BEFUWTSOK THEGRY NYE-'1‘
REPUSMENS ”EC GHEMQXL EEQCTEOH
EQUAEGNS '

 

 

This is to certify that the

thesis entitled

BIFURCATION THEORY WITH
APPLICATIONS TO CHEMICAL REACTION EQUATIONS

presented by
Ms . Nancy Theresa Waller
has been accepted towards fulfillment

of the requirements for

P114). degree in Mathematics

 

 

'éL'wrxL UL

Major professor .

 

0-7639

 

  

? “3m av
F 5- “W
0mm IND:

LIBRARY BIND? ES
:rmaun were 3“

   
    

   

||||l|llllll|||||llllllllllllllllllllllllllllllllllllllllll :2”th

3 1293 10530 9474

ABSTRACT

BIFURCATION THEORY WITH APPLICATIONS
TO CHEMICAL REACTION EQUATIONS

BY

Nancy Theresa waller

This thesis concerns families of nonlinear differential
equations in a Banach space which depend on one or more para—
meters. At certain critical values of the parameters, non-
trivial equilibrium states may bifurcate from the trivial
solution. ‘we consider two cases.

In the first case, the generalized null space of the
linear part of the system is one-dimensional at the bifur-
cation point, and the system depends on a single parameter.
We determine the number and magnitude of the bifurcating
solutions and their stability properties.

The second case involves dependence on two parameters.
We consider the situation where there are two "bifurcation"
curves in the parameter plane which intersect transversally.
The linear part of the system which corresponds to these
curves has a one-dimensional generalized null space, except
at the intersection where it is two-dimensional. We develop
analytical methods which can be applied to study the number

and magnitude of the bifurcating solutions as a function of

Nancy Theresa waller

the parameters near the bifurcation point. We then apply
these techniques to a system of partial differential equa-

tions which arises in the study of chemical reactions.

BIFURCATION THEORY WITH APPLICATIONS
TO CHEMICAL REACTION EQUATIONS

BY

Nancy Theresa waller

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Mathematics

1976

In memory of my grandmother,
Amelia Zegaren

ii

ACKNOWLEDGEMENTS

I am indebted to my thesis advisor, Professor Shui—Nee
Chow, for his stimulating mathematical discussions. I am
especially grateful to my academic advisors, Professors
Shui-Nee Chow and Lee M. Sonneborn for their encouragement
and the confidence they had in me.

I extend my appreciation to the faculty of the Mathe-
matics Department of Michigan State University, to the
secretaries and my friends; and to my husband for his love
and support.

Finally, I would like to thank Ms. Jill Hagan for her

time and her careful preparation of the final text.

iii

CHAPTER I:
§1.
92.

§3.

§4.

TABLE OF CONTENTS

PRELIMINARIES . . . .
Bifurcation . . . . . . . . . . .
Differential Equations and Stability

Analytic Functions and the Implicit
Function Theorem . . . . . . . . .

Remarks . . . . . . . . . . .

CHAPTER II: BIFURCATION AND STABILITY-ONE

§1.
§2.

§3.

§4.

DIMENSIONAL NULL SPACE . . . .
Preliminaries . . . . . . . . . . .

Reduction to the Finite-dimensional
PrObleln O O O O O I O O O O O O O O

Bifurcation [One Dimensional Null
Space] . . . . . . . . . . . . . .

Stability . . . . . . . . . . . .

CHAPTER III: A TWO-DIMENSIONAL NULL SPACE

g1.
§2.
g3.
g4.
§5.
§6.

Introduction . . . . . . . . . . .
The Bifurcation Equations . . . . .
An Appropriate Change of Scale
Solutions of TYpe I-Fold curves
Solutions of Type II. . . . . .

Solutions near (XO’YO’YO’

iv

0) = (0,0,0,0).

Page

12

15
23

30
3O
34
37
45
54

71

Page

CHAPTER IV: APPLICATION TO A SYSTEM OF
CHEMICAL REACTION EQUATIONS . . . . . . 77

§l. Introduction . . . . . . . . . . . . . . . 77

92. Eigenvalues, Eigenvectors, and
Projections . . . . . . . . . . . . . . . 80

§3. Calculation of the Bifurcation
Equation . . . . . . . . . . . . . . . . . 84

§4. Analysis of the Bifurcation Equation . . . 92

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 104

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure
Figure
Figure

Figure

Figure

10.

11.
12.
13.

14.

15.

LIST OF FIGURES

Transversal crossing of zero
eigencurves . . . . . . . . . . . .

Dependence on (T,n) of the number of
nontrivial solutions of (3.1) . . . . .

Correspondence between (T,n)-plane and
(n,Y)eplane . . . . . . . . . . . . . .

Dependence on (n,y) of the number of
nontrivial solutions of (3.2) .

Behavior near a fold curve in the
(§,y)-plane . . . . . . . . . . . . . .

Behavior near a fold curve in the
(T , 7") -plane a o o o o o o o o o o 0

Behavior near a cusp . . . . . . . . .

Bifurcation diagram in the

(T,Y)-plane . . . . . . . . . . . . . .
Bifurcation diagram in the

(T, n)-plane o o o o o o o o o o o o o 0
Behavior of nontrivial solutions of
system (6.2) . . . . . . . . . . . . .
The curve Red; = O . . . . . .

A typical curve of neutral stability .

The sectors S1 and S

2 O O I I O

Bifurcation diagrams for the system
(3.10) in the (T,n)-plane . . . . . . .

Intuitive representation of bifurcation
When he < O and c/a < O. . . . . . .

vi

Page

33

38

39

4O

53

53

68

69

69

72
82
82

92

101

102

CHAPTER I

PRELIMINARIES

§l. Bifurcation

As a simple example of bifurcation, consider what is
observed experimentally when a compressive axial thrust
is applied to a thin elastic rod [12]. As the thrust T
is gradually increased from zero, the rod first becomes
thicker and shorter, but its center line remains straight.
The classical linear theory of elasticity predicts this
straight state to be the unique equilibrium state of the
rod for all values of T. However, when T readhes a
certain critical value To, the rod is observed to buckle
into a bent state which becomes more pronounced as T in-
creases. The classical theory is inadequate to describe
the phenomenon of buckling. When nonlinear effects are no
longer neglected, one may construct a model which predicts
both the straight state and the bent state as possible equi-
librium states for T greater than To. Intuitively, the
straight state ”loses its stability" to the bent state at
this critical value.

Let X, Y, and A be Banach spaces, and let F: AxX-pY

be continuous. We will say that x

O is an equilibrium

state corresponding to do in A if x0 is a solution
of F(ao,x) = 0. we will say that (ao,x0) is a bifur-

cation point for T if and only if

(i) F(a0,x0) = 0

(ii) For every neighborhood V of (a ,xo), there exist
a in A and.:xl,x2 in. X. with x1 # x2, such that
(a,x1) and (a,x2) are in V, and F(a,xl) = F(a,x2) = 0.
That is, we can always find a arbitrarily close to do
such that there is more that one equilibrium state near
xo that corresponds to a.
§2. Differential Equations and Stability

First we shall mention some standard results, from the
theory of ordinary differential equations [£3]. Let
A: £514 If} he a continuous linear map, and let N: If!» 151
be continuous, with N(O) = O, DN(O) = 0 [where DN(O) de-
notes the Fréchet derivative of N at x = 0]. Consider

the ordinary differential equation

(2.1) %%=Ax+N(x)

and the associated linear differential equation

:13:

(2.2) at

= Ax.
Definition 2.1. The zero solution of (2.1) (or of
(2.2)) is said to be uniformly asymptotically stable if

and only if there exists 6 > 0 such that lxbl < e

implies |x(t)| 4 0 as t 4 m where x(t) is the solu-
tion of (2.1) (or of (2.2)) with x(0) = x0. A necessary
and sufficient condition that the system (2.2) be uniformly
asymptotically stable is that all the eigenvalues of A

have negative real parts. If this is the case, there exist

positive constants, K and a, such that

-a(t-t0)
lx(t)\ < ‘xol Ke
where x(t) is the solution of (2.2) with initial condition
x(to) = x0. If N(x) is O(|x|2) as x approaches zero,
the asymptotic stability of (2.2) implies that the zero
solution of (2.1) is asymptotically stable.

‘We will now consider differential equations in a Banach
space X. Let A be a closed linear operator whose domain
is a dense subspace D(A) contained in x. It is useful
to consider D(A) as a Banach space Y with the so-called
graph norm: |z|Y = lzlx + |Azlx. The injection j: Y 4»X
is continuous with dense range, and A: Y 41x is then a
continuous linear map. Let N: Y 4.x. be continuous, with
N(O) = O and DN(O) = 0. Consider the differential equa-

tion

(2.3) %% = Az+N(z)

and the associated linear differential equation

dz _
(2.4) 3-1:- — AZ.

Here

dz

dt

z(t+h)-z(t)

lim
hHO h

where the limit is taken in the norm l-lx.
Motivated by the results for ordinary differential

equations, we make the following definition.

Definition 2.2. we shall say that the system (2.4)
is stable if there exists a 5 < 0 such that whenever
A is in the spectrum of A, Rex < 5. We shall say that
the system (2.4) is unstable if part of the spectrum lies
to the right of the imaginary axis.

In many cases, given that (2.4) is stable in this
sense, it is possible to prove stability results analogous
to those for ordinary differential equations [10], [l3],
[14]. That is, if the linear system (2.4) is stable in the
sense of definition (2.2), then the zero solution of the
nonlinear system (2.3) is stable. These results depend on
the particular properties of the operator A, and we will
not go into them here. Instead we make the following hy-

pothesis.

Principle of Linearized Stability. Let 2 = 20 be

an equilibrium solution of (2.3). If the linearization of

(2.3) about 2 given by

O)

dz _
(2.5) at — Az+DN(zo)z

is stable in the sense of definition (2.2), then the solu-

tion 2 = z of (2.3) is stable.

0

In what follows, we shall say that 20 is stable, if
(2.5) is stable in the sense of definition (2.2), and that

z is unstable, if (2.5) is unstable. Statements about

0

stability of 2 will actually be statements about the

0
location of the spectrum of the operator A-PDH(zO).

§3. Analytic Functions and the Implicit Function Theorem
For the sake of completeness, we mention the following
results, which may be found in Dieudonne's Foundation 2:

Modern Analysis [ 5 ].

Definition 3.1. Let D be an open subset of KP,
where K = I! or C. We say that a mapping f of D into
a Banach space E over K is analytic if, for every point
a e D, there is an open polycylinder P = [z E Kpllzi—ail<(ri,
l g i g p}, such that in P, f(z) is equal to the sum of
an absolutely summable power series in the p variables

(Zk-ak), l < k < p. The fbllowing are true:
(3.2) The power series in (3.1) is unique.

(3.3) Let A.c Cp be an open connected set, f and 9
two analytic functions in A with values in a com-

plex Banach space E. If there is a nonempty open

subset U of A such that f(x) = g(x) in U,
then f(x) = g(x) for every x in A. Let U
be an open subset of A, b a point of U, and
suppose that f(x) = g(x) in the set U{1(b:+lé5,

then f(x) = g(x) for all x in A.

(3.4) Let E be a complex Banadh space, A an open sub-
set of fig), f an analytic mapping of A into E.
Then there is an open set B c GP such that B nIIRP=
A and an analytic mapping of B into E into which

extends f.

(3.5) A continuously (Fréchet) differentiable mapping f
of an open subset of cp into a complex Banach
space is analytic. [Henceforth, differentiable will

mean Frechet differentiable].

Implicit Function Theorem. Let E, F, G be three
Banach spaces, f a continuously differentiable mapping
of an open subset A of ExF into G. Let (xo,yo) be
a point of A such that f(xo,yo) = O and the partial de-
rivative D2f(xo,yo) be a linear homeomorphism of F onto
G. Then there is an open neighborhood U6 of XO in B
such that, for every open connected neighborhood V of
x0, contained in Ub, there is a unique continuous mapping
V into F such that u(x0) = yo, (x,u(x)) e A and

f(x,u(x)) = O for any x e V. Furthermore, u is contin-

uously differentiable in V, and its derivative is given by

(3.6) u(x) = -[D2f(x,u(x))]_1[le(x,u(x))].

If f is p times continuously differentiable in a neigh-

borhood of (xo,yO

differentiable in a neighborhood of x

), then u is p times continuously
O . The fol lowing
also hold

(3.7) If E, F, G are finite dimensional and f is an-

alytic in A, then u is analytic in a neighborhood

of x0. (Here Act):p or AcRp).

(3.8) If E = mp, then u: E 4.F is continuously differ-

entiable, hence analytic by (3.5).

§4. Remafiks
In this thesis we will be interested in two special
cases of the following problem which we briefly outline

here. Given a family of differentiable equations

Oslo
rrN

= A(a)z-+N(a,z) (as in 92),

which depend on a parameter a in Cn, describe the set
of equilibrium solutions near a bifurcation point (aO,O)
of the operator F(a,z) = A(d)z-+N(a,z). In particular,

we shall be interested in the number of real equilibrium
solutions which correspond to a in is). ‘We shall intro-
duce the hypotheses we need and make our notions more pre-

cise in the chapters that follow.

In Chapter II, we use the Liapunov-Schmidt method [7 ]
to reduce the problem to a finite system of “bifurcation
equations" on a finite dimensional space. We shall then
specialize to a case where a is in C and generalized
null space of A(ao) is one dimensional, and determine the
set of bifurcating solutions along with their stability
properties. This situation arises in fluid dynamics and.has
been studied by Kirchggssner and Sorger [13] in the context
of the Taylor problem, and by Kirchgassner and Kielhdfer
[14] in a general survey of bifurcation in fluid dynamics.
Sattinger [17] has used Leray-Schauder degree to study the
stability of bifurcating solutions, and has obtained results
whidh overlap those of Chapter II. The technique which we
employ is different, and shows how the sign of the critical
eigenvalue is related to the leading terms in the bifurca-
tion equations.

Chow, Hale, and Mallet-Paret [(4] have studied a two
parameter bifurcation problem which concerns the buckling
of a rectangular plate. In one of the situations they -
studied, the generalized null space is two-dimensional. In
Chapter III, we shall develop methods for analyzing the bi-
furcation set in a different general setting Where the gen-
eralized null space is two-dimensional and the system depends
on two complex-valued parameters. ‘We apply these results to

a system of chemical reaction equations in Chapter IV.

