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VIB‘E'SYS

 
     

  

LIBRARY
Michigan state '

University

w t w
, ‘—

This is to certify that the

dissertation entitled

INVARIANT CURVES FOR NUMERlCAL METHODS
AND THE HOPF BIFURCATION

presented by

HAI THANH DOAN

has been accepted towards fulfillment
of the requirements for

PH.D. degreein MATHEMATICS

 

 

SHUI N . CHOW

Major professor

 

Date 10’12’82

 

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

MSU

LIBRARIES
a

 

 

 

RETURNING MATERIALS:
Place in book drop to
remove this checkout from
your record. FINES will
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returned after the date
stamped below.

 

 

 

 

 

 

INVARIANT CURVES FOR NUMERICAL METHODS
AND THE HOPF BIFURCATION

BY

Hai Thanh Doan

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1982

ABSTRACT

INVARIANT CURVES FOR NUMERICAL
METHODS AND THE HOPF BIFURCATION

BY

Hai Thanh Doan

We consider the problem of tracking a family of

periodic orbits using numerical methods.

First it is shown that if an one parameter family
i = f(o,x) has a family of periodic orbits bifurcating
from a stationary solution then when i = f(d,x) is
approximated by a convergent single—step method, the
resulting difference equation possesses a family of
invariant curves bifurcating from the same stationary
solution. The result is then extended to convergent,
strongly stable, linear multistep methods. The results
also show that the rates of convergence toward the
invariant curves are roughly the same for all these

methods. Finally, we discuss the time delayed equations.

 

To
My Parents
and

anh Son, Long, Van, Lam

 

ACKNOWLEDGEMENTS

The writer is sincerely grateful to Professor
S.N. Chow for his patient counsel and guidance in the
preparation of this thesis. He wishes to offer special
thanks to Professors David Yen, Sheldon Axler and
William Sledd for their teachings and encouragements
during his career as a graduate student. He is also
deeply appreciative of the friendship and moral support
given him by Dr. Le Tien Thu, Mr. and Mrs. Trinh Van Trac,
Mr. and Mrs. Tran Tan Tanh, Mr. and Mrs. Nguyen van Thanh,
Dr. and Mrs. Le Thanh Minh Chau and their families.
Finally, he also likes to thank Ms. Cindy Smith and

Ms. Tammy Hatfield for typing this manuscript.

TABLE OF CONTENTS

CHAPTER
INTRODUCTION . . . . . . . . .
1. SINGLE STEP METHODS . . . . .

1.1 Preliminaries . .
1.2 Hopf Bifurcation in RP .

1.3 Existence and Continuation

LINEAR MULTISTEP METHODS .
2.1 Preliminaries . . . . . .
2.2 The Hopf Bifurcation . .

2.3 Existence and Continuation
TIME DELAYED EQUATIONS . . . .
3.1 The Hopf Bifurcation . . .

3.2 Existence and Continuation

BIBLIOGRAPHY . . . . . . . . .

iv

PAGE

49
49
50
58

85
85

104

INTRODUCTION

Many physical and biological systems are governed by
differential equations depending on a parameter. The

state variable x satisfies an equation of the form
(*) >'<<t) = f(a,x(t))

where a e Ii, x 6 some Banach space, usually I51.

Of particular interest to us is the case when (*) possesses
a family of periodic orbits bifurcating from some stationary
solution. The global behavior of these orbits have been
studied by several authors (see [1], [3], [12]). Generally
speaking, since f is nonlinear it is impossible to compute
such behavior explicit. Often one has to resort to some
numerical procedure. Typically (*) is then approximated
by some difference equation of the form

(*H §M1=Eme

It is reasonable to expect that under favorable circumstances
the solutions of (**) stay close to the periodic orbits of

(*).

In this present work we will show that under
appropriate conditions, (**) possesses a family of

invariant curves.

In chapter 1 we consider the case when (**) arises
from some convergent, single—step method. First we show
that if x Eimz and (*) has a Hopf bifurcation of
periodic orbits at do then generically, (**) has a
Hopf bifurcation of invariant curves at d04-O(h). The
case x e IUI, n.2 3 follows from the center manifold
theorem. We then establish the existence and continuation
of invariant curves for (**) away from the bifurcation
point. This means that (**) has a family of invariant
curves which is at least piecewise continuous in d. These

curves may be thought of as the approximation to the family

of periodic orbits of (*).

In chapter 2 we extend the above results to convergent,

strongly stable, multistep methods.

Finally, in chapter 3 we concentrate on time-delayed
equations. Some numerical implications will also be

discussed.

l. SINGLE—STEP METHODS

1.1 Preliminaries

In this section we shall provide some basic definitions
that will be needed throughout the chapter. Mainly we will

be concerned with a system of O.D.E.

(1.1) S: = f(x): x 6 R“,

f is sufficiently smooth,

together with a finite approximation arising from some
explicit, single—step method. This yields a difference

equation of the form

(1.2) x = xm4-h¢(h,xm)

m+l
where h is the step size of the method employed.

Definition 1.1.1: (1.2) is said to be a convergent
approximation of (1.1) if the solution of [xn} of (1.2)

with starting point xO satisfies

lim xn = x(t)
hao
nhat

where x(t) is the theoretical solution of the initial

value problem

 

{cm = f(x(t)); x(0) = x0

Definition 1.1.2: (1.2) is said to be a consistent

 

approximation of (1.1) if lim ¢(h,x) = f(x).
h40
The following theorem is standard and can be found,
for example, in [9].
Theorem 1.1.3: (1.2) is convergent iff it is
consistent.
In this chapter we are primarily interested in obtaining

the invariant curves for (1.2). Thus we need the following.

Definition 1.1.4: A curve T c If} is said to be

 

invariant under (1.2) if T(T) = T where T is the mapping

from JRn to JRn defined by (1.2).

Definition 1.1.5: An invariant curve T of T is
said to be attracting under T if Tk(x) spiral toward
T as k 4 m for all x sufficiently close to F.

1.2 fippf Bifurcation in RP

 

Consider an one—parameter family of O.D.E.
(2.1) x(t)=f(d,x(t)); GEJR; xelR; n22.

Suppose that f(d,0) = O for all a near 0 and that

f(d,x) is as smooth as needed in a and x. Let

A(o) = af/ax(a,0) and g(d,x) = f(a,x)-A(d)x

Assume further that A(d) has a pair of complex conjugate
eigenvalues x(a) and x(a) satisfying the Hopf bifurcation

conditions

(2.2) Re x’(0) > 0; Re x(o) = 0; Im x(o) ;! 0

while all other eigenvalues of A(d) have negative real
parts and stay uniformly away from the imaginary axis for all
a near 0. It was shown by Hopf ([10]) that there exists

a family of periodic orbits emanating from the zero solution

as d varies across 0.

Now suppose we approximate (2.1) by the Euler method.

We then obtain a difference equation of the form

(2.3) xm+l = xma-hf(d,xm)
= xm+ hA(d)xm+ hg(d,xm)
Assume temporarily that x E R2, then A(d) is a

2 x2 matrix with eigenvalues x(a) and 1(a), then there

exists a family of nonsingular matrices P(a) such that

-1 Re 1(G)-Im X(G) def
P(d) A(d)P(d) = = B(d)
Im Ma) Re Ma)

As usual we may assume that P(d) is as smooth in a as

needed.

Set x = P(a)y and xm = P(d)ym, then (2.1) becomes

(2.4) §z= B(d)y+gl(c1,y)
and (2.3) becomes
(2.5) ym+l = ymd-hB(a)ym+-hgl(a,ym)

where gl(a,y) = P(a)_lg(a,P(d)y) is as smooth in a and
y as needed. Also gl(a,y) = O([y|2), i.e., gl(a,y) has

a second order zero in y at y = 0.

Following Lanford [11], we now identify 2R2 with
the complex plane by writing 2 = ul-l-iu2 to denote the

vector [ul,u2]T in It denotes transposition).

Equation (2.4) then becomes
(2.4’) é: x(a)z+gz(a,z)

where g2(a,z) = O([z[2).

Similarly (2.5) becomes

(2.5') 2 = (14-hx(a))zm4-hgz(a,z)

m+l
We now bring (2.4’) and (2.5') to their normal form.

Lemma 1.2.1: Let ¢a be the family of mappings from
C to C defined by (2.5’). Then for h > O sufficiently
small there exists an d—dependent change of coordinates of

the form

z = w+y(h,a,w) where Y(h,C(,W) = O(lw|2)
such that in the new coordinates ¢c(w) = wl where
l

(2.6) w = (l+h).(cx))w-ha(h,a)|w|2w+hg3(h,d,w)

where g3(h,d,w) = O([w[5).

Similarly there exists a change of coordinates of the form
~ ~ 2
z = w+y(d,w) where y(c1,w) = O(|w| )

so that (2.4') becomes

(2.7) w = x(a)w-a(o)|w|2w+g4(d,w)

where 5(a) = 1im a(h,d) and g3(h,c1,w) converges uniformly
haO

to g4(d,w) on a neighborhood of (a,w) = (0,0) as h 4 0.

Proof. We rewrite (2.4’) and (2.5’) as

(2.4") é= x(a)z+A2(c,z)+g5(a,z)

(2.5”) zl = [1+h),(c1)]z+hA2(d,z) +hg5(d,z)

where A2(d,Z) is homogeneous of degree two.
A (or z) = a (d)22+a (d)z§+a ((3)32
2 ’ O 1 2

and 95(d,z) = O(|zl3).

8

For (2.5”) we consider a change of coordinates of the form
2 = w+y(h.a.W)

where y(h,a,w) = yo(h,c1)w2+yl(h,d)WW+y2(h,a)w2. Substitute

in (2.5”) we obtain

(2.8) wl+y(wl) = [1+h).(c)JIW+Y(W)]

+ hA2[w+y(w)] +h0(|w|3)

where for convenience we have suppressed or and h.

Since wl = [1+h).(a)]w+h0([w]2) we have
Y(wl) = Y[(l+hx(c))W] +h0(|w|2)

Also A2[w+y(w)] = A2(w)+0(|w|3). Thus (2.5”) yields

(2.9) wl= [l+h).(c1)]w+[l+h).(a)]y(w)+hA2(w)

-Y[(l+h).(a))W]+hO(|Wl3)
We now choose y so that
(2.10) [l+h).(d)]y(w)+hA2(w) —y[(l+h).(a))w] '=' 0
This can be done by choosing Y0’ Y1’ Y2 so that

(2.11a) [l+h).(d)]yo(h,or)+haO(h,d)

- [1+hx<a)12yo<h,a) = o

(2.11b) [l-thx(a)]yl(h,a)-+hal(h,a)

- [l-thk(d)][l-thx(d)]vl(h.a) = O

(2.11c) [14-hx(d)]Y2(h,d)+-ha h,d)

2(

— [l+h>.(a)]2Y2(h,cx) = 0
or

(2.12a) yo(h.a) = a0(h.a)/[x(a)(14-hi(a))]
(2.12b) Yl(h.a) = al(h.a)/[>.(a)(l+h(a))]

(2.12c) y2(h,a) = a (h.a)/[2).'_(a_)+h).(a)2—>.(d)]

2

We note that since Im x(O) # 0 the right hand side of
(2.12) are well defined for all a near 0 and h small.
(2.5”) then becomes

(2.13) w = [1-thx(d)]w-thg6(h,d,w)

1

where g6(h,d,w) = 0(lwl3).

From (2.12) we have

~ d ~
1im Yo(d,h) = ao(d)/X(X) 3f YO(G)

hao
~ ————- def ~
1im Y (o,h) = a (a)/x(a) = y (a)
h40 l l l
. ~ ____ def ~
11m y2(d,h) = a2(a)/[21(G)-k(0)] = Y2(0)

hao

lO

Rewrite (2.5”) as
(21 -2)/h = ).(c)Z+92(a.z)

and by taking the limits as h a 0, it is easy to see that

under the change of coordinates

z = w+-§O(d)w24-§l(d)ww+-§2(d)wz
(2.4”) becomes
(2.14) 4:: x(ct)w+g7(d,w)

where g6(h,d,w) converges uniformly to g7(d,w) on some

neighborhood of (a,w) = (0,0) as h 4 0.

Assuming that we have made such changes and rewrite

w by 2 so that (2.13) and (2.14) become

(2.15) zl= [1+h>.(a)12+h96(h,a.z)

(2.16) 2: ).(a)z+g7(a.z)

We now try to eliminate all terms of degree 3 in (2.15)
and (2.16). First we rewrite them as

(2.15’) z = [1+h)((a)]z+hA

l h,d,z)-+hg8(h,a,z)

3(

(2.16') 2 = x(a)2+5s3(a,2)+99(a.z)

11

where

_ 3 2— “—2 -3

A3(h,d,z) — ao(h,d)z -tal(h,d)z z-ta2(h,d)zz -+a3(h,d)z
~ _ ~ 3 " 2— ~ “—2 " -—3
A(d,z) — ao(d)z +-al(a)z z4-a2(a)zz 4-a3(d)z

1im a.(h,a) = 5(a); j = o, 1, 2, 3;
haO 3 J

and g8(h,d,z) converges uniformly to g9(d,z) on some

neighborhood of (a,z) = (0,0) as h a 0.

As before we consider a change of coordinates of the

form
2 = W-FY(h,d,w)
where now

y (h,o:,w) (h,d)w3 + yl(h,a)w2§;+ ”(11,01)sz + Y3 (11,0063

(2.15') then becomes

(2.15”) Wl+y(wl) = [1+h).(a)l[W+y(W)]

+ hA3[w+ y(w)] +h0(|w|4)

(2.15”) w = [1+-hx(a)]w-t[l-thx(a)]y(w)‘thA

1 (W)

3

— Y[(l+h).(a))W] +0(\w\4)

12

where again we have suppressed a and h for convenience.

