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dissertation entitled

BIFURCATION OF SYMMETRIC
PLANAR VECTOR FIELDS

presented by

Hyeong-Kwan Ju

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mathematics

 

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BIFURCATION OF SYMMETRIC PLANAR VECTOR FIELDS
By

Hyeong—Kwan J u

A DISSERTATION

Submitted to
Michi an State University
in partial ful illment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1988

 

(‘0
fl“)

q

\Fx

ABSTRACT

BIFURCATION OF SYMMETRIC PLANAR VECTOR FIELDS
By

Hyeong—Kwan J u

The pitchfork homoclinic bifurcation of symmetric planar vector fields
and some codimension three bifurcation of symmetric planar vector fields with
nilpotent linear part are studied.

The set of symmetric planar vector fields with an equilibrium point of
pitchfork type and a symmetric homoclinic orbits at this equilibrium point is
a oodimension two submanifold. This is shown using Melnikov's integral
around the homoclinic orbit and studying the asymptotic behavior near the
equilibrium point in R2. The bifurcation diagram is also obtained.

The bifurcation diagrams of generic 3—parameter families of symmetric
planar vector fields with linear nilpotent part is analyzed and described on
the sphere using abelian integrals. These integrals generate the Picard—Fuchs
equations which in turn gives the number of limit cycles. The t0pological
equivalence of the bifurcation diagrams is shown to be determined by the
number of limit cycles. This can then be used to determine the topological

equivalence classes of the bifurcation diagrams.

 

 

 

To

Rockjune Ju
January 21, 1904 - June 13, 1969

He
always
wanted the world to be
. a mathematical place
without mathematics,
and
whenever and wherever
he touched it,
it was.

He
always
tried to make his son to be
a strong man
without strength.

iii

 

 

 

ACKNOWLEDGMENTS

I wish to express my deep gratitude to my thesis advisor Professor
Shui—Nee Chow for his consistent assistance and patience as well as frequent
comments and encouragement. My thanks also go to Professor Chengzhi Li
for his many useful suggestions during his visit at Michigan State University.

I wish to thank Professors Wei Eihn Kuan, Kyung Whan Kwun,
Konstantin Mischaikow, and Wellington Ow for their patient reading of my
thesis and attending my defence.

I am grateful to Ms. Barbara Miller and Ms. Sterling Tryon—Hartwig
for their kind help with things organizational, Mr. and Mrs. Henry and June
Huber for their sincere assistance, my brother Hyeong—Soo for his continuing
support, and Ms. Cathy Sparks for her most efficient and quick elegant
typing of my rough manuscript.

Finally, a special thanks go to of course my only lovely daughter Amy
and my wife Eunice for their understanding and motivating me through the
Spartan life. I always felt so sorry for Amy because I could not spend

enough time with her.

iv

 

 

 

Chapter

TABLE OF CONTENT

1 PRELIMINARIES

231.

§2.

Basic Definitions and Structural Stability
on the Plane

Center Manifold Theorem for the Vector Fields

2 PITCHFORK HOMOCLINIC BIFURCATION

§1.
§2.
§3.
§4.

Introduction

Assumptions

Statement of the Main Theorems
Proofs and Remarks

A. Proof of Theorem (2.3.2)
B. Proof of Theorem (2.3.3)
C. Proof of Theorem (2.3.4)

3 GENERIC 3—PARAMETER FAMILIES OF THE
SYMMETRIC PLANAR VECTOR FIELDS WITH
NILPOTENT LINEAR PART

§1.

§2.
§3.
§4.

§5.

Introduction

A. Nonsymmetric Case

B. Symmetric Case

Versal Deformation (a 96 0, b = 0 case only)
The Case a > 0

The Case a < O

A. 61 > 0

B. el<0

The Case b at 0

Page

toque

11
14
15
24
35

38
38
39
40
41
45
65
68

92

97

 

 

APPENDIX
FIGURES
BIBLIOGRAPHY

vi

 

103
110
148

 

Figure

1

\lQCfidkww

coco

LIST OF FIGURES

Page
110
110
111
112
113
114
114
115
117
117
117
118
118
119
121
121
121
122
123
127
129
129
130
131
132

 

 

ilfi
S27'
5223
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III)
231
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231}
fill
2353
13(3
13'?
ZIEB
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110
ll 1

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1:313
1.11 1
1111
11122
1.11i2
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1:41}
1.11:}
11411
1111?
1.4'7
1417
1417
114’?
114'?

CHAPTER 1. PRELIMINARIES.

In this chapter we will introduce basic definitions, some fundamental
results concerning the structural stability on the plane, and the center manifold
theorem. Theorems will be stated without proofs. We restrict our interests to

the autonomous vector fields through the dissertation.
§1. Basic Definitions and Structural Stability on the Plane.

Let M be a two—dimensional smooth manifold and .3 r(M) a set of
Cr—vector fields on M. xocM is an equilibrium point of felt” r(M) if

f(xo) =0. Let fc$r(M) and x an equilibrium point of f in M. x0 is

o
generic if ReAtO for AcSpec(Df(xO)) = {A1,A2}. x0 is a m (resp. source)

 

if ReAi < 0 (resp. Reai > 0) for i = 1,2, a saddle if A1A2 < 0, a node

 

if A1A2 > 0, and a focus if Im()\i) at 0 for i = 1,2.
Let cp(t;x) be a flow of fez r(M) with cp(0;x) = x. Then
w(f,X) = n U <p(t;X), a(LX) = n U 90(t;X),
selR tZS sclR tgs

and are called w— and o—limit set respectively. I‘ is a closed orbit g g

 

 

(periodic orbit with period T) if there is T > 0 such that

 

w(t+T;x) = w(t;x) for th (and T is a smallest positive such number
possible). If I‘ is a closed orbit whose a— and w—limit set are only a
point {x0} where x0 is an equilibrium point of f, then P is called a
homoclinic or_bit a_t 50. Let two points {x0,y0} be equilibrium points in M

with x0 at yo. Then F is called a heteroclinic orbit between 50 ml yo

 

 

if there exists an orbit F whose only a—limit set is {x0} and w—limit set

is {yo}-

 

A family Of vector fields leRIn —i .5 r(M) is a universal unfolding of
fee? r(M) if m is a minimal dimension of a stable family F with
F(0) = f. For {1, f2c$r(M), fl is topologically equivalent to f2 (and we
denote it *f1 - f2) if there is a homeomorphism h: M —» M such that for
each ’xcM, 6 > 0 and ‘0 < t < 6, h(w§(x)) = w;(h(x)) for some 6 > 0
with O < s < 6. Note that - is an equivalence relation. If f is an
element of an interior of a topological equivalence class, then f is called to be

structurally stable.

 

Now let A be a manifold. We want to define the topological

equivalence between two families of vector fields.