CHAPTER II

BIFURCATION AND STABILITY-ONE DIMENSIONAL
NULL SPACE

§l.- Preliminaries

This chapter is divided into two parts. The first two
sections are largely introductory: we consider the problem
of determining nontrivial equilibrium states which bifurcate

from the trivial solution of

23% = A(a)z +N(a,z),

and give basic hypotheses under which this problem can be
reduced to a finite-dimensional problem. In §3, we consider
the case where the generalized null space of A(O) is one-
dimensional. We shall assume that for real a, A(a) has
a simple real eigenvalue x*(a) which crosses the imaginary
axis as a moves through zero. The number of bifurcating
solutions and their dependence upon a will be discussed.
In g4, we shall study the stability properties of these
solutions and show how stability is related to the leading
terms of the bifurcation equation.

We now give our basic hypotheses and discuss their con-

sequences.

10

(H1) Let A be an open set in Tn, and let X be
a complex Banach space with dense subspace D(A). Let
A: A xD(A) 4 X be a closed linear operator with domain
D(A) fer each a in A, and let A be analytic in A
in the sense that A(a)z has a Taylor expansion at each
a in A which converges in a disc la-—a I < r indepen-

O 0
dent of 2. Assume the origin is in A.

It follows from (H1), that if Y' is the Banach space

consisting of D(A) endowed with the norm
|z|Y = |z|x + [A(O)z[x,

then for some neighborhood U of the origin in Tn, we
may regard A: U)(Y-+X as a continuous map for each a

in U.

(H2) Let N: U)(Y-ox be a continuous map such that
N(a,O) = O and D2N(a,0) = O for all a in U; [i.e.,
N has zero linear part at (a,O) ]. We assume that N
is continuously Fréchet differentiable in a neighborhood

V of (0,0).

According to Nachbin [15], this implies that for every

V0 in V, there is a p > O and a power series

itfla

1 m
ET'Qm‘V"Vo)
that converges to N uniformly for [v-v0[ < p. D is a

m

symmetric m-linear form on [CnVXY]m. In fact, Dm is the

11

th

m Fréchet derivative of N at v denoted by DmF(vO).

0)
In most applications, this series is finite.
In most physical problems, we deal with real spaces

and real parameters. Thus we assume

(H3) A and N are extensions of "real" operators

in the sense that A(a)z = A(a)§ and N(a,z) = N(5,z).

Now consider the differential equation

(1.1) 3% = A(a)z+N(0L,z)

and the steady state equation
(1.2) O = A(a)z+N(a,z).

The trivial solution is always an equilibrium state for

a near a = 0. If A(O) is a linear homeomorphism of Y

onto X, then the implicit function theorem guarantees that
the only solution of (1.2) in a neighborhood of (0,0) is

the trivial solution. Thus we may expect nontrivial equi-

librium solutions to bifurcate from the trivial solution

at (0,0) only if A(O) is not a linear homeomorphism;

i.e., A = O is in the spectrum of A(O). We now assume

(H4) A = O is an isolated eigenvalue of A(O) with
finite dimensional generalized null space. For our purposes,
we shall assume that the null space is equal to the gener-

alized null space.

12

§2. Reduction to the Finite-dimensional Problem

we shall show that the problem of determining equilib-
rium solutions of (2.1) corresponding to a near a = O,
can be reduced to a finite system of equations on the gen-
eralized null space of A(O). In our case the generalized
null space is the null space and is finite-dimensional.

Let Px denote this null space.

By standard results from theory of closed operators we

have

Proposition 2.1. There is a continuous projection

P: XJ+P which commutes with A(O) in the sense that

X
A(O)Pz = PA(O)z for z in D(A) = Y.

By means of this projection, X may be decomposed

into two complementary subspaces, Px and Qx = range (I-P).

Each element 2 in X can be written uniquely as a sum

2 = b-+w where b is in PX and w is in Qx. By setting

P = erjY = P and QY = Qxij, we have a corresponding

Y X
decomposition for Y'c X. These spaces are invariant under

A0 in the sense that A(O): PY-on

Furthermore, the spectrum of the restriction of A(O)

and A(O): QY-on.

to QY does not contain A = O, and we have

Proposition 2.2. The restriction A(O): QY-on is

a linear homeomorphism.

13

The Liapunov-Schmidt method.['7] consists in using
the projections P and I-P given by proposition (2.1)

to decompose the equation
(2.1) O = A(a)z+N(a,z)

into an equivalent system of two equations as follows. Let
z in Y’ be rewritten as 2 = w-+b where w is in QY
and b is in PY; b may be considered as a point in Cm
where m is the dimension of Py. Equation (1.2) is then
equivalent to the system

0 (I - P) {A(O) + [A(a) — A(O)]](w +b) + (I - P)N(a,w +b)

(2.2)

0
II

P[A(O) + [A(a) -A(O)]}(w+b) +PN(a,w +b)

Since P and (I-P) commute with A(O), we have

r(2.3) (a) o Fl(a,b,w) = Ao(w) + (I -P)[A(a) -A(0)] (w+b)

< + (I-P)N(a,w+b)

 

[(2.3)(b) O F2(a,b,w) = P[A(a)-A(0)](w~+b)-+PN(a,w-bb)
where

Fl: Ux‘lflluxQY 4 QX
and

F2: Ux<lmeQY «Cm

are continuously differentiable. Since

l4

D3F1(0,0,0)[a,b,w] = A(0)w + (I -P)D2N(O,0)[w] = A(O)w,

and the restriction A(O): QY a Qx is a linear homeomor-
phism, the implicit function theorem allows us to solve
(2.3)(a) for a unique w(a,b) for (a,b) near (0,0),
such that Fl(a,b,w(a,b)) = O and w(0,0) = O. The func-
tion w(a,b) is analytic in some neighborhood of (0,0)
in Cn'XCN‘ and may be expanded in a convergent power series
in that neighborhood.

we will refer to (2.3)(a) as the auxiliary equation and

(2.3)(b) as the bifurcation equation. Upon substitution of

 

w(a,b) into (2.3)(b) we have

Proposition 2.3. Under the hypotheses (Hl)-(H4), the
problem of finding equilibrium solutions of (1.1) near the
trivial solution for a close to zero, is equivalent to

solving the finite dimensional problem
(2.4) O = F(a,b) = F2(a,b,w(a,b))

for (a,b) near (0,0). [F(a,b) is analytic in a neighbor-

hood of (0,0).]

The simplest case occurs where a is in C, and the

subspace p is one-dimensional. It is this case which we

Y
shall consider in the remainder of the chapter. In the

following chapter we shall discuss a case where PY is

two dimensional.

15

93. Bifurcation [One Dimensional Null Space]

Let A and N satisfy the hypotheses (H1)-(H4),
with A an open subset of C. A and N are to be thought
of as extensions of "real“ operators defined for a real par—

ameter. Let

_ 2
A(a)z — Aoz-+aAlz-+a Azz-+... .

Recalling the example of the elastic rod in chapter I,
we see that we intuitively expect bifurcation to occur when
the trivial solution "loses its stability“. In keeping with

this intuition we make the following assumptions

(H5) A = O is a simple real eigenvalue of A(O)

with eigenfunction go.

By results from analytic perturbation theory, (H5)

implies that A(a) has a simple isolated eigenvalue
x*(a) = A (1+). a2+~~
l 2

which is an analytic function in a neighborhood of zero.
Since A(a)z = A(a)2, A*(d) is real for real a. In order
to assure that A*(a) actually crosses the imaginary axis

as a passes through zero, we assume

(H6) x1 7‘ 0.

Thus in the case where the remaining spectrum of A(a)

initially lies to the left of some line ReA = 6 (5 < O)

16

and A*(a) < 0 for a < O, the trivial solution is stable

for a < O and becomes unstable for a > 0.

Example (3.1). Let A(u) = L-+pB for u in C,

where L and B are densely defined closed linear operators
on a Banach space X, with D(L) c D(B). Suppose that L
has a continuous compact inverse and that L-lB has a con-
tinuous compact extension to all of X. The spectrum of A
then consists of isolated eigenvalues with finite multipli-
cities. Suppose A = O is a simple eigenvalue of A at

“O #'O, with eigenfunction go. Then we have analytic ex-

pansions for the eigenvalue

Mu) = “(u-HO) +A2(H_HO)2+...

and the eigenfunction

(9(a) = (904-291((4-(40) Ham-(1&2 +---

which are valid, in a neighborhood of p = “0 [11]. we
claim that (H6) is satisfied.
. l -1
Since 0 = (L-—pOB)¢b, we have 0 = [ES I-L B]¢b, and
go is an eigenfunction for L-lB corresponding to the simple
eigenvalue SL-. As in proposition 2.1, there is a projection

P: X » span[qo} that commutes with L-lB. Then

A(u)w(u) = (L-HB)m(u),

and

x(u)PL-lcp(u) = PL’1 (L - (whom).

17

Differentiating with respect to p gives

PL"l (L - “oBN’l + PL-chpO

-1 -1
-1 _ -1
or AlPL m0 — L 890.
. —l -1
Since 0 =(I-L B)cpO and m0 #'0, L Bmo #'0. Thus

A17’0-

Many of the linear operators encountered in mathematical
physics fall into this category.
The aim of this section and its successor is to prove

the following

Theorem 3.1. Let A and N satisfy (H1)-(H6). In
addition, suppose that the remaining spectrum of A satisfies
[ReA[ > 5, for some 5 > 0. Then one of the following occurs

(i) There is an infinite number of real nontrivial
solutions of (2.2) near the zero solution for a = O ["ver-
tical bifurcation"]

(ii) For each real a near 0, there is a real non-
zero solution z(a) of (2.2) whidh may be expanded in a
fractional power series about a = O. 2(a) is the only
nontrivial solution such that zl(a) tends to zero with a.

(iii) There are two real nonzero solutions zl(a) and
22(0) for (2.2) for a > O (a < O) and none for a < O
(a > 0). These solutions may also be expanded in fractional

power series.

18

Theorem 3.2. Let L(a) denote the linearization of

(2.2) about zl(a) [or 22(a)] given by
L(a) = A(a)-+D2N(a,zl(a)).

Under the hypotheses of Theorem 3.1, the spectrum of L(a)
remains near that of A(O). For a near zero there is a
simple real eigenvalue A+(a); A+(a) and A*(a) have oppo—

site signs in a neighborhood of a = 0.

Corollary 3.3. If the spectrum of A(a) (with the
exclusion of A*) satisfies ReA < 6 < O for a near zero
and A*(a) < O fer real 0 < O, and there is no vertical
bifurcation, then real nontrivial solutions are unstable

for a < O and stable for a > O.

Sattinger [17] has proven a similar result using Leray
Schauder degree when the linear operator A(a) is given as
in Example 3.1. Gavalas [£5] has also applied degree theory
to the stability of bifurcating solutions. In some cases the
stability for particular systems which arise in fluid mechan-
ics has been determined by perturbation methods [13], [18],
[19]. The proof we give here is different and directly re-
lates the sign of A+(a) to the derivative of the bifurca—

tion equation.

19

We now return to the analysis of the system

['(3.l)(a) 0

Fl(a,b,w) = A(O)w + (I -P)[A(oz.) -A(0) 1 (w +b)

4 + (I-PO)N(OL,w+b)

 

[(3.1) (b) o F2(a,b,w) = P[A(a) -A(O)] (w+b) +PN(a,w+b)

where Fl: de: xQY 4 QX’ F2: U x03 xQx 4 C. We have already

shown that for (a,b) near (0,0), there is a unique analytic

w(a,b) such that F1(a,b,w(a,b)) = 0. Let w(a,b) = bloc-r
2 L
bolb-+bllab-+b20a -+ ---+bkzakb .
Since w = O is a solution for (a,b) = (a,O), and

w(a,b) is unique in a neighborhood of (0,0), we must have

w(a,0) = 0. Thus bk0 = 0 for all k.

We now determine blo = g—% (0,0). Since F(a,b,w(a,b)) =
O and D2N(O,O) = O, implicit differentiation shows that

Em = - - - as =
A(O) ab (0,0) 0. Since b01 IS in Qy, ab (0,0) 0,

and we have

Proposition 3.1. There exists an analytic function
w(a,b) which solves (3.1)(a) in a neighborhood of (0,0).

This function has the form

2

w(a,b) = b ab+b0 b +O(a2b+b2a+b3).

ll 2

Upon substitution of w(a,b) into (3.1)(b), the bi-

furcation equation becomes

20

(3.2) O = F(a,b) = aPAlb-+aPA1w(a,b)

+ P[A(a)-aAl-Ao](w(a,b)-+b)-+PN(G,b'+W(G,b)):

where PN(a,b-+w(a,b)) is analytic and has a power series
expansion in a neighborhood of (a,b) = (0,0). Recall that
N(a,0) a O and D2N(a,0) E O for all a in a neighborhood
of zero. Thus D§N(a,0) = O and D§D2N(a,0) = O for all
a near zero, and we have PN(a,b-+w(a,b)) = b2N1(a,b)
where N1(a,b) is analytic in a neighborhood of (0,0).

It is useful to calculate the term PAlb. Let m(a) =
eb-+aml-+a2m2-+-o- be the eigenfunction corresponding to

x*(a) = 00.1 +a2A2 +---

Then P[A*(a)I-A(a)]m(a) = O. Differentiating with respect

to a at a = 0 gives
P[AlI-Al]¢b-+P[-Ao]ml = O

or Almo = PAlqO.

Thus PAlb = Alb. The bifurcation equation then becomes

_ 2
o — Alab+aPOA1[b11ab+b02ab +... ]
+ P [azA -+a3A -+---](w(a b)-+b)
0 2 3 ’
2

Thus a and b are related by a power series in two vari-

ables with real coefficients:

21

b-+c a2-+0(ab-+b2)].

(3.3) O = b[Aloz+clo 02

The trivial solution b = O is a solution for all a near

1 2 .
Clo to be EPDZN (0,0) \ Cpo,¢po>

where wb is the eigenfunction corresponding to A = 0,

zero. One may calculate

with [¢b[ = 1. If c10 is not zero, the implicit func—
tion theorem gives a unique solution b(a) of (3.3) which
depends on a analytically in a neighborhood of zero.
b(a) is real for real a and b(O) = O.

A more general analysis of (3.6) may be carried out
by using Newton's polygonal method [ 1]. Consider a power
series in two complex variables that converges in a neigh-

borhood of (0,0).

2

_ ii
G(a,b) — clOb-+cola-+c20b -+ -+cijb a + .

Suppose the coefficients are real. We wish to find real
solutions of G(a,b) = 0 corresponding to real a in a
neighborhood of a = 0. Let ck0 be the first of the co-
efficients c.

10
lemmas [ 1] apply:

that do not vanish. The following two

Lemma 3.7. Let k be even, Col #’O. Then if
col/ck0 < O (col/ck0 > O), the equation G(a,b) = O has
two different real roots b1(a), b2(a) for a > O (a < O)
which are simple real roots, and has no real roots for

a < O (a > O).

22

Lemma 3.8. Let k be odd and c01 # 0. Then
G(a,b) = O has precisely one real root for both a > O

and a < 0 and this root is simple.