Ideally we would like to choose y so that

(2.17) [14-hx(a)]y(w)-+hA3(w)-y[(14-hx(a))w] a O

or
(2.18 a) [1+h1(a)]vo(h,a)+ha0(h,a)

— [1+h>.(<1)]3yo(h,a) = o
(2.18b) [1+h).(a)]yl(h,cx)+hal(h,a)

— [1+h1<a)12[1+h'1_(a)1vl(h.a) = o
(2.18c) [l+h).(c1)]y2(h,d)+ha2(h,cx)

- [1+h1(a)1[1+h1(a)12y2<h,a) = o
(2.18<i) [l-thx(a)]y3(h,d)-+ha3(h,a)
- [1+h1(a)13y3<h.a>= 0

However the equation (2.1813) can not be solved in

general because

[1+h).(a)] - [1+h).(d)]2[1+h).(d)]

= [1+h).(a)l[l— |1+h1<a>|21

and for h small there exists an a near 0 such that

|1+h>.(a)| = 1.

13

Thus we choose

(2.19a) Yo(h:0) = ha0(h,cx)/[(l+h).(d))3- (1+h1(a))1

ao(h.a)/[21(a) + 3h).(a)2 + h21<oo31

(2.1910) yl(h,d) = o

(2.19c) y2(h.a)

 

= ha2(h.a)/[(1+hx<a))2(1+h1(a)) - (1+h>.(a))1

= a2(h.a)/121(a' ')'+ fuel—)2 + 2h|1(o()|2 + h21(a)‘1'(_a)21

(2.19d) Y3(h.a)

= ha3(h,a)/[(1+m‘(a_))3 - (1+hx(a))1

= a3 (mm/131??) + strut—)2 + h21("a‘)3 - 1(a)]
then (2.15’) becomes

(2.20) wl = [1+h).(cz)]w+hal(h,d)[w|2w+h0(]w|4)
Since

def ~
= Yo(d)

1im y (h,d) = 5. (on/21m)
h40 O O

. ~ ———— def ~
lim y (h,a) = a (d)/2)(d) = y (a)
hao 2 2 2

1im (301.01) = 53(a)/[31(a)-x(a)1 = (3(a)
haO

 

14

under the change of coordinates
z = w+ (70(c:()w3 + ;2(a)ww2+ ;3(a)w3

(2.16') becomes

(2.21) v} = x(d)w+al(d)|w|2w+0([w|4)
where 51(a) = 1im a1(h,d).
h»0

Repeating the above process we can eliminate all terms of
degree 4 in (2.20) and (2.21). Set a(h,d) = -al(h,d)

and 5(a) = —al(d) we obtain the lemma.

We now introduce the polar coordinates by setting

r = [w] and r = [w Since

1 1"

|l+th)—hamnw|wfi|

=[1+hxunl-hbaudflw]2+haxlwfi)

where b(h,d) = [14-h Re x(o)]a(h,d)/|l4-hx(d)|, (2.20) becomes

(2 22a) r1 = [14-h1(a)1r_1umh,a)r3+-hfl(h,a,r,e)
5
where fl(h,d,r,e) = 0(r ).
Similarly we set 9 = arg(w) and 61 = Arg(wl) then since

Arg[1-thx(a)-—ha(h,d)lw]2] = Arg[l-thx(d)]-th0([w]2)

we have

15

(2.22b) = e+hcp(h,a)+hf2(h,a,r,e)

91

where hcp(h,a) dgf Arg[l+h).(d)] and f2(h,a,r,e) =O(r2).

By taking the limits as h 4 0, (2.21) becomes

(2.23a) r = Re x(d)r-b(a)r34-fl(a,r)

(2.23 b) 6: Im 1(a) +f2(d,r)

NOW suppose that b(0) > 0. Let T > 0 be fixed and
let p(t) and m(t) be the solutions of (2.23) satisfying

the initial conditions
p(O) = r7 o(0) = 6
Define a mapping f :(r,e) a (5,?) where
d

F = pm; 3 = cw)

Note that
(2.24 a) E = eRe "(O‘Hr -bl(a)r3+Fl(a,r,e)
(2.24b) E: e+Im 1(a)T+Gl(a.r,e)

where

l6

13(ot)eRe “a” (e2 Re “a“ -1)/(2 Re 1(a))

bl(d) = for a # 0

F1(G:r.9) = 0(r

Gl(c1,r.e) = 0(r2)

NOW set
(2.25 a) V(h,a) = 11+hx(a)\2-1
= 2h Re 1(a)+h2l1(a)\2
(2.25 b) Vl(h,c1) = 2 Re x(d)+h[x(d)|2
then Vl(0,0) = 0 and aV/aa(o,o) = 2 Re x’(0) > o. By

the Implicit Function Theorem there exists an ah such that
Vl(h.ah) = V(h,ah) = O
In fact a little calculation shows that
2 , 2
(2.26) ah: -h|).(o)\ /2 Re 1 (O)+0(h)

Let be the mapping as defined by (2.22). Define
a

17

”Q = $5 where N = [T/h] = the greatest integer g T/h .

Using the fact that ¢a is the Euler approximation of

(2.23), it is easy to see that

¢a(r.e) = (rN.eN)

where

(2 27a) r = ll+hx(a)lNr-b (h a)r3+F (h a r e)
‘ N 2 I 2 I I I

(2.27b) 9N = e+—Im x(d)T+-G2(h,d,r,e)

where 1im b
haO

converge uniformly to Fl(d,r,e), Gl(d,r,e) on some

neighborhood of (d,r) = (0,0) as h a 0.

2(h,a) = bl(a) and F2(h,a.r.e). G2(h.a.r.e)

We need the following

Theorem 1.2.2 (Ruelle-Taken): Let ¢H :(r,e) a (rl,el)

be a family of sufficiently smooth mappings from 1R2 -oIR2
where
_ 3 5
(2.28a) rl - (ll-g)r-bl(u)r 4—O(r )
2 4
(2.2813) 91 = e-+a(u)-tb2(u)r ~+O(r )

If bl(0) > 0 and a(O) # 0 then there exists 5 > 0 such

that ¢H has an attracting curve for each U satisfying

18

0 < H < 5. This curve is unique on a neighborhood of

r = 0 and depends continuously on u.

Proof: See [11] or [13].

Since Re 1(0) = 0; Re x'(0) > 0 and b1(0) > 0,
we could apply Theorem 1.2.2 to obtain an attracting
invariant curve for (2.24) for 0 < a < c. It is easy to
see that this curve is invariant under the flow of (2.23)

and hence is in fact a periodic orbit of (2.23).

Similarly as 1im b2(h,d) = bl(d) and bl(a) is
haO
continuous for a near 0 we have b2(h,dh) > 0 for h
sufficiently small. Theorem 1.2.2 then yields a family of

attracting invariant curves for (2.27) for a satisfying
thageh

a is given in (2.26) while eh is such that

h

where 5h > 0 is the constant provided by Theorem 1.2.2.

Since F2(h,d,r,e), G2(h,d,r,e) converge uniformly

to Fl(d,r,9), Gl(a,r,9) on some neighborhood of

(d,r) = (0,0). An easy application of Ascoli—Arzela theorem

implies that the partial derivatives of F2 and G2 converge

uniformly to the corresponding partial derivatives of F1

and G1 as h 4 0. This means that we could choose 5h to

be independent of h provided that h is sufficiently small.

19
We then choose eh to be independent of h. Thus Fa
possesses an invariant curve for each a satisfying
oh < a < e where ah is given by (2.26) and e > O is
some constant independent of h. This result is needed
for the continuation of the invariant curves toward a Hopf

bifurcation which will be discussed in the next section.

NOW suppose that Ta is invariant under ”a then
so is ¢G(TG) for

7rd(¢a(1‘a)) = ¢§+lwa> = (drama) = (Mfg)-

Uniqueness implies that ¢a(ra) = rd and hence Ta is
invariant under 0a. Thus 00 has a family of invariant

curves depending continuously on a for ah S'G < e.

In case b(0) < 0 we could apply Theorem 1.2.2 to

f to obtain a family of periodic orbits for (2.1) which
Cl

bifurcates subcritically from the zero solution, i.e., there
exists a nontrivial periodic orbit for (2.1) for each a
satisfying -e < d < 0 for some 6 > 0. These orbits are
repelling. Similarly, we could apply Theorem 1.2.2 to

Val to obtain a family of invariant curves depending
continuously on d for -e < a < oh where e > 0 can

be choosen to be independent of h. These curves are invariant

and repelling under DC.

We now turn our attention to higher dimension cases,

i.e., x 6 I2 ; n12 3. We thus have a system of 0.D.E.

20
(2.29) i: = f(a,x) = A(o)x+g(o,x); x 6 R
and the approximation

(2.30) x = xm4-hf(d,x )

nwl m

By making an a dependent change of coordinates, we can

decompose (2.29) into
(2.31a) g, = B(d)y+gl(d,y,z)

(2.31b) z

C(a)z+gz(a.y.z)

and (2.30) into

(2.32a)

ym+l ymi-hB(d)ym+-hgl(c,ym,zm)

(2.32b) z

zmi-hC(d)zmi-hgz(d,ym,zm)

m+1
where
2
y, Ym' ym+1 62m
n—2
z, zm, zm+1 62R
B(d) = 2 x2 matrix with eigenvalues 1(a)
and x(a)
C(a) = (n-2) x(n-—2) matrix with spectrum

in the left half plane for all a near

0.

 

gl: R.xIR XII ».R
2
g2: R xiR le 41R
Also g1 and 92 have second order zeroes in (y,z)

at (0,0).

We note that the eigenvalues of Idi—hB(d) are 14-hk(a)

and 14-hx(d) which cross the unit circle when a crosses
oh (given in (2.26)). While the eigenvalues of Idi-hC(d)

stay inside the unit circle.
Again we let T > 0 be fixed and N = [T/h].

Let y(t), z(t) be the solutions of (2.31) with

initial conditions

We then define f (y,z) = (§;E) where
0
§ = Y(T) 7 E = z(T)

Also let ¢a be the mapping given by (2.32) and define

N
7d - ¢d
As before we have
(2.33 a) y = eB(a)T y+Gl(O(,y,z)
(2.3316) ‘2' = emu“

Z+ G2(GIYIZ)

22

and since ¢a is the Euler approximation of (2.31) we

have

Wa(y.z) = (lezN)
where
(2.34 a) yN = B(C()y+ G3(h,a,y,z)
(2.34 b) zN = C(d)z+G5(h,d,y,z)
where

§(o) = [1.1+ hB(d)]N ,

[Id+hC(a)]N ,

()1
0
ll

G3(h,a,y,z), G4(h,d,y,z) have second order zeroes in (y,z)
at (0,0) and converge uniformly to Gl(d,y,z), G2(d,y,z)
on some neighborhood of (d,y,z) = (0,0,0) as h a 0.

Note also that

1im "3(a) = e1“OUT 1im 5(a) = eC‘O‘)l

h-vO ' h-vo

in some matrix norm.

We need the following

Theorem 1.2.3: Let W be a mapping of a neighborhood of
0 in a Banach space X into X. We assume that w is Ck+l,
k 2 l and that ¢(O) = 0. We further assume that D¢(O)

has spectral radius 1 and the spectrum of D¢(0) splits

 

23

into a part on the unit circle with generalized eigenspace
Y of finite dimension and the remainder which is at a
non-zero distance from the unit circle with corresponding
generalized eigenspace Z. Then there exists 6 > 0 and
a Ck mapping u from [Y E Z :ly] < a] into Z with

a second order zero at zero such that

a) The manifold I‘u = [(y.2) [z = My): [y] < e}
c Y e Z, i.e., the graph of u is invariant
under t in the sense that if [yl < e and
if ¢(y,u(y)) = (yl,zl) with |yll < c then
z

1 = 11(yl).

b) The manifold Pu is locally attracting for

t in the sense that if |y| < 6, [Z] < e

. n
and if (yn,zn) = t (y,z) are such that [ynl < a,
[Zn] < e for all n > 0 then 1im lzn-u(yn)| = O.

n-Oeo
Proof: See [11] or [13]. We also remark that c
depend on |D¢3(x)|, j = O,...,k+—l for x in some

neighborhood of 0 in RF.

Let SO be the mapping

SO 3 (GIYIZ) d. (Gigi-Z.)

where y, E are given by (2.33). Assuming that f(d,x)
is Ck+l k+l

for some k 2 1 then S0 is a C mapping

and satisfies all the hypotheses in Theorem 1.2.3. Thus

it has a Ck invariant manifold

24
230 = {(a.y.Z) [z = u0(a.y); lyl < e: [a] < e]

for some 6 > 0.

Similarly, let Sh be the mapping

sh :(a,y,z) 4 (a.wa+ (y,z))

ah

then Sh is also Ck+l and satisfies all the hypotheses

in Theorem 1.2.3. Thus it also has a Ck invariant manifold

2% = ((a.y.z) [z = uh(c.y): ly] < ch: [0] < eh}

for some ch > 0.
Since all: 0(h), Sh converges uniformly to S on some

0
neighborhood of [d,y,z| = (0,0,0) as h 4 0. Using the

Ascoli—Arzela theorem, we can show that all the derivatives

of Sh converge uniformly to the corresponding derivatives

of SO as h 4 0. Thus we could choose eh
sufficiently small h, i.e., Eb and 2% are well defined

= e for

on a common neighborhood of 0.

