(t) r(t)
Y(t'“'0'0) ' cut) f(t)
where q and r are given by (2.7) and (2.9). The

characteristic multipliers of the linearized equation are
the eigenvalues of Y(k,a,x,u). Therefore, the charac-

teristic multipliers satisfy

02 — A(a,x.u)o + D(a,x,u) = 0

where A(a,x,u) = tr Y(k,a,x,u), D(a,x,u) = det Y(k,a,x,u).

Lemma 4.1. D(a,x,u) = exp(—Xk).

I!

Proof: D(a,X,u) det Y(k,a,x,u)

k -
det Y(0,a,x,u) - exp I tr A(s)ds
0

exp(—kk).

 

21

Lemma 4.2. If (H1) holds, then

 

(4.5) A(a,x,u) = a1(a) + a2(a)x + a3(a)u + h.o.t.
where
_ - _ . _ 4(k)
a (a) — 2, a (a) — -k, a a — - h a , C — —7————
l 2 3( ) C ( ) b(0)2
h.o.t. = 0(|>.|2 + Iulz), as x,“ a 0

Q(k) is given by (2.7) and

k -1 k ..
n = (J0 p2 dt) , h'(a) = n [0 p(t)f(t—a)dt.

grogfz Since both yl and y2 are solutions of (4.4)
and ¢(t,a,x,u) is twice continuously differentiable with
respect to a,x,u, A(a,x,u) = y1(k,a,x,u) + y2(k,a,x,u)
is twice continuous differentiable with respect to a,x,u.
Hence A(a,x,u.) has Taylor series expansion as in (4.5)

with x,“ in a neighborhood of x = u = 0.

a1(a) = A(a,0,0) = y1(k,a,0,0) + y2(k,a,0,0)

l1

q(k) + f(k) = 1 + 1

2.

Let b1(t) = %% (t,a,0,0) then b1(t) is a solution

of the problem

ll
O

hence b1(t)

22

Let b2(t) = 5% ¢(t,a,0,0) then b2(t) satisfies
[ § + 8'(p)z = f(t-a) - h(a)p

z(a)p(a) + 2(a)§(a) = o

where h(a) is given by (2.18). Let P be the projection

operator as in (2.12), since P(f(t—a) — h(a)p) = 0, it

follows from Lemma 2.2, that b2(t) is a k—periodic
solution.

Let b3(t) = 5% y1(t,a,0,0), then b3 is a solution
of

{ 2 + g'(p)z = -q
z(O) = 2(0) = 0

The variation of constants formula implies

t t
b3(t) = q(t) I0 r(S)Q(S)ds - r(t) f0 q(S)Q(S)ds

Let b4(t) = 5% y2(t,a,0,0) then b‘ satisfies

{ § + q'(p)z = -f
2(0) = 2(0) = O
k
Since I p(—r)ds = 0, b4 is a k—periodic solution.
0
Again by the variation of constants formula

t t
b.(t) = q<t> f0 r(s)f(s)ds — f(t) f0 q(s)f(s)ds

Therefore
6A
a2(a) = 5: (aroro)
6y1 ay2
= “—3—": (kra'O'O) + “a—x (kya'O'O)

 

23

b3(k) + B.(k)

II
“—1
H
.9
Q:
m
l
“—5
.Q
’1
Q:
(D

ll
‘
O k
.0
H.
I
’1
.Q
Q;
(D

By
Let b5(t) = —§i (t,a,0,0), then b5 is a solution

of

[ § + 8'(P)Z = ~g”(p)b2q
z(0) = 2(0) = 0

Applying the variation of constants formula, we have

t t
bs(t) = q(t) forg"(p)b2qu — r(t) foqs"(p)b2qu

6y
Let b (t) = ——3 (t,a,0,0), then b satisfies
6 a“ 6

l E + g'(p)z = —g"(p)b2r
2(0) = 2(0) = 0

by the variation of constants formula

t . t
b6(t) = q(t) I0 rg"(p)b2rds - r(t) J0 qg"(p)b2rds

Therefore
8A
aa(a) — 5; (a,o,0)
6y1 6y

I!
W
R
O
O
+
O)
J.
7"
9
O
O

 

24

b5(k) + 56(k)
k k
= I0 rg"(p)b2qu + q<k) J0 rg"(p)b2rds

k
- I0 qg"(p)b2rds

k
4(k) I0 rg"(p)b2rds

 

. k
= §?étl f0 s“(p)b2b2ds

k k 0 k
c - ntg'(p)pb2 IO— fog'(p>pb2ds — Jog'(p)pbzd51
k .. k
= —c - ntfo g'(p)pb2ds + f0 g'(p)pbzds1
k '0 k k D.
= —c - nEIO g'(p)pb2ds + g(p)bz l0 - I0 g(p)b2dS]

k u k
= -c -n[I0 8‘(p)pb2ds + J0 g(p)[g'(p)b2 - fa
+ h(a)p]ds

= —c - ntf: g'(p)b2[§ + g(p)]ds

k k
_ f0 g(p)fads + f0 h(a)8(p)PdS]
k k
= —( ~ "[IO pfads — h(a) I0 ppds]

= ”C ' n[IJ( fifads]
Jo

= —c ' h'(a)

25

Since

h(a)

k
V I0 p(t)f(t-a)dt

—a
D fk p(s+a)f(s)ds

-—a

it follows that

-a
n p(k)f(k-a) — n p(0)f(-a) + [k fi<s+a)f(s)ds

h'(a) =
—a

k-a

= f p(s+a)f(s)ds
—a

= fk p(t)f(t-a)dt
o

= fk fi<t>fa(t)dt
0 Q.E.D.