Under the above conditions, these solutions are given
by fractional power series in ¥/a . The leading coeffients

are given by ilf/Icol/Ckol f°r k even, and by k\""301;Cko
for k odd.

 

Since A1 ¥ 0, these lemmas apply to (3.6). The number
and nature of the real roots b(a) will be determined by
the first term ck0 #'O, the sign of Al/ck0 and the parity
of k.

we now conclude the proof of Theorem 3.1.

(i) If no ci0 is nonzero, then (O,b) is a solu—
tion of (3.3) for arbitrary b, and "vertical bifurcation"
occurs.

Therefore, we assume that C01‘ is the first such non-

zero coefficient and apply the lemmas:

(ii) If k is odd,

n
K’"
I

(3.4) bl(u) = k(/ -)‘17ck0 p + , where u

and zl(a) = w(a,bl(u))-+b1(u)

(111) Let k be even. If -Al/'ck0 > O

r'bl((..L) = §/-A17Cko H + -°° , where H = +}(d
(3.5) (

 

k k
kb2(u) = ‘/-Al7cko H + --- , where H = - a .

23

If -Al/'ck0 < O, replace a by -a, and use Al/ck0 under

the radical sign. In either case,

21(0)) = W(a,b1(u)) +bl(u)
and
22(0) = W(a,b2(u)) +b2(u).
In fact, the analytic function
(3.6) z(()) = w(uk,b(u)) +bm)

is a solution of
(3.7) o = A(pk)z+N(pk,z)

for all (complex) H in a neighborhood of zero. Here

b(p) is any of the functions given by (3.4)-—(3.5) and

fik = a. [In the case fik = -a, 'we replace H3 by -u¥

in (3.6)-—(3.7)]. For each a near zero, there correspond
k distinct zi(a) (i = 1,...,k) corresponding to the k
roots of a. These zi(a) form a cyclic system of solutions
of the system (1.2). If a moves along some Jerdan curve
about zero, the values zi(a) undergo a cyclic permutation

when we return to the starting point. We also note that

2(p) may be expanded in a power series with real coefficents.

94. Stability
In this section we give the proof of Theorem 3.2. Re-

call the differential equation

24

(4.1) gi- = A(a)z+N(a,z)

and the stationary problem
(4.2) 0 = A(a)z-+N(a,z).

We have shown that real nontrivial solutions 2(a) of
(4.2) bifurcate from the trivial solution for real a
near zero. These solutions may be found by substituting
an appropriate real kth root of a (or -a) into an
analytic function z(p) = Ylu-byzp2-+--- where Y1 is
given in (3.4)-(3.5). For all p in a neighborhood of

zero, 2(p) is an equilibrium solution of

(4.3) %_zt_ = A(pk)z +N(p,k,z) if pk = a
or
(4.4) 3%;- = A(-uk)z +N(-uk,2) if (3‘ = -a.

we wish to discuss the stability of the equilibrium solu-
tions z(u) where u is a real kth root of a (or -a).
Without loss of generality, we shall use p} = a in our
calculations.

Consider the linearization of (4.3) about z(p) for

fixed p:

(4.5) 3% = A(pk)y+D2N(pk,z(U))Y = L(u)y.

For each p, L(p): Y 4 X is a continuous linear map. L(p)

depends on H analytically, and LZHSy = L(fi)y. For H

25

sufficiently close to zero, L(u) may be regarded as a
closed operator with dense domain D(A) in X; thus we
may refer to Kato [11] for information about the dependence
of its spectrum on u. Since L(O) = A(O), the spectrum
of L(p) satisfies the following when H is sufficiently

close to zero.

(4.6) There is a simple eigenvalue A+(u) such that
A+(O) = O and A+(p) is analytic in a neigh-
borhood of zero. A+(p) is real for real p

near zero .

(4.7) The remainder of the spectrum satisfies

[Rex] > 6/2 > 0.

Our aim is to relate the sign A+ to that of A*
when a is real and close to zero.

Let y in Y be written as y = x-+§ where x E QY
and g e PY
z = x-tg. The eigenvalue problem for L(H) may be written

and consider N(a,z) as N(a,x,§) where

as the system

r
(4.8)(a) Ax

[A(O) + (I -P)[A(p.k) -A(O) +DxN(pk,z(p))]x

+ (I-PO)[A(uk) -A(O) +D N(pk,z((_())]§

§

(4.8)(b) A5

L + pmm“) -A(O) +D§N(uk,2(u))]§

P[A(u") - A(O) +DXN(uk, z (u) ) 1x

 

or more simply

26

(4.9)(a) O

[B(u)-l]X + C(u)§

(4.9) (b)

0
ll

D(ulx + [EM -k]§

where B(u), C(u), D(p), and E(u) are linear maps for
each u, and depend on H analytically in a neighborhood

of the origin.

Lemma 4.1. For A and p sufficiently small, B(p)

and A-B(p) are invertible and

1

2 -1 n+1

+°°'+An[B([J.) ] +...

(3(a) -)()'l = B(u)-1+A[B(u)- ]

converges uniformly in the operator norm for A and p

in a neighborhood of the origin.

Proof. Since the set of linear homeomorphisms in
L(Qy,0x) is open [ S] and B(O) = (I-P)A(O) is a linear
homeomorphism, by proposition 2.1, B(p) and B(a)-—A
are invertible for H and A sufficiently small. Also
B(p) is close to A(O) in norm. The rest follows from
the fact that if Tn are elements of Banach space and

Q Q
Zl)[lTnH < co, then ZETn converges.l’3

we may now solve (4.9)(a) for x in terms of g and

substitute into (4.9)(b) to get

0 = [E(u) -D(u) (Em) - X)-1C(u) - x]:

or

27

(4.10) 0

an.) -D(u)B(u)‘lcm) - A

- D(u)[A[B(u)-1]2 + (A)2[B(u)-l]3 + - --]cm)

f(u) -).->.9(u,>.)

where f is an analytic function u in a neighborhood of
zero, and g is an analytic function of H and A near
the origin. A Newton's polygon argument shows that for
real u there is a real solution A(p) = dluP-+d2p2p-+---
where d1 is the coefficient of the lowest power of p
in f(p) = x(u) -D(u)B(u)-1C(u) and p is that lowest
power.

We now make the following observation. Recall that

z(u) = w(px,b(u))-+b(u) satisfies

0 (I -P)[A((_(k)[w +b] +N(pk,w,b)] = Fl(u,w,b)

(4.11)

0
ll

P[A(uk)[w +b] +N(uk,w,b)] .-.- F2 (u,w,b).

Let p near 0 be fixed. Differentiating F2(H,w,b) with
respect to b at b(p) gives
k k
(4.12) P[A(u ) +D3N(p ,w(pk,b(u)),b(u)]
+ P[A(uk)%%+D2N(uk,w(uk,b(u)),b(u)]'g%(uk,b(u)).
Since by I (3.6),

35% = _[DZF1(p,w(uk,b(u)),b(u)]-1[D3Fl(u.W(uk,b(u)bbhfll:

28

(4.12) is exactly E(p)-D(H)B(p)_1C(u). We have proved
Lemma 4.12. For H sufficiently close to zero, we may
determine the leading term of X+(H) by differentiating
F2(p,w,b) with respect to b at b(u). The leading term
of this derivative will be the leading term of A+(p).

[Nete that F2(u,w,b) = O is the bifurcation equation.]

Recall that the bifurcation equation may be written

in the form

_ 2 2
(4.13) F(a,b) — Alab+c02a +clob +
where a = pk. Let cokbk+1 be the first term of the form
ciob1+1 that does not vanish. In this case

r
k . .
./[A7cko| H + --- if k is even

b(u)= <

 

(k1/‘Vcko A if k is odd, uk=a.

[For k even, pk = a if -A/ck0 > O and pk = -a if

-A/'ck0 < 0.] Thus

Ala+c02a2k +.-- + (k +1)cko[b(u)]k+

§-§(a,b(u))

. . k
Ala-(k-+1)Ala-+ higher order terms in (A;

-kAla-+higher order tenms in 5%;

Thus d1 = -kA, p = k, and X+(H) is actually an analytic

function of a: A+(a) = -kA1d-+--- . Recall that

29

1*(0) = Ala-+x2a2-+--- . Thus for real a near zero,

A+ and A* have opposite signs. This completes the

proof of Theorem 3.2.

CHAPTER III
A TWO-DIMENSIONAL NULL SPACE

§l. Introduction
Our aim in this chapter is to develop tools which will
allow us to describe the set of nontrivial equilibrium so-

lutions which bifurcate from the trivial solution of

3% = A(a,a)z +N(a,r3,z)

at (0,0), where (a,8) is in C2. Part of the problem
will be to choose suitable hypotheses so that the dependence
of the bifurcating solutions on the parameters may be studied
in a full neighborhood of (a,B) = (0,0).

In the case which we shall study, A(0,0) has a two-
dimensional null space, and A(a,8) has two eigenvalues
A(a,B) and u(a,8) which pass through zero as (a,8)
passes through the origin. The curves A(a,8) = O and
“((1,8) = 0, defined for ((1,8) in R2, are assumed to cross
transversally at the origin. This last assumption will allow
us to change coordinates to T = A(a,B), n = “(0,5). we will
formalize these hypotheses and discuss their implications at

the end of this section.

30

31

In §2 we discuss the bifurcation equations and the
coordinate change mentioned above. In §3 we give a simple
example which demonstrates that the behavior of the bifur-
cation set depends on the ratio Y = n/T (or T/nl- Moti-
vated by this observation, we make a change of scale which
allows us to introduce the ratio Y in place of one of
the parameters ¢,n. Section 4-—6 discuss methods for
analyzing the scaled equations and the interpretation of
these results in the (¢,n)—space.

‘We now list the basic hypotheses of this chapter.
(Hl)-(H3) are from chapter II and are repeated for conve-

nience

(H1) Let A be an open set in C2 that contains the

origin and let X be a complex Banach space with dense sub-

space D(A). Let A. ijD(A) 4.x be a closed linear operator
with domain D(A) for each (a,B) in A, and let A be
analytic in A in the sense that A(a,8)z has a Taylor
expansion at each (ao,Bo) in A which converges in a disc

[(b,a)-—(b0,ao)[ < r independent of 2.

It follows from (H1), that if Y is the Banach space
consisting of D(A) endowed with the graph norm, then for
some neighborhood U of C2 we may regard A: U)(Y 4 X

as a continuous linear map.

(H2) Let N: U;(Y 4 X be a continuous map such that

N(a,6,0) = O and D3N(a,B,O) = O for all (a,B) in U.

32

We assume that N is continuously Fréchet differentiable

in a neighborhood V of (0,0,0).

(H3) A and N are extensions of "real" operators
in the sense that A(a,B)z = A(5,5)Z and N(a,8,z) =

mass).

In addition, we make the following additional assump-

tions about A(a,8) = Aoz-+aAlz-+5Azz.l...

(H7) Zero is an isolated eigenvalue of A0, and the
generalized null space is two-dimensional. In addition,
we assume that there are two distinct branches of eigen-

values that depend on the parameters analytically and take

on the value zero at (0,0).

and

“(0,6) H10a+polB+°°°

are eigenvalues of A(a,8). Moreover, there are two dis-

tinct eigenfunctions wb and Y corresponding to A(0,0) =

O

O = p(0,0), and we have the expansions
m(a,8) = $b-leOa-tmblB-+ooo
Y(a,B) = Yo'kyloan+y013.+...

which converge in some neighborhood of (0,0).

33

(H8) For real values of the parameters, A and p
are simple real eigenvalues (except when g(a,B) = A(G,B)),
and ¥(a,B) = Y(a,3), $(a,B) = $(a,B). Furthermore, the
curves A(a,B) = 0, u(a,3) = 0, defined for real a and

B, cross transversely at the origin (see Fig. l); i.e.,

 

 

r- M ii 1
aa 38
det # O at (0,0)
QB. he.
aa 65
1dir

Ahufiéffl)

 

 

Figure l. Transversal crossing of zero eigencurves.

This last assumption is important as it will allow us
to make a local change of coordinates about the origin in
JR2 in which the curves A(a,B) = O and p(0,5) = 0 be-

come the axes.

34

Example 1.2. The hypothesis (H7) is necessary as
the analytic dependence of the eigenvalues and eigenfunc-
tions upon the parameters does not necessarily follow from

that of A. Let

 

 

L_B -a

map C2 into C2. The eigenvalues are

A = a2-+82 and H = -./ 2-+82

which are not differentiable at (a,B) = (0,0). The problem

is that the eigenvalues are given by different branches of

the same multivalued function a2-+82 and (0,0) is the

branch point.

§2. The Bifurcation Equations

Let P be the projection given by proposition II 2.1,

and PX’ Py, QX’ Qy. be the subspaces described in §2 of

chapter II. In our case Px = PY = span[qo,Yo} and is

homeomorphic to C2. The projection P: X 4 PX is the sum
of two projections, P1 and P2. P1: X 4 Span{¢o} = P;
2 1 2

and P2: X.4 span[YO} = Px where Px = Px:® PX‘ If 2 is

in X, we may rewrite 2 as 2 = w-+x-+y where w 6 0x,

2

x 6 P; and y e PX’ As in §2 of chapter II, the equation

(1.3) is equivalent to the system

35

”(2.1) (a) o Aow + (I -P)[Ao-A(a,f3)] (w+x +y)

'l" (I " P)N(G,B,X,Y,W)

(2.1)(b) 0 = Pl[aAl-+BA2-+-°-](w-+x-+y)-+P1N(a,B,x,y,w)

 

[(2.1) (c) o = P2[aAl+BA2 +---](w+X+y) +P1N(a,f3,x,y,w)

Furthermore, we have shown in chapter II, §2, that there

is a unique w(a,8,x,y) such that w(0,0,0,0) = O and
(a,8,x,y,w(a,B,x,Y)) is a solution of (2.1)(a) for
(a,B,x,y) near (0,0,0,0); w(a,8,x,y) can be expanded

in a power series in some neighborhood of the origin. Also
§(a,f3.x,y) = “5.5.5350.

As before, we may show that
g(o,o,o,0) = %Ԥ(o,o,o,0) = 0.

Moreover, since w(a,B,0,0) = 0 for all values of (a,fi)
near (0,0), the uniqueness of w implies that this power
series will contain no terms which contain only a or 8.

Thus w(a,B,x,Y) is of the form

W(G,5,X,Y) = Clax-fczfix-+c3ay-+c48y-+c5x2

+ cexy-+c7y2-+higher order terms in x,y,a,B.

we may substitute this expression for w into the bi-
furcation equations (2.1)(b) and (2.1)(c), and obtain two

power series in 4 variables:

(2 2) o P1[aAl-+BA2]x +-Pl[dAl-+BA2]y-+-.-

O

P2[aA1-+BA2]x + P2[aA1-+8A2]y-+-o- .