Suppose Fi(i=1,2): A -i .Zr(M) are Cr. F1 is topologically equivalent
to F2 if there exists a homeomorphism g: A —i A and for each AcA, F1(A)
is topologically equivalent to F2(g(/\)) in the previous sense. Also a family

of vector fields F is structurally stable if F is an element of an interior of

a topological equivalence class in Cr(A,$ r(M)). fez r(M) is called a
bifurcation poi—nt if it is not structurally stable.

Next we want to define the bifurcation point of degree n by induction
which is crucial in the understanding of the bifurcation diagram.

f is a bifurcation m o_f m 9 if it is structurally stable.

f is a bifurcation poin_t 9_f m 1 if (1) f is not a bifurcation point
of degree 0, and (2) there is a neighborhood U of f such that for every g
in U, g - h for some h of degree 0, or g ~ f.

f is a bifurcation PM o_f @g‘rfi 2 if (1) f is not a bifurcation point
of degree 0 or 1, and (2) there is a neighborhood U of f such that for every
g in U, g ~ h for some h of degree 0 or‘l, or g ~ f. Similarly we can

define a bifurcation point of degree n.

 

From ‘the above definitions we have the following theorem which is

fundamental in the characterization of the bifurcation point of degree 1 on

M=IR2.

Theorem (1.1.1). A vector field f is a bifurcation of degree 1 in
$7M), r z 3, if and only if there is a neighborhood U of f and a

submanifold A of codimension one in U such that hle -+ .2” r(M) is

continuous with h(0) = M and h(a) eU—A for a a! 0 is structurally

stable but h(a1) is not tapologically equivalent to h(aQ) if ala2 < 0.

For h(0)cA, only one of the following occurs:

(1)

(2)

(5)

at a saddle point x0. h(a) for a < 0 has a saddle near x

h(0)cA has an elementary Saddle—node at X0 in M. There are no
equilibrium points of h(a) near x0 ' if a < 0 and a saddle and
a node near xO if ‘ a > 0.

h(0)cA has an elementary focus at. x0. There is no periodic orbit of

*h(a) ‘near xO if 3 a <. O and a periodic orbit near xO if

a > 0.

h(0)cA has a periodic orbit 7 which is stable from one side and
unstable from the other. h(a) for a < 0 has no periodic orbit
near -7 and h(a) for a > 0 has two hyperbolic periodic orbit
near 7.

Trace (-g)£((xo)) at 0 and} h(0) = feA has a homoclinic orbit 7

o
and no periodic orbit near. 7, Mai) for a > 0 has a saddle point
and a unique hyperbolic periodic orbit near 7 which coalesce as
a -i 0+. I

There is a connection between distinct saddle points (heteroclinic

orbit). '

For the proof, see Sotomayor [16] or Andronov et a1 [1].

Remarks.

(i) Phase portraits are for (1) — (5) in the Theorem (1.1.1) are
given in Figure 1.

(ii) This is the generic situation which arises in the case of one
parameter families (codimension one) of vector fields.

(iii) Schecter [15] described the bifurcation of codimension two which
occurs (1) and (4) simultaneously and showed the bifurcation
diagram of saddle—node homoclinic bifurcation.

(iv) Let sign?) = {f6$r(lR2)l f(-x) = —f(x) for deQ}. Then

fa?)f :(IR2) implies f(0) = 0, and we have the following in
Theorem (1.1.1) in a neighborhood of 0:

(6) There is a continuous map hle a .3 :(IRZ)

and spec(§§(o)) = {on}, A 9e 0, o is an only equilibrium

such that h(0) = f

point which is a saddle for h(a) if a < 0, and there are two
saddles and 0 is a node for h(a) if a > 0. (See also

Theorem (2.3.1) and Theorem in Appendix.)

The phase portraits for (6) are given in Figure 2.
(v) We are interested in the bifurcation diagram of an equilibrium
point which occurs (4) and (6) simultaneously (pitchfork

homoclinic bifurcation) and we will describe it in Chapter 2.

 

§2. Center Manifold Theorem for the Vector Fields.

Center manifold theorem is one of the most important and necessary
techniques for the nonlinear analysis and the bifurcation problem. It provides
us a benefit of the dimension reduction to a certain number (dimension of the
eigenspace of eigenvalues whose real parts are zero). We introduce the center
manifold theorem of the finite dimension for vector fields (we can get the same
theorem for maps by the discretized version which is the outside of our
interests here). In an obvious way it can be generalized to the infinite
dimensional problem. For details and proofs, for example, see Carr [3], Chow

and Hale [4] or Vanderbauwhede [19].

Let fee? r(an), r 2 1, and x(t;x0) be a. solution of x = f(x) with
x(0;x0) = x0. We say that a set A C [Rn is an invariant manifold of
x = f(x) if for every xocA, x(t;x0)cA for all telR. Also a set A is called
a loLal invariant minim of x = f(x) if there exists 6 > 0 such that for
erA, x(t;xo)cA for all tc(—c,e).
Let x = 0 be an equilibrium point of an fafi l.(an), r 2 1, and let
the spectrum of Df(0) be Spec(Df(0)) = SP + U SP 0 U SP_, where
SP+ == {AcSpec(Df(0))|Re)i > 0}
SP0 = {AeSpec(Df(0))|ReA = 0}
SP_ = {AcSpec(Df(0))|ReA < 0}.
Let E +(reSp. E0, E_) be the generalized eigenspace for SP +(l‘eSp. SP 0, SP_)

so that [Rn = E + e E0 69 E_. Now we state the theorem.

Theorem (1.2.1). With above notations and assumptions, there exist local
invariant manifolds Wu, Wc and WS tangent to E +, EO and E_ at 0
respectively, Wu and W8 are Cr and unique, however, WC is Cr—1

and not necessarily unique.

In Theorem (1.2.1) WC is called a loom center manifold. Usually we
prove the so—called global center manifold for the bounded vector field and then
using a cut—off function in the neighborhood of the equilibrium point of the
(not necessarily bounded) vector field and applying the global center manifold
theorem, we prove the local center manifold. Proof of the existence of the
global center manifold is required to use the implicit function theorem or a

contracting mapping theorem.

CHAPTER 2. PITCHFORK HOMOCLINIC BIFURCATION.

In this chapter we will give a bifurcation diagram and its explanation.
Next we will state formal assumptions for the pitchfork homoclinic bifurcation

and describe main theorems that we sh0uld prove. Then proofs will be given.
§1. Introduction.

Let fed? :(D), r sufficiently large (will be determined later), and consider
the following:

(2.1.1) it = f(x), xelR2,

such that 0 is a pitchfork and there are a pair of homoclinic orbits in D,

I‘ and —F which are stable. Then the set of all such vector fields is a
codimension two submanifold with an appropriate smoothness in .2” r(D), where
D is a symmetric neighborhood of I‘ (D is said to be symmetric if

—D = D in R2).