Lemma 4.3. If h'(a0) = 0, then for every small

 

xo,uo that satisfy the bifurcation equation, we have that
l is a characteristic multiplier of the linearized

equation of ¢(t,a0,x0,u0).

groof: Suppose ¢(t,a,x,u) == ¢(t,a,x,u,x,y) and
0 < a < 1, where (x,y) is the initial condition of
¢(t,a,x,u), and ¢(t,a,x,u) is a solution of (4.1).

Let w1(t) = %% (t,a0,x0,u0,x0,y0), then w1(t) is

the solution of

H
O

{ i + g'(¢)x + xi
x(0) = 1, 2(0)

II
D

 

 

 

26

a
Let w2(t) = 5% (t,a0,x0,uo,x0,y0), then w2(t)

satisfies

[ i + g'(¢)x + xi = 0
X(0) = l, 2(0) = 1
Let
W1(t) W2(t)
W(t) =
w1(t) W2(t)

 

then W(t) is the fundamental matrix of
x + xx + g'(¢)x = 0
If none of the characteristic multipliers of ¢ is one,
then
det(W(k) — 1) ¢ 0
Let H(x,y.a.k,u) = (¢(k.a.X.u,x,y) - x,
¢(k.a,x.u.X.y.) - y). Since H(xo.yo.ao.xo,uo) = 0 and

3H
det(57§7§7 (xoryoraorkotuo)) = det(W(k) -1) ¢ 0 .

it follows from the Implicit function theorem that there

exists 6 > (3 and three unique solutions x*(x,y),
u*(x,y), and a*(x,y), such that H(x,y, a*(x,y).
1*(x,y), n*(x,y)) = 0 for Ix — xol < o, Iy — yol ( 5.
Therefore, for (a,x,u) near (a0,x0,u0), there is £1

unique k-periodic solution of (2.1), which contradicts the
result we obtained above which says that near (a0,x0,u0)

either there are at least two solutions or no solutions.

27

Therefore, 1 must be the characteristic multiplier of the

linearized equation. Q.E.D.

Without loss of generality, assume q(k) > 0, then

C > 0. So we have the following theorem.

 

Theorem 4 . 4 . Let x and u. be small . Let
¢(t,a,x,u) be the k—periodic solution of (2.1) from
Theorem (3.1) and loll < Iozl be its characteristic

multipliers. We have,

(I) If h'(a)u > 0, then
(i) if x > 0, then either 0 < 01 < 02 < l
(stable node) or loll = lozl < l (stable focus)
(ii) if A < 0, then either 1 < 01 < 02
(unstable node) or loll = lozl ) l (unstable focus)
(iii) if x = 0, then both characteristic multi-

pliers are complex and simple and have modulus 1.
(II) If h'(a)u < 0, then 0 < 01 < 1 < 02 (saddle)

(III) If h'(a) = 0, then the characteristic multi-

pliers are l and e_XK.

Proof: Let 01,02 be the characteristic multipliers
of the linearized equation, then by Lemma 4.1 and 4.2,

they are the solutions of the following equation.

a2 — A(a,x,u)o + exp(-kk) = 0.

 

2 _ . -

 

28

Case I. If h'(a0)u > 0, then there exists a

5(a0) > 0, such that if III,Iu| < 5(a0), we have either
-1k
2 exp(—§—) < A(a,x,u) < l + exp(—Xk) or
-Xk
.A(a,x,u) < 2 exp( —§—) for Ia - a0! < 6(a0). If
-Xk
2 exp(—§—) < A(a,x,u) < l + exp(—kk), then 01 < 02 < l
for x ) 0 and l < 01 < 02 for x < 0. If

-2k), then 01,02 are complex conjugate

 

A(anuu) < 2 WM

with modulus greater than one or less than one according to

x < 0 or x > 0. The above argument holds for la - a0! <

6(a0).

Case II. If h'(a0)u. < 0, then there exists a
5(a0) > 0 such that if lk|,|ul < 5(a0) we have A(a,x,u)

> 1 + exp(—1k) for Ia - aol < 6(a0). Then 01 < l ( 02.

Case III. If h'(a) = 0, Lemma 4.3 shows that

01,02 = l, exp(—1k).