36

We now determine the terms listed in (2.2). Differ—

entiating the expression

0 = P1[A(a,f3) -A(a,B)]cp(a,t3)

. - .QA _
With respect to a at (0,0) gives 80(O’O)¢0 — PlAlmO.

A similar argument shows that

$3“), 0)ch = P1A2°~°o

31*. = 3E. _
a“(dome P224140 and 63(0’0)Y0 P2112110

Also P2A1cqO = PzAchO = PlAlYO = PIAZYO = 0. Thus the bi-
furcation equations become

(

 

o = abs-g}: +B%%X+P1[aAl+fiAl]w
+ Pl[A(a,B) —A0 -aAl -BA2][w+x+y] +P1N(c,8,x,y,w)
(2.3)( H
o = a%§X+B§-5x+P2[aA1+BAl]w
t + P2[A(a,8)-Ao-aAl-8A2][w-+x-+y]-+P2N(a,8,x,y,w)

Henceforth, we will seek real solutions (x,y) of (2.3)
corresponding to real parameters (a,B). we may as well
assume that the norm is Euclidean.

It is convenient to make a change of variables in the
parameter plane so that the curves A(a,8) = O and u(a,8)==0
become the axes. Define the transformation T: (a,B) 4 (T,n)

by T = A(336): T] = “(aaB)o Since A(a;B) = 0 and “(0,5) =0

37

cross transversely at (0,0), T is a homeomorphism in a

neighborhood of (0,0). Then

 

 

 

... PM m'fi—lr- 1
a.) ad OB T 2
= +0!(T,n)| .
314 A!
[34 _aa 36‘ [TL

 

 

 

Thus terms which are first order in a and B, are first
order in T and U7 terms which are second order in a
and B, are second order in T and n; etc. The bifurca—

tion equations take the form

0

Tx 2nd order terms in x and y + higher

(2 4) + order terms in x,y,T,n
O = ny

There are no terms which contain only powers of T and
n. Terms which contain only powers of x and y are de-
rived from the nonlinear term which we refer to as

N(T, ngx3Y:W(x:Y: Tafl))°

§3. An Appropriate Change of Scale

As motivation for what we are about to do, consider

the following.

Example 3.1. Consider the system

O = TX + %y2+x2
(3.1)
l 2 2

0 = ny +-Z>c +37

38

The numbers of nontrivial solutions of (3.1) corresponding

to each (T.n) are given schematically in Figure 2.

l
in

l
\\\‘

 

l

‘A

‘7’

3

 

Figure 2. Dependence on (T,n) of the number

of nontrivial solutions of (3.1).

As we can see, the number of nontrivial solutions depends
on the ratio T/n or (n/%). Also note that it is impos-
sible to apply the implicit function theorem to obtain
solutions of (3.1) of the form (x(T2n):Y(T:n):T:n) in a
neighborhood of (x,y,¢,n) = (0,0,0,0).

New consider the following change of scale. Let T =
Yn, and replace x by fix and y by ny in (3.1). After

division by n2, (3.1) becomes

39

r
O = YX +%Y2+X2

02) (
2

 

y.%a+y

L.O

Sectors in the (T,n)-plane and the corresponding re-
gions in the (Tby)-plane are represented in Figure 3. Points

on the Y-axis correspond to different slopes through the

origin in the (T,fi)-plane.
// 43$ 47
- ,,__.:\\ Rs
45/ /
4‘
$4, // 4

Figure 3. Correspondence between (T.n)-plane

 

 

7
//

 

 

 

 

 

 

and (n,Y)-plane.

The numbers of nontrivial solutions of (3.2) correspond-

ing to each (U,Y) are given in Figure 4.

4O

 

:////////W//’/4

 

 

///////3////A

 

Figure 4. Dependence on (n,Y) of the number

of nontrivial solutions of (3.2).

It is easy to check that at most of the solutions (XO’YO’YO)
of (3.2), the implicit function theorem applies. The points
at which it fails correspond to the lines in Figure 4 which

indicate a change in the number of solutions.

In what follows we shall give hypotheses under which
a similar scaling may be determined for the bifurcation
equations (2.4). Solutions (x,y) of (2.4) which tend to
(0,0) with (T:n) will correspond to solutions of the
scaled equation which remain bounded as (7,n) approaches
(0,0). In order to accomplish this, we shall require the

bifurcation equations to be of a certain form so that we

41

can obtain an a priori estimate of the magnitude of solu-
tions of (2.4) which tend to (0,0) with (T,n). This a
priori estimate will determine the scaling we choose.

Let T = n = O, and suppose that the first terms of
(2.4) in x and y which do not vanish have the same de-
gree q in each equation. The terms of degree q form a

continuous q-linear form Mq: 1R2q4 ZIR2

 

 

' n n-1 n-1 nf
(3.3) aox -+a1x y-+--w+an_1xy -+any
Mq((x,y),..., (x,y)) =
n n-1 n-1 n
box -+blx y +-.-+bn_lxy -+bny‘d

we say that Mq is nondegenerate if

(3-4) Mq((X:Y):-°-: (XJY)) # (0:0) for all

(x,y) 7! (0,0) in 122.

Since the unit ball in R2 is compact and Mq is contin—
uous, (3.4) implies there are positive constants c1 and

c2 such that
(3.5) all (x,y))q _<. Mq((x,y),..., (x,y)) 3 c2 [(x,y) (‘1

for (mil) in R2.
(H9) We assume the bifurcation equation has the form
(3.6)

0 7x

+S(T:T]:X:y) +M ((X,Y),..., (X,Y)) +]R(T:T]:X3Y)
O 111/ q

42

where S(0,0,x,y) s O, S(T,n,x,Y) contains terms in x
and y of order less than q and there is a positive con-

stant c3 such that

OJ) [Nnm&W\g%HnmLH&W|

when (T,n,x,Y) is sufficiently close to (0,0,0,0). We

also assume that

IR(Tk: le: xk’ Yk)l

(3.8) 4 0 for all sequences

Hampfl

 

[7k]: [nk}, [Xk}, {Yk} that tend to zero with [(xk,yk)[ #‘o

for all k, and that Mq is a nondegenerate q-linear form.

Proposition 3.1. Let (H9) be satisfied. Then for any
(T,n,x,y) sufficiently close to (0,0,0,0), such that
(x,y) is a solution of (3.6) corresponding to (Tan) # (0,0),

we have

Hmw(ngVWRJT
where m is a constant independent of ¢,n,x,y.

Proof. Suppose not. Then there exist sequences [Tk],

ink}: {Xk}, [yk] that converge to zero, with (Xk’yk) #‘o,
(7k3nk) # (0,0) and

Hflwflfld
1(Tk, nk)‘

 

>k.

43

We may assume that Tk’ nk’ xk, and yk are small enough

so that the estimates (3.5) and (3.7) hold. Consider

1 Tkxk
| (xk, yk) lq nkyk

 

O + S(Tk: fik’xk’yk)

Mq((xk:yk):---: ()ck’yk» R(Tk’nk’xk’yk)
+ +
‘(x-k: Yk) ‘q )Xk’ yk) ‘q

 

 

As (Tk’nk) and (xk,yk) approach the origin, so does the
last term. Estimate (3.5) implies that there are positive

constants m1 and m2 such that

|(xk,yk) (‘1

 

ml‘g S.m2

Tkxk
leYk + S(Tk’ 'flk: xk’ yk)

for k sufficiently large. Thus by (3.7),

 

 

 

m 2 [(xkdknq
2 )(TRXJQ, myk)‘ +c3lTk: TR) " ‘ (xk: YR”
((kak) (‘1
Z (1 +C3) ‘ (Tk: 71k) ‘° ‘ (xk’YkH
and
-l

[(inyk) [‘1

‘1 +c3"”“3 2 wkmkn

which is a contradictioan

44

Using the a priori estimate of proposition (3.1), we
will now exhibit appropriate scalings of the system (3.6)
for different sectors of the (¢,n)-plane near (0,0).

Let 9p = UT,TQ: [n] > plwl} for p > 0. Let (x,y)
be a solution of (3.6) corresponding to (T,n), that tends

to (0,0) with (T,n). Then

x and -——JL--

q‘lff‘n) q‘IJ—I) n

remain bounded as long as (T,n) remains in Qp. Thus

none of the bifurcating solutions corresponding to (T:fi)

in Qp are lost if the following changes of scale are made.

Case 1. If q is even, let n = gq'l, T = ygq'l for
|y| g %- and replace x and y by gx and gy respec-

tively. Then (3.6) becomes

0 yx

(3.9)(a) = + Mq((x,y),...,(x,y))-+§§(x,y,Y,g)
0 Y

where R is analytic in (x,y,Y,§). [The transformation of

the sector Op in the parameter plane under this change of

scale was given in Figure 3].

Case 2. q is odd. In this case, we must distinguish

between D > 0 and U < O.

45

To study solutions corresponding to n > 0, we use
the same change of scale as in case 1. Here, however, we
must keep in mind that each U corresponds to j;§, and
there will be an "extra set of solutions" for g g 0.

To study solutions corresponding to U < 0, we let
n=-§q-1, 'r = -Y§q-l
gy for |y[ g %=. Then (3.6) becomes

, and replace x by gx, and y by

Y O x =
(3.9)(b) 0 = - M (0,0,(x,y),---,(x,y)%+§R(X,Y,Y,§)
O l y q
where R is analytic in (x,y,y,§). Again, we will disregard
the extra branch of solutions corresponding to g < 0.

To study solutions of (3.6) for (T,n) near the T-axes,

the roles of T and n are reversed.

In all cases, we must now determine the behavior of
real solutions (X,y) corresponding to real (y,§) for
systems of the form

-1
YX+aan+an_1Xn 17+". +aOYn+§Nl(X,Y,Y, g)

0

(3.10)

n n-l n
y-+bnx -+bn_lx y-+----+boy -+§N2(x,y,Y,§).

0
ll

§4. Solutions of Type I - Fold curves
Consider the system
(4.1)
n n-l n
O = F(x,y,y,§)==yx-+anx -+an_lx y-+--o-+a0y -+gNl(x,y,y,§)

n n-l n
o = G(x,y,Y.§)==y-+bnx -+bn_lx y +°°°-+boy -+§N2(x,y,v,§)

46

where

Mn((x,y),...,(x,y)) =

n n—l n
b bnx +bn_1x y +--- +b0y .1

 

 

is a nondegenerate continuous n-linear form, Mn: It 4 It;

and N1 and N2 are analytic in (x,y,y,§). Let

F F
x y

(4.2) J(X,y,v,§) = det at (x,Y.Y,§).
GX Gy

For g = O, (4.1) and (4.2) become

n n-l n
O = P(x,y,y) = yx-ranx -+an_1x y-+----+aoy
(4.3)
n n-l n
0 = Q(x,y,y) = y-kbnx -+bn_1x y-+-o--+boy
and
(4-4) 0 = J(X,Y,Y,O) = PXQy-QXPY at (X:Y:Y:o)-

Let (XO’YO’YO) be a solution of (4.3). If J(xo,yb,Yo,O)7¥
0, the implicit function theorem guarantees a unique solution
(x(y,§),y(y,§),y,§) of (4.1) in a neighborhood of (yo,0)
such that x(y0,0) = x0, and y(yo,0) = yo. J(x,y,y,§) #’O
for (x,y,y,§) near (xo,yo,y0,0). Thus the solution

(XO’YO’YO’O) determines the behavior of nearby solutions

of (4.3).

47

In the event that J(xo,yo,yo,0) = O, we wish to impose
conditions on (4.1) and (4.2) so that we may still determine

the local behavior of solutions near (xo,yo,yo,0):

(H10) We assume that for all (XO’YO’YO) satisfying

(4.3) with (x0,yo,y0) # (0,0,0), we have

rank = 2 at (xo,yo,yo,0).

 

 

(H10) guarantees that about each nontrivial solution
(xo,yo,yo,0), there is a neighborhood of solutions to (4.1)
that is homeomorphic to an open set in IR2 . [The point
(0,0,0,0) is a special case which corresponds to studying
solutions which bifurcate from the trivial solution near

the T or U axes. This case is banished to section §6.]

(H11) we assume that for all (XO’YO’YO) which satisfy

both (4.3) and (4.4), the matrix

 

 

has rank three at (xo,yo,y0,0).

A solution (x0,yo,y0,0) of (4.1) and (4.2) for which

either

48

F F F
X y Y
4.5 d t G G G o
( )(a) e x y Y 3’
J J
L X Y Y.

 

 

 

or
F F F
x y *3
4.5 b d t G G O
( H) e ex y g 9‘
J J J
-x y 5,

 

will be said to be of type I. When (4.5)(a) holds, these
solutions have the pleasant feature that local behavior of
nearby equations is completely determined by the polynomials

(4.3) and (4.5), since (4.5)(a) is exactly the condition

 

’ '1
Px PY }{
(4.6) det Qx Qy O 7’ O
J
L.x Y QY‘

 

Note that (4.5)(a) implies

Gx GY
det #'O and x #’O.
Jx Jy

Theorem 4.7. If (XO’YO’YO) is a solution of (4.3)
and (4,4),and (4.6) holds, then there is a curve y(§) and
a solution curve (x(§),y(§)) such that y(O)

(x(O),Y(0)) = (X

= YO,
O,yo). When y crosses y(§), the number

49

of solutions of (4.1) near (x(g),y(§)) changes by two.

we will refer to y(g) as a fold curve.

 

If (4.5)(a) fails and (4.5)(b) holds, we may prove a

similar result. In this case the solution (x O)

0: YO: YO:
of (4.1) determines curves (x(y),y(y)), g(y) with

(x(yo),y(yo)) = (XO’YO) and g(yo) = 0. In the following
we shall concentrate our attention on (4.5)(a), as results

corresponding to (4.5)(b) are similar. We isolate a special

case in the following lemma.

Lemma 4.8. If (XO’YO’YO) is of type I, the number
of solutions (x,y,y) of (4.3) near (XO’YO’YO) changes

from zero to two as y passes through YO'

Proof. Without loss of generality, we will assume

that

det y #’0.

Thus PY and Qy are nonzero at (xo,y0,y0), and we may

use the implicit function theorem to guarantee the existence
of y(x), y(x) for x near x0,
is a solution of (4.3) and y(xo) = y0 and y(xo) =

such that (x,y(x),y(x))

Yo'
we shall show that y may be written in the form

y(x) = Y0+a2(x-xo)2 +O(x-x0)3

..l
where a2 — 2-y”(x0)

50

(i) Computation of y’(xo). By the chain rule,

 

 

 

 

 

 

 

 

 

 

"' I 1 r 1 ’- fi
1! (x0) QY —PY Px
= l
’ PYQY - PYQY
Ly (XOL _-QY Py __ L- Qx
r- m
QYPX -— Pny
= 1
P - P
YQY YQY P P
- +
t QY x YQX_.
I
Qny — Pny
- 1 at x
_ P _ ,
YQY Pyoy O
o
t .1

 

I =
Thus y (x0) 0.