Suppose we have a two—parameter unfolding family 0f (2.1.1)
(2.1.2) x = f(x,al,a2)

where I(-,al,012)€.$;(D) for each (a1,a2)elR2, such that f(x,0,0) = f(x).
We would like to find a computable condition on the transversality of the
family (2.1.2) to Jim) at (01,012) = (0,0).

If the transversality condition is satisfied, we have certain smooth

nonsingular parameter coordinates changes (a1,o2) —» (71,72) —i (”1’M2)’

 

preserving the origin, and let f(x,a1,a2) = g(x,rl,r2) = f(x,,u1,u2), then
x = f(x,p1u2) has the bifurcation diagram of Figure 3 in a sufficiently small

and He’ in the

neighborhood of (#1412) = (0,0). We have two curves, H01

left hand ‘side of the pitchfork bifurcation curve P(=p2—axis). They meet P
and H 0 (= the

at (u1,a2) = (0,0) with quadratic tangencies. H 2

positive 111—axis) are homoclinic bifurcation curves of codimension one, while

01

He is a heteroclinic bifurcation curve cf codimension one. H 02 meets P

transversally.
The phase portraits of x = f(x,u1,a2) in a neighborhood of —I‘UI‘ is as
follows: ' i
A. #1 = 0 (The origin is a pitchfork.)
1. p2 = 0: two homoclinic orbits at the origin (figure eight).
2. 112 > 0: two stable closed orbits inside the stable manifolds.
3. p2 < 0: one stable closed orbit surrounding the origin.
B. a1 > 0 (The origin is a saddle.)
4. a2 > 0: two [stable closed orbits inside the stable manifolds.
5. a2 = 0: two homoc1inic orbits at the origin (figure eight).
6. a2 < 0: one stable closed orbit enclosing the origin.
C. ,u1 < 0 (The origin is a node, and there are two saddles in the
opposite side of the origin.)
7. p2 above H01: two stable closed orbits inside the stable
manifolds.
8. a2 on H01: two homoclinic orbits at saddles.
9. #2 between H01 and He: the flow of the vector field with
the initial x in the unstable manifolds of saddles tend to the

origin as t —> oo.

 

10. ”2 on He: two heteroclinic orbits which join one saddle to
the other. .

11. #2 below Hez' unique closed orbitsurrounding three equilibria.

Our techniques can be applied to the investigation of the vector field with
double heteroclinic orbits joining a saddle—nOde equilibrium point and a saddle
equilibrium point as in Figure 4.

We can extend our results to the more general problem by dropping "the
symmetry condition" on the vector field which will lead to the codimension
three problem. “Also we can consider this on the higher dimensional manifold

(of dimension greater than two) in a similar way as in Chow and Lin [5].

§2. Assumptions.

We consider a vector field it = f(x) with fez ;(D) where r > 3 for
a moment and DCIR2 a symmetric neighborhood of l", satisfying the following

conditions at the origin in 112.
(I) Spec(Df(0)) = {0,41}, )1 > 0.

Let u be a right eigenvector of the eigenvalue 0 and w be a left

eigenvector of the eigenvalue 0 such that u'w > 0.

(11) w-D3f(0)(u,u,u) > 0.
(III) x = f(x) has a homoclinic orbit 1‘ (hence —I‘) at 0 which

is hyperbolic. (Here we assume that I‘ is stable.)

 

 

 

A homoclinic orbit is hyp_erbolic if any Poincare return map from a
transversal section of the homoclinic orbit into itself (if defined) doesn't have
eigenvalues with absolute value 1 at a fixed point which is an intersection of

the homoclinic orbit and the transversal intersection.

Remarks
(i) fc.$;(D) implies w-D2f(0)(u,u) =
(ii) AssumptiOns I and II imply that x = f(x) has a pitchfork at
x = 0 with one negative eigenvalue. (See Appendix.)
(iii) We may assume that u is a tangent vector to I‘ at 0.

(Otherwise replace it by —u.)

For z,wdR2, we denote zAw by' Jz-w where J = [(1) 1%]. Note that
wAz = -—zAw and .12 is a rotation of the vector 2 by 7r/2 to the positive
direction in angle. 1

Now let v be a right eigenvalue of Df(0) corresponding to —/\ such
that v is tangent to F at 0. .We may assume that uAv > 0 without
loss of generality since if not, one can consider the reflection of the vector field
with respect to the x—axis ((x,—y) -1 (x,y)) .or y—axis ((—x,y) —i (x,y)).

Let x = f(x, £11,012) be a two parameter family of vector fields on IR2

such that f(-a, a2)6.2’ §(D) for each ((11,02)cIR2 and f(x,0,0) = f(x) as
well as '
(IV) w-D( x53; Moon )>> 0,

(V) f is sufficiently smooth (at least Cu).

 

11

From Assumptions (II) and (IV), perturbation in the positive L11
direction makes the origin a saddle from-a pitchfork, while perturbation in the
negative 011 direction produces a new. pair of saddles from the pitchfork

point(0,0), which becomes a node. '
§3. Statement of the Main“ Theorems.

We state the theorems whose proofs will be given in the next section
except. the proof of Theorem (2.3.1). First 'we state the following. Its proof

will be shown in the Appendix.

Theorem (2.3.1). Under the assumptions (I), (II), and (IV), there is a Cr—1
function p(a2), with p(0) = '0, such that for (011,012) near (0.0)
x = f(x,a1,a2) has an equilibrium of pitchfork at 0 if and only if

011 = p(c12).

From the Theorem (2.3.1), we change the coordinates in the parameter
space, say,
T1 = 0‘1 ‘ p(0‘2)

T2 — “2
and let

30971172) : f(xaa'1(71172)1 0201172))
Then '

(2.3.1) x = g(x,rl,72)

is CPI. x = 0 is an equilibrium of pitchfork in (2.3.1) if and only if

 

 

 

12

r1 = 0. ’ If Tl < 0, 0 is a node of (2.3.1) and there are a pair of saddles
near 0 but opposite sides of 0 each other, and if r1 > 0, 0 is a unique
equilibrium point which is a saddle.

' Let x0 be a point -I‘ and let ”cp(t; x0) be a flow of (2.3.1) with

<p(0; x0) = x0 for (71,7'2) = 0. We will denote cp(t; x0) simply. by 1,0(t)
if there is- no confusion.

Let

-oo

00 t .
(2.3.2) 11 = J eXP("l div g(so(S),0,0)dS) g(r(t),0,0) A 35—(r(t),0,0)dt.
0 T2
Then we have the following theorem.

Theorem [2.3.2].

(1) 11‘ converges.

(2) If I1 ,e 0, then-there exists a Cr—l’function q(rl), with q(0) = 0,
such that for (r1,r2) near (0,0) (2.3.1) has a pair of homoclinic orbit at 0
if and only if 72 = q(rl) for r1 2 0, where q(rl) = mr1 + 0(r1) for

some constant mclR.