Now note that uO and uh satisfy

(2.35) uo[on,eB(c‘)T yi—Gl(a,y.uo(d.y))]
= eC(G)T uo(GrY)*'G2(a'y’uO(a'y))

(2.36) uh[d,B(d+oh)y+-G3(h,d+ah,y.uh(o,y))]

= C(d+dh)uh(a,y)i-G4(h,d+dh,y,uh(d,y))

25

In general the center manifold ZN) is not unique.
Hence we would not expect that uh converges uniformly to

u On the other hand, Wan has shown in [14] that the

o.
kth order Taylor expansions are the same for each manifold.
This means that Djuo(0,0) are unique for j g k.
Similarly Djuh(0,0) are also unique for j g k. Since

h and its jth derivatives for j g k are uniformly

u
bounded on a neighborhood of (d,y) = (0,0) for all h
sufficiently small, using the Ascoli-Arzela theorem we
can show that Djuh(0,0) converge to Djuo(0,0) for

j g k-l.

From (2.33a) and (2.34a) we have

(2.37) §= eB(Q)Ty+Gl(a.y.uO(a.y))

(2.38) yN = B(d+cxh)y+G3(h,d+dh,y,uh(c1,y)

Note that the change of coordinates for ¢a in Lemma 1.2.1
depend only on Dj¢a(0) for 0 g_j g 4 and that the

(k-l)th order Taylor expansion of (2.38) at (d,y) = (0,0)
converges to that of (2.37). Thus assuming that k 2 5,

we could transform (2.37) into

(2.39a) E: eReMO‘Mr_bl(o)r3+Fl(o,r,e)
(2.39b) 3: e+Im1(o)¢+F2(o,r,e)
and (2.38) into
N 3
(2.40a) rN = lli—hx(a)| r-b2(h,d)r -+F3(h,a,r,e)

26

(2.40b) 9N = e+Im)\(c1)vr+F4(h,CI.r.e)

l(a,r,e) = 0(r5); F2(a.r.e) = 0(r2):
F3(h,a,r,e) = 0(r5); F4(h,d,r,e) = 0(r2). But F3 and

where b2(h,a) 4 bl(a); F

F need not converge to F1 and F .

4 2
Assuming that bl(0) > 0 We then can proceed as
before to show that (2.40) possesses a family of

invariant curves for all a satisfying ch 3 a g 5h

where ch = O(h)-

We now show that there exists a 5 > 0 independent
of h such that 5h 2 5 for all h sufficiently small.

Suppose the contrary then there exists a sequence [hn]

where hn a O as n 4 a such that 5h a a g 0. By the
n
Ascoli—Arzela theorem, the sequence [uh (d,y)] has a
n
subsequence, which we again call [uh (a,y)], converging
n

uniformly to some function u(a,y) on [Id] < a) [Y] < e)-

The function u(a,y) satisfies (2.35) and hence defines a

center manifold for SO. The equation (2.37) and (2.38)
becomes
(2.41) y = eB(a)Ty-+G1(G,y,u(d,y))

= ” (
(2.42) yN B(d+dhn)y+-63(hn,o+dhn,y,uhn d,y))

where now the right hand side of (2.42) converges uniformly
to that of (2.41) on [lo] < e, [y] < e)- This means that
(2.42) possesses a family of invariant curves for a

satisfying ah gIG g 5 for some 5 > 0 independent of

27

h. This contradicts the assumption that 5h 4 o.
n
Hence there must exist 5 > 0 so that 5h 2 5 for all

h sufficiently small.

Now let To be an invariant curve for (2.33) as
obtained above then it is easy to see that ¢G(Td) is also
invariant under (2.33). Moreover since 2%] is locally
attracting, rd and ¢G(F0) must lie on 23h and hence
induce two invariant curves for (2.38). Uniqueness implies

that these two curves are the same and hence To = ¢G(Ta).

This means that To is invariant under 0a as expected.

Also note that the invariant curve for (2.38) induced
by TC is attracting under (2.38). This means that
Ta is attracting under (2.33) on the manifold 23h.
Using the fact that 2M1 is locally attracting it follows

easily that Ta is attracting globally.

The case bl(0) < O can be treated similarly.

Thus we have

28

Theorem 1.2.4: If the term bl(d) in (2.39)
satisfies bl(0) > O (bl(0) < 0) then the difference
equation (2.3) possess a family of invariant curves depending
continuously on a for ah < a < e (—e < a < ah) where
e > O is independent of h provided that h is sufficiently
small and ch = 0(h). Also in the case bl(0) > 0, the

invariant curves are attracting under (2.3).

Remark 1.2.5: Thus far we have assumed that (2.3)
arises from the Euler method. However all the arguments
given still hold true if some other explicit, convergent,
single—step method is employed. Implicit single-step

methods will be discussed in Remark 1.3.4.

Remark 1.2.6: Theorem 1.2.4 also holds if some of the
eigenvalues of A(c) have positive real parts. What we
need is a center Manifold theorem more general than the

one in Theorem 1.2.3. We now have a mapping of the form

t :(X.y.z) » (xl.yl.zl)

N
ll

Axi—fl(x,y.z)
y1 = By+—f2(x,y,z)

z = Czi-f3(x,y,z)

where HAH < l, HB_1H < l in some matrix norm and the

spectrum of C lies on the unit circle. Also f1' f2, f3

Ck+1

are in x,y,z and have second order zeros at 0.

 

29
First, we could construct a center-stable manifold

for i of the form
(17 = u(x,Z) l [XI < e, [z] < e}

for some 6 > 0. The existence of such manifolds can be
proved in the same manner as in the proof of the Center
Manifold Theorem given in Theorem 1.2.3 (see [13]). We

then have

x1 = Ax4—fl(x,u(x,z),z)

zl = Cz-tf3(x,u(x,z).z)

Now we could use Theorem 1.2.3 to obtain a center manifold

for w.

1.3 Existence and Continuation

We now prove the existence of the invariant curves
away from the Hopf bifurcation. First we temporarily fix

a and write (2.1) as
(3.1) x=f(x); XCJR7n22

Assume that (3.1) has a w—periodic orbit T whose

characteristic multipliers pl = 1, H2”"’Hn satisfying
lug-[<1 for Zgjgk and lujl
Suppose (3.1) is approximated by some convergent, explicit,

> 1 for k4-l g j gyn.

single—step method. We then obtain a difference equation

3O

2
(3.2) x — xmi-hf(xm)4-h F(h,xm)

m+1

We now show that (3.2) has an invariant curve near T for
h sufficiently small. The case k = n or k = l was
proved in [2]. Thus we are primarily interested in the

case 1 < k < n.

From Hale [7], there exists a local coordinate system

along T of the form (p,e) so that (3.1) is equivalent to

(3.3a) (3=A(e)p+fl(e.p)
(3.3b) é: l+f2(e,p)
where fl(e.p) = o<|p(2); f2<e,p) = o<|p|). Am,

fl(e,p), f2(e,p) are m-periodic in e.

The variational equation for (3.3) is
(3.4) dplde =A(9)p

which has characteristic multipliers “2"°"Hn' Let X(e)
be a fundamental matrix solution of (3.4) then so is
X(e-+w). Thus there exists a nonsingular matrix D such

that

x(ei-w) = X(G)D

Let B be a real matrix such that D2 = eZBw. Define

 

31

P(e) = x(e)e‘Be

then P(84-2w) = X(e+-2w)e‘B(e+2w) = e(e).

Also since X(9) is a fundamental matrix solution of

(3.4) we have
(3.5) dP(e)/de+P(e)B=A(e)P(e)
Set p = P(9)r then

b = d(P(e)r)/dt

=duwmnaeér+mefi
which gives using (3.5) and (3.3)

AWHRMr=[Amnwm-pwnnr+mefi+ourfi)
or

i: Br+f3(9,r)

Thus (3.1) is equivalent to

(3.4a) r=Br+f3(e,r)
(3.4b) é: l+f4(e,r)
where f3, f4 are 2w—periodic in e, f3(e,r) = 0(|r|2);

f4(e,r) = O(|r[); B is a (n-l) x(n-—1) matrix such that

32

eZBw has eigenvalues u:,...,u:.

Let u:Rn an, v:]Rn 41Rn be such that

r = u(x); e = v(x) for all x near T .

As usual we can assume that u and v are as smooth as

needed. We note that

i: = du(x)/dx -$: = du(x)/dx f(x)

9 = dv(x)/dx '3'; = dv(X)/dx f(x)
Set r1 = u(xl)7 91 = u(xl) where
x1 = x+hf(x) +h2F(h,x)
then
(3.5 a) r1 = u(x+hf(x) +h2F(h,x))

= u(x) + du(x)/dx[hf(x) +h2F(h,x)] + 0(h2)
= r+hr+0(h2)

=r+hMHJfiyem)+h%3m,mr)
where in general F3(h,e,0) # 0.
Similarly

(3.516) 91 = v(x+hf(x)+h2F(h,x))

- e+—hé+-O(h2)

e+h+hf4(e,r) +h2F4(h,e,r)

 

33

where in general F4(h,e,0) F 0.

Let 12"'°'*n be the eigenvalues of B. By rearranging

if necessary we can assume that

In particular this means that Re xj < 0 for j = 2,...,k

while Re kj > 0 for j = k4—1,...,n.
Let Y = generalized eigenspace of B corresponding to
the eigenvalues 12""'Xk
SE = generalized eigenspace of B corresponding to

the eigenvalues xk+l""’kn'

By projecting to Y and Z we can write (3.5) as
(3.6a) y1 = (Id+hC)y+hf5(e,y,z)+h2F5(h,e,y,z)

(Id+hD)z+hf6(e,y,z) +h2F6(h.e,y,Z)

(3.613) 2

(3.6c) = e+h+hf7(e,y,z)+h2F7(h,e,y,z)

61

where f5(e,y,z) and f6(e,y,z) have second order zeroes
while f7(e,y,z) has first order zero in (y,z) at (0,0).

All the functions are as smooth as needed.

Also there exist c, d > 0 such that

(\Id+hcu g 1 —hc

 

34

1] (Id + hD)-1H g 1 — hd

for h sufficiently small.

We have the following
Lemma 1.3.1: Let

B = {continuous 2w—periodic functions y(G) with
values in Y such that [y(e)| g 5 and

[y(el) -y(82)[ g alel-ezl]

B = [continuous Zw-periodic functions z(e) with
values in Z such that [2(9)] 3 5 and

[z(el) -Z(82)l g 6181-921}

where 5 > 0 is to be determined.

Then for every z(e) E 82 there exists a unique

y(e) 6 BY such that

(3.7) y[8+h+hf7(e,y(e).z(e)) +h2F7(h.e.y(e).z(e))]

= [Id+hc1y(e) +hf5(e,y(e),z(e))+h2F5(h.e.y(e).z(e))

Similarly for every y(e) 6 BY there exists a unique

2(9) E BZ such that

(3.8) z[e+h+hf7(e,y(e).z(e)) +h2F7(h.e.y(e),z(e))]

= [Id+hD12<e) +hf6(e.y(9).2(e)) +h2F6(h.e,y(e).2(e))

 

35

Proof: We first prove (3.7). So assume that
2(9) E 62 is given. Define a mapping Tz :BY 4 BY as

follows:

Let g(e) 6 By. Fix 9 and let 3 be such that

(3.9) e = ‘6+h+hf7(8‘.g(§),z(§) +h2F7(h.§,g(E),z('5))

(mod 2w)

such 3 exists and is unique (mod 2w) since the right
hand side of (3.9) is strictly increasing and covers an

interval of length 2m as 3 varies from 0 to 2m.

Tz g is defined to be the function satisfying
(3.10) Tz 9(6) = [Id+hc1g(§)+h£5(§.g(‘e‘).z(6))

+ h2F5(h.6.g(§) ,z(§))
Clearly Tz 9(9) is Zw-periodic. Also

(T2 9(9)] 3 (1161+th (9(3) 1 +h|f5(3.g(6),z(8))|

+ h2|F5(h,€,g(§),z(§))

Now HId-thCH g l-hc.

For convenience we let N denote a bound for the

f f F F and some of the first

5' 6' 7'F5' 6' 7

and second derivatives of these functions for [y] g 5;

norms of f

lzlgé.

 

36
Since f5(9;y,z)has a second. order zero in (y,z)

at (0,0), we have

[f5(e.y.z)l 3 N52

Also [F5(h,e,y,z)| g N.

Hence

|Tz 9(9)] g(1-hc)5+hN25+h2N

= 5[l—hc+hN5 +h2N/5]

Thus if we choose 5 = Kh where K 2 2N/c then for h

sufficiently small
l—hc+hN5+h2N/5 < 1

which gives \Tz 9(9)] 3 5.

To see that Tz g(e) is Lipschitz continuous, let
61. 92 be given and let 81' 92 be such that
(3.11a) el = el+h+hf7(el.g(el).z(el))
2 _ _ _
+ h F7(h,el.g(el).z(el))
(3.11b) 82 = ez-th-thf7(92.g(62).Z(82))

+ h2F7(h,62.g(§2),z(§2))

NON

 

37
If7(61.g(§l),z(31)) -f7(§2,g(§2),z(§2))|
3 N6 [81 -§2| +ng(§l) -g(§2) l +le(§l) - 2(32)|

The first estimate follows from the fact that as f7(e,y,z)

has a first order zero in (y,z) at (0,0) so does

af7(e,y.z)/ae.
Similarly
(F7(h,61.g(§l) .z('e‘l)) -F7<h.§2.g(§2) ,z(32)|
g NIIEl—Ezl + (gal) -g('e'2) | + (z(Fl) -z(§2)l1

g N(25 + 1) [‘51—'62]

Substract (3.1J.b) from (3.1J.a) and take absolute values,

we obtain

(91-92] g (El-621+[3hua+h2N(25+1)1 (81-82]
Hence

[81—62] g [l-hN(35+2h5+h)]-l [91—92]

Therefore

 

38
sz 9(91) ‘Tz 9(62)|
g HIdd—hCH |g(El)-g(§2)l
+ h|f5(61.9(81).z(31))-f5(32,g(€2).z(§2))|
+ h2|85(h.§l.g(al).z(§l)) —F5(h.€2.g(€2),2(§2))|

2

g [(1-hc)5+3hN5 +Nh2(26+1)] (51-521

3 [(1 -hC)b +3hN62+Nh2(25 + 1)]

[l-hN(36+2h6 +h)]‘l [el‘ezi

golel-ezl

if we choose 5 = Kh where K 2 2N/c and if h is

sufficiently small.
Hence T2 9 6 BY.