(ii) Computation of y”(xo). Further implicit dif-

ferentiation plus part (i) gives

P +2P 91+P QXZ+P d—2-X+P d—2-I--O at (x )
xx xy dX yy(dx y dxz y de _ O’YO’YO '

dx2 y yx

if and only if

2
d 1 2
Also __Y.= Q- [Qxx-ZQ (gfi + Q Q2) ]. Thus y”(xo) = 0

(4.9) o = Qy[Pxx +2ny

_ 92
Py[Qxx-+ZQ + Q

51

By (4.6),

O # Jny--JYQx

Q

= [Pxey'H‘Dnyx"Pquxuprxx:l y

-[p +PQ -PQ-PQ]QX

nyy x YY YY x Y xY

2
=P +2 P -2P — P
xey Qxy ny nyny Qxx yQy

2
P Q

+ P Q2 - Q _Y_§.
YY X YY Qy

(2x (2x 2
QY{QY[PXX ' ”mu-2;) + PYYKQ) ]

(2x ox 2
' Pymxx ' 201675;) + (INK-6;) J}

2

BY; .22
QY[QY[Pxx +2ny dx + PYYKOX) ]

2
+°yy\%) 1}.

filfi‘

- pymxx + ZQxy

Thus y”(x0) #'O.[]
Proof of Theorem 4.7. Once again, we assume

P x
det Y #’O.
Q 0
Since (4.6) holds, there is a unique curve (x(g),y(§),y(§),§)
of solutions of (4.1) and (4.2) with x(0) = x0, y(0) = yo,
Y(0) = YO' For E sufficiently close to zero, (4.6) still

holds. Moreover,

52

det Y Y = O

at (x(E),y(§),y(E),§). Fix E and consider the functions

 

r-§(x.vY.v'Y) = F(X:Y:Y:§)
(4.10) ' 4 6(X3Y3Y) = G(XJYJYJE)
L3(X:Y:Y) = J(X,Y,Y,E)

(4.10) satisfies

r' 1

'UI
"Ul
WI

(4.11) det

0|
(0|
0|

x y Y ‘r’0 at (x(E),y(E),y(E)).

 

cll
c-ll

q:

 

L" Y Y.)

‘we now do some calculations similar to those in Lemma 4.8
and use (4.11) in place of Jka-Jny #’O to reach the
desired conclusion. Thus for fixed E close to zero,

y(x,§) has the form
- — — 2 — 3
((095) = y(g) +a2(x-X(§)) +0(x-X(§))

where a2 depends continuously on E and hence has con-
stant sign for g near 0. Thus there are two solutions
of (4.1) near (x(g),y(§)) on one side of y(g) and no

nearby solutions on the other.C)

53

We have the following local picture when (y,§) is

near (yo,0) [Figures 5 and 6].

 

 

 

 

4A
“1v

 

Figure 5. Behavior near a fold curve

in the (g,y)-plane.

 

 

Figure 6. Behavior near a fold curve

in the (T3n)-plane.

54

These pictures are local representations. It is en-
tirely possible to have other solutions (xl,y1) that
correspond to YO' However, the solutions near (x1,yl)
will correspond to a different branch which does not in-
tersect the branch determined by (x0,y0) in some neigh-

borhood of (y0,0).

§5. Solutions of Type II

Let (H10)-(Hll) hold. Solutions (XO’YO’YO’O) of
(4.1) which satisfy neither (4.5)(a) nor (4.5)(b) will be
called solutions of type II. In general, the local behavior
near solutions of type II is not as easy to determine as
that near a solution of type I. waever, we can give a
complete description of this behavior when the original bi—
furcation equations contain a nondegenerate bilinear form
when T = n = 0. We shall determine some general properties
of solutions of type II.

If (XO’YO’YO’O) is a solution of type II, we must

have
Px Py
(5.1) det Q Q = 0 at (xo,yo,vo).
X Y
Proposition 5.1. If Px = Py = 0x = QY = 0 at a solu—
tion (XO’YO’YO) of (4.3), then either (XO’YO’YO)

(0,0,0) or the n-linear form is degenerate.

55

Proof. At (xo,yo,y0), we have
_ _ n n-l ... n
O - xOPx-tyOPY - yoxo-+n[anxO-+an_lxo yO-t -+aoyo]

n-l

n n
+bn-lx0 YO + - . - +boy0]

O = XOQx-tyOQ = yO-+n[bnxO

Y

Since P(xo,y0,yo) = Q(xo,yo,yo) = O, we have (n--l)yO = 0
and (n--l)yoxO = 0. Thus yo = 0. If Y0 = O and x0 #
0, then an = bn = O and a degeneracy occurs in the n-linear

form. If x0 = 0, PX = 0 implies Y0 = 0.C)

Recall that (0,0,0,0) is not considered in (H10)-
(H11), and therefore cannot be a solution of type II. Be-

fore we go on, we shall consider two special cases.

Case 1. x0 = 0. If a0 #'0 in (4.1), then y = O

and J(0,0,yo,0) = YO cannot be zero unless Y0 = 0. If
a0 = 0, then
.1.
_ n-l
YO - (-1A0) o
[If b0 is also zero, the n-linear form M.n is degenerate].
Suppose yO is real. Then the matrix in (H11) becomes
' W
YO -al/b0 O 0 N1 (031(0) Y030)
(5.2) «bl/bO -(n-l) 0 N2(O,yo,yo,0)
J J - n - 1 J
_ x Y ( ) g A

 

 

Now J(0,yo,yo,0) = 0 if and only if Y0 = al/bo. Case 1

can only occur if a0 = O and b0 # O. In this case

56

(H10)-(Hll) are satisfied if and only if Nl(O’YO’Y0’O) #

 

 

0. Then
r- 7
det (1 ’n) 0 N2 (0:170: Y020) 7! O
J (l-n) J
L Y s -
and
W
O N1(O’YO’ Y030) i
det 1 #VO.
(1 ‘11) N2 (0: Yo: YO’O)_)

Case 2. ox = Qy = 0 at (x ) 7‘ (0,0,0). As in

o’yo’ *0
the proof of proposition 5.1, we have that y0 = 0. If

 

 

bn #'0, then x = O and J(0,0,yo,0) = YO' Thus as before,
we assume bn = O and an #'0. Then (XO’YO’YO) # (0,0,0)
implies
__L_
_ n-l .
xO — (-YO/en) Wlth Y0 #’O.
The matrix of (H11) becomes
(5.3)
F ]
Yo‘nYo -an-1YO/ an xo N1("o’ 0’ Yo’ 0)

O l—Yobn-l/an O N2 (X0, 0, YO’O)

2 n-2
[:Y0(n-l) bn-le Jy l-YObn—l/an JE J
QY = 0 implies Y0 = an n-1 and bn-l #’0. Thus Case 2

occurs only when bn = 0, an #’O and bn-l #’O. In order

57

for (H10)-—(Hll) to hold, we must have N2(xo,0,yo,0) #'0,

x0 #'0, and Y0 # 0.

Example 5.1. Case 2 may be reduced to Case 1 by a
change of variables. we shall demonstrate this in the case
where M.n is a nondegenerate bilinear form. If we recall
the form of the nonlinear term before the scaling of §3,

we may write

2 2

o = YX+a2x +alxy+aoy +N1(XJY)Y§3§)

(5.4)
2

O = y +blxy +boy +N2 (X:Y:Y§:§)
Then
(5.5) x = _ fiL Y = aZ/bl’ y = O is a solution.

1

Now consider

pu-Tbouz-+blvu-+Nl(v,u,o,po)

0
II

(5.6)

O = v+aov2 +a1uv+a2u2 +N2 (v,u,o,p0)

If (xo’Yo’Yo’go) is a solution of (5.4) with y #'0, then
u = y/&, v = x/y, p = l/y, o = y§ is a solution of (5.6).

The solution (5.7) becomes
11 = 0, v = -l/a2, p = bl/aZ’

This correspondence arises from interchanging the roles

of T and n in the scalings discussed in § 3.

of

58

Proposition 5.2. Let (x 0) be a solution

03 Yo) Yo)

type II with x0 #'0. If 0x = O and QY #'0, then

 

 

 

 

Px = J2 = 0. If 0x #'0 and Qy = 0, then PY = Jy = 0.
Proof. PXQy--QXPy = 0 and (4.5)(a) fails if and only
Since (4.5)(a) and (4.5)(b) both fail for a solution

of type II, we have

Px x F§
(5.7) det Qx o G); a" o
J J J
t x Y 5.)
or
PP x F '1
Y 5
5.8 det O Gr 0
( ) Qy g 7‘
J J J
_ Y Y 5..
Proposition 5.4. Let (XO’YO’YO’O) be a solution of
type II. Then one of the following occurs.

(a) (5.7) holds and'

Px F PX x
det § # 0 or det #’0
OX G5 (.0X 0
(b) (5.8) holds and
P F P x
det Y 5 a! 0 or det Y 7! 0
Q G Q 0

Y E Y

59

Proof. This follows from Case 1, Case 2 and proposi-

tion (5.2).:)

Definition 5.1. Let T: R24 R2 be given by (u,v) 4

(f(u,v),v). We say that (uo,vo) is a cusp point for T

 

if
i: if.
(i) Bu(u0,v0) = 0 and all2(u0,v0) = 0
2
.. a f
(ii) auav(uo’v0) #’O

(iii) §3§(uo,vo) #'0.
au

Proposition 5.4. Let (x0,yo,y0,0) be a solution of
type II. Then

(a) The implicit function theorem enables us to solve
(4.1) for at least one of the following in a neighborhood
of (XO’YO’YO’O): y(x,§), y(Y,§), g(x,y), gum).

(b) At least one of the following maps exists and also

satisfies (i) and (ii) of definition (5.1) at (XO’YO’YO’O)

(x,§) 4 (y(x,§),§)
(ing) 4 (y(y,§),§)
(x,y) 4 (§(X,y),y)
(Y.y) 4 (g(y,y),y).

Proof. We will only give the proof in the case that

Py X Fg
(5.9) det Qy 0 Gg 7! 0
J,Y Jg

6O

Py x
and det # O at (x

Qy 0

031,0: YoJO) .
All other cases are similar.

Since XOQy #’O, there exist unique real analytic func-
tions y(x,§) and y(x,g) such that (x,y(x,§), y(x,§),§)
satisfies (4.1) near (x0,0) and y(x0,0) = O, y(xo,0) = 0.
Recall that x0 #’0 implies Qny-Qka = O for a solution

of type II. As in Lemma 4.8, we have

(a) fl(x0,0) = O and %§(x0,0) = -Q /Q

 

 

ax X Y

3.3!
(b) 8x2(xO,O) = 0 if and only if Qny-Qny = O

3.1 31

(C) sz (x ’0) = 3y aa+ Jy at+ J):

axag 0 -XOQY

_ Jy[xOG§] -Jv[PyG§ -:§Qy] +JE[xOQy] .
(xOQy)

2
C '
Thus (5.9) implies g§§%(xo,0) #'O.D

Definition 5.2. If (XO’YO’YO’O) is a solution of

type I, we will call (x 0) a fold point. If

0) Yo) Yo)

(x O) is a solution of type II, and all conditions

0: YO: Yo:

of definition (5.1) are satisfied for one of the maps guar-

anteed by proposition 5.4, we will call (x 0.0) a

o’YOJV
cusp point.

61

In general, we shall not be able to verify the third
condition of definition (5.1), as it depends on the order
of contact between the two curves P(x,y,y0) = O and
Q(x,y,yo) = 0 at (x0,yo). On the intuition that two
conics cannot "touch too much“ without some form of degen-

eracy, we now specialize to the case of a nondegenerate

bilinear form

Ax2-+Bxy-+Cy

M2[ (x,y), (x,y)] =

axz-tbxy-Tcy

where A, B, and C are not all zero. Consider the system,

0 F(x,y,y,§) Yx+Ax2+BXY+CY2+§N1(X:Y:Y:§)

(5.10)

0 = G(x,y,v,§) y+ax2 +bXY+CY2 +§N2 (3910165)

As an example of what to expect, we will look at Case 1.
In this case C = O, c #’0 and (XO’YO’YO) = (O,-l/c,B/c).

The matrix (5.2) becomes

 

 

1

[ o o 0 N1 (0,-1/c,B/c,0)
-b/c -1 0 N2 (0,-1/c,B/c,0)

L-2A + Bb/c -B -1 J: 3

For a type II solution, we must have -A-+Bb/c = 0.

62

By the implicit function we may solve (5.10) for
g(x,y) and y(x,y). If we eXpand g(x,y), y(x,y) and
the equations in (5.10) in power series and then equate

coefficients, we find that

31 — _
(5.11) ax‘XO’Yo) - Qx/QY
331 -1 41 31 2
(5'12) ax2(xO’YO) = -Qy [Qxx-+Qxy ax + QYYKBX) ]
_ 2 21 ex 2
‘—- Qy[a-+b ax + CKBX) ]
2
(5.13) 6x2(xo,yo) = 0 if and only if Px-Pny/QY = 0
2
(5.14) 5—125-(x0,y0) = o if and only if

8X

2
° = [Paws £26 + 54%)]

2
_ '1 .31 31
PYQY [Qxx'l'Qxy BX + QYYKBX) ]

2
31 31 _ - gr
2[2r>.+sax + CKax) ] 2PyQy1[a +b x+c[§1x)2]

3
(5.15) 'i-§(x ,y ) = 0 if and only if
3 0 O
ax
-Qx 52 -Q 2
= _ _.1_ _1 + J El—
0 [ny nyK :flax PyQy [Qxy QYYKQY flaxz
2 2
if and only if, J = 0 or h—2 = 0. However, Q—x(x ,y )==
y 2 2 0 0
Bx 8X
0 implies that (x,y) = (l,%§) is a nontrivial solution of

‘I‘l'll

63

O = axz-I-bxy-i-cy2

O = sz-i-Bxy-i-Cy2

Since the bilinear form M2 is nondegenerate, this implies
Jy = 0. Since 0y #'0, we must have JX = O.
This gives the following

-B=O
-2An+B(b/c) = 0

and -An+Bb/c = 0

Hence A = O = B.D

Because of example 5.1, we may neglect Case 2. It
remains to examine the situation where x0 #'0, and one of
0x or Qy is nonzero. By proposition 5.3, one of the

following must hold

 

 

 

 

r- a-n
Py X F§ Py x
(5.16) det Qy O Gg 710 and det 7’0
Q 0
J J J
L y Y 5. y
or
i- -1
PX x Fg Px x
(5.17) det Qx C) Gg # O and det #'0
Q 0
J J J x
t x v t,
at (x0) Y0) YO, 0) 0

64

we shall only consider (5.16), as essentially the same
result holds for (5.17). We use the implicit function the—
orem to obtain y(x,§) and y(x,§) such that (x,y(x,§),
y(x,§),§) is a solution of (5.10) in a neighborhood of

(x0,y0,y0,0), and y (x0,0) = yo, y(xo,0) = YO' The analogues
3

of (5.11) - (5.15) now hold, and if L40:
5X

yx-tsz-thy-+Cy2

0,0) = 0, the system

0 = P(x,y,v)

(5.18)

y+ax2+bxy+cy2

0 = Q(X:Y:Y)

must satisfy J(xo,yo,yo) = PxQ --QXPy = 0, Jx = J = 0 at

Y Y
(x0,yo,yo). ‘we will take care of this situation with the

following lemma.