The way to find a neighborhood V0 of x = u = 0 such
that the stability arguments hold uniformly is similar to
the proof of Theorem 3.1. It follows from Remark 3.3. that
for (a,).,u.) in the region of existence, 1 can be

written as a function of u,a, say x*(u,a). Let

INCL“) = A(arx*(aru)ru-) _ (l + exp(—X*(ar“)k))

29

ll

- c h'(a)u + 0(Iulz)

u'(-C h'(a) + 0(|#|))o

Let G(a,,u.)

- ( h'(a) + 0(Iul). For those aj's in

Hypothesis (H2) we have G(aj,u) = 0. Since
8G _ __ ,,
55(arfl) — C h (a) + 0(‘fl')

and h"(aj) # 0. Choose €(aj) > 0, such that h"(a) is
bounded away from zero for Ia — ajl < €(dj), then there

exists a 5(aj) > 0 such that if lul < 5(aj) then

6G
a; (arfl) ¢ 0!

in particular

as .
55 (ajrfl) ¢ 0-

Which shows that G(a,u) changes sign as a varies from

one side of aj to the other side of :13. That is

+

A(a,x,u) < 1 exp(-kk) ( => node) on one side of aj and
A(a,X,u) > 1 + exp(—1k) ( => saddle) on the other side of

aj. Let

C
I

[0,1] ” (a; la — ajl < 6(aj)}

IICZZ

j 1

then U is compact and h'(a) # 0 in U. Apply the same

argument as in Theorem 3.1. There exists 50 > 0 such

that the stability result holds in

v0 = {(141) ; lM,lu| < 50}.

 

30

Remark 4.5. By continuity of the eigenvalues of the
matrix Y(k,a,>.,u), we can see that when h'(a) is near
0, we have a node and when h(a) is near 0 we have a

focus.

To illustrate the theorem, consider the following

example.
Example: Suppose h(a) has one maximum and one
minimum on [0,1] which occurs at an and am

respectively. Since h(a) has period one, that identify
£1 with (L It follows from Theorem 3.1, there are two
curves ). = C* (u) , x = C: (u) which are respectively
tangent to x = h(aMm, x = Mam)”. at x = u. = 0, and
which divide a neighborhood of x = u. a 0 into two
disjoint open sets 81 and S, (see Fig. 1), such that
(2.1) has two solutions if (Lu) e 81, no solution if

(x,u) e S; and one solution if (x,u) is on either curve

 

 

 

 

 

Fig. 1 Fig. 2

 

 

31

Pix 1:. ) 0, let x vary from greater than C*(n.) to
less than Cam) then the number of solutions of (2.1)
varies from 0,1 then 2 and back to 1 then 0. Let
y - B (we used )1 = Bu. in section 2.3) be a horizontal
line in the parameter plane, as B varies from greater
than ham) to less than h(am), the number of
intersections of the line y - fl and h(a) changes again
from 0 to 1 then 2 and back to 1 then 0. We can
see how the two solutions of (2.1) change by looking at

when 3 changes from Man) to less than h(am).

To see how the characteristic multipliers of the
linearized equation move when (1 moves along h(a). Let
loll < lazl be the characteristic multipliers and label
some points on h(a), see Fig. 2. Then we obtain the
corresponding 01,02 situated near the unit circle in the

complex plane (see Fig. 3).

( (B)

 

/
\x

'01' ' I02| a l |01|-|02|< l

" K ,
\

 

 

32

(C) (

 

KN
VJ

Q

 

 

 

 

 

 

 

 

 

 

a1 < a2 < 1 a1 - e“k, .02 - 1
(E) (F) .
01 < 1 < a2 a1 = 1, a2 - e‘*k
(G) (H)
m / x
_—/
l < 01 < 02 '01|=|02| ) 1

Fig. 3

Remark 4.6. Note that the above results hold only when

Lu. are small. If My. are not small, then some

 

33

interesting phenomena might occur which will be illustrated

later by numerical experimentation.

Remark 4 . 7 . We have analyzed the stabil ity of the
solution of 5': + g(x) = —).x + afa(t) . To discuss the
stability of the solution of SE + g(x) = —).x + uf (t) ,

let's consider the following two systems

(1) i = A(t)x

(2) 3" = A(t+a)y

where A(t) is periodic with least period k. Let X(t,s)
be the matrix solution of (l) with X(0,0) = I, and let
Y(t) = X(t+a,0). Then Y(t) is a.nmtrix solution of (2)
with Y(0) := X(a,0). Define Z(t) == Y(t)X(0,a). Then
Z(t) is a matrix solution of (2) with 2(0) = I.

First, we claim that X(k+a,k) = X(a,0). Let W(t,s) =
X(t+k,s+k). Then W(0,0) = I and a; = A(t+k)W = A(t)W.
Therefore W(t,s) = X(t,s) for any t,s, in particular

X(a+k,k) = X(a,0).

Since Z(T) Y(k)X(0,a)

X(k+a,0)X(0,a)

X(k+a,k)X(k,0)X(0,a)

X(a,0)X(k,O)X(0,a)

34

and X(a,0) = X(0,a)’1, it follows that Z(k) is similar
to X(k,0). In particular they have the same eigenvalues.

Thus the stability result of (1) can be applied to (2).