We now show that T2 is a contraction map from

BY to By.

Let gl(e), 92(8) 6 B . Fix 9 and let and

Y 81

92 be such that

(3.12 a) e = eli-hi-hf7(el,gl(el),z(el))

2
+ h F7(h'91'91(91)'2(91))

(3.12b) e = ez-th-thf7(92,g2(ez),z(92))

2
+ h F7(h,62,gz(92),z(62))

Substract and solve for [el-62l we obtain

 

 

39
[81- 82] g h|f7(61.gl(el),2(el))—f7(92.92(62).2(62))l

+ h2|F7(h.el.gl(el).z(el)) -F7(h.92.92(62).z(62))I
Note that
|f7(el.gl(el).z(el)) -f7(ez.gz(92).z(62))|
g Natlel- 62] + [91(91) -92(62)l+ |z(el) -Z(62)|
and
(91(91) -gz(ez)l g I91(el) -g2(el)| + [92(61) —92(82)|
C

where Hgl-g2HCO = MEX Igl(e)-92(9)|.

Also

[F5(h.el,gl(el) 'z(91)) -F5(h,ez,92(92).z(ez) )l

g N[]gl(el) “92(92)|+ (z(el) -z(62)|+ '91‘ e l]

2

Hence

[el-82| g [el-92|[h(N524-N54-N52)-+h2N(25-+1)]

+ Hgl - g2“ Quins + hZN]
C
Thus

2
iel"92| S Nl(h5‘+h )Hgl-92HCO

 

 

40

for some constant Nl > 0 independent of h and 5. NOw

(3.13a) Tz gl(6) = (Idi—hC)gl(el)-+hf5(el,g(el),z(el))

2
+ h F5(h,el,gl(91).z(el))

2
+ h F5(h,82.gz(62).z(62))

Therefore

sz 91(e)--Tz 92(6)] 3 HIdi—hCH \gl(e)-92(e)l
+ h | f5(el.gl(el).Z(el))-f5(62,92(92).2(92))l
+ h2|F5(h,el.gl(el),Z(el))-F5(h.82.92(62),Z(62))l

Note that

[f5(el,gl(el).z(el))-f5(ez.gz(ez).z(ez))|
g Nozlel-82|+-N6[lgl(el) -92(92)l-+[2(el)-Z(92)[]

2
i N5Hgl"92HCQ'I3N5 lel"92)

Similarly

lF5(h,el.gl(el).z(el))-—F5(h,82.92(82),z(ez))|
g N[lel-ezl+-lgl(el)-gz(82)|+-[z(el)-z(82)l]

g.NHgl—ggHCO+(2N6+rn1el—ez1

Hence

41
(T2 gl(e)-Tz 92(e)| g (l-hC)[Hgl-92HC04'5[91"92)]
+ h[N5Hgl -g2HCO+ 3N62[Gl- 62]]
+ hztnllgl -92HCO+ (21% +N> lel - 9211

g [1 _ hc + N(h5 + hangl - 92llco

2

+ [6(l-hC)+N’h(35 +2h5 +h)]l61—62i

N = 1-hc+N(hé+h2

2 )

+ [6(1—hc)+Nh(352+2h5+h)]th(5+h) < 1

if 5 = Kb and h is sufficiently small.

By the Contraction Mapping Principle, TZ has a fixed

point which is the solution of (3.7).

The existence of a function z(e) which solves (3.8)
can be obtained in exactly the same fashion. First we
assume that y(e) 6 BY is given, we then construct a

mapping T :B a 8 given by
Y z 2

(3.14) 9(6) = [Ic1+hD]Ty g(e)+hf6(e.y(e),'ry 9(6))

+ h2F6(h,e,y(e),Ty 9(a))

where 5 is given by

42

(3.15) ‘e‘ = e+h+hf7(e.y(e).g(e)) +h2F7(h.e.y(e).g(e))

We can then proceed to show that Ty is well defined and
is a contraction map and hence has a fixed point which is

the function 2(9) that solves (3.8). Q.E.D.

If all the nontrivial characteristic multipliers
“2""’“n satisfy (pm) < l (uj > 1) then by letting
2(6) 2 0 (y(e) 5 0); Lemma 2.3.2 provides an attracting

(repelling) invariant curve for (3.2).

In the more general case, i.e., when not all “j
satisfy [Ujl < 1 (or |uj| > 1) we actually need a
stronger result than Lemma 2.1. More precisely, let
zl(e), 22(8) be two functions in 32 , they then define
two mappings T1’ T2 from )BY to By. with fixed points

yl(e) and y2(e). We need to show that

“Y2"Y1HCO g Kluzl-z2HCO for some Kl < 1.

We have
(3,155.) 111(91): [Id+hc]yl(e)+hf5(e,yl(e),zl(e))
2
+ h F5(h;9:Yl(9):Zl(e))
where 81 is given by

(3.16 b) 81 = e+h+hf7(e.yl(e).zl(e))

2
+ h F7(h,e,yl(e).zl(e))

43

and

(3.17 a) y2(92) = [Id+hC]y2(e)+hf5(e.y2(e).zz(e))

2
+ h F5(h.e,y2(e).zz(e))
where

(3.17b) 92: e+h+hf7(e.y2(e),zz(e))

2
+ h F7(h.6,Y2(e),22(6))
From (3.16 b) and (3.17 b) we obtain
2
[el- 82] g N2(h +6h)[llyl-yzllco+Hzl-zzllcol

for some constant N2 > 0 independent of h and 5.

From (3.16a) and (3.I7a) we obtain
[yl(el) -y2(82)| g (l'hC)HY1‘Y2HCo
+ N(h6+62)[\1yl-y2|l O+l|zl-22l| o]
C C
Hence

3 Iyl(61)-Y2(62)|+6]61-62l

This yields

44
2
Hyl-yzllco S Hyl -y2||CO[l-hc+ (11+ 6N2)(h5 +6 )1
+ “21—22” O[(N+5N2)(h5+(52)l
C

or

y-y K z-le
lll zucoilnl 2C0
where

2

2)

K = (N+5N2)(h6+5 )1'1 < l

l [hc—(N+5N2)(h6+6

if 5 = Kh and h is sufficiently small.

Similarly given two functions yl(e), y2(9) in By,

the solutions zl(6). 22(6) of (3.8) then satisfy

“21"22HC0 g KZHYl"Y2UCO

where K2 < l for h small.

Now we can prove that the mapping defined by (2.6)

possesses an invariant curve. First we set

(0)

y (e) = 2(0)

(8) E O .

(k)

Inductively given 2 (e) in 82, we lEt y(k+l)

(e) be

the function which is the fixed point of the mapping Tz(k).

We then let z<k+l)(e) be the fixed point of T

y(k+l)'
We then have

 

45

k l k
+ )_y( )H

((.<k+1>_.<k>uco g Kzuy‘

CO

3 K2Kle(k)--z(k_l)||CO

Similarly we also have

(k+1)

(k) (k)_ (k-l)
Hy - 17 “Co _<. KzKlHy y NCO

(k)

Hence the sequences y(k)(e), z (6) converge uniformly

to some functions y(e), 2(6).

The curve F = [(e,y(e).z(e)) [0 g e g 2m} is the
desired invariant curve.

Note that F lies in a 5-neighborhood of P where

5 = Kh. Thus if h 4 0, we have P 4 T in the CO norm.

Now suppose we allow (3.1) to depend on a real parameter

(3.18) S; = f(c1,x)

Suppose (3.8) has a periodic orbit To at a = do whose

nontrivial characteristic multipliers satisfy ‘Uj‘ # l

for j = 2,...,n. Using the fact that the Poincare map

for (3.18) has a nonsingular Jacobian at d = do, we obtain
a family of periodic orbits To depending continuously on

a for all a near go.

Let

 

46

2
(3.19) xm+1 - xmi-hf(c,xm)i-h F(h,d,xm)

be a difference approximation of (3.18) which arises from
some explicit, convergent, single-step method. We now could
repeat the arguments given earlier to construct the
sequences y(k)(d,e), z(k)(d,e) depending continuously on
a for a near do. As before these sequences converge
uniformly to y(d,e), z(a,e) which give a family of
invariant curves for (3.19) depending continuously on a

for a near 0.

We have proved:

Theorem 1.3.2: Suppose that the system

(3.18) :2 = f(c1,X); x 6. in“,

f is sufficiently smooth in a and x

I

has a periodic orbit T at d = a whose characteristic

0
multipliers “1 = 1, uz....,un satisfy iHjl # l for
j = 2,...,n. Then for h sufficiently small, the approximating

difference equation

_ 2
(3.19) xm+1 — xmi-hf(o,xm)+-h F(h,d,xm),

which arises from some explicit, convergent, single—step
method, has a fmily of invariant curves depending continuously
on a for a near do. As h 4 0, these curves converge

to the periodic orbits of (3.18). Also if lujl < l

(lujl > 1) for j = 2,...,n then these curves are

attracting (repelling) under (3.19).

 

 

47

Remark 1.3.3: Suppose that (3.18) has a stationary
solution for all d near do and that it satisfies the

Hopf bifurcation conditions at d = d Suppose also

0'
that the quantity bl(0) as in (2.24 a) is positive,
then there exists an dl > dO such that (3.18) has a
non-trivial attracting periodic orbit for each d satisfying

do < d < d1. From Theorem 1.3.2, (3.19) has a family of

invariant curves depending on d for d2 < d < d where

3
d2--dO = 0(h) and d3--dl = 0(h). By ChOOSlng h small
we can make d2 close to dO and then using Theorem 1.2.4,

we could trace along the invariant curves back to the
stationary solution. Thus we also have continuation of

invariant curves for (3.19) near the Hopf bifurcation.

It is well known that the periodic orbit of (3.18)
could undergo some secondary bifurcation if one of the
nontrivial characteristic multipliers crosses the unit
circle (see [12]). We were unable to establish similar

results for the invariant curves of (3.19) however.

Remark 1.3.4: Suppose that an implicit, single—step
method is employed. We then have a difference equation of

the form

_ 2
(3.20) X — Xmi-hBlf(xm)4-hBOf(xm+l)+-h F(h,X )

,x
m+1 m m+1

Fix x and consider the mapping

T : y 4 x+hBlf(x) +hBOf(y) +h2F(h,x,y)

48
Let [x] < M for some M > 0, by choosing h sufficiently
small, T is a contraction map and hence has a fixed point
x1. We thus have

x = x-thBlf(x)4-hBOf(xl)-+h2F(h,x,x

1 1)

Also if f and F are sufficiently smooth then xl

will be smooth in h and x. Since x1 = x-+O(h) we

have

(3.21) x1 = xi-hBlf(x)+-hBOf(x)i—O(h2)

Assuming that Boi-Bl = l, (3.21) becomes

(3.22) x1 = X+hf(x)+h2Fl(h,x)

for some function Fl(h,x). We now would apply Theorem 1.2.4

and Theorem 1.3.2 to obtain invariant curves for (3.22)

which are also invariant curves for (3.20).

2. LINEAR MULTISTEP METHODS

2.1 Preliminaries

In this chapter we will approximate
(1.1) 5: = f(x); x 6 ]R

by some linear multistep method. We thus obtain a difference

equation of the form

k
+h Z) 8. f(X
j=o 3

(1.2) x d

.X .
l J m+1-J )

m+1 =

(tax

+1—‘
3 m 3

We will assume throughout the chapter that the method

employed is convergent. This means that

dj = 0 (d0 = —1)

k
3:

k
E B- 2’0
3:0 3
. . . kn n
Since (1.2) defines a mapping from Ii 4 Ii we need

a new definition for invariant curves.

49

50

Definition 2.1.1: A curve T c I91 is said to be
invariant under (1.2) if for every point x e T, there
eXist k-l pOints on T X-l""’X-(k-l) such that the
points generated by (1.2) with starting points,

x-(k—l)""’X—1’X all lie on T.

Definition 2.1.2: A curve F c I91 which is
invariant under (1.2) is said to be attracting under (1.2)
if given any point x sufficiently close to T, there
exist k-l points X—l""’X-(k-l) such that the points
generated by (1.2) with starting points X—(k-l)""'X—l’x

Spiral toward T.

2.2 The Hopf Bifurcation

As before we have a family of 0.D.E.

(2.1) 5: = f((l,x) = A(u)x+g(u,x); x e 1Rn
where f(u,x) is as smooth as needed in u and x,
g(u,x) = 0(|x|2) and A(u) satisfies the Hopf bifurcation

conditions at H = 0, i.e., A(u) has a pair of complex

conjugate eigenvalues, X(H) and (u) such that

(2.2) Re 1(0) = 0; Re X'(O) > 0; Im X(0) # 0

 

51
Assume further that all other eigenvalues of A(H) stay

uniformly away from the imaginary axis for all M near 0.

Now suppose we approximate (2.1) using some explicit,
convergent linear multistep method. We then obtain a

difference equation of the form

 

 

k k
(2.3) Xm+l j2=31 orj xm+1-j+h jg: Bj f((l,xm+l_j)
k k k
where Zd.=l and Zjd.=Z)(3.¢0.
i=1 3 i=1 3 i=1 3
. _ T T T kn
Define ym — [Xm+l-k . xm ] e I! (T denotes
tranposition). The difference equation (2.3) is then
equivalent to
(2.4) ym+1 = B ymi-hG(u,ym)
where
0 I 0 0
0 I 0
B =
0
0 0 I
de GlIJ
and
T T k T T
G(u,ym) = [o . o ( 5:71 ij(u,xm+l_j) ]

52
where in B, 0 and I denote the n)<n zero and
identity matrices while in G(u,ym), 0 represents the

n-dimensional zero vector.