Theorem (2.3.2) gives us an analysis for the bifurcation on the right hand
side of the pitchfork bifurcation curve. Once we have the transversality
condition (11 at 0), we can change the coordinates in the parameter space
again as in the following. i

Let

I ”1 =671
#2 = 72 — (1(71),

13

and let

. f(xiflliflg) = 309710111112), 7'20‘14‘2»:
Then

(2.3.3) _ Sc = fin/11.112)

is a vector field of Cr—1 since q is Cr—l.

In (2.3.3) 0 is an equilibrium point of pitchfork if' and only if [11 = 0.
Also p1 > 0 and 112 = 0 if and only if 0 is a saddle and there exist a
pair of homoclinic orbits at 0 near —FUF.

To considerthe bifurcations on the left hand side of the pitchfork
bifurcation curve (pl = r1 < 0), we need‘the following.
Let-

(2.3.4) 12 = gain—[tow f(cp(s),0,0)ds) f(ip(s),0,0) A %(o(t),o,0)dt.

Then

Theorem (2.3.3).

(1) I2 converges.

(2) If I2 7% 0, then there exists Cr“4 curve ((#2), [(0) = 0, such
that for (111412) sufficiently near (0,0), (2.3.3) has a pair of homoclinic (resp.
heteroclinic) orbits near —PUF if and only if ,u1 = £7012) and 12412 2 0
(resp. S 0) where ((112) = mflig +Ao(,u§), m1 at 0 a constant in R.

In the expression of I1 and 12, since g(x,0,0) = f(x,0,0),

 

14

div g(x,0,0) = div f(x,0,0), and gig—(hop) = gig—(x103), we have 11 = 12.
2 2

So we denote I1 = I2 simply by I.

I Next, we consider the transversality condition of the family of vector
fields it = f(x,al,a2) to the vector fields which satisfy the assumptions
(I) — (V). Let D be a symmetric open neighborhood of (n_ot necessarily
—I‘UP, M) 0. We denote 20 by the space of all symmetric vector fields in
D with Cr—topology, r z 11, i.e., 230 '= $;(D), r 2 11. Let
21' = {feEOIf satisfies (I) and (II) at 0, and all other-equilibria of f in D
are hyperbolic}.. The Appendix says that 21 is a CI—1 submanifold of

codimension one in 20. Also we let E2 = {ftEl|f satisfies (III), and

—1“uI‘cD}.

Theorem (2.3.4). The family x = f(x,al,a2) under assumptions in the
previous secton is transverse to 22 at (ol,a2) = (0.0) if and only if

ItO.

We are interested only in a neighborhood of —I‘UI‘. When the
transversality condition (I it 0) is satisfied, Theorems (2.3.1) — (2.3.4) and
Poincare—Bendixon Theorem (plus the hyperbolicity of F) gives our diagram

and its corresponding phase portraits of Figure 3.
§4. Proofs.

Without loss of generality we may consider the neighborhood of P by

the symmetry property.

A. Proof of Theorem [2.3.2].

Assume that 72 2 0, and consider the system

it = g(x,71,r2) .
(2.4.1) '71 = 0

T2 —
r—1

Center manifold theorem shows that there exists a 3-dimensional C

c
loc

Let <p be a flow of (2.3.1), and let

local center manifold W at (0,0,0) tangent to the center space in (2.4.1).

0 wheels,»

= U U
tclR (x,rl,r2)cW10c

Then each (11,72)—section of W is a curve and (0,0)—section of W is
TI‘UI‘. Let W(r1,r2) be a (r1,r2)—section of W. Then for (71,7-2)
sufficiently near (0,0), W(rl,r2) is a curve in the neighborhood of —FUP.
Let L be a transversal line segment in R2 perpendicular to F at x0.
Then for (r1,r2) near (0,0), W(rl,r2) intersects L transversally near x0.
Hence there exists x(r1,r2) such that _x(0,0) = x0 and

x(‘rl,r2) £W(Tl,72)rlL. Also x(7'1,r2) is CPI. Now we have a Cr—1
family ofsolution of (2.3.1) 1pc(t; x(rl,r2),rl,r2) (simply cpc(t,rl,r2)) for
(rl,r2) small, such that wc(0,rl,r2) = x(rl,'r2), and so

cpc(t,0,0) = wc(t; x0,0,0): E w(t), (pc(t,rl,r2)cW(rl,r2) for th.

Note that (pc(t,0,7'2) _ is a branch of the local unstable manifold of the
pitchfork 0 of x = g(x,0,r2) and for T1 > 0, (pc(t,rl,r2) is that of the
saddle 0 of (2.3.1). _

Next we define a parameter dependent Cl?“1 change of coordinates on

R2

 

16

(2”42) Y(X1T117-2) = (2110971172), y2(xaTlaT2))

in such a way that for (x,rl,72) near (0,0,0)
(1) y(-X,T1,T2) = -y(x,r112),
(2) y2(x,‘rl,r2) 5 0 for (x,‘rl,72)t(Trap—section of w‘focw), and
(3) y1(ip(t; x,rl,r2), r1,1'2) = constant for every x in IR2 near 0
with y1(x,rl,r2) = constant and for each fixed 01,72).
(2.4.2) is possible because of the symmetry property and the center manifold
theorem. By the change of coordinates (2.4.2), (2.3.1) becomes the following

Cr“1 differential equation in ‘ y:

o 2
(2-4-3) 3’1 = Y1 a(y1,71,72)
5'2 = Y2 b(y11y217'117'2)

where a is independent of y2, b iseven in y = (y1,y2). We also

assume that
(2.4.4) ny(0)u = (1,0) and ny(0)v = (0,1).

f(-,al,a2)e$; implies g(i,r1,r2)e.$ ;, and
6 " 6 .
W°(Dx 30—1 f(0)u) = w°(Dx “57—1 g(0)u). So from our assumpt1ons (I), (II),

(IV) and (2.4.4), (2.4.3) becomes

" 3 2
(2°45) y1 = ¢0(72)Y1(1+yl¢1(yl172)) + 71y1902(y117'1a72)
5’2 .= “903(y1a7'1a72)yQ(1+y2‘P4(y1iy2aTl172))

 

 

 

17

where
(pom), 711(0), and cp3(0) = A are positive, 901,902 and 1,03 are even in
yl, and (p4 is odd in y, 904 = O(|y|). Let v(rl,r2) = Dyx(0,rl,r2)(0,1).
Then .v(0,0) = v and (2.3.1),has an invariant curve at 01is2 tangent to
v(r1,r2). Note that this invariant curve is Cr._1 in (r1,r2) and contains 1‘
for (71,12) = (0,0). 7
In a similar way as in tpc(t,r1,72), we can get a CF1 family
cps(t,rl,r2), (r1,r2) small, such that (ps(t,r_1,r2) is a solution of (2.3.1) with
cps(t,rl,r2) -1 0 as t -i 00 along the negative v(rl,r2) direction and
cps(0,1'1,r2)£L so that cps(t,0,0) = W) (See Figure 5.)
Define diam) = g(so(0).0,0) A {tac(0,r1,r2) — do»
3
diam) = g(¢(0),0,0) A [v2 (0.5.5) — n01], and
c s
d1(rl,r2) = d1(rl,r2) — d1(r1,r2).
It is easy to see that d1(rl,r2) = 0 if and only if there exist a pair of
homoclinic orbits of (2.3.1) at 0.
(9d1

We will show that I = (0,0) since, if had shown it, by the implicit
1 3‘1;

function theorem II at 0 implies that there exists a CF1 function q with
(1(0) = 0 such that d1(rl,q(r1)) = 0 for 0 5 T1 << 1.