Lemma 5.5. Given A, B, C not all zero, there does
not exist (x,y,y) # (0,0,0), x #'0 and a, b, c such
that M2 is nondegenerate and the system in (5.18) satisfies

)==J =J =0.

Proof. we must satisfy the equations

(1) o = YX+AX2+BXY+CYZ
.. 2 2
(ii) 0 = y-Tax -+bxy-+cy
(iii) 0 = J(X,y,Y) = y-fybx-tZycy-tZAx-tZsz

+ 4Acxy+By+2ch2--2an2--4any--2bCy2

(iv) 0 = J

x yb-TZA-t4Abx-+4Acy-4an-4aCy

(V) o = Jy

2yc-T4Acx-PB-+4ch-4aCx-4bCy

65

We may use the fact that xe +yJY = 0 to replace (iii)

bY

(vi) 0 = y - 2Abx2 - 4Acxy - 2ch2 + 2an2 + 4any + 2bCy2 .

Since x 7’ O, we may multiply equations (iv), (v) and (vi)
by x and use (i) to eliminate Y. We now have four equa-

tions which are linear in a, b, and c,

F O = ax2 +bxy+cy2 +y

3 3

0 = a(2Bx +4Cx2y) +b(-2Ax +2Cy2x) +c(4Ax2y -ZBy2x)

J + (-Ax2 - Bxy - Cyz)

O = a(-4Bx2 - 4ny) +b(-Ax2 -Bxy -Cy2 +4Ax2) +c(4Axy) +2Ax

 

k 0 = a(-4Cx2) +b(-4ny) +c(2Ax2 +2Bxy-2Cy2) +Bx.

This system may be reduced by the usual row operations to

the following

[o

0

ax2 +bxy + cy2 +y

3 2

b(Ax +Bx y+ny2) +Ax2 +Bxy-Cy2

3 2

0=b(3Ax +3Bx 2

(5. 19) 1

y + 3ny2) + c (4Ax y + 4By2x +4Cy3)

+ 2Ax2 + 4Byx + 4Cy2

 

k0 c (2Ax2y + ZBxy + 2Cy2) + Bx + 40y.

An appropriate linear combination of the last three equations

shows tha t

66

3[Ax2-+Bxy-Cy2]-+2y[Bx-+4Cy]

- [2Ax2 +4Bxy +4Cy2]

0
ll

sz-TBxy-TCyZ.
Now (5.19) implies

O = Ax2+Bxy--Cy2

0 = sz +2Bxy +2Cy2

0
ll

Bx-T4Cy.

Thus Cy = O = Bx = Ax. Since x y'o, we have A = B = 0.
If C #’O, we must have y = 0. Then (ii) implies a = 0

and

CY
M2[ (x,y) (x,y)] =
bxy+cy2

has the nontrivial solution (x,0), and therefore is degen-
erate.

we have just proved the following:

Theorem 5.6. If the bilinear form

Ax2-+Bxy-(-Cy2

M2[(x,y), (x,y)) =

ax2+bxy+cy2

is nondegenerate and A, B, and C are not all zero, then

all solutions (XO’YO’YO’O) #’(0,0,0,0) of

67

F0 = yx+Ax2 +BXY+CY2 +§N1(X’Y’Y’ g)

(o

[_0

y-Tax2-+bxy-+cy2-+§N2(x,y,y,g)

 

J(XJY: Y: S)

are either cusp points or fold points, provided that (H10)-

(H11) are satisfied.

we shall use the following example to demonstrate be-

havior near a cu3p point.

Example 5.2. Consider the system

(5020) (a) O YX+XY+TN1(X:Y:Y:T)

(5.20)(b) o

2 2
y+x +y +TN2(X:Y:Y:T)

where N1(O,-l,l,0) #'0 At the solution (O,—l,l,O), the

matrix corresponding to (5.2) is given by

O -1 0 N2 (0,-1, 1,0)

 

O -1 -1 J
L 5 3

 

and has rank three. By theorem 5.6, (0,—1,l,0) is a cusp

point. If we solve (5.16)(b) for y(x,y,T), we obtain

y(x,y,T) = -1-+x2-+higher order terms.

substitution into (5.20)(a) gives

(5.21) o = (Y-1)X+X3+TN1(X:Y(X,Y,T);Y:T)-

68

we make the change of variables § = -(y-1) T =
-TN1(x,y(x,y,T),y,T) in a neighborhood of (O,-l,l,0).
Since Nl(O,-l,l,0) is nonzero, this change of variables
is a homeomorphism by the inverse mapping theorem. Now

(5.21) becomes

The mapping T: R 4 1R2 given by

>4
:4
l
.<
N
41

(5.22) 4

<1
<1
<1

is the standard form for the cusp singularity [ 3]. we

have the following local picture, [Figure 7]

‘N

 

 

T\

 

 

 

 

Figure 7. Behavior near a cusp.

69

The number of real solutions (x,y) corresponding to
each (Y,T) in a neighborhood of (1,0) is given schema-

tically by the following bifurcation diagram [Figure 8]:

 

 

 

Figure 8. Bifurcation diagram in the (T,Y)-plane.

If we change back to the (T,n)-plane by setting U = YT:
the corresponding diagram is given by Figure 9. The curve

r in Figure 8 has become a curve f tangent to the line

n=T-

A
q
4

 

 

Figure 9. Bifurcation diagram in the (T,n)-plane.

70

Note that this is only a local picture, and is valid
only in a neighborhood of the curve T. The picture can
be completed only when we have analyzed the behavior of

solutions near every (x which is a solution

0’ Yo, YO: To)
of

2 2

{0==yx+yx
O=y+x +y .

In general, if T: R24 1R2 satisfies definition (5.1),

there are local changes of coordinates in the domain and in
the range of T such that the local picture is given by
Figure 7 and T has the form (5.22) [ 3].

The hypotheses (H10)-(Hll) were chosen to insure that
the set of solutions (x,y,y,T) of (4.3) is a two manifold
in the neighborhood of (xo,y0,y0,0) #'(0,0,0,0). That is,
there is only one solution branch passing through
(XO,YO:YO,
always happen in applications. If two solution branches pass-

0). As we shall see in Chapter IV, this does not

through (x0,yO,Yo,O), both hypotheses (H10) and (H11) of

§4 are violated. In this case,iJ:is necessary to find a

way to "factor out" a branch so that the methods of §4 and

§5 apply. One is then faced with determining how the branches
intersect. we shall demonstrate how this may be done in an
application to chemical reaction equations discussed in

Chapter IV.

71

Q6. Solutions near (XO’YO’YO’O) = (0,0,0,0)
In this section, we shall informally discuss some ways
of determining the behavior of solutions of (4.1) near

(0,0,0,0). At (0,0,0,0), the matrix

 

 

 

 

and does not have rank two. Thus the hypotheses (H9)-—(HlO)

of §4 do not apply. The system

F n n-l n
(6.1)(a) o = F(x,y,y,§) = Tx+anx +an-1x Y+"‘ +‘J‘oY

+ §Nl (x: Y: V) g)

n-l n

n
(6.1)(b) O G(x,y,y,§) = y-tbnx 'kbn-lx y-+°°'-+boy

 

k + §N2(X:Y:Y:§)

is similar to the l-dimensional case of Chapter II as y
passes through 0 at g = 0, except for the extra parameter

1;.

Example 6.1. Consider the system

2

0 ny-x

(6.2)

0 = Tx-Tnxz-Tyz.
After the previous scaling, we have

0

y-—x
(6.3)

0 = YX+T1X2 +y2

3 T YT)-

72

Solutions near (x,y,y,n) = (0,0,0,0) correspond to solu-
tions which bifurcate from the n-axis near (0,0). If
(x,Y) is a nontrivial solution, elimination of y shows

that x must satisfy

0 = y+fix+x3
The discriminant of this cubic is given by —4n3-27Y2.
Thus along the curve 4n3 = ~27y2 in the (y,n)-plane
(or 4fi5 = —27T2 in the (T,n)-plane) the number of non-

trivial solutions changes by two.

 

 

 

A

 

”‘4

Figure 10. Behavior of nontrivial solutions

of system (6.2).

Figure 10 shows the projection of the "solution space" unto
the (T,n)—p1ane. This example shows the advantage of con—

sidering a full neighborhood of (T,n) = (0,0) in our

73

studies. It does not suffice to fix n and vary T to
study solutions which bifurcate from the n-axis.

This example shows that we may eXpect to find a cusp
on curves tangent to the axes in the (T,n)—plane. These
curves and cusps play a role similar to the fold curves
discussed earlier.

Since FY = l at (0,0,0,0), we may use the implicit
function theorem to solve for a unique y(x,y,g) such that
F(x,y(x,y,§),y,§) = 0 and y(0,0,0) = 0. Since (0,0,y,§)
is a solution for all (y,g) sufficiently close to (0,0),
y(O,Y,§) = O and y(x,y,g) = xH(x,y,g) where H is ana-
lytic in x, y and g. Substitution of y(x,y,g) in F1,

and division by x gives an equation of the form
(6'4) 0 = F(X:Y:§> = Y+[f1(Y:§)]X+[fZ(Y,§)]x2+~~

and we have successfully "divided out" the solution (x,y) =
(0,0).

At this point we may take the following approach:
solve (6.4) for y(x,g) and analyze the singularity of the
map T: (x,§) 4 (y(x,§),g) at (x,§) = (0,0). We shall

sketch how this may be done for the following special case

(6.5) (a) O

yx-Talx2-+b1xy-+cly2-+§Nl(x,y,y,§)

(6.5) (b) o

2 2
y-Tazx -+b2xy-+c2y -+gN2(x,y,y,g)

'We now find that

2 3
y(x,y,§) = -a2x -+b2a2x -+xH(x,y,g).

74

substitution of y(x,y,§) in (6.5)(a) and division by x

yields,
_ 2 3

where fl(0,0) = a1, f2(O,O) = -bla2, and f3(0,0)
b1b2a2-c1a2.
If a1 #’O, we may solve for a unique X(y,§). Since

ala2 = 0 would make the bilinear form

F 2 2
alx -+blxy-+cly

M((X:Y): (X:Y)) =<

 

2 2
Lazx -+b2xy-+c2y

degenerate, we now assume that a1 = 0, a2 #’0. As before,

we may solve for y(x,§). ‘we find that %%-= a1 = O and
2
h—% = —2b1a2, at (x,g) = (0,0). If b1 #'0, there is a
BX 3
"fold" [ 3] tangent to the axis. If bl = 0, i—% = -6c1a2.
2 Y"
If c1 # 0, and in addition,-§§§% #’O at (x,§) = (0,0),

there is a cusp [ 3] at (0,0). The situation is similar
to Example 6.2.

'We now outline an alternative approach which is more
analytic in nature and does not simply entail applying class-
ifications from singularity theory. This is useful for
problems which do not fit the standard classifications.

The main idea is to find curves (x(g),y(§),y(g),§) such

that x(0) = y(0) = y(0) = O and the Jacobian of system

75

(6.1) has zero determinant along these curves. These curves

are determined by the system

 

r0 =F1(X:Y(X)Y:§)1Y)g) = Yx+fl(Y:§)x2+f2(Y)§)3+"°
(6.6%
K0 = §F1(X,Y(X, Y: ELY: g) = Y +2fl(Y: §)X + 3f2 (Y: §)X2 +’ ' '

since

a}? as
a = ._1 ___l. ax =
axF1(x’Y(x’Y’§)’§) ax + By ax

aFl 5F aFZ AFZ

=__ 1(-_

ax ay. ax ay =0

at (x,y(x,y,§),y,g) if and only if

 

 

1- ‘1
asl aFl
ax ay
det = 0.
532 .532
t. ax ay .—

One such curve is x(g) = 0 = y(g). This corresponds
to the g-axis and the trivial solution (x,y) = (0,0). We
are interested in curves where x(g) #’O for g #'0. Let

xG(x,y, g) = Fl(x,y(x,y,g),y, g). G(x,y,§) = 0 implies

523141 (X,Y(X,y,§),v.§) = X%G(X.Y,§).

Thus nontrivial solutions of (6.6) are found by studying

(6.7)(a) O = G(x,y,g) = y+f1(y,§)X+---
(6.7)(b) o = 35,23 (X,y,§) = f1(Y:§)+2f2(Y:§)X+'°'

76

Since %§-#'0 at (0,0,0), we may use the implicit
function theorem to solve for y(x,§) with y(0,0) = 0.
y(x,§) is analytic in (x,g) for (x,§) in a neighbor-
hood of (0,0). Substitution of y(x,§) in (6.7)(b) gives
a power series C(x,§) which converges in some neighbor-

hood of (0,0). The curves x(g) may be determined by

Newton's polygonal method.

CHAPTER IV

APPLICATION TO A SYSTEM OF
CHEMCIAL REACTION EQUATIONS

91. Introduction

In this chapter we shall apply the method of the pre-
ceeding chapter to a system of equations that arises in

the study of chemical reactions [ 2]:

2
(1.1) 33-:A-(B+1)x+x2y+D-a—’5, O<r< 1, t) 0
at arz
a 2 2
g%=Bx—xy+w31%, 0<:r<l, t>0
3r
with boundary conditions
X(O,t) = y(l,t) =A,

y(o,t) = y(l,t) B/A.

A and D are constants, and D = Dx’ vD = DY’ where
Dx and Dy are diffusion coefficients for x and y
respectively. B and v are taken to be the parameters.