Assuming that the method employed is strongly stable,

i.e., the roots of the polynomial

k .

p(e) = Z a. 8‘”
j=‘o 3

are l, €1"“’€k where [ejl < 1 for j = 2,...,k.

Note that B then has 1 as an eigenvalue with
multiplicity n while all other eigenvalues of B have

absolute values less than 1. Let

Z: = n—dimensional eigenspace of B corresponding

to the eigenvalue 1.

W — n(k-—l) dimensional generalized eigenspace of B

corresponding to the remaining spectrum of B.

We then can decompose Rkn as:

y = Pyi—(I-—P)y for y e Rkn

where P is a projection matrix along Z: such that I-P

is a projection matrix along W.

We now rewrite (2.4) as
(2.5 a) z = zmi-hPG(u,zm4-wm)

(2.5 b) w = Ewm+h(I—P)G((_l,zm+wm)

53

where zm=Pym, zm+l=Pym+lez

wm = (I-P)ym, w

m+1 = (I-P)Ym+l E W

Note that the spectrum of E lies inside the unit

circle.

Although we are interested in the case b > 0, the
equation (2.5) however is well—defined for [h] < 5 for

some 5 > 0. Thus (2.5) defines a mapping

S 2 (hfilpzmtw ) '3 (hILlIz )

m m+l’wm+1

on some neighborhood of (0,0,0,0).

It is easy to see that S satisfies all the hypotheses
in the Center Manifold Theorem (Theorem 1.2.3) and thus

has an invariant manifold of the form
w=u(h.u,2)7 \hl<o; |u[<o: [z\<5

for some 5 > 0.

Note that u(0,g,z) a 0 and hence we can write
u(h,u,z) = h v(h,u,z)
where v(h,u,z) is uniformly bounded on
)9 = ((h.u,z) l lhl < 67 [u] < 6; (2| < 6)

From (2.5(a) we obtain

54

(2.6) 2

m+1 zmi-hPG(u,zmi-hv(h,u,zm))

‘ 2
— zmi-hPG(u,zm)+-h PF(h,u,zm)

where F(h,g,z) is uniformly bounded on 3. Now suppose

that
_ T T T
zm — [x x ]
where x e.Rn, then
_ T T T T
G(.Lllzm) " [O 0 X1 ]

where 0 is the n-dimensional zero vector and x c.R

is given by

Assuming temporarily that the roots ej of

p(e) = Z) d. ek—J are all distinct. Note that W

i=0 3

is spanned by the columns of

 

 

55

where I is the n)(n identity matrix.

Set
I I . . I
I €2I . . ekI
D = O O O U 0
k—l k-l
I 82 I . ck I

Since ej are distinct, D is invertible. Thus there
kn

exists b e R. such that
G(u.zm) = Db
Now if
T ... T T
b — [bl bk ]

n

where bj 6 R then
T T T
P G(U:Zm) — [bl ~-- bl ]

To find out what bl is we note that the n xn block in
the upper right corner of D-1 is of the form dI where

I is the n)<n identity matrix and d is given by

where

 

 

 

1
e2
dl=
62
and
1
1
d2:
1
Hence
Therefore

56

 

 

1
€k
k-2
6k
1
ek
k—l
6k
k
(-1)k'l/ r1 (8
j=l
k
1/ '11 (l—ej)
1/ p’v(l)
k
l/ 23 (3
i=0 3
k
d .23 Bj f(u,X)

3:0

(e-

= . n
j > i
zgi,jgk
k
II (8 —l)
i=2
«1)
f(u.X)

3

“€-

)

l

57

which gives

 

 

 

T T
(2.7) P G(u.zm) = [f(u.x)T f(u.x) 1
In case ej are not all distinct, e.g., 62 = 83.
If 62 # 0 we use
"I I O I I W
I 621 €2I €41 ekI
D =
k-1 k-l k—l k_1
LI 32 I (k—1)5 184 I . 6k I“
while if €2 = 0 we use
I I 0 I I l
I O I 841 ekI
0
D =
k-l k-l
I 0 O 64 I ek I-
(2.7) is still valid in either case.
Thus (2.6) induces a mapping on I91 of the form

_ 2
(2.8) Xm+l - Xm+hf(Ll,Xm) +h Fl(hILlIXm)

58
where Fl(h’“’xm) is a component of PF(h,u,zm), here
T TT

Z =[xm ... xm]

Proceed as in Chapter 1, we obtain a family of invariant
curves for (2.8) which bifurcates from the zero solution.

This implies a similar result for the difference equation (2.3).

Thus we have

Theorem 2.1.1: If the system (2.1) is approximated by
a strongly stable, convergent, explicit, linear multistep
method then we also obtain a family of invariant curves

bifurcating from the zero solution as in Theorem 1.2.4.

Remark 2.1.2: As will be discussed in Remark 2.3.5,

some of the requirements in Theorem 2.1.1 may be weaker.

2.3 Existence and Continuation

Suppose that the system
(3.1) 5: = f(X); x 6 ]R

has a periodic orbit F with characteristic multipliers

“1 = 1, u2,...,un satisfying [ujI < l for 2 g j g 2

and [ujl > 1 for z-+1 g j g n. If (3.1) is approximated
by some convergent, explicit, multistep method, we would

obtain a difference equation of the form

k

.-+h _Z) Bj f(X

k
(3.2) x = Z) d. x _
= j m+1 3 3:1

)

m+1-j

59
Let p = [p(e) )0 g 6 g_w} be the periodic orbit of (3.1),

i.e., p(e) is a w-periodic function in e and satisfies
dp(e)/d6 = f(p(6))

We now imbed F in Pkn by define

1"“= [P(9)|Ogegw)

where 5(9) = [p(e-(k-1)n)T --- p(e)T]T. Similarly given
a sequence xm—k+l""’xm in 351 we define

z = [x T ... X T]T

m m-k+l m

The difference equation (3.2) then defines a mapping
kn kn

T :R. * R- by
sz = m+1

where z - [x T "' X TlT

m+1 - m—k+2 m+1 '
Define
(3 3) f(6) = aT/az (5(6))
Then for

_ T ... T T n
z — [yl yk ] where yj 62R

we have

60

T T T T
f (e) z = [y2 ---- yk yk+l 1
where

(3.4) y

k
= Z) d. y .
j=l i m+1-j

k
j—O
We now develop a local coordinate system around f. Assuming

that “j are distinct then the variational equation
(3.5) dy(e)/de = af(ax(P(e))y : y eznn

has n linearly independent solutions of the form

ql(6) = dP(6)/d(e) dgf vl(e)

qj(e) = e j Vj(6) for 2 g jig n

where Vj(9) are 2w-periodic in e and xj are real
2). .(n

numbers such that e j = ujz. We have choosen vj(6)

to be 2w—periodic in order to ensure that kj are real.

Also as [qj(e) [1.3 j g k] are linearly independent

at any a, so are [vj(e) [l g j g k].

As vj(e) form a moving coordinate system around
T, we would like to obtain a moving coordinate system

around I based on these functions.

l-B
Set 11 = 0 so that qj(6) = e j Vj(e) for l g j g n.

Define

61

_ T TT kn
(3.6) zj(6) — [ujl(9) ujk(e) ] E I?
-)..(k-z)h n
where u. (e) = e 3 v.(e-(k-z)h) e Ii .
33 3
Note that
-A-6
zj(8)=e 3 [qj(e-(k-1)h)T ~ qj(e)T1T

Hence using the facts that the local truncation error
induced by a convergent method is of order 0(h2) and

that qj(9) is a solution of (3.5), we obtain

.7 .
(3 ) f(e)zj(e)

‘1-6
= e 3 [qj(e-(k-2)h)T ~qj(e)qu(e+h)T1T+0(h2)

h).
e szm+h)+om2)

Now assume that the method employed in (3.2) is strongly
k .
stable, i.e., the polynomial p(e) = Z) d. ek—J has

k roots 1, 32,... where [ejl < l for j = 2,...,k.

lek

Also for simplicity we will assume that ej are all distinct.

For m = 2,...,k we define
_ T T k—l T T
for j = 1,...,n. We then have
(3.9) f (6) emj(8) = em emj(6)-+O(h)
= cm emj(e+lfl-+0(h)

 

62

assuming that vj(e) are at least Cl.

Now note that when h = 0, z.(e) becomes

3
_ T , T T
zj(6) - [VJ-(6) vj(e)]
Hence if
n k n
, . . . . = 0
3'51 all 23(9) + “:2 i=1 am? em”)

then the linear independence of vj(e) implies that

i
a . + Z) 8 a . = 0
13 m=2 m mj
for j = l,2,...,n and i = O,l,...,k-l. This in turn
implies
l l l alj
k-l
1 82 62
= 0
k-l
1 6k . . . ck akj
which yields amj = 0 for l g m g k; 143 j g n. Let
D(h,e) = determinant of the (kn) x(kn) matrix
whose columns are 21(8),...,zn(e),
e21(e)lotole2n(e)locoolekl(e)Inc-Iekn(e)

We have just shown that D(0,e) # 0 for all 8. Since
D(h,e) is continuous in h, e and periodic in 6 there

exists a 5 > 0 such that

63

D(h,e) # O for 0 g h < 5 for all 6

Hence the vectors 21(9),...,zn(e), e21(6))...,e2n(e),
., ek1(e)""'ekn(e) are linearly independent at any
8 if h is sufficiently small.

In the general case when not all “j are distinct,

(3.5) has a fundamental matrix solution of the form

X(e) = V(e)eAe where v(e) is zw-periodic and A is a

real n)<n matrix whose eigenvalues are x ,-..;1 where
Zx.w 2 l n

e 3 = uj. We simply choose vj(e) to be the columns of

V(e) and qj(e) the columns of X(e).

Similarly if ej are not all distinct, e.g., 82 = 83

we choose

e . = as before
23
T k—l T T .
[0 €2Vj(e) (k--l)62 Vj(6) ] 1f 82 T 0
[o vj(e)TO'°'O]T if 62:0

We then can proceed as before to show that the vectors
21(9).....zn(e), e21(e),...,e2n(e)......ekl(e),...,ekn(e)

are linearly independent at any 9.

64

Thus we have the following

Lemma 2.3.1: There exists a neighborhood of I in

Rkn such that for any 2 in this neighborhood, there

exists a unique (e,r,w) such that

(3.9) z=§(6)+Z(6)r+E(6)w
where 2(a) is the (kn) x(n -1) matrix whose columns are
22(6)....,zn(6)) E(6) is the (kn)x[(k—1)n] matrix
whose columns are emj(e); j = 1,...,n; m = 2,...,k.
Also r e Rp_l and W e Rik—l)n.
Proof: Set F(e,r,w,z) = z-[p(e)+-Z(e)r+-E(e)w] then
F(e.0.o,§(e)) = o
and
aF/66(6,o,0.§(e)) = 21(6)
aF/ar(e,o.o,{a(e)) =z(6)
aF/aw(e.o.o.§(e)) = E(6)

Since [21(9),Z(e),E(e)] is invertible at all 6, Lemma 3.1
follows from the Implicit Function Theorem and the fact

that f is compact. Q.E.D.

We note that since Z(6) depends continuously on
h and E(e) is independent of h. The local coordinates
(3.9) is valid for all [r] < 5, [w] < 5 where 5 > 0

can be chosen to be independent of h provided that h > 0

65
is sufficiently small. Also by assuming that the system
(3.1) is sufficiently smooth, 9, r, w will depend as

smoothly on 2 as needed. Let v1, v2. v3 be appropriate

functions such that
6 = v1(2): r = v2(6); w = v3(z)

for all z sufficiently close to I.

Let 2 = p(e)+-Z(e)r+-E(e)w be sufficiently close

to F, then
(3.10) T2 = T(§(e))+ ]‘ (6)[Z(6)r+E(6)W] +hgl(h.e.r.W)

where gl(h,e,r,w) has second order zero in (r,w) at

(0.0).

We have
(3.11a) T(f>(e)) = §(e+h)+0(h2)
(3.11b) ((6)2(6) = Z(e+h)A+0(h2)
(3.11c) ((6)8(6) = E(e+h)D+O(h)
where

z(e) = [21(6) zn(e)]

3(8) = [e21(6) ... e2n(8) .... ekl(6) ... ekn(6)]

A = eAh

and D is a [(k-—l)n] x[(k-—l)n) matrix satisfying “DH 3

Q

<

1.

 

66

Set 3 = p(e+h)+Z(8+h)Ar+E(e+h)Dw then

Tz-—E = hgl(h,e,r,w)i—hGl(h,e,w)+-h2F (h,e,r,w)

1

the first term arises from (3.10), the second from (3.11c)
and the third from (3.1J.a) and (3.]J.b). The function
gl(h,e,r,w) has second order zero in (r,w), Gl(h,e,w)
has first order zero in w while Fl(h,e,0,0) e 0 in

general.

Thus if T2 = p(el) +z(61)rl+E(el)wl then

(3.12a) = vl(Tz)

el
= v16) + avl | 52 (E) [Tz-E] +O(h2)

e+h+hg2(h,e,r,w) +h82(h,e,r,w) +h2F2(h,B.r,w)

where g2(h,e,r,w) has second order zero in (z,w) at

(0,0) while G2(h,e,r,w) has first order zero in w.

Similarly we have

(3.12 b) r1 = Ar+hg3(h,9,r,w) +hG3(h,e,r,w) +h2F3(h,e,r,w)

(3.12c) w = Dw+hg4(h,e,r,w) +hG

2
l 4(hlelrlw)+h F4(hlelrlw)

where g3 and g4 have second order zeroes in (r,w) while

G3 and G4 have first order zeroes in w.