Hence
0d 6d

73—71 (0,0) + Fri-(0’0) q'(0) = 0, and so
1 2
ad1 adl
Q'(0) = - 3?; (0,0)/ a}; (0,0). 5 II].
It follows that q(rl) = mr1 + 0(r1), and our proof will be completed.

Claim. I =

1 (0,0).

.31 n33

 

 

 

 

18

G

Let 1:201 = 14001001 A a, (1.0.0).

€010

12320) '= station) ,A ‘3, 0.0.01.

[0

Note that .
EEO-1172) = g(<p(0),0,0) A (#0171172) — 3%(017'11T2D
so
old
1 _ c _ 3
3720.0) - [972(0) PT2(0)-
While, by definitions of we and cps,

Eaflaam =Dx,g()tp(t )0,)0 )%(t,0,)0+ +33-(,,,,o(t)00)

S

a a
3533th =Dg(<p(t),0,)0 330,00) +B§5(),<p(t ),0,)0

2

From these we have equations

Saga) = div g<at1.0.01 11:201th 013%001A0M1MO
an): 2—(t) — div g(<p(t)0,:0)p 2(0 + g(cp(t) )A 0)),)3%(o(t ),),00

and their solutions are

0
(2-4-6)c 0:2(0) = 02201) exp 1 div s(<e(t).0,0) dt
t1
+ 10ex1>(—1d1v aids) 00) ds) show 001 353000100) d1
t1
I

,0 1 + 11011,

 

19

(2.4.01, 19:2(0) = 1132011 exp (31 div nation) at)

0 .t

+] exp (—(1) div g(go(s),0,0) ds)g (,,)go(t)00A 135—(o(t),,)dtoo
1

i a Is(t1).+ II(t1)

 

respectively. We will show later that
lim I (t) =lim IS(t).

t-* —00 ' t-ioo

 

First, consider the following.

 

Lemma (2.4.1 ].

(2.4.7)c lim BLMtOO )=0and
1H -00

(2.47) lim35ۤ(,,)=t00

Proof. Let
"c .
‘P 03,72) = Y(900(t,0,72),0,72): = (y1(t,r2),0) for t << —1
then y1(t,r2) > 0 for t << —1 and 4y1(t,r2) is a solution of the scalar

equation

(2.4.8) at = <p0('r2)z3 (1+z2wl(z,r2)).

(See equation (24.5).)

‘po and zzwl are Cr_4. Let

Z_3(1+Z2‘P1(Z172))-1 = Z—3 + H1(72)Z—1 + H2(Z17'2)'

 

20

Z
, and hence H3(z,r2) = g H2(s,72)ds

0

Then H1 is Cr‘6 and H2 is of—7

is Cr-7.
Separation of variables in (2.4.8) gives

- ' 1—2
(2.4.9) — Q-z + H1(r2)lnz + H3(z,r2) = (p0(72)t + H4(r2)

where H402) depends on the value of y1(t,7-2) at some t = to. Hence
H4 is Cr_7' From (2.4.9) we have

2 2 2 ' —1 —1
22 (1 — 2 Hl(r2)z lnz — 2z H3(z,r2)) = —-(<p0(r2)t + H4(T2)) .
Let w = Z2 and let 2 > 0.

Then
(2.4.10) 2w(1—H1(r2)wlnw — 2wH3(t/v_v,r2))-1 = —(tp0(r2)t + H4(T2))‘1.

Let ¢(w,r2) be the LHS of (2.4.10). Then by the assumption (V), (I) is at
least C1 and 3%(0+,72) it 0 for 72 near 0. By the implicit function
theorem we can solve ¢(w,r2) = v for w, and let w = 3 + R(v,72) be
the solution to ¢(w,72) = v. Then R is C1 and R = O(v2).

From (2.4.10)

(2.4.11) y? = — %(w0(r2)t + H4(r2))_1 + R((<,00(r2)t + H4(T2))_1, 72).
SO we: have

'y1 =-— 3(w0021t — H4ir211‘1/2 + R1((100(72)t + H4(r211‘1,r21

' 2
where R1(v,r2) = O(v3/ ).

21

6y ' ‘ .

So 37—;(tfl2) —1 0 as t —1 —oo for 72 near 0. This implies that
a“ 3Y1
632(th = (aT—2(t,r2),0) a (0,0) as t a — a and

c _ -
31,30,001 = Dyx(ac<t.01.o.01%%:(t01e Dyx<0.0.01(0.01 = (0.01

aSt-l-co.'

Hence we proved (2.4.7) c'
Next, to prove (2.4.7)S, let
~s s
90 03172) : Y(W(t10172)10172) = (013’2(t172))-
Since (ps(t,0,r2) is tangent to v(0,r2) and ws(t,0,7'2) —1 0 as t —> 00 for
each 72 near 0, ws(t,72) > 0 for t >> 1, and y2(t,r2) satisfies a

differential equation

(2412) dz — — (r)z(1+z2 (27))

. ' ' at — $3 2 ‘P4 7 2

' _ 2 r—2
where (p3(0) —— A. tp3 and z (04 are C .
(See equation (24.5)).

Note here that we set r1 = 0 so that ([13 and (p4 may be different from
those of (2.4.5). It is easy to get the following from (2.4.12)

(2.4.13) z epo5(z,72) = H6(72) exp(—tp3(r2)t)

where H5 and H6 are CF3 and H6> 0. Since
%(2 epo5(z,r2))| # 0, we can solve the equation
z=0

(2.4.14) 2 epo5(z,1'2) = v

 

22

for 2 if z and v are near 0. Let 2: R(v,r2) be the solution to

(2.4.14). Then R is (Jr—3 and
(2.4.15) R(0,72) a 0.
Hence we get the following solution from (2.4.13)

(2.4.16) y2 = R(H6(r2) exp(—<p3(7'2)t), r2).

(2.4.16) and (2.4.15) imply that

WZ-(tflé) -+ 0 as t‘ -1 oo.
2

From the definition of 33,

~

8 3y
. 3%“,7’2) = (0, 373W) 4 0 as t .-. so.