SECTION 5. UNIFORMITY

By Theorem 3.1, for each integer k > 1, we obtain a
neighborhood Vk of x = n = 0 such that if (x,u) e Vk,

there exists at least 2k k-periodic solutions of

(5.1) i + g(x) = -xi + “f(t)
where g(x) is defined as before and f(t) has least
period 1. It is interesting to know whether there exists

a neighborhood \lg; n Vk such that the existence theorem
holds. In other words, is there a neighborhood \I such
that if (Lu) 6 V, there exists k—periodic solution of
(5.1) for every k? If the operator G as defined in
Lemma 2.3 is uniformly bounded for every k, then such
neighborhood exists. We will show that sometimes this is
true (Theorem 5.4) and in some other cases, it is not true
(Theorem 5.1). To discuss this, we first consider the

following equation

(5.2) i =

X
I
5’
X
IV
F"

35

 

36

Let ék be the operator as in Lemma 2.2 corresponds to

pk(t), where pk(t) is the k-periodic solution of (5.2).

Theorem 5.1. The Green's function Gk for equation

(5.2) satisfies

3 k _
(5.3) lakfl _ cosh(t) 1

for all k = l,2,°°°

Proof: Let ¢(t) be any k-periodic function, it

follows from Lemma 2.2 that ék¢ is the solution of

SE-x=¢(t)
x is k-periodic
x(0) = 0

Since SE — x = 0 has cosh t and sinh t as linearly

independent solutions, we obtain (see (2.7) and (2.9)),

 

sinh t 0 s t 5 ¥

"-' -k k4 43}.

r(t) - 51nh(t 3) : — t _ 4

i sinh(t - k) 3% e t s k

and

( cosh t 0 5 t 5 g
— . _ _ k _ - _ k k ‘ ‘ 3k
q(t) — cosh(t ;) c1 Sinh(t 3) I _ t _ ’7
L cosh(t - k)+ c sinh(t — k) 35 e t 5 k

 

 

37

 

where
k
2 cosh I 4 cosh 7
Ci: .T' c2=—.—_F
Slnh T Sinh I
Take ¢ = l and 0 £ t 5 ¥. Then
15
G¢(t) =—c‘_’—:h—t (I‘ cosh ds
2 0
3k

‘7 k . k
+ I E ( cosh(s 3) c1 51nh(s :))ds
4

k
+ I15 (cosh(s - k) + c2 sinh(s - k))ds)
4

t t
— cosh t I (sinh 5 ds + sinh t I cosh 8 ds)
0 0

cosh t

‘C

. k . k . k k
2 (Sinh T 0 — Slnh(:) + 51nh( 4) c1 cosh(z)

+ c cosh(—E) + 0 — sinh(—E) + c — c cosh E)
1 4 4 2 4

2
— cosh t(cosh t — 1) + sinh t:sinh t

= —cosh t(l — cosh é) + cosh t — l

= cosh t cosh ¥ — 1

Set t = 0, then n5k¢g é cosh I — 1, therefore

 

 

 

38

~ 3 k—
'Gk“ _ cosh T 1

Next, it follows from Lemma 2.3 that

N

Gk¢ = Gk¢ + ak(¢)bk

where

k — ” k 2 —1
ak(¢) = —n f0 pka¢ dt and n = ([0 pk dt)

Set t = 0, we obtain Gk¢(0) = gk¢(0) + ak(¢)Pk(0) =

Gk¢(0), therefore Gk satisfies (5.3). Q.E.D.

On the otherhand, consider the following equation

H —x x 5 l
5.4) x =
( x - 2 x a 1
For equation (5.4), the equilibrium point (0,0) is a

saddle and the other equilibrium point (2,0) is a center.
Also, the global stable and unstable manifolds of (0,0)
coincide. That is equation (5.4) has a homoclinic orbit

which crosses the x-axis at (0,0) and (4,0).

Let pk(t) be the k-periodic solution of (5.4) with
Pk(0) = bk and pk(k/2) = ck. Since equation (5.4) admits

the first integral

 

39

.2 x
E = g x + I g(s)ds
= g i2 - g x2 if x é l
g x2 + g x2 — 2x if x a 1

Therefore the period of the periodic orbit pk(t) is given

by the formula

Ck
dx + I dx )
k VX - bi 1 V—(x — 2)2 + 4 — bi

 

1
T(bk) = 2(I
b

where ck = 2 + «4 — bi , therefore

 

(5.5) T(bk) — 2 ln b k + n — 2 sin‘l(———:l———)
k «4 — bi
= 21k + 20k
= k
Lemma 5.2. a = q(k) a 2, as k a w.
Proof: It follows from Proposition 2.1 that q(k) =
T'(bk)°g(bk). Hence
-2b 4b
- k 2 k
010:) = ( — — ————)(‘bk)

5‘
«1 — bi (1 + «1 - bi) k «4 — bi «3 — bi

2 2 4
k( + ) + 2 .
«1 - bi (1 + «1 — bi) «4 — bi «3 — bi

= b

 

40

As bk —. 0, out) _. 2. Q.E.D.