67

We now show that (3.12) possesses an invariant

curve.
hx.
Rearrange the eigenvalues l4—e 3; j = 2,...,n of
A so that Re Xj < 0 for j = 2,...,z and Re xj > 0
for j = z+-l,...,n. Let
Y = generalized eigenspace of A corresponding to
hx.
the eigenvalues l+-e 37 j = 2,...,n.
Z: = generalized eigenspace of A corresponding
hi.
to the eigenvalues l+-e 3; j = 2+—l,...,n.

By projecting to Y and z: we can write (3.12) as

(3.13a) 61 = e-th-thg5(h,e,y,z,w)i—hG5(h,e,y,z,w)

+h2F5(h,e,y.z,W)

(3.131)) yl BY+hg6(hIGIYIZIW)+hG6(hIGIYIZ:W)

+hZF6(h,e.y.z.w)

(3.13C) Z = CZ+hg7(hpe:Y;Z:W)+hG7(ha9:Y:ZIW)

-+h2F7(h,e,y,Z.w)

(3.13 d) wl

DW-th98(h.6.y.z,W)-+hG8(h,e,y.Z.W)

+h2F8(h.6,y.z.W)

where g5, g6, g7, g8 have second order zeroes in (y,z,w)

at (0,0,0); G G , G , G have first order zeroes in w.

5' 6 7 8

68

For h sufficiently small we also have

HBH _<. 1 -hb
)(c‘ln _<. 1 -hc

“DH is < l

for some constant b, c > 0.

As in Chapter 1, we introduce the spaces

BY = [continuous Zw—periodic functions y(e)

with values in Y such that [y(e)[ g 6

and [y(el) -y(62)[ g alel- ezl

E
II

[continuous 2w—periodic functions z(e) with

Z
values in Z such that lz(6)| g5 and
[z(el) —Z(62)[ g Mel-62])

6W = [continuous Zw—periodic functions w(e) with

R(k-l)n

values in such that lw(e)| g_ h5

and 1w(el)-W(62)lgh6l91 -92U

Lemma 2.3.2: Given 2(8) e 82, there exists a unique
pair of functions y(B), W(e) in BY and BW respectively

such that

(3.14a) y(61)= By(6)+h96(h,6.y(e).2(6).W(6))

+hG6(h,8,Y(9)IZ(9)IW(9))

+h2F6(h,e.y(9),Z(e),W(e))

69

(3.14b) w(el) =Dw(6)+h98(h,6.y(6).z(6).w(6))
+hGB(h.6.y(6).Z(6),W(6))

.+h2F8(h.e,y<e).z(e).w(e))

where is given by

91
(3-15) 91: 6+h+h95(h.6.y(6).z(6).w(6))

+hG5(h.6,Y(6).Z(6)oW(6))

+h2F5(h.e.y(9).Z(6).W(6))

Similarly given a pair of functions y(e),w(e) in BY and

B respectively, there exists a unique function z(e) in

W
82 such that

(3.16) 2(61) = CZ(9)-thg7(h.6.y(6).Z(6).W(G))
+hG7(h.6.y(e),Z(6),W(6))

-+h2F7(h,e,y(e),z(e).w(e))

where is given by (3.15).

61

Proof: Let 2(6) 6 B be given. We then construct

Z;
a mapping Tz :By.xBW 4 By.x6w as follows:
Let u(e), v(e) be a pair of functions in BY and

BW respectively. Fix a and let 3 be such that

70

(3.17) e = 6+h+hgs(h.6.u('e').z(6).v(6))
+hos(h,§.u(§).z('e').v(§))
+h2F5(h,3,u(§).z(‘é‘).v<6)) (mod 2..)

such 3 exists since the right hand side of (3.17) is

strictly increasing and covers an interval of length 2m

as E varies from 0 to 2m.

We then define Tz(u,v) = (6,?) where
(3.18a) me) = Bum+hg6(h.€.u(8),z(§).v(6))
+ho6(h.e.u(6),z(6),v(e))
+h2F6(h.6,u('6').z(6),v(e))
(3.18b) 3(9) = Dvfi)+h98(h.6.u(6),z(6).v(‘5))

+ hoB(h,€.u(6) .z(8) .v(e))

+ h2F8(h.'e‘,u(6) .z(6) ,v(6))

Let N denote a bound for the norms of g6, g7, 98’ G6’
G7, G8, F6, F7, F8 and some of their first and second

derivatives on the neighborhood [y[ g 5; [2[ g 5;
[w[ 3 h5. Using the fact that g6(h,e,y,z,w) has a second
order zero in (y,z,w) at (0,0,0) we have

‘2

[g6(h.§,u(§).z(€),v(‘e')| g N1|u(6) [2+ lz(8)|2+ [v(6) 1

2

g 2N5 +h2N52

71

Similarly since G6(h,e,y,z,w) = 0([w[) we have

[G6(h,6.u(6) .Z(6) ,v(6)[ g N]V(6)[

g hN5

and of course

[F6(h.3.u(6).z(3).v(5)[ g N

Hence (3.11a) gives

[fi(6)[ g (1-hb)[u(§)[-t2hN52-th3n52-th2N5-th2N
g 5 [1 —hb+ 2hN5 +h3N5+h2N+h2N/5]

36

if 5 = Kh where K 2 2N/b and h is sufficiently small.

Similarly we also have

[5(6)[ 3 e[v('6) [ + 2hN52+h3N52 +h2N5+h2N
g h5[€+ 2N5 +h2N5+hN+hN/5]

Shes

if 5 = Kh where K > N/(l-—e) and h is sufficiently

small.

To see that 6(a) and 3(6) are Lipschitz continuous,

let 91, 62 be given and let 81’ 62 be such that

72

(3.19a) 91 = §l+h+hg5(h,§l,u('e'l),z(§l),v(81))
+hG5(h.61.u(61).2(61).V(61))
+h2F5(h.El.u(81).z(61).v(81))

(3.19b) 62 = 62+h+hgs(h.62.U(6Z).z(62).V(_6_2))
+h85(h.62.u(§2).z('e‘2),v(§2))
+h2F5(h.62.u(62).z(62),V(62))

We have

|gs(h.§l.u(61),z(6l).v('e'l)) —g5(h.‘e'2.u(62),z(§2).v(62))l
g N62I6l-62l +N6IU(61) —u(62) [ +N6 [2(61) -2(62)[
+N6IV(61) -V(62)[

g (3N62+hN62

The first estimate follows from the fact that as
g5(h,e,y,z,w) has second order zero in (y,z,w) at (0,0,0)

so does 895(h,6.y.ZaW)/89-

Similarly since G5(h,e,y,z,w) = O([w[) we also have

aG5(h,6,y,2.W)/oe = O([W[): oG5(h,e.y.Z.W)/ay = O([W[) and

aG5(h.6)y,Z.W)/az = O([w[). Hence

73

[65(h.61.u(61).z(61),v(6l))-G5(h,62.u(62).z(62).v(62))

g Nho [[6l -62| + [u(61) -u(62) [ + [2(61) -2(_e_2)[]
+N[V(6l) -V(-e_2)[

g Nh6(2+26) [61—62]

Also

[85(h,‘§l,u(61),z(61).v(61))-85(h.62.u(32).z(62).v(‘e'2)|
g MEI—6“ + [u(61) -u(62) [ + [Z(61)- 2(62)[
+|v(61) —v(62)|1

g N[l+25+h5] [31'§2)

Hence
[61-6213 Yl[91_ e2l
where
11 = [l-h(3N62+hN62) -h2N5(2+ 26) -h2N(l+26 +h6)1'l

Note that the estimates for gj, Gj’ Fj are the same as

those for g5, G5, F5 when j = 6,7.

Therefore

74

[13(91) —fi(ez)l g [[3]] [11(31) ”(32)l

+ hlg6(h._e_l.u(6l) .Z(6 ).V(61))

H

— g6(h,62.u(62).z('e‘2).v(62))1
+ hlG6(h,'e'l.u(€l),z(81),v(61))
- G6(h,62.u(62),z(62),v(62))[
+ hle6(h,61,u('6'l),Z(-6'l).v(6l))
- F6(h,62,u(62).2(62),V(62))l
g[(1-hb)5+h(3N52+hN52)+h2N5(2+25)
2 _ _
+ h N(1+25+h6)] [61-62]
go Y2Y1 [61-62]
where

Y2 = l -hb+h(3N5 +hN5)+h2N(2+26)+h2N(1+25 +h6)/6

Note that
YZY]. S 1
if 5 = Kh where K 2 2N/b and if h is sufficiently
small.
Hence u 6 By.

Similarly

75

[3(61) -\7(62)[ 3 e[V(-§l) -v(62)|
+ hlgs(h,‘e'l.u(§l).z(§l).v(31))
-' 98(h.82:u(§2)'Z(82):V(_9-2))[
+ h|88(h.§1,u(‘61).z(61).v(§l))
- 88(h.62.u(62).z(62),V(62))l
+ h2188(h,§1.u(81).z(§l),v(el))
- F8(h:_6-21u(-9_2):Z(_9.2):V(82))[
g [eh5 +h(3N52+hN52) +h2N5(2+ 25)
+h2N(1+25+ho)] [61-62]

S hé Y3Y1 [91"92[

where

Y3 = e+3N5 +hN5 +hN(2+25)+hN(1+25 +h5)/5

where Y3Yl g 1 if 5 = Kh where K >N/(1-e) and if h

is sufficiently small.
Thus 6(6) E SW.

We now show that TZ is a contraction map. To that
end let (ul,vl) and (u2,v2) E By.xBW. Fix 9 and let

and e be such that

9l 2

(3.20a) e = el+ll+hg5(h,el,ul(81).Z(61).Vl(61))
+—hG5(h)el.ul(el)12(81),Vl(91))

2
+h F5(h,el,ul(el)12(91),Vl(el))

76

(3.201)) e = 62-th-+hgs(h,62.u2(62),z(92),V2(62))

-+h2F5(h.62.u2(92).Z(92).V2(62))

Note that

[95(h.el,ul(el),z(el),vl(el)) -gs(h.ez.u2(62).Z(62).V2(82))|
g Nazlel '62[+N6[[ul(el) -u2(62)[+|2(61) -Z(62)[
_ + [Vl(el) 'Vl(82)[]
3 [61-62] [3N62+hN62]
+ N5[[[u -u + v -v ]
. 1 211C. 11 1 211C.

Similarly

[G5(h,el.ul(el),z(el).vl(el)) -G5(h,92,u2(82),z(ez),v2(ez))[
g Nh5[[el -92[-t[ul(el) -u2(62)[-+[z(61) -z(92))]
+N'Vl(91) ‘V2(92)[
g (Nh5 + 3Nh52) [a1 - 62[ +Nh6Hul -u2[[CO

-+NHvl ‘VZHCO

77
[F5(h,el,ul(el).Z(el),vl(el)) —F5(h.921u2(92),2(92),v2(ez))[
S N[[61 ‘62[+ [111(61) -u2(c_)2)[+ [2(91) -Z(92[
g N[1-t25-+h5][gl —62[+-N[Hul —u2HCO-huvl -v2HCo]

Hence

[91 - 62[ 3 N11 (ha + hza +h2) nul -u2uco

+ (h5+h+h2)[[vl -V2[[ 0]
C

where N1 > 0 is some constant independent of 5 and h.

(3.213) {31(9) = 3111(61) +h96(h,el:ul(el) :Z(91) .Vl(61))
+hG6(h,el.ul(el).Z(el),vl(91))

2
+h F6(h,el,ul(el) ,z(el) .v1(gl))

(3.21 b) 62(4) = 8112(92) +hg6(h,92,u2(92).z(62).V2(92))
+hG6(h,ez,u2(92),2(92).v2(62))

+h2F6(h,92.u2(92)12(92),V2(92))

We have

[131(6)-1'32(6)[

78

g (1 -hb) ([[ul -u2[[co+ a [61 - ozp

+ hN[(352 +h52) [91 - 62) + 6 ([[ul -u2[[co

+ v -v ) ]
Ill 2Hc°
+ hN[(h5-+3h52)[el -92[
+lmHu -u +Hv -v ]
.1 zuco .1 2[[Co
+ h2N[(1+25 +115) [:31 -92(
+ Hu —u H -+ v -v ]
l 2"c0 [[1 2Hco

g (1 -hbl)[[ul -u2HC04'hK1HV1‘_V2HC0

for some constants b , Kl > 0. Hence

1

[[61 -fiz[[co g (1 -hb1) [[ul -u2[[co+hKl[[vl —v2[\co

Similarly from

(3.22 a) 51(6) =

(3.22 b) (72(9) =

We have

-+hG8(h,qllul(el).Z(el).vl(el))

2
+—h F8(h,gl,u1(el).2(el),vl(el))

Dv2(92)-th98(h,92,u2(ez).z(92).v2(92))
-thG8(h,92,u2(ez).Z(ez).v2(92))

2
-th F8(h,82.u2(92),z(92).V2(92))

79
+ hN[(352-+h52)[al -82[
+ 6(Uu1 'u2HCo*'HV1‘-V2HC0)1
+ hN[(h5+3h62) [61‘82[
+1mnu —u + V -V 1
1 21¢. 11 1 11¢.
+ h2N[(1+25 +h5) [91 —92[
+ [[ul-u +[[vl-v

211C. 111C111

S e1[["1 ‘V2[\co+h2K2Hul ‘u2\[co

for some 0 < 61 < 1, K2 > 0, assuming that 5 = Kh and

h is sufficiently small.