Now

a 3 ~s as 0X ~s

£0,001 = Dyxw 0.01.0.0) 31,120.01 + 93(0) (1.01.0.0) .. 0 as t a a,
and we complete the proof of Lemma (2.4.1). 13

Now back to the previous problem,

0
Ic(t) = g(cp(t),0,0) A_35g(t,0,0) exp ] div g(cp(s),0,0)ds.

(2”417) g((p(t)1010) 7) 0 as t’ '7 _ 001

(2.4.18) div g(o(t),o,0) = -1 + O((go0(72)t + H4(T2))‘1) as t a - a

23

from (2.4.5) and (2.4.11). Hence from (2.4.17), (2.4.18), and (2.4.7) c of Lemma
(2.4.1), we have
3 lim I c-(t) — 0.

t-tfiioc

While, from the definition of its, (2.4.15) and (2.4.16),
20) ‘= 0(expi—At11 as t-» a,
$230.01 = (0.4: eta—10 + o(exp(—At111
for t >> 1 and a constant C > 0.
Hence g(vlt),0,0) = (at)
= IDXY(‘P(t),0i0)l_1 $30.01
= {[I>,,y(0.0,011‘1 + 0(exp(-At11} . (0,—c expi—At) + mean—1011.
Also div g((o(t),0,0) = div g(0,0,0) -l- Dx(div g(0,0,0)) + O(<p(t)2)
= —1 + O((p(t)) = —1 + O(exp(—/\t)) as t a a.

Now,as t-ioo

IS(t)= )A 03332 t, 0, 0) )exp[—]t div g( (0(s ),0,0)ds]

MA

t
= — g::L:(t,0,0) A g(cp(t),0,0) expl—(l) H + O(exp(—As)))dsl

= _ 3352““) A {[D,,y(0,0.01]"1 .+ O(exM-MD} °

(0,—C exp(—)it) + O(exp(—/\t))) exp(/\t) exp ([tO(exp(—As))ds.

Thus lim 1 S(t ) = —0 A [ny(0,o,0)]'1 . ((),—0 exp ([OOO(exp(—/\s))ds) = 0

t-too

since the integral converges.

Now, back to (2.4.6) the LHS of (2.4.6)c s are constant, so as t —) co, II(t)

c,s’
and II(—t) converge, and

 

 

24

'0 t . I
.p$2(01 = 1 exp<—1 div g(to(s1.0.01ds1 g(WLOiO) A gigs/tonne.

, - “:2 ”t . ,

p,2(01 =(1) each—(1), div 'e(to<s1.o,01ds1 30201.00) A 3%(dt1001dt,
since lim Is(t) = 0 = lim Ic(t). i

t-ioo t—l =00

Hence I1 in (2.3.2) is convergent] and is equal to p: (0) — p: (0), and so
' 2 2

6d .
equal to 371(00). So we proved Theorem (2.3.2). :1
2 .

m. Theorem (2.3.2) gives us a homoclinic bifurcation curve of
codimension 1: {71,7'2)|r2 = q(r1), T1 > 0}. After change of coordinates in
the parameter space

((71,721 2 (141,142): #1 = 711 142 = T2 - 0(a)) (866 page 12),
if we consider (2.3.3), for ”1 > 0 we have two structurally stable connected
components. If I1 - p2 > 0, then the phase portrait is topologically
equivalent to that for region 4 in Figure 3, and if I1 - 112 < 0, then it is
topologically equivalent to that for region 6 in Figure 3. (Note that Figure 3
is desciibed only the case I1 > 0.)

Existence of limit cycles in phase portraits for region 4 and region 6
is immediate from the hyperbolicity of F, the Poincare—Bendixon Theorem plus
the symmetry property of 'the vector field. We again emphasize that our

interests are only in the neighborhood of —I‘UI‘.

B. Proof of Theorem (2.3.3).

Basically this proof is almost the same as the previous one. However, for

111 < 0 new equilibrium points arise and this makes the proof complicated.

25

Through the proof, we assume that al 5 0.

Again let us consider the system

, it = foe/41,22)
(2.4.19) [11 = 0'

1.42:0

which is CPI. Using the center manifold theorem in (2.4.19) and so on, we
can define (006,111,112) of CF1 family. (See-equations (2.4.1) and (2.4.2).
Also see Figure 5.) [Noteb we will use the same notations even through they
may be different from those in the proof of Theorem (2.3.2)]

Under the same assumptions and change of variables (from x— to
y—coordinates), finally we get to a CF1 equation which is almost the same as

(2.4.5):

. 3 2
(2.3.20) y1 = ¢0(u2)y1(1+y1) + ”1y1902(y1’”1’”2)

where <p1,(02,<p3 are even in yl, (04 = O(|yl) is odd in y = (y1,y2), and
<p0(0), (02(0), (p3(0) = /\ are positive.
Equilibria of (2.4.20) near (0,0,0) consists of
E1 = {0} x [R2 and
E2 = {(ylt0),01,02)lul = #101,142»
where [11 = p1(y1,)t2) is as follows:

l‘1 . 2 2 2
—2(0,02) i 0 so that 141 = -y-1(¢0(142) + y1¢1(y1,02))
aY1
for some $0012) > 0 and $103,112) by the implicit function theorem.

 

26

' 2 . —4
Note that 2110012) + y1w1(y1,/12) ls Cr . For a1 < 0, E1 does not
give usany further information concerning the topological qualitative features
since 0€R2 is a node. So we will consider E2 only.

For ”1 g 0,_ setting ”1 ‘= —62, we have
' , 2 1 2
(2-4-21) 6 = 21030012) + y1¢1(y1’/‘2)) / -

' 86 ‘
So 0.
5. 51'3'120 #
By the implicit function theorem, we can solve (2.4.21) for yl, and we let the

solution to (2.4.21) be
(2-4-22) Y1 = 1105,02)-

Since (2.4.21) is odd in yl, p in (2.4.22) is odd in 6, i.e.,

hating) = —p(6,)l2). Also n(a,n2) >' 0 if s > 0. (00(0) > 0 in (2.4.20).
So (2.4.22) implies that the equilibrium point (p(6,,a2),0) of (2.4.20) with
111 = —62 is a saddle if 6% 0, and (p(0,,u2),0) = (0,0) is a pitchfork.

Now define

(2.4.231 pals) = x(<n(d.t21,01,—62.121.

Then p(6,/12) is a CF4 mapping such that for (6,)12) near (0,0),
(0,0) is a pitchfork of it = f(x,0,p2),

p(§,/l2) = a saddle of it = f(x,—§2,)12) if a > 0,
another saddle of it = ion—52,112) if 6 < 0.

(See Figure 5.)

 

27

From (2.4.23) 330,0) = D yx(0,0,0) (3%(0,0),0), by the definition of u
u = (ny(0.0,011‘1(1.01 = 0,300.01 (1.01.