For all values of B and v, x0 = A, y0 = B/A is an
equilibrium state. Substitution of x = An+u(r,t), y =

B/An+v(r,t) into (1.1) gives

77

 

78

2
.52: [3-1]u+A2v+D3—1—1+§u2+2Auv+u2v
at arz A

(1. 2) 2
A! = -Bu -A2v +\)D L2: —-B-u2 - 2Auv - u2v
A
ar
with boundary condition

u(0,t) = u(l,t) = v(0,t) = v(l,t) = 0.

The associated linear system is

f' 2
53 = [B-l]u+A2v+D L-u-
at 612
(1.3) < 2
31:- _ 2 5.1
Lat Bu AV+VD 2

 

hr
with the same boundary conditions.

Since we are interested in steady state solutions, we

will consider u and v as functions of r, and study the

stationary problem

 

F 2
O = [B- l]u+A2v+D 5L; + guz + 2Auv+u2v
dr
(1.4) 2
0 = -Bu-A2v+vD 534% -% 2- 2Auv-u2v
k ar
with boundary conditions
(1.5) u(O) = v(O) = u(1)= v(l) = 0

Let Y be the Banach space of twice continuously dif-

ferentiable functions from [0,1] to IR2 which satisfy

(1.5) and

 

 

 

79

(1.6) U”(0) = V” (0) = l1”(l) = V”(l) = 0.

Let 1(u:v>1y= 111111+11v11+11u'11+11v'11+11u”11+11v”11 where
“-H denotes the sup norm. Note that if (u,v) is a
solution of (l.4)-(l.5), then (1.6) is automatically

satisfied. Let

 

 

 

 

2 7 ' T

(B-l)+DL A2 u

A(V,B)[u,v] = dr 2

-B -A +vD-g—‘2- V

- dr .1 - .1
and
~ 1
l 2

BA- u + 2Auv + uzv

N(B,u,v) =

L-BA_lu2-2Auv - uzv‘

 

 

If (u,v) is in Y, then A(vgB)[u,v] and N(B,u,v)
satisfy (1.5) for all v and B. Let X be the Banach
space of continuous functions from [0,1] to JR2 which

satisfy (1.5). Let
[(u,v) )x = ”u“ + “V“

Then A(v,B): Y 4 X. is a continuous linear operator for
each (v,B) in 1R2, and N: IRx Y 4 X is continuous.

The stationary problem (l.4)-—(l.5) is equivalent to

(LH o=Awmnmw+mhmw.

80

If we replace R2 by C2 in the above, we see that
.A(v,B) and N have continuous complex extensions. A(v,B)
depends on v and B analytically and N depends on B,

u, and v analytically.

§2. Eigenvalues, Eigenvectors and Projections

The spectrum of the linear operator A(v,B) consists

of the eigenvalues

+
(2.1) a; =-§1-[B-l-A2-n21r2D(l-1-v)

: [[B -1 +112 +n21T2D(\) - 1)]2 -4A2B]1/2}.

The corresponding eigenfunctions are

sinrmm:
+
(202) Q; = + .
M; Sinlnn:

+
where M; satisfies

+
(2.3) 0;'- (B-l)-+n2w2D - A2 = 0.

5l+

+
For real 05, the projection unto the linear subspace spanned

+
by 6; is given by

1

I u .i _ 2 . —- . .i

(2.4) Pn[v:)§n - ii J‘ us1nn7rr + Nn vs1nn1rrdr 6n
1-+M N 0
n n
i
where Nh satisfies
+ 2 2 .i

(2.5) o;- (B-1)+n7rD+BNn=O.

81

sin mrr
(2.65)

UI+

.i
N sin n'rrr
n

is the solution of the adjoint equation

 

 

 

 

 

 

YT ’YB-l)+D-JL -B 7 VA
1 2 1
(2 7) '1 ar
. o' =
n 2
)1 A2 -A2 + vD 3— ‘11
2 2 2
L _, L_ ar _, L. _a
with boundary condition Y1(O) = Y2(0) = 1111(1) = Y2(l) = 0.
+ +
We can see that P;[ 6;] = 1.

We shall now determine a curve in the real (V,B)-plane

such that all the eigenvalues of A(B,v) have negative
real part when (v,B) lies below this curve [ 2 ]. This
curve will be called the grave 9_f_ neutral stabilifl. We
can see that Rec; 2 Red; for all n. If 0

+ .
n is complex,
then the curve Red; = O is given by the straight line

(2.8) 2

B = 1+A +n2'11'2D(1+V).

If 0; is real, Rec; = O is given by the hyperbola 1%.
(29) B-l+n22D+Ail+——l—D
o — 1T \){ 2 2 30
n 1r
. +
A typical curve Reo '-

n — O is given in Figure 11.

82

 

 

+
Figure 11. The curve Reon = 0.

The curve we seek is found by joining portions of the curves
‘Reo; = 0 which lie lowest for each (v,B). Red: = 0 lies
lowest for sufficiently small v and for sufficiently large
\n For values in between, a finite number of the hyperbolas

lie lowest. A typical curve is "scalloped" and is given by

Figure 12.

 

 

V

Eimnelz. A typical curve of neutral stability.

83

'mhe intersection of Hn and Hn+1 is given by

A2
n2 (n+1) 2 (T212) 2

 

B = l-+n2w2D-+n2(n-+l)2(w2D)2[1 + 212 }
n W D

2 2

= l-Tn W 2

D-+(n-+l)zw D-l-n2(n-+1)2(1T2D)2

2

= (l-tn WZD)(l-+(n-+l)2w2D)

The number of such intersections which lie on the curve
in Figure 12 depends on the values of A and D. we will

study bifurcation in the neighborhood of (Vc’Bc) for

n 2.1.

Proposition 2.1. If 0; # a; for n ¢'m is real,

then it has multiplicity 1. That is, if [A(B,v)-o;I]k§ = 0,
+

then s = K§n'

Proof. We proceed by mathematical induction. It is
true for k -1. Suppose [A(B,v) -o;I]k§ =0. Let 3 =
+ k-l .. + + +
[A(B,v)-onI] Q. Then 6 = cgn and Pn[cqn]§n =

Pn[[A(B,v)-U;I]k-l§]§;. Since Pn commutes with A,

c4; = [A(B,\)) -O;I]k_1[Pn[¢]§;] = 0.

Therefore c

0 and the induction hypothesis implies
+
6 - KQno
Thus we have exactly the situation described in Chapter

III.

84

§3. Calcuation of the Bifurcation Equation

Let (u,v) in X be decomposed as follows.

u sin nTrr sin (n+l)7rr W1
(3. l) = x + y +
v Mn 811') mm Mn+l Sln (n+l)7rr sz
= xgn + Y§n+l + w
where
u u w1
x = Pn , y = Pn+l , and w = 18 in Qx.
v v w2
M N M N c rres ond to 0+ = O - 0+ and
n’ n’ n+1’ n+1 o p n - n+1

are given by

(3.2) Mn = -A‘2[Bc+1-n21rzn]
(3.3) Mn+1 = -A'2[Bc+1 - (n+l)21r2D]
(3.4) Nn = B;1[Bc-l-n21r2D]
(3.5) NM1 = B;1[Bc-1- (n-l)21r2D].

The subspace Qx is given by

wl w
{(wlwz) e XIPn

ll
'1!

II
0
9...)

n+1
w2 w2

<2Y = Ynox.

85

The auxiliary equation is given by

(3.6) o = [I-Pn-Pn+1][Aow+(B-BC)A1(u,v)+(v-vc)A2(u,v)]
+ N(B,u,v)
where A0 = A(VC,BC),
1 O O 0
A1 = ’ A2 = dz
-1 O 0 ‘——5
dr
A-lu2
and N(B,u,v) = N(B ,u,v)-+(B-B ) .
c c
-A-1u2

Proposition 3.1. There exists a unique real analytic
function w(x,y,v,B) such that XQD'FY§n+1'*W(X,Y,v,B) is
a solution of (3.6) for all (x,y,v,B) in a neighborhood
of (0,0,0,0) and w(0,0,0,0) = 0. Moreover, w(x,y,V,B)
can be expressed by a convergent power series in this neigh-
borhood. [This power series contains terms of second order
or higher in x,y,v,B and all terms contain at least an

x or y.]
Proof. we need only show that the restriction

(3.7) A

is a linear homeomorphism, and then apply the implicit func-

tion theorem. The rest follows from the remarks of I§3 and

86

equating coefficients when the power series for w(x,y,v,B)
is substituted into (3.6).

Since there are no eigenvalues of H0 in Qy, the map
(3.7) is one-to-one. It is onto by a "Fredholm alternative"

type theorem for boundary value problems [‘9]. Hence by

the open mapping theorem it is a linear homeomorphiamdj

The bifurcation equations are given by

(3.8) (a) 0 Pn[(B-BC)A1(U,V) + (v-vC)A2(U,V) +N(B,u,V)]

(3.8) (b) o = pn+1[ (B-Bc)Al(u,v) + (v 'Vc)A2(“’V) +N(B,u,v)].

Substitution of w(x,y,v,B) into (3.8) gives a system

of the form

0 = F(X,Y,\),B)

0 = G(X,Y,V,B)

where F and G are power series that converge in some
neighborhood of (0,0,0,0).
From this point on, we shall assume that n is odd.

F and G are actually the coefficients of

§ = ' and Q +1 =

O O W

8111 mrr 1 8111 (n + l)1rr '
- n

Mn 81!) mrrj J

Mn sin (n + l)1rr

respectively. 'We Shall demonstrate how symmetry properties
can be used to obtain information about the bifurcation

equations.

87

Recall the original system for the stationary problem

0 = A(B,v)[u,v] +N(B,u,v) .

Let 9 be the transformation given by

e(u(r),v(r)) = (u(l-—r),v(l-—r)).

It is easy to see that g is an involution and that it
commutes with the operators A(B,v) and N(B,u,v) for

all v and B. ‘We have

9 (Mn +Y§n+1 +w) XQn - ”n+1 + 9w

and
QW(XJY:V:B) = W(X, 'Y:V:B)°

Thus F and G must satisfy

F(X,-Y,\),B) F(X,Y,\),B)

-G(X,-Y,V,B) G(XJY:V:B)’

for all x,y,v,B in some ball about (0,0,0,0), and we have

the following.

Proposition 3.2. If n is odd, the bifurcation equa-

tions (3.2) can be reduced to the form
(3.9)(a) O = F(x,y,v,B) = Tx-+ax2-+by2-+(higher order terms)

(3.9) (b) O

G(x,y,v,B) ny-+cxy-+(higher order terms)

88

where n = n(v,B), T = T(v,B). Here F(x,y,v,B) contains
only even powers of y and G(x,y,v,B) contains only odd

powers of y.

As in the previous chapter, we may show that the first

terms of (3.9)(a) are

+ +
on aon
'?fi§(vc’Bc)(B"Bc)x + ifir(vc’Bc)(V"vc)x'+°°' .

Those of (3.9)(b) are

 

+
80 ac
—a‘;3-‘fl<vc,ac) (B-Bcw + anvflwdsc) (v-vc)Y+°°°

Thus T(v,B) and n(v,B) have the same linear part at

+ + .
(Vc’Bc) as that of on and Un+l reSpectively.

Since bifurcation can occur at (0,0,V,B) near (0,0,

. . + +
Bc) if and only if on or °n+l
only if the jacobian of (3.3) at (0,0,v,B) vanishes, we

v is zero, and if and

c)
conclude that T = (V,B) = 0 if and only if a;(v,B) = 0

. . +
and n(v,B) = 0 If and only if Un+1(v,B) = 0. Also
7(v,B) will have the same sign as o;(v,B), and “(u,B)
will have the same sign as o;+1(v,B) for (v,B) near
(Vc3Bc).

The second order terms in x and y that appear in
the bifurcation equations and which do not depend on (B-Bc)
or (v-—vc) are easy to compute. For the sake of complete-
ness, we give them both fOr n-odd and n-even. In (3.8)(a)

these terms are given by

.2...» 4
n

 

FF'
2

 

 

rr'

 

 

F 2

yM

 

F'

 

. 2
x Mn sm nvrr

L- (XZMn sin2 n1rr)

2 . 2
y Sin

2 .2
-y51n

.1

 

.J

89

x2 sin n21rr + 2xy sin n1rr sin (n + l)1rr 1

K L- (.7:2 sin2 nvrr + 2xy sin mrr sin (n + INTI)

(n-+1)wr

(n +1)7rr

 

L.

sin2 (n + l)1rr

 

_J

1\

 

 

xy(Mn +Mn+1) sin m1rr sin (n + 1)1rr

-xy (Mn +Mn+1) Sln mrr an (n + 1)1r'

 

 

n+1
+
_ 2M sin2 (n +1)7rr
y n+1
L J
The coefficient of x2 is
r
0 if n is even
1- N n]
—[Bc A l+2 [ if n is odd.
3"" AMnL—_1+MnNn
The coefficient of xy is
l-N n] Bc
2 1 l 1 [
W[(n+l) . 6n+2 _ 2-2n H[1+M N n] [IT + 2A(Mn'+Mn+l)]
< if n is even
if n is odd

 

‘

 

J

90

The coefficient of y2 is

r

 

 

 

0 if n is even
B
2 l l l [1-N ] c
f— — +-—-——- n f—— + 2AM ]
k W n 6n+4 2n+4 [lmnNn] A n+1

if n is odd.

For (3.8)(b), the coefficient of x2 is

 

 

 

 

 

r'2 1 1 1 [l-N ] Bc

'rr[n+1 _ 6n+2 - 2—2n:I 1+M 11:11 1 [A + ZMn:I

[ n+1 n+1
< if n is even
(_0 if n is odd.
The coefficient of xy is

('0 if n is even

2 1 l l [l-N ] B

--— -'—-- +-————] n+1 c
L7r[n 6n+4 2n+4— [— + 2AM ”4 )1

[1+Mn+an+i] A n n+1

if n is odd.

 

 

The coefficient of y2 is
r 8 [1“Nh+1] Bc
[-—-+ 2AM ] if n is even
3 (n+l)1r [1+Mn+an-l-1] A n+1
k 0 if n is odd.

we will carry out the analysis of the bifurcation equa-

tion for n odd. The case when n is even is similar. We

91

now make the change of variables from (B-Bc), (v-vc)
+ + .
to T = On(v,B), n = on+l(v,B). The point (Vc’Bc) becomes

the origin and (3.9) now has the form

(3.lO)(a) 0 TX-tax2-+by2-+higher order terms in x,y,n,T

(3.10)(b) O

nY'tcxy-thigher order terms X,Y,n,T.

we will assume that A and D are such that a, b, and c

are nonzero. Then the bilinear form

ax2+by2

B< (x,y). (x,y) > =

 

cxy
L

is nondegenerate, and we may use the change of scale given
in III §3. Recalling the consequences of symmetry given in
proposition 3.2., the bifurcation problem is equivalent to

studying the two systems

(3.ll)(a) O = x-+ax2-+by2-+TF1(x,y2,y,T)

(3.11) (b) o = yy+cxy+TyF2(x,y2,Y,'r)
and

(3.10)(a) 0 = yx-+ax2-+by2-+nGl(x,y2,y,T)

(3.10) (b) 0 = y +cxy +71sz (x, yzm, 'r)

where all terms in F1, F 61’ 62 are second order or

2)
higher terms in x and y.