Hence
~ ~ 2
“V1 'Vzuco i €1HV1 ‘Vznco'Th KzHul 'uznco
1 —hbl th
Since the matrix 2 has spectrum inside
h K2 61

the unit circle, TZ is a contraction map. The fixed point
of T2 is the pair of functions (y(g),w(g)) in BY'XEW

which satisfy (3.14).

The proof of the second part of Lemma 2.3.2 is similar

and hence omitted. Q.E.D.

If the nontrivial characteristic multipliers

“2"'°’Hn of (3.1) satisfy [“j[ < l for j = 2,...,n,

 

80
then by letting 2(9) 5 0, Lemma 2.3.2 provides an
attracting invariant curve under T which then yields an

attracting invariant curve for (3.2).

In the more general case, i.e., when not all “j
satisfy [Uji < l, we actually need a stronger result
than Lemma 2.3.2. More precisely, let 21(9) and z2(e)
be two functions in BZ' Let (yl,wl) and (y2,w2) be

the fixed points of T2 and T2 as provided by Lemma 2.3.2.
1 2

Let 9 be fixed and 91, 92 be such that

(3.23 a) o = el+h+hg5(h,el.yl(el),zl(el),wl(el))
+hG5(h.el.yl(el),zl(el),wl(el))

2
+h F5(h’91’yl(91)'zl(91)’w1(61))

(3.23 b) e: 92+h+hg5(h,92.y2(62).22(92).w2(92))

+ hc;5 (h, 92.y2(92) 122(62) .w2(62))

2
+h F5(h,92,y2(92) 122(62) ,w2(ez))

Solving for [31 -92) we obtain
- N [(h -+h2 4-h2)H — +(h -+h2 -+h2) -
[61 62[ g 2 6 6 ..Y1 Y2HCo 5 6 H21 zzHco
2
+ m5+h+h le-wfl[d
C

Hence from

 

81

(3.24 a) yl(9) Byl(el)+h96(h’91’y1(91)’zl(91)'wl(el))
+hG6(h’el’y1(el)'zl(el)'wl(el))

2
+h F6(h,el.yl(el) ,zl(el) .wl(el))

(3.2418) y2(e) = By2(e2) +hg6(h,92.y2(92),z2(92).w2(92))
+hG6(h'92'Y2(92)'22(92)'w2(92))

2
+h F5(h'92'yz(92) .z2(ez) ,w2(ez))
We obtain

”Y1 ’quco 3 (1‘-hb1)HY1 ‘anco'FhKIle 'wzHCo
2
+ h K3|[zl -z2[[co

for some constants bl’ K1' K3 > 0.

Similarly we also have

le--w2[[CO g 51“ l -W2Hco'thK2HY1 ‘YZHCO

2
+h K3[[zl — ZZHCO

Conversely, suppose two pairs (yl,wl) and (y2,w2) in
BY XBW are given then Lemma 2.3.2 provides two functions

2 22 in 5% which are the fixed points of mappings

1!
defined by (yl’wl) and (y2,w2) respectively. We then have

82

[[21 " z2[[Co S 0* "hC1) [[21 ' Zz[[co+ hK4[[w1 ‘w2[[co

2
+ h KSH-yl ‘YzHCo

As in Chapter 1, we can proceed to construct a sequence

(K)

(y(K)(e). Z<K)(e),w (9)) which converges uniformly to
(y(a),z(e),w(e)). These functions provide a curve invariant
under (3.2). Thus we have

Theorem 2.3.3: The difference equation (3.2) possesses
an invariant curve which lies inside an 0(h)—neighborhood

of the corresponding periodic orbit of (3.1).

Remark 2.3.4: As in Chapter 1, if (3.1) depends on
a parameter d, we also obtain the continuation of the
invariant curves for (3.2) away from the bifurcation points

and near the Hopf bifurcation.

Remark 2.3.5: Thus far we have assumed that the method
employed is strongly stable, convergent, explicit, linear
multistep method. We now try to weaken some of the above
conditions. First suppose some nonlinear method is used and

yields a difference equation

(3.25) xm+1 = dj Xm+1-j'[

1W

2
+ h G<Xm+l—k"‘° m

where the term h2 G(x .,xm) corresponds to the

m+1-k"'

nonlinear part ofthe method employed. Assuming that

K
Z) d. = 0
j=o3
K K
D Bj= E jet-#0
j=l i=1 3
K K_.
and the roots of the polynomial p(e) = Z) dj 8 J are
i=0
1, $2,...,€K where [€j[ < 1 for i = 2,...,n. Then all

the arguments given in this chapter still hold true since

2 . .
the term h G<Xm+l-k""’xm) is very small when h is

small. Thus we do have invariant curves for (3.25)

provided that h is sufficiently small.

Now suppose an implicit method is used to yield

K
(3.26) Xm+l = [:1 )

3

Clj Xm+1—j+h .23 Bj f(xm-l-l-j

+ h2G(x )

... x
m+1—K’ ’ m’ m+1

then by writing the above equation as a mapping from

:m‘n .olflxn and apply the argument given in Remark 1.3.4,

we obtain an equivalent, explicit difference equation of

the form
K K
= v‘ _
(3.27) Xm+1 jél dj Xm+l-j4'h j:1 (Bj Bodj)f(xm+l_j)
2
+ h G1(Xm+l-k" ’Xm)
Note that
K K K K
E (B —ch.)= z B--B 2: o.= z 8.

 

84

thus we can proceed to obtain invariant curves for (3.27)
which are also invariant under (3.26).

The requirement that the method employed is strongly
stable is important, however. For example suppose we

study the system
(3.28) x = ax; x e Ii 7 a < 0
using the midpoint rule which yields the difference equation

(3.29) x = x

m+1 +2h a xm

m—l

Note that x = 0 is a solutiOn which is attracting under
the flow of (3.28). On the other hand since the characteristic

equation for (3.29) is
2
A -2h a x-tl = 0
which has two roots

ha+ [1+ (ha)2]l/2

X1
X2 = -1/Xl
where [11[ < 1 and [12[ > 1. It is easy to see that the

degenerate invariant curve x = O is not attracting under

(3.29).

3. TIME DELAYED EQUATION

3.1 The Hopf Bifurcation

Consider a time delayed equation of the form
(1.1) x = f(d,x(t -l))
= a(d)x(t -l) +g(d,x(t -l))

where 1(61R, g(d,x) = 0([x[2) and f(d,x) is as smooth

in d and x as needed.
The linearized equation for (1.1) is
(1.2) y= a(a)y(t-l)
with characteristic equation
(1.3) 1-a(d)e_x = 0

Assume that (1.3) has a pair of complex conjugate solutions
x(d) and x(d) satisfying

(1.4) Re ((0) = 0; Re )’(0) > 0; Im 1(0) # 0

while all other solutions have negative real part and

stay uniformly away from the imaginary axis. It is then
well known that (1.1) possesses a family of periodic

orbits bifurcating from the zero solution as d crosses

0 (see [4] or [8]).

85

 

86

Now suppose we approximate (1.1) using the Euler method
with step size h = 1/n where n is a positive integer.

.We then Obtain a difference equation

(1.5) x = xK+ha(d)xK_n+hg(d,x )

K+l K-n

Since (1.5) requires ni-l starting points, it is

natural to think of it as a mapping from Iin+1 to
Iin+l. To that end we set
_ T
zK - [XK_n ...XK]
_ T
zK+l - [Xx-n+1 '°°xx+1]
where T denotes tranposition. (1.5) then defines a

1 l

mapping U : IRn+ 4 Iin+ by

(1.6) UzK = zK+l = A(d)zK+hg(d,zK)
where A(d) is an (nmtl) x(nutl) matrix
1
F 0 1 0 . . . 0
0 0 l 0
A(d) = . . . . . 0
l 0
0 0 0 1
ha(d) 0 . . . 0 O‘J

 

 

and G(d,zK) is a (ni-l) vector.

87

_ T
G(G,ZK) — [0 ...0 9(d’xx-n)]

The characteristic equation for A(d) is

u.n JHl-wn—ham)==o

hB .
Set w = e then (1.7) can be written as
(1.8) (eh’3 -1)/h -a(d)e-B = 0

We have the following

Lemma 3.1.1: Let x be a solution of (1.3)
satisfying 1 ¥ -1. Then for h > 0 sufficiently small
there exists a solution B(h) of (1.8) such that

lim B(h) = 1
n40
Proof: Set

(ehB -l)/h if h a o
V(h,B)=

then v is continuously differentiable in B and h.

Define

F(hIB) = v(hIB) -a(u)e-
then

no.1) = 1-a(cx)e‘x = o

 

88

and
aF(O.).)/a). = “a(d)e": = 1+1 14 o

The lemma follows from the Implicit Function Theorem.

Generically we may assume that a(O) # -e-1 so
that -l is not a root of p(x) = x-a(d)e_x for d
near 0. Since p’(x)-= 1-+a(d)e'x, the above
assumption implies that all the roots of p(x) are

distinct.
Now note that every solution x of (1.3) satisfies

—Re

Re x = a(d)e 1 cos (Im1)

which implies that Re 1 < M1 for some constant Ml'

Also

-Re R

Im x = a(d)e sin (Imx)

which yields [Inlxl < M2 for some M2 > 0.

Since (1.3) is entire in x, it follows that given
any real number b, there exist only1afinite number of

solutions of (1.3) say, Xl""'kx with real parts > b.

Similarly for h sufficiently small, (1.8) also has a

finite numbers of solutions with real parts > b. These
solutions, being bounded in a compact set, must converge
to k., j = 1,...,K as h 4 0. Thus there are K such

3
solutions 81,...,BK and

89

[Bj -xj[ g.Mh for j = 1,...,K

for some constant M > 0.

Let 11(0). 12(d) be the pair of solutions of (1.3)
satisfying (1.4). Then (1.7) has a pair of complex

conjugate solutions ml(G), w2(0) such that

(18) UUj(o): 1+hpflo)+oa3) j= L2

These wj(d) cross the unit circle as d crosses some

value dh = 0(h).
From the argument above it also follows that all other
solutions of (1.7) satisfy

[w[ 3 l —hb

for some b > 0. We now could proceed as in Chapter 1
to obtain a family of invariant curves for (1.5) which

bifurcates from the zero solution as d crosses 0.

To see that these curves approximate the periodic

orbits of (1.1) we introduce
5 =[Lipschitz continuous functions on [-1,0]]
then 6 is a Banach space with norm

[o[= [o(0)[+ sup [1129(8 ) -co(6 ))/[e -6 [1
4361.6ng 1 2 1 2
el#62

 

90

_ T n+1 .
For a vector 2 - [21...zn+l] 61R we define In(z)
to be a function in 5 with values at -l, -l+h, . . ..0
equal to 21,...,zn+1 and is linear on each of the
intervals [-l+-jh, -li-(j4-l)h] for j = 0,...,n-l.

Conversely given a function @658. ‘we define Jn(m) to
be a vector in Iin+1 whose components equal to the values

of m at —l, -li—h,...,0.

Let @658 ‘be given with [m[ g.5 for some 5 > 0.

Let x(t) be the solution of

3&(t)

f(x(t-l)) for tZO

x(t)

cp(t) for -1§_tgo

let T :5’4 5 'be the mapping defined by

ww(0) = x(l-te) for -l g_e g.0

Similarly define Fn :5)4 5 'by uh = Int] Jn

82
then ‘mm(92)-mm(el) = [ f(m(s))ds

91
while

62

vncp(62) -1rnso(el) =j‘ En f(tp(S))ds

6l
where En f(w(s)) is the step function which equals to
f(m(-14-jh)) on [-l-tjh, -l-t(j-+l)h] for j = 0,...,n -1.

Since a(s) is Lipschitz continuous with Lipschitz

constant 5, f(m(s)) is also Lipschitz continuous with

 

91

Lipschitz constant N5 for some N > 0. This implies
that
[f(w(S)) -En f(m(s))[ g N5h

and hence
[vnm -vw[ g N5h
or fine converges uniformly to Fm as n-:w for

[:0] S5-

Let ca 65 be such that [m[ g 5 then for any cal 68

we have

6
F(w-ttm1)(6) = p(O)-+tml(O)-+[ f(o(s)+-tml(s))ds
—1
which gives for —1 g 61 g 92 g 0

DW(w)ml(62) - DW(u)wl(6l)
92
= [ f’(o(S))ol(S)ds
91
Similarly we also have
62
= [ En f’(m(s))ml(s)ds
61
This implies

[DTrn(CD)CDl *DTF(<o)ml[ g Nlhalell

for some N1 > 0. Hence

 

92
Hflfin(o)-DH(T)H g_Nih6

Let X(d) = eigenspace of DH(0) correSponding to the

eX(a) ex<a) , then there exists a

eigenvalues and
subspace Y(d) of B such that B = X(d) 0 Y(d). By
choosing some appropriate basis for X(d) and projecting

to X(d) and Y(d) we can decompose II as II : (x,y) 4 (§,§)

where
(1.10 a) 3:- = A(d)x+F(d,x,y)
(1.1013) y = B(d)y+G(d,x,y)

where x}:- 6 Ii2

: 17.3? e Y(d); A(d) is a 2 x2 matrix with
eigenvalues X(d) and 'TTET: B(d) is a linear mapping
from Y(d) to Y(d) with spectrum inside and away from
the unit circle. F : JR le2 xY(d) 4 R2; G : JR )(ZIR2 xY(d) 4Y(d).

Also F and G have second order zeroes in (x,y) at

(0,0).
Similarly we can decompose Hn’ as fin :(x,y) 4 (xn,yn)
where
(1.11 a) xn = An(o)x+hcn(a)y+Fn(a.X.y)
(1.11 b) yn = Bn(d)y+th(d)x+Gn(d,x,y)

The extra terms hCn(d)y and th(d)x appear due to the

fact that X(d) and Y(d) are not the eigenspaces of

H .
n

93

Using the fact that Tn(o) and Dwn(¢) converge uniformly
to v(m) and Dr(¢) respectively for all m satisfying
[¢[ $.5, we have An(d) 4 A(d) as bounded linear Operators
from X to X, Bn(d) 4 B(d) as linear Operators from Y
to Y. Also Fn(d,x,y), Gn(d,x,y) and their derivatives
converge uniformly to F(d,x,y), G(d,x,y) and the
corresponding derivatives on a neighborhood of

(a.X.y) = (0.0.0).