Hence

(2.4.24) 330,0) = 330,0) 11.

Now (2.4.20) with 711 = —62 has an invariant curve {(ylty2lly1 = p(6,u2)}
at the equilibrium point (p(6,p2),0). For 6 = 0 (so 121 = 0), this invariant
curve is the stable manifold of the'pitchfork (0,0), and for 6 at 0, it is the
stable manifold of the corresponding saddle (p(6,u2),0).

Let v(6,,u2) = Dyx((p(6,,a2),0), —62,u2)(0,1). Then it = rec—152,112) has
an invariant curve at p(6,p2) tangent to v(6,ll2) and this invariant curve is
CF4 in (6,112). For (6,112) = (0,0), this invariant curve contains 1‘.
Similarly (see the paragraph below (24.5)) we have'a Cr—4 family
(ps(t,6,a2), (6,;t2) small, such that 3180,6422) is a solution of
x = f(x,—62,/12), (ps(t,6,,u2) -) p(6,,u2) as t —) 00 along the negative v(6,,u2)
direction, (ps(0,6,;12)cL and cps(t,0,0) = cp(t).

(ps(t,0,u2) is a branch of the stable manifold of the pitchfork
p(0,_p2) = (0,0) of x = f(x,0,u2) and if 6% 0, ws(t,6,,u2) is a branch of
the stable manifold of the corresponding saddle p(6,/12) of x = f(x,—62,u2).

We define

(15041.02) = f(¢(0),0,0) A ((pcwal‘laHQ) - 10(0)),
(13(51112) =i(d(01,0.01 A (1030,0321 — 2011

From now on, we assume that 6 2 0, and define
d303,) = dgwiwp - dawns)

respectively. Hence d; is CF4t since (15 is CIT1 and d; is Cr_4.

28

From the definition,

($05,112) = 0 (resp. d5(6,p2) = 0) if and only if there exists a pair of
homoclinic (resp. heteroclinic) orbits of x = Ion—62,712) at p(t6,p2)(resp.
from 00%) to paid?»

We will show the following:

(2.4.25) ‘ 333mm < 0 (resp. 332mm > 0) and

(2.4.26) (0,0) = I

2.

silica?

Suppose we have shown (2.4.25) and (2.4.26). Then (2.4.25) and I2 9!: 0
imply that {(6,u2)|d;(6,u2) = 0} is a Cr"4 curve through (0,0) of the

form .

‘5 = [Hi/12 + 0(H2)
where

4
1: 2 ‘

m = ‘ I2/W(0,0) ii 0-
(Note that = m+ a2 + 0012) and —-6 = -m_a2 + 0012).)
Hence p1 = —(mi)2ag + 0013):; [(112).

Also, 0 S sgn(6) = sgn(m+a2) = sgn(12op2), so» I2 . [12 2 0.

(Resp. 0 2 sgn(-6) = sgn(—m—p2) = sgn(I2 . 112), so I2 . a2 5 0.)
To show (2.4.25) and (2.4.26) ,,first we Construct variational equations for
(peat—62412) . . and ¢S(t1i§1#2)-

Since,

 

 

('5

6”2 (t,0,0) =¥ D xf(<p(t),0,0) gigaph) + gfjdw),

go.

(i ‘9 s(t,,)00)— _ D x,,=tf(<p(t)00)( 3538“,,”0
S

gfiO—(t, 0,0) .=Dxf(¢(t),o,0) 35—(t,0,0) + gift—Zedong),

”2 2

m

94‘; fi“

if we define
C
12,320) = mama) A 3536,09),
psi“) = f(so(t),0,0) A (i 3553mm
p;2(t)_ = memos) A 33%qu

we have the following variational equations:

(2.4.27)c % pizza) = div f(go(t),0,0) [$20) + f( m ),,00) A6 317} (t),,,00)
(2.4.27)Si gf p35“) = div f(<p(t),0,0) pfsfit),

d s _ . t 3 8f
(2.4.27)S Elf pfl2(t) _ le f(<p(t),0,0) pfl2(t) + f(<p(t),0,0) A %(¢(t),o,0).

If we solve the equations (2.4.27), we have the following:

0
(2.4.28)c 612(0) = p;2(t1) exp { div f(go(t),0,0)dt
1
0 t
+ { expl-(g div f(<p(s),0,0)ds]' KW )00) A %M(<p(t)00)
1

.1 t
(2.4L28)S# p%i(0) = p36i(t1) aim-(J) 1 div f(<p(t),0,0)dt],

30

t
(2.4.28)S p3 (0) = ps2(.t1) dim—(J) 1 div f(gp(t),0,0)dt]

[‘2 It
0

exp Ht div f(<p(s);0,0)ds] f(go(t),0,0) A %(¢i),o,0)di.

+1
.t 0

1

Let Ic(t1) be the first term and II(tl) the second term in the RHS of
(2'4'28)c' Also let Is(t1) be the first term in the RHS of (2.4.28)S. Then

' p;2(0) =Ic(t1) +1101),

p320» =Is(t1) .+H<t1>.

For a moment,. we prove the following:

Lemma (2.4.2).
(2429) 1‘ 53¢} 00) 0
.. 1m t, , = , ‘
C t-l—oo #2

a 3* ‘
(2429M lim —§3—(t,0,0) =- i gimp), and
t-loo

S .
(2.4.29)S :im 3%0’0’0) = 0. ~
400

Proof. For (2.4.29)c, see the proof of (2.4.7)c in Lemma (2.4.1). (Note that

5 = 0 corresponds to ”1 = —62 = 0.) For (2.4.29)

Si, we define

(2.4.30) ws*(t,6,u2) = y(w3*(t,6,u2), #52412) = (pew), y2(t,i6,u2))

31

for t >> 1, for each (i6,p2) near (0,0). By the relationship between the
x—coordinates and the y—coordinates and from (2.4.20),

y2(t,t6,u2) > O for t >> 1,
and y2(o,:k6,p2) satisfies a differential equation of the form

(2.4.31) 4, = -<P3(*5,u2) z(1+22v4(z..~=6.u2)).

.Here <p3 and z2¢4 are Cr—5.

If we solve (2.4.31) by the separation of variables,

(2-4'32) Z eXpJ1(Z,d:5,/12) = J2(*6i#2) exP(_903(i6i/‘2)t)

for some J1 and J2 > 0, both Cr_6. Since %(z epr1(z,i5ifl2))l 0 i‘ 0
z:

for (£6,112) near (0,0), by the implicit function theorem, we can solve

(2.4.33) 2 epr1(z,i6,p2) = v

for z. Let z = R(v,t6,p2) be the solution to (3.4.33). Then R is Cr—6

and
(2.4.34) R(0,i:6,p2) s 0.

Thus from (3.4.32), if we let 2 = y2 and v = the RHS of (3.4.32), then we

get

(24°35) yQ(ta*6iN2) = R(J2(i6,u2) eXp(_903(i51#2)t)i $6,,u2).