Note that pk(t) is given precisely by the following

 

 

(5-6)
bk cosh t 0 5 t é T
p (t) = « sinh T cos(t — E) + 2 T e t e k — T = T + 20
k cosh T sin a“ 2
bk cosh(t — k) k — T s t 5 k
where, bk cosh T = 1.
Lemma 5.3. qub is uniformly bounded in k, where
¢ 6 5 = {¢° is a 'k-periodic characteristic function and is

symmetric with respect to k/2}.

Proof: For simplicity, we will drop the subscript k.
First choose k large enough, such that 3fl/4 ( a < 5n/6,
where o is given by (5.5). Let r(t), q(t) be the

solutions of the linearlized equation (see also (2.7) and

(«Z-9)):
x — x = 0 0 e t s T
x + x = 0 T 5 t s k — T
x - x = 0 k — T 5 t s k

It follows from (5.6) and p(t) = br(t) that

41

sinh t 0 s t s T
(5.7) r(t) = F(t) T e t s k - T
sinh(t - k) k — T e t s k
_ sinh T . _ k
where F(t) — :Eifi—E 51n(t ;) and
cosh t 0 s t e T
(5.8) q(t) = mm 1 s t s k - T

cosh(t — k) + a sin h(t - k) k - T 5 t 6 k
where a = q(k) and

2 cosh T — a sinh T k a sinh T
= cos(t — —) — ———4————
2 cos 0 2 Sin 0

Q(t)

V

NIW

 

sin(t —

It follows from Lemma 2.2, that

 

' q(t) t
(5.9) G¢(t) = a 0 q(s)¢(s)ds - q(t) f r(s)¢<s)ds
0
t
+ r(t) f0 q(s)¢(s)ds

For ¢ 6 P, there exist constants 81, 82, 33 such that

1 31 5 t e 32 , k - 32 s t s k — Bl
¢(t) = 1 Ba s t s k - 33
0 otherwise

Therefore we only have to consider the following forms of

¢

 

1
(5.10) Mt) =
0 otherwise

where 31 < 32 é T, and

l B s t s k - 33
(5.11) Mt)

0 otherwise

where T e 33 s

NIX

Substitute (5.7), (5.8) and (5.10) into (5.9), we

obtain the following,

for 0 é t e 31
~ 2
G¢(t) = Eflgg—E [f8 cosh 5 ds

31

+ [”1

(cosh(s - k) + a sinh(s - k))ds
k—Bz

_ cosh t . _ _ -
— ——:5—— [(2 Sinh 32) a cosh 32 2 s1nh 31

+ cosh Bl]

_ cosh t

— "‘iE“ [(2 - sue"2 - (a + 2)e”32 + (a — 2)eBl

+ (a + 2)e“31)

It follows from Lemma (5.2) that a — 2 = 0(e’27) and

Hence, there exists M1 > 0

Id}(t)| 5 M1 for 0 e t s 31. Next, for 31 s t 2 32,

t e 31 < 32 s T. such that

we
have

43

G¢(t) = 2222.2 [2 sinh 32 - a cosh 32 — 2 sinh 31

+ a cosh Bl] - cosh t(cosh t — cosh £1)

+ sinh t(sinh t - sinh £1)

:5 [(2 — a)et+32 + (a - 2)et+B1 - (2 + a)et'32
+ (2 — a)e32—t
+ (a ‘ 2)efll_t - (a + 2)e’(t+32)
+ (a + 2)e’(t+81)]

+ (—g + net-B1 + Zeal—t
-a

Since a — 2 = 0(e‘27) and Bl é t 5 82 5 T, there exists

M2 > 0, such that Iéh(t)| 2 M2 for 51 s t s 32.

Consider now the interval £2 5 t 5 k - 32. We have

c¢(t) Q(t) [:§(sinh 32 — sinh 31) + (cosh 32 — cosh 31)]
- Q(t)(cosh 32 - cosh Bl)

+ F(t)(sinh £2 — sinh Bl)

 

 

cos(t - E)
= [(2 cosh T - a sinh T) —
-a cos a
sin(t - E)
— (2 sinh T — a sinh T) . ]
Sln o

(sinh £2 - sinh 81)

since a — 2 = 0(e’27), 377/4 5 o s 577/5 and —o ‘ t -
k/2 é. 0, there exists M3 at. 0, such that lG¢(t)l 5 M3

for B; 5 t é k — 82.

44

For k - 32 s t 5 k — 31 and k - 31 5 t s k, the
computations are similar to the first two cases by
replacing cosh t by cosh(t - k) + a sinh(t - k) and
sinh t by sinh(t — k). We obtain for some M4 a 0

IG¢(t)l e M‘ , k — 32 e t e k .

One can see that the constants M1, M, M3 and M‘ can

be chosen independent of k.