92

If (x,y,y,T) is a solution of (3.11), then
(Tx,Ty,T,yT) is a solution of (3.10). If (x,y,y,n)
is a solution of (3.12), then (nx:nY:Yn:n) is a solu-
tion of (3.10). Equation (3.11) will be used to study
solutions of (3.10) corresponding to (T,n) in a set of
the form S1 = {(T:fi)| |T‘ 2.p1|n‘} where pl > O. Equa-
tion (3.12) will be used for sectors 82 = {(T,n)\ 'n‘IZ

PZIT‘}- S1 and 82 are given in Figure 13,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 13. The sectors S1 and 82.

(Of course p1 and p2 have been chosen so that the

sectors overlap).

94. Analysis of the Bifurcation Equations
Recall that n is odd and that we assume A and D

are chosen so that a, b and c are nonzero.

93

When T = O, (3.11) becomes

0 = x-i-axz-i-by2
(4.1)

O

yy + cxy

which has the nontrivial solutions
1 _
(4.2)(3) X=--a-, y—O

and

y(1 -%Y)
(4.2)(b) x=-3cfi, y=: cb .

These solutions coincide when y = c/a.

 

 

 

 

The matrix of (H10) at a solution (x0,yo,yo,0) is
given by b
r 7
l + 2axO 2byO 0 Fl (x0, yo, YO’ O)
L Cyo Y+cxo yo F2(xo’yo’Yo’0U
. _ 2
The determinant J(xo,yo,yo,0) — (l-+2ax0)(y+cxo)--2bcyo

1
3.903%: 0) and (030,030)

is zero only for the solution (-
at T = 0. Thus if (xo,yo,yo,0) is any other real solu-
tion, there are unique x(y,T), Y(Y,T) such that x(yo,0)
x0, y(yo,0) = y0 and (X(y,T),y(y,T),y,T) is a solution
of (3.11) in some neighborhood of (yo,0). It remains to

determine the behavior of real solutions of (3.11) near

(_ 1

C
530,530) and (0,0,0,0).

 

At the solution (- %, O,n§,,0), the matrix given in
(H11) becomes
1
l
O O O O
J J —l J
L X Y T J

 

and therefore does not satisfy the hypotheses (H10)-(Hll)

of Chapter III. However, we shall still be able to decribe

l

the behavior near (- 3, O , g, 0) . Recall that the bifur-

cation equations have the form

(4.3) (a) 0 x +ax2 +b)/2 +TF1(X:Y2: Y, T)

(4.3)(b) o y(y+cx+TF2(x.yz.y.T))

after scaling.

‘We may apply the implicit function theorem to (4.3)(a)

to obtain:

Proposition 4.1. There is a unique solution branch
of (4.3) of the form (x(y,T),O,y,T) in a neighborhood of
(— i, O , g, 0), with x(g, O) = - i. This branch is deter-

mined by (4.3)(a) and y = 0.

We shall refer to this branch as branch I. We shall

see that another branch is determined by

(4.4) (a) O

2 2 2
x+ax +by +TFl(x,y ,y,'r)

(4.4) (b) o

Y'+CX'*TF2(X,Y2:Y,T)

95

Let 3(x,y,y,T) denote the Jacobian of (4.4) at
(x,y,y,T). The matrix corresponding to (4.6) in Chapter

III is given by

 

 

r- '1
-1 O O
c 0 l
O 2bc O

L e

and has nonzero determinant. By Theorem III there is a

unique fold curve §(T), with corresponding solutions

(anion) such that $40) = -§-, Em) = - i, 37(0) = o,

o = 3(i(7),§(T),§(T),T), and(4.4) determines a solution
1 c

branch near (- 3, O, 3, O).

The direction of the fold is determined by the number
of real solutions of (4.4) for T = O and y near Eu
These solutions are given by (4.2)(b). we have the follow-

ing.

Proposition 4.2. The system (4.4) determines a solu-
tion branch of (4.3) which satisfies

(a) If be > 0, there are two real solutions of (4.4)
for Y < Y(T) and none for Y > Y(T).

(b) If bc < 0, there are two real solutions of (4.4)

for y > Y(T) and none for y < Y(T).

Note that this result is local and only holds in a

9,0).

neighborhood of (x,y,y,T) = (--%,,O, a

96

We shall refer to these solutions as branch II. Branch
II is determined at T = 0 by (4.2)(b) and in this sense

is distinct from branch I which is determined by (4.2)(a).

l c

At (--;, O, 3’ O), brandh I and branch II coincide.

Proposition 4.3. The only intersection of branches I

and II is the set of solutions (2(T),§(T),§(T),T).

Proof. For any (x,y,y,0), we have

 

 

1+2ax+T§ 2by+TW
J(X: Y: YJO) = det 5F2 3F2 = O
+ —— _—
.. C T ax T ay 4
when y = 0, because
—a-y—(X,O, Y,T) = O = $090, Y, 1') = 0.

Since (4.4) determines x(T) and y(T) uniquely when
y = O, the solution corresponding to the unique fold curve
§(T) has the form (imp) with x(0) = - %. Hence
(x(T),y(T),§(T),T) is part of branch I.

Secondly, the two branches can intersect only when the
Jacobian J(x,y,y,T) of the system (4.3) is zero and y = O.

'we have

_ BF]. .1
1+2ax+T —— 0
5X

J(x,0,y,T) = det

 

 

2
_ O y+cx+TF2(x,y ,Yfl-L.

97

we see that J(x,0,y,T) = O at a solution (x,0,y,T) of
(4.3)(fbr 1'sufficient1y small), if and only if (x,0,y,T)
is also a solution of (4.4). There is a unique solution
of (4.4) of this form and it is the one corresponding to

the fold curve.

The situation just studied is completely different
from that considered in Chapter III. we may view branch
II as bifurcating from the solution (i(t),0) of branch I
at (7%), T).

we now determine the behavior of real solutions of

(4.3) near (0,0,0,0).

Proposition 4.4.(a) If cb > O, and T and y are
sufficiently small, there are two nontrivial real solutions
(x(y,T),y(y,T)) of (4.3) for y'> 0 such that x and y
tend to zero with y. There are none near (0,0,0,0) for
y(O.

(b) If cb < 0, there are two nontrivial real solutions
for y < O, and none for y > 0. At T = 0, these solutions

are given by (4.2)(b).

Proof. we may apply the implicit function theorem to
(4.3)(a) to obtain x(y,y,T) such that (x(y,y,T),y,y,T)
is a solution of (4.3)(b) in a neighborhood of (0,0,0)
and x(0,0,0) = O. substitution into (4.3)(b) gives an

equation of the form

98

3 3
o = YY-C'bY +y g(y,Y,T)

where g(0,0,0) = 0 [Terms in g(y,Y,T) which do not con-
tain T or y as a factor will be at least first order

in y]. Division by y takes care of the trivial solution
that exists for all y. The results follow from a consider-

ation of

2 2
(4.5) 0 = Y—cby +y gum/n)
for T and y sufficiently small.D

In fact, the curve Y(T) = O is a fold curve for the
system consisting of (4.3)(a) and (4.5), and the solution
correSponding to this fold curve is (x(T),y(T)) = (0,0).
The line Y(T) = 0 corresponds to the T-axis in the T’fi
plane. Thus branch II bifurcates from the trivial solution
at the T—axis, and disappears at the curve of solutions
(T§(T),O) of branch I when y reaches ?(T); i.e., (T,“)
reaches (T,?(T)T) in the T-n plane. In the (T,n)-

coordinates, branch II solutions have the fbrm

x
ll

—TY/c+ooo =—n/c+ooo

/y 11 -a/cY1 +cb/nU -a/cn)
y=irr Cb +ooo=_ +...

All that remains is to determine the behavior of solu-

 

 

 

 

tions for Y near zero in (3.12). [This corresponds to
Y near "a“ in (3.11); or the n-axis in the T-n plane].

When T = 0, (3.12) becomes

99

O = Yx-i-axz-i-by2

(4.6)

O y-Tcxy

Nontrivial solutions of (4.6) are given by

 

 

(4.7) x=-§. y=o
and

(X-C/afi
(4.7)(b) X=-%, y=i/ be

The solutions (4.7)(b) are real for y near 0 if
and only if bc and c/h have opposite signs. The solu-

tion (4.7)(a) approaches the origin as Y tends to zero.

Proposition 4.5. There is exactly one nontrivial
solution (x(Y,n),y(y,n)) of (4.6) in a neighborhood of the
origin, such that x(Y,n) and y(y,n) tend to zero with

y. This solution is given by (4.7)(a) when n = 0.

Proof. As in proposition (4.4), y(x,Y,n) is deter-

mined by (3.12)(a) and the problem reduces to
2 2
(4.8) 0 = Yx+ax +x h(x,y,n)

where h(0,0,0) = 0. Since a #’O, the result follows

from the implicit function theorem.U

Thus branch I bifurcates from the trivial solution at

the n-axis. In (T,n)-coordinates branch I solutions have

100

the form
x = - Yn/é-t--o = -T/h'+°" : Y = 0-

We now choose the sectors S1 and S2 mentioned in
§3. Let S1 = [(T,n)| |T\ 2.91ln\} ‘where 0 < p1 < [a/c|.
p2 is chosen so that it overlaps 81, but does not contain
the line n = (c/a)T.

we summarize our results by the following bifurcation
diagrams in the (T,n)-plane (Figure 14). The numerals de-
note the number of nontrivial solutions of (3.10) which
correspond to (T:n) in each sector. The curve §(T)
given by proposition (4.2) has become the curve F(T) =

(T,§(T)T) which passes through the origin with slope c/a.

 

lOl

 

 

 

h n
F
I \f\\ \\\\ . - 1e
‘ \\\ 1" f 17
F

A
Case 1. bc > O, c/a > 0 Case 2. be > O, c/a < O

n “7,

F

 

 

”1

 

Case 3. be < O, c/a > 0 Case 4. bc < 0, c/a < 0

Figure 14. Bifurcation diagrams for the system

(3.10) in the (T,n)-plane.

Figure 15 is an attempt at a more geometrical repre-
sentation of the situation. We are only considering Case

4 in Figure 15.

102

 

  

 

 

B

 

Intuitive representation of bifurcation

and c/a < O.

103

Figure 15 is meant more as an aid to the intuition
rather than an actual representation of the solutions
(x,y,T,n). One must interpret the drawing in the following
spirit.

Let T2-+n2 = c be a sufficiently small circle in the
T‘fi plane. As we pass through point A, solutions of branch
I (broken line) bifurcate from the trivial solution at the
n-axis. As we move around the circle counterclockwise the
corresponding solution of branch I passes through branch II
at B, proceeds to bypass the trivial solution at the T-axis,
and finally passes through the trivial solution when it
reaches the n-axis again at C; etc. Solutions of branch
II (double valued) bifurcate from the T-axis and continue

until they disappear into branch I at the solution (T§(T),O)

corresponding to r.

[1]

[2]

[3]

[4]

[5]

[5]

[7]

[8]

[9]
[10]

[11]

BIBLIOGRAPHY

Andronov, A.A., E.A~ Leontovich, I.I. Gordon, and
A. G. Maier, Theory Lf Bifurcations of Dynamic
Systems Ln a Plane, John Wiley & Sons, Inc.,
New Ybrk, 1973.

Boa, J.A., and D.S. Cohen, Bifurcation of Localized
Disturbances in a Model Biochemical Reaction,
SIAM J. Appl. Math., 30, no. 6 (1976), 123-135.

Br5cker, T.H., and L. Lander, Differentiable Germs
and Catastrophes, Cambridge university Press,
Great Britain, 1975.

Chow, S.N., J.K. Hale, and J. Mallet-Paret, Appli-
cations of Generic Bifurcation I, Arch. Rational
.Mech. Anal. 59 (1975), 159-188. [Applications
of Generic Bifurcation II to appear].

Dieudonné, J., Foundations 9§_Modern Analysis,
Academic Press, New Ybrk, 1960.

Gavalas, G.R., Nonlinear Differential gguations Lf
Chemically Reacting Systems, Springer, New York,
1968.

Hale, J. K., _pplications Lf Alternative Problems,
Lecture notes 71- 1, Division of Applied Mathe-
matics, Center for Dynamical Systems, Brown
University, Providence, R.I.

Hale, J.K., Ordinary Qifferential Equations, in
"Pure and Applied Mathematics", Vbl. XXI, Wiley-
Interscience, New YOrk, 1969.

Hartman, P., Ordinary Differential Equations, John
‘Wiley & Sons, Inc., New York, 1964.

Henry, D., Geometric theory of semilinear parabolic
equations, unpublished lecture notes.

Kato, T., Perturbation Theory for Linear Operations,
Springer-Verlag, New YOrk, 1966.

104

[12]

[13]

[14]

[15]

[161

[17]

[18]

[19]

105

Keller, J.B., and S. Antman, Editors, Bifurcation
Theory and Nonlinear Eigenvalue Problems,
Courant Institute of Mathematical Sciences,
New YOrk University, New Ybrk, 1967.

Kirchggssner, K., and Sorger, P., Stability analysis
of branching solutions of the Navier-Stokes equa-
tions, Proc. TWelfth Internat. Congress Appl.
Mech., Stanford, 1968, 257-268.

Kirchgassner, K., and H. Kielhcfer, Stability and
bifurcation in fluid dynamics, Rocky Mtn. J.
Math., 3, no. 2 (1973), 275-318.

Nachbin, L., pr01092 22.Spaces g; Hblomorphic
Mappings, Springer-Verlag, New Ybrk, 1969.

Rabinowitz, P.H., Existence and nonuniqueness of
rectangular solutions of the Bénard problem,
Arch. Rational Mech. Anal., 29 (1968), 32-57.

Sattinger, D.H., Stability of bifurcating solutions
by Levay-Schauder degree, Arch. Rational Mech.
Anal., 43 (1971), 154-166.

Yudovich, V.I., On the origin of convection, PMM
(Jour. Appl. Math. and Mech.) 30, no. 6, (1966),
1193-1199.

Yudovich, V.I., Free convection and bifurcation, PMM
31, no. 2, (1967).

        

      

 

HICHIGRN STGT

u [[1 [g

3 2 31

    

 

 

 

 

 

 

 

 

 

 

 

 

E UNIV. LIBRnRIES
[IIIIIWIWIIHIIIHHINIIW\l
05309474