We now could proceed as in Chapter 1 to obtain a family
of invariant curves for Wn(d) for dn < d < e or
-e < d < dn in the generic case where dn 4 0 as

n 4 0 and e > 0 is independent of n.

Theorem 3.1.2: If (1.1) is approximated by the
Euler method then the resulting difference equation
(equation (1.5)), in the generic case, possesses a family
of invariant curves depending continuously on d for
either dn < d < e or -e < d < dn where dn 4 0 as

n 4 0 while 6 > 0 is independent of n.

Remark 3.1.3: As in Chapter 1, Theorem 3.1.2 still
holds true if some convergent single step method other
than the Euler method is employed or when some solutions
of the characteristic equation (1.3) have positive real
parts but stay uniformly away from the imaginary axis
for all d near 0. It is also true if (1.1) has a

more general form such as

 

1': = t(o.x(t) ,x(t-1)); x e 13“

provided that the corresponding characteristic equation
still has a pair of complex conjugate solutions
satisfying the Hopf bifurcation (1.4) while all other
solutions stay uniformly away from the imaginary axis for

d near 0.

3.2 Existence and Continuation

 

As in Chapter 1, we now show the existence of the
invariant curves away from the bifurcation points.

More precisely we now have a time delayed equation
(2.1) x = f(x(t -l))7 x 61R

which has an w—periodic solution p(t). This defines a

periodic orbit r in C = C([-1,0],IR) by
I‘=[PS[OSS§(1)[:
where PS EC is given by

Ps(e) = P(s+—e) for —l g e g 0

We assume that no nontrivial characteristic multipliers
of r lie on the unit circle. Thus by rearrangement if
necessary, we may assume that the multipliers

“1’“2""’“L"“ of r satisfy

 

 

u1=171ujl>l 1.1 221.41:
and [“j[ < l for j 2_1-+l
Now suppose we approximate (2.1) using the Euler method

with step size h = 1/n for some positive integer n.

This yields a difference equation

(2.2) xK+l = xxi-hf(XK_n)
. . . n+1 n+1 .
which induces a mapping U :Ii 4 Ii given by
U :2K 4 zK+l
_ T
where zK — [Xx-n x ]
- T
z((+1 _ [Xx—n+1 "'Xx+l]
Set N = [w [h] and define
S = UN

Also let 5, In, Jn be as in section 3.1. The mapping

S then yields a mapping from 5 4,6 given by

W' = I SJ
n n n

Let F :6 4 B be the period map of (2.11),i.e., given

a function a(ga, we solve the initial value problem
{:(t) = x(t—l); x(t) = (p(t) for —l g t g o

and define

Wo(e) = X(wl-e) for -l g 6 g 0

 

96

By repeating the argument given in section 3.1 several

times if necessary we can show that
[fine -W@[ 3 Nlh
HDwn(m) -DW(®)H g N2h

for all m near r, i.e., @658 and [m-Ps[ is small

for some 5 e[0.w).

Note that DW(PS) has eigenvalues “1,...,u£,... for

all 5. Thus we can decompose 3 as
B = E(s)<3K(s)

where E(s) is the one dimensional eigenspace of Dv(Ps)
corresponding to the eigenvalue “1 = 1. It is easy to

see that PS is a basis for E(s).

Let \7 be a neighborhood of P 1J1 8. Define

0

H(s,¢,z) = Ps'*¢ —z
for 56112, ¢6K(s), zEV. Then

H(0,0,P = 0

o)
and the derivative of H with respect to s, 0 evaluated
at s = 0, ¢ = 0 and the pair (0,5), 0 eIi, [ EK(0) is

0

are linearly independent, the above derivative has a

POO+-¢. Since P is a basis for E(O) and E(O), K(O)

bounded inverse. The implicit function theorem implies

 

97

there is a 5 > 0 and unique 5(2) and O(z) continuously
differentiable with respect to z for [2 —PO[ < 5 so
that H(s(z),¢(z),z) = 0. Since r is compact we can

apply the above argument a finite number of time to Obtain

Lemma 3.2.1: There exists a neighborhood W of 1"
in B such that for any z eW there exists unique

se[0,w) and 06K(s) such that z=Ps+q>.
Let 2 = Ps-+¢ 6W. Then
whz = wz+hF(h,z)

where F and its derivative with respect to z are

uniformly bounded for z eW and
lTZ = v(PS) +D1r(Ps)q)+g(s,q))
= PS+D7r(Ps)q)+g(s,q>)
where g(s,¢) = 0([¢[2). Note also that Dv(Ps)¢(EK(s).

Suppose VnZ = Pg~t¢ where O(EK(s) then Since 5 and
T depend continuously differentiably on vnz and hence

on 2, we have
(2.3a) 's‘ : s+gl(s,q)) +hFl(h,s.<p)
(2.3b) 6 = DTr(Ps)Q+g2(S,C[)) +hF2(h,s.<p)

where gj(s,¢) = 0([¢[2) for j = 1,2 while Fj(h,s,Q)

and their derivatives are uniformly bounded for j = 1,2.

 

98
Assuming that [pj[ < l for all j Z 2 then
”DF(PS)” g p < l for all 5. Define

u = {continuous w-periodic functions u(s) with
values in B such that u(s) EK(s), [u(s)[ g 5

and [u(sl) -u(sz)[ g 5[sl -52[]
where 5 > 0 is to be determined.

As in Chapter 1, given u.eu we define :u to be

the function satisfying

(2.4a) 311(5) = DTT(PT)U(T) +gz(¢.u(7))

+ hF2(hITIu(T))
where T is such that
(2.4b) s = T+gl(¢,u(1)) +hFl(h,¢,u(T))

Since HDF<PS)H g p < l we can proceed as in Chapter 1 to
show that in 6Q if 5 = Mh for some constant M > O

and h is sufficiently small.

Now given ul,u2 eu. We have

(2.5a) Jul(S) = DTT(PTl)ul(Tl) +92(Tl'u1(3'1))

+ hFZ (thlIul(Tl))

(2.5b) Ju2(S) D1r(PT2)u2(T2) +gz(¢2,u2(T2))

+

th (hrTzru2(T2))

 

99

where T1 and T2 are given by
(2.6a) S = Tl+gl(T1:ul(Tl))

+hF1 (hITlIul(Tl))

(2-6b) S = T2+gl(T2Iu2(T2))

From (2.6a) and (2.6b) we have

[T1 -.2[ g Nl(5+h) nul 412”

where Hul -u2H = 52p [ul(s) —u2(s)[ and N1 > 0 is some

constant independent of 5 and h. Hence (2.5) gives

for some constant N2 > 0 if 5 = Kh and h sufficiently

small.

Hence for h sufficiently small, I is a contraction map
which has a fixed point. This yields an invariant curve

for (2.3).

In the general case when [“j[ > 1 for 2 g j g z
and luji < l for j > L where 1 2 2 we decompose
B as B = Kl(s)(3K2(s) where Kl(s) is the generalized
eigenspace of Dw(PS) corresponding to 52,...,u£. We then

define

 

100

'24]. = [continuous (u-periodic functions u(s) with
values in B such that u(s) 6Kj(s) , [u(s)] g 5

and [u(sl) -u(sz)[-g5[sl-sz[} for j= 1,2.

Given a function u:L 61(1) we define a mapping .7 14:2 4-742

by

(2.7a)ul(s)+.7u2(5) D1r(PT) (ulbr) +u2(7))
+ 92(Toul(‘T) +u2(T))
+ hF2(h,Tpul(T) +u2(¢))
for u2 61242 Where T is given by

(2.7b) s = T+gl(7.ul(T)+u2(T))

+hFl (hITIu1(T) +u2(T))

As before we can show that 3’ is a contraction map and

hence has a fixed point u2 63:42 such that

(2.8a) u1(s)+u2(s) DTT(PT) (u1(..) +u2(7))
+ 92(Trul(T) +u2('r))
+ hF2(h,q-,ul(rr) +u2(T))
where

(2.81)) S = T+gl(TIul(T)+u2(T))

+hF1(h,1-,ul('l') +u2(T))

 

101
Conversely given 1.1.2 652 there exists u.l eul such that
(2.8) is satisfied.

As in Chapter 1 we now construct sequences of functions

uik) and uéx) in In, and .fl: by defining

u{0)(s) a uéo)(s) = o

Inductively given ufK) we construct a contraction map
:2 :u2-4u2 and let uéK+l) be the fixed point of 72.
Using uéx+l) we then construct a contraction map

(K+l)

:1 :ul 431 and let 111 be its fixed point. As in

Chapter 1 we can show that

(K+1) _u(1<)H
1

”“1 K+l) _uéK)”

i Klflué

( +1) ( ) ( —l)
(1122" 1“ -u1" H

_uéx)” g_K2Hu

where K1,K2 < 1 if 5 = Mh and h is sufficiently small.

(K) (K)

Thus ul and u2 converge to some functions ul and
u2 as K 4 a. These functions define an invariant curve

for (2.3).

Now let P be an invariant curve for 7n then so is

IntJJn(F). Uniqueness implies that IntIJn(f) = f or
f is invariant under IntIJn. This means that Jn(f),
as a curve in Iin+l is invariant under U.

Thus we have

 

102

Theorem.3.2.2: For h = 1/n sufficiently small,

1 which, when

(2.2) has an invariant curve in Iin+
imbedded in B, converges to the periodic orbit of (2.1)

as h 4 0.

Remarx 3.2.3: As in Chapter 1, we also obtain the
continuation of invariant curves away from the bifurcation
points and near the Hopf bifurcation. Also Theorem 3.2.2
is still true when a more general system such as
{c = f(x(t),x(t-1)); xean

is approximated by some convergent single—step method.

Remark 3.2.4: In [5], Georg reported that in his

 

numerical studies, the Euler method provided the best
result in tracking the periodic orbits. Also decreasing
the step size in many case did not improve the rate of
convergence. These observations can be explained by
noticing that under any convergent single-step method or
linear convergent, strongly stable multistep method the
rates of convergence to the invariant curves of the
resulting difference equations are roughly the same as
the rates of convergence to the periodic orbits. Thus
for example it is erroneous to assume that the Runge-Kutta
methods would yield the invariant curves more quicxly

than the Euler method.

 

103

In case all the nontrivial characteristic multipliers
are inside the unit circle, the invariant curves are
attracting if the step size h is sufficiently small. Thus
if the round-off error is small compared to h, we could
obtain approximations to those curves numerically using
only some standard numerical method such as the Euler

method.

If one or more of the multipliers cross the unit
circle, the corresponding invariant curves are no longer
attracting. Thus some special procedure is required. One
such procedure involves transforming the problem of finding
the periodic orbits into the problem of solving the equation
F(d,x) = w(d,x)-—x = 0 where W(G,X) is the period map
(or Poincare map). v(d.x) and its Jacobian can be
approximated using some standard numerical method. The
Newton method or some of its modified form can be used to
generate the approximations to the solutions of F(d,x) = 0.
Discussions and numerical results of the above procedure

can be found in [5] and [6].

BIBLIOGRAPHY

 

BIBLIOGRAPHY

[1] Alexander, J.C. and Yorke, J.A., Global bifurcations
of periodic orbits, Amer. J. Math. 100 (1978),
263-292.

[2] Braun, M. and Hershenov, J., Periodic solutions of
finite difference equations, Quart. Appl. Math.
35 (1977). 139-147.

[3] Chow, S.N. and Mallet-Paret, J., The Fuller index
and global HOpf bifurcation, J. Differential
Equations 29 (1978), 66—84.

[4] Chow, S.N. and Mallet-Paret, J., Integral averaging
and bifurcation, J. Differential Equations 26
(1977), 112-159.

[5] Georg, K., Numerical Integration of the equation,
'Numerical Solutions of Nonlinear Equations',
Springer Lecture Notes, vol. 878 (1981).

[6] Hadeler, K.P., Computation of periodic orbits and
bifurcation diagrams, Numer. Math. 34 (1980),
439-455.

[7] Hale, J.K., 'Ordinary Differential Equations', Wiley-
Interscience, New York (1969).

[8] Hale, J.K., 'Theory of Functional Differential Equations',
Applied Math. Sci. vol. 3, Springer-Verlag,
New York (1977).

[9] Henrici, P., 'Discrete Variable Methods in Ordinary
Differential Equations', John Wiley and Sons,
New York (1962).

[10] HOpf, E., Abzweizung einer periodischen Lolung von einer
stationaren Losung eives Differential systems,
Ber. Verh. Sachs. Akad. Wiss. Leipsig Math. - Nat.
94 (1942). 3-22.

104

 

[11]

[12]

[l3]

[14]

105

Lanford, 0.E., Bifurcation of periodic solutions into
invariant tori: the work of Ruelle and Takens,
'NOnlinear Problems in the Physical Sciences
and Biology', Springer Lecture Notes, vol. 322
(1973).

Mallet-Paret, J. and Yorke, J.A., Snakes: Oriented
families of periodic orbits, their sources
and sinks, J. Differential Equations 43 (1982).
419-450.

Marsden, J.E. and McCracken, M., 'The HOpf Bifurcation
and Its Applications', Applied Math. Sci. vol.
19, Springer—Verlag, New York (1976).

Wan, Y.H., 0n the uniqueness of invariant manifolds,
J. Differential Equations 24 (1977), 268-273.

 

 

 

 

 

 

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