Now, repeat the above procedure for ¢ defined by

(5.11), for 0 s t 5 Ba and B3 a T, we have

.. k—Ba
G¢(t) = flégl (f3 Q(S)d6)
3

 

 

 

= cosh t(2 cos; T — a Sinh T(_2 cos 0)) t é T
- a cos a
Q(t) (2 cosh T ; a sinh T) T 5 t 5 33
Again, since a — 2 = 0(e’27), there exists N1 ) 0, such
that
|é¢(t)l s Nl , for 0 s t s 33
For 83 5 t 5 k - 83 , we obtain
” 2 cosh T — a sinh T t
G¢<t> = Q(t)(- a ) — Q(t) f F(s)ds
33

t
+ F(t) I Q(s)ds
33

~..----——-—- -

= Q(t)(

45

2 cosh T - a sinh T
a )

+

(2 cosh T — a sinh T)
2 cos a

(EEEE—I) {-1 + cos(t — §)cos()33 — g)

sin a

+ sin(t — %)sin(83 — §)]

Since a — 2 = 0(e‘27), —o 5 t - k/2 s o and 3n/4 é o 5
5n/6, there exists N2 > 0, such that |G¢(t)| é N2 for
33 s t s k — Ba. For k - B3 6 t s k, by the similar
arguments, we can choose N3 2 0 which is independent of
k such that G¢(t) is bounded by N3.
It follows that there exists M a 0 independent of k
such that
max Ié¢(t)| s M
0sték
Since
-_ . l k . t
G¢(t) = 01¢an q(s)¢(s)ds1 — q(t) I r(s)¢(s)ds
0 0
. t
+ r(t)] q(s)¢(s)ds ,
0
one can see by similar arguments that there exists N a 0
which is independent of k such that
max Ié$(t)l 2 N

Oéték

46

Since
54 = —g'(p)e¢ + ¢
we have
max léil é max ig’(p)l max [£81 + l
Oéték Oéték Osték
5 JM + l
where J = 0mgxktg'(p)l. Since p(t) is uniformly bounded,

therefore J can be chosen independent of k.

We have shown that

u
N

flé¢fl é max(N,M,jM + 1)

Theorem 5.4. The operator Gk is uniformly bounded in

 

PS, where Gk is defined by Lemma 2.3 and P3 = {¢ is a
continuous k-periodic function and is symmetric with

respect to k/2}.

Proof: We first show that Gk is uniformly
bounded in PS. For simplicity, we will drop the subscript

k. For any ¢ 6 PS, there exist ¢i 61;, where P is

defined by Lemma 5.3, such that

a ¢.(x) » ¢(x) uniformly in [0,k]
l 1 1

IIPJZ

i

47

Let E1 = {x : ¢i(x) = 1}. Then ¢i(x) = X31: the
characteristic function of E1. Let x 6 Bi be fixed. We

have

C¢(x) — C

i ai¢i(X) = G¢(x) - aiG¢i(x)

1

"P12

It follows from Lemma 5.3, that

ue¢fl 2 Koi¢u .

where K0 is independent of k.
For any ¢ 6 PS, since 45 is symmetric with respect

to k/2, P¢ = 0. It follows from Lemma 2.3, that

G¢ = C¢ + ap ,

such that P(C¢ + ap) 0. Therefore

a(¢) -P(é¢)

-nb fk b(t)e¢(t)dt
0
k :2 . - .

where n = ([0 p dt)’1. Since, p,p,p are uniformly bounded
and is near 0 for very long time, hence n is uniformly
bounded too. Similarly

k c
I lp(t)ldt ,

0

is bounded. Therefore

 

 

48

ua(¢)n s Kllé¢l

lk

K1K0fl¢fl

hence a(¢) is a uniformly bounded operator. This shows

that G is a uniformly bounded operator. Q.E.D.

 

 

SECTION 6. NUMERICAL STUDY

In this section, we will give a numerical scheme to
find the periodic solutions of (2.1). Let ¢(t,x,y) be
the solution of (2.1) with initial condition (x,y) at
t = 0. In order for ¢(t,x,y) to be a k—periodic
solution of (2.1), the following equations have to be

satisfied.

¢(k,x,y) - x
$(k7x,y) - y .
Let
¢(k.x.y)
(6.1) F(xry) = ,
¢(k,x.y)

The problem is reduced to finding the fixed points of

F(x,y). We apply the Newton's method to find the zeros of

x
(6.2) G(x,y) = F(x,y) ‘ [ y 1

We obtain the following scheme:

49

50

 

 

 

xn+1 n _1
(6.3) = - DG(xn.yn) G(xn.yn)
yn+1 yn
X
n
_ _ _ '1
— (DF(xn.yn) I) G(anyn)
yn
where
a¢ a¢
‘6'; (kIXIY) 'a—y (kIXIY)
(6.4) DF(x,y) = . . .
a¢ a¢
5i (kIXIY) E'y" (kIXIY)

and a¢(t,x,y)/ax, a¢(t,x,y)/ay satisfy respectively

i + g'(¢)x = 0
(6-5)

x(0) = 1, x(0) = 1
and

i + g'(¢)x = 0
(6.6) .

x(0) = 0, x(0) = 1

Let

 

51

 

 

 

 

' 21(t) ‘ ' ¢(t.x.y)
22(t> $(t.x.y)
23(t) g; (t.x,y)
z(t) = z‘<t) g g; (t.x,y) '
25(t) g;- (t.x,y)
. 26<t) J . g; (t.x,y) J
then it follows from (6.3), (6.4), (6.5) that
(6.7)
* 22(t) xo ‘
-g(zl(t)) - xé2(t) + uf<t> Yo
z‘(t) l
i(t) = with 2(0) =
~g'(zl(t))23(t) — xz4(t) 0
25(t) 0
_ “8'(21(t))25(t) - x26(t) '1

 

 

Now we use Runge-Kutta method to solve

We obtain

-l

value problem.

(6-8)

X
n+1 ]

Y

X

n 23(k) - l

25(k)

 

 

 

Y Z‘(k) 25(k) - 1

n+1 n

 

 

the above initial

1

H

21(k) — xn

 

22(k) - Yn

52

Note that if ¢(t,x,$') is the k-periodic solution of
(2.1), then the characteristic multipliers of ¢ are the
eigenvalues of the Jacobian matrix DF(x,y). Hence, if
both eigenvalues of DF(x,y) have modulus less than 1,
then the fixed point (x,y) or the periodic solution
through (x,y) is stable and if one of the eigenvalues has
modulus greater than 1 then it becomes unstable.
Therefore, when we compute the fixed points, we determine
the characterist multipliers simultaneously.

Our experiments will be performed on the forced
pendulum problem. Let pk(t) be the k—periodic solution

of
(6.9) x + sin x = 0

Consider the perturbed system:
(6.10) x + sin x = —xx + uf(t)

where f(t + l) = f(t). Since the initial conditions of a
k-subharmonic solution of (6.10) are near (p(a),p(a)) and
x ~ h(a)u for small A and u (see Section 2), we can
choose our initial guess to be (p(a),p(a)) with h(a) = 0
and x = 0.

In Figures 6.1 and 6.2, we show the Poincare map under
iterations for (6.10) with f(t) = sin 2nt, x = 0, u = .2.
Note that in Figure 6.2, we easily observe the subharmonic

motions. Figure 6.3 is the magnification of the square box

53

in Figure 6.2. We observe that there are saddle
connections and the stable periodic solutions are enclosed
by invariant tori.

For fixed x = 0 and f(t) = sin 2nt, we increase u
and follow numerically the subharmonics of order 9. We do
not observe bifurcations of these solutions. See Table
6.1.

In order for us to observe bifurcation phenomena for

these solutions, we let f(t) cos(.67t) and look for

subharmonic solution of order 1. For this f(t) =

cos( .67t) , we have maxih(a) I .61681. It follows from
Theorem 3 . l, we can only have per iodic solutions if k
lies between - . 61681 and . 61681. Table 6 .2 shows how

the characteristic multiplies of the periodic solution vary

as x goes from —.61681 to .61681 and u = .2. In
Figure 6.4, we fixed u. = .2 and plot x against r,
where r = «x02+y;§ and (x0 ,y0) is the initial

condition of the periodic solution.

From now on we would like to consider the Poincare map
F(x,y) defined in (6.1). First, we will show how the
stable fixed point of F loses its stability through
period doubling. Again we fixed 7. = 0 and vary u. At
each stage, we apply (6.7) and (6.8) to obtain the fixed
point for the next u by using the present fixed point as
the initial guess. We continue this process - until one of
the characteristic multipliers of the fixed point passed

through —1. Let's assume it occurs at u = no, and the

54

fixed point is (XuOrYuo)~ Let (xwyu) be the fixed
point of F(x,y) for a slightly larger than no, then
(xfl,yu) is a unstable fixed point of F2(x,y). There are
also two stable fixed points of F2(x,y) which are not the
fixed point of F(x,y).

We find numerically the unstable fixed points (xu,yu)
by our scheme (6.7), (6.8). To find the stable fixed
points, we first locate numerically the saddle connections

and choose an initial guess in the interior of the

connections. 'The :method allows us to continue and to
detect the bifurcat ion po ints . In F igure 6 . 5 we f ixed
x = 0 and increase u. , one can see how the per iodic

solution changes from one loop to two loops then 4 loops in

the sinx x—plane. This shows the period of the solution
doubles and. doubles again» IFigure 6.6 shows the same
pheonomena with graph u against r, where r = «x02+yo2

and (xo,yo) is the initial data-

Now we start with the fixed point (XuOrYuo) and
increase I. t0v obtain the period—doubling curve, see
Figure 6.7. That is if (x,u) goes through the curve with
increasing It then the stable periodic solution loses its
stability and another stable periodic solution is created,

with least period 2 times the original one.

\oooxlmmpwml-To

HHHHI—‘i—‘HH
\Immwar—‘O

2.18106292
2.18093417
2.18036890
2.17940921
2.17805323
2.17629843
2.17414159
2.17157875
2.16860526
2.16521572
2.16140393
2.15603471
2.15248489
2.14736114
2.14178208
2.13573716
2.12921478
2.12220229

55

-.28080103
-.35920429
—.51601111
".67282215
-.82964198
~.98647512

-l.l4332606
-l.30019926
-l.457099l4
-l.61403012
-l.77099662
-l.96726l43
-2.08505389
—2.24215356
-2.39930658
-2.55651749
-2.7l379091
-2.87113151

TABLE 6.1

01'

.99483581
.98968713
.97947406
.96943533
.95964335
.95016770
.94107458
.93242616
.92428013
.91668919
.90970662
.90187488
.89769104
.89273472
.88850988
.88503265
.88231246
.88035202

l-i-l-bl-i-H-H-HH-H-HH-Hl-l-HHH-H-HH

02

.101497251
.l4324583i
.201570201
.24534694i
.281219901
.31173919i
.338199091
.36136054i
.381714851
.39960094i
.415264611
.431997321
.44062544i
.45058263i
.4588574Si
.465528941
.470664121
.474320891

 

 

56

4 x0 Yo 01v 02
.12336274 .28465940 1.77898547 .99789666, .31512796
.1233624 .27953712 1.78025094 .98806257, .31802605
.123361 .27177950 11.78213926 .97491078, .32256313
.123330 .22472892 1.79286449 .88629559, .35491738
.12320 .14854213 1.80758616 .70526800, .44656136
.12315 .12855802 1.81090718 .62271744, .50599698
.12314 .12484698 1.81149919 .59330936, .53112713
.12313 .12121602 1.81207094 .56012761 i .037540781
.123 .07952532 1.81810551 .53627383 1 .167173761
.12 -.36144268 1.82265190 .29791170 1 .485580651
.11 -1.01698498 1.63434569 -.02793731 1 .5963784li
.10 -1.40960328 1.41440422 -.21776604 i .586575201
.05 —2.30221653 .56463577 -.69586196 1 .376124661
.0 -2.50801521 -.00000000 -.95303355 i .30286473i

-.05 —2.30221653 -.56463577 -l.11214486 i .601132301
—.10 -1.40960328 -1.4l440422 -.55624572 i 1.498304991
-.11 -1.01698498 -1.63434569 -.07837704 i 1.673116151
-.115 -.74595905 -l.73982553 .30663149 i 1.687043851
-.12 -.36144268 “1.82265190 .91795078 1 1.496212231
—.123 .07952532 -1.81810550 1.69956093 t .529807621
-.1231 .11075522 -l.81367677 1.75775532 i .287181371
—.1232 .14854215 -1.80758616 2.39334066, 1.41790061

-.1233 .20103788 -l.79779243 2.66166160, 1.19404065

.123361 .27177964 -l.78213923 3.10016907, 1.02573461
.1233626 .28169078 -l.77972067 3.15658155, 1.00741843

TABLE 6.2

 

 

 

 

 

 

 

 

 

 

 

 

57

 

 

- . .
......

 

 

Figure 6.1

58

 

 

 

 

 

 

 

 

 

 

 

Figure 6.2,6.3

 

 

(7
U
W

 

Figure 6.4—a

 

60

J”

sinx

 

sinx

Rfl (\i

 

Figure 6.4—b

 

 

 

 

Figure 6.4ec

62

 

sinx

 

x

Figure 6.4-d

 

 

63

 

 

Figure 6.5

3

 

 

 

Figure 6.6

64

 

 

.25

Figure 6.7

.25

LI ST OF REFERENCES

 

[l]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

LIST OF REFERENCES

S.N. Chow, J.K. Hale, and J. Mallet—Paret, An

example of Bifurcation to Homoclinic orbits,
J. Diff. Equ., Vol. 37, No. 3, pp. 351-373,
September, 1980.

S.N. Chow and J.A. Sanders, On the number of

critical points of the period, J. Diff. Equ.,
to appear.

S.N. Chow and J.K. Hale, Methods of bifurcation

theory, Springer-Verlag, New York, 251.

M.J. Feigenbaum, Universal Behavior in Nonlinear

Systems, L.A. Science, pp. 3-27, Summer, 1980.

J.K. Hale and P. Toboas, Interaction of damping

and forcing in a second order equation, J. Non—
linear Analysis 2 (1978), pp. 77-84.

B.A. Huberman and J.P. Crutchfield, Chaotic states

of Anharmonic Systems in Periodic Fields, Physical
Review Lett., Vol. 43, No. 23, pp. 1743-1747,
(3) 1979.

B.A. Huberman, J.P. Crutchfield, and N.H. Packard,

Noise Phenomena on Josephson Junctions, Appl.
Phys. Lett. 37(8), pp. 750-752, October, 1980.

D‘Humieres, M.R. Beasley, B.A. Huberman, and A.
Libchaber, Chaotic states and notes to chaos in
the forced pendulum, Physical Review A, v.25,
No. 6, pp. 3483-3496, December, 1982.

Ito, Successive Subharmonic Bifurcations and
Chaos in a Nonlinear Mathieu Equation, Progress of
Theoretical Physics, Vol. 61, No. 3, pp. 815—824,
March, 1979.

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