32

(2.4.34) and (2.4.35) imply that
' W2 W2
2!: ”(t,i6,fl2), Ell—2(t,i6,fl2) —’ 0 as t "i (I).

From (2.4.30)

~Si
g%_(t06,fl2) = (i%(i67”2)3i i%_(i6’”2)t))

So

9330,43) . (4 3%(oo,o2),0) and
(2.4.36)

—%:—:,,)(t§p2 (3L(45,,p2)0) as to...

st - ~s:h
Since 33—(000) = Dyx(<ps*(i,0,0),0,0) Q3fli,0,0), from (2.4.20) and (2.4.36)

lim £953-3);(t,0,0) = Dyx(0,0,0)(i 33(0,0,),0) = 4 330,0).
t-loo

Finally for (2.4. 29)S , we consider
W“ 6,#2)— " X(‘ps i,“ (Sis/‘2) “'62 #12)

6 6x ‘st —+
So 45—:(t00) Dyx<Zo (t,o,0).o.o>(pi’9,,2<o,o),0) + p—u2(o (t.o,0),o,o>
A ax _ 4
Dyx(0,0,0)(g£—2(0,0),0) + 315(0’0’0) _ 0 as t do

. 613 _
Since 372(0’0) = 0 and x(0,0,p2) = 0.

So we complete the proof of Lemma (2.4.2). :1

By (2.4.30)

1.

(2.4.37) os*(t,6,o2) = nywsfitooz), —o2,o2) WW2)

33

Let' 6': p2 = 0 in (2.4.37). Since (3314,30) = (p(t) = f((0(t),0,0),

More») = [ny(o(t),o.0)1‘1 romeo).
From (2.4.30), (2.4.34) and (2.4.35), we obtain (00.) = O(exp(—At)) and

(2.4.38) ps*(i,0,0) = (0,43 exp(-At) ,+ O(exp(—/\t))) as t —+ e
where C > 0. Therefore

(2.4.39) f(o(t1),o.0) = {[ny(0,0,0)]’1 + 0(exp(—At))} -
(0,—C exp(—/\t) + o(exp(-At))).

(2.4.38) also gives
div f((p(t),0,0) = -x\ + O(exp(—/\t)).
Hence

(2.4.40) exp[——(()t1 div f(<p(s),0,0)ds] = exp()\t1)oexp (1:1 O(exp(—)As))ds.

Then, from (2.4.39) and (2.4.40) we have

(2.4.41) :im f(cp(t1),0,0) dim—(1:1 div 'f(<p(s),0,0)ds] =
1"“0

[DXY(0,0,0)]_1 ° (0,-C exp ((00 O(exp(-/\S))dS)o

Now back to (24.28)“, by (2.4.29)S of Lemma (2.4.2)

i:

 

’34

' =4 . t
offs» = f(so(t1),0,0) A (4 gigs—31.0.0» exp {—3 1 div 4443.034]

1:
= —(4 3540.0» A Iim f(o(t1),o,0) «~2po 1 div f(o(s)o,o.)ds1
131-100 0
where the limit exists by (2.4.41).

Note that
(1) from definitions of d; and psgh,

aid:
2 :l:
73(020) = — pg (0))
(2) 3%(09) is a. positive multiple of v by (2.4.41) from similar
assumption as (2.4.4),

(3) lim f(<p(t1),0,0) is a negative multiple of u.

1"°°

By above (1), (2) and (3),

8+ 6d; 3 s— 6d2
706 (0) = —a—5(O,0) (resp. —p5 (0) '= m— (0,0)) is a negative (resp.

positive) since u A v > 0 which shows (2.4.25). The proofs of

(2.4.42) lim 10(4) = 0 = lim 18(4)

t-l —oo t-ioo

are immediate by Lemma (2.4.2), and are the same as those in Theorem
(2.3.2). It is easy to see that (2.4.42) shows our final claim (2.4.26).
The only thing we have to notice ‘is the smoothness of f we need. In

Cr_4 instead of Cr-l, we

the proof- (2.4.29) c in Lemma (2.4.2), since (as is
need the smoothness of Cr.10 instead of Cr—7. So r has to be at least 11.

We finished our proof of Theorem (2.3.3). o

 

35

Remark. Theorem (2.3.3) gives us a homoclinic bifurcation curve of
codimension 1: {(—(m"')2p§,,u2))|u2 > 0} = H01 and a heteroclinic bifurcation
curve of codimension 1: {(—(mj2p§,,u2)| 112 < 0} = He' -

For “1 < 0, we have three structurally stable connected components as
in Figure 4 which is the case I2 > 0.

Existence of limit cycles in regions 3, 7, and 11 of Figure 4 is again from
the same reasons, hyperbolicity of I‘, Poincare—Bendixon Theorem, and the
symmetry property of the vector field.

Note that I1 = I2 = I.

C. Proof of Theorem (2.3.4).

Consider a Cr—mapping ‘11: U C IR2 —A JISXD) with
\Il(a1,a2) = f(~,al,a2) and \IJ(0,0) = f(-), where U is a neighborhood of
0.

We want to show that ‘1! is transversal to 22 at f if I # 0.

First, we will show that 232 is a CF1 submanifold of 21 of
codimension one.

Let ch2 with pitchfork 0 and let L be a line segment perpendicular
to I‘ as in the Theorems (2.3.2) and (2.3.3). For grill near f, 0 is
again pitchfork and the stable and center manifolds of 0 are CPI—dependent
on g. Thus their intersection with L are Cr—1 function of g. Therefore
the function d(0,n2) is Cr—1 even though di(6,p2) is only CF41.

d(g) = 0 if and only if gc22. It is easy to find a perturbation
f + ch in 21 such that g;|€~=0d(f+ch) # 0 and this says that 22 is a

36

Cr—1 submanifold of 21 of codimension one. By Theorem (2.3.1) (or
Appendix), 21 is a Cr—I‘ submanifold of 20 = .2” g(D) of codimension one.
Now let ¢(p1,p2) = f(-,pl,p2). Then it is enough to show that 4) is
transverse to 22 at 041442) = (0,0) if and only if I at 0.
Since ¢(0,fl2)£21 for |p2| << 1,

(2.4.43) $509) is tangent to 21.

Also, if I at 0, then we have

i -
(2.4.44), 331.00) = 33430) — q'(0) 33: (0,0)

1

332400) = 332(00)

 

from the transformation (71,72) —» (#1442) (p1 = T1, #2 = 72 — q(rl)) and

(2.4.45). 3%(03) = 3§I(-,0,0)

 

from the transformation (021,422) -+ (71,72)('r1 = a1 — p(a2), 72 = 02).

By assumption (IV) and Theorem (2.3.1),

6‘1! . '
(2.4.46). 531— IS transversal to 21 at f.

So (2.4.43), (2.4.44) and (2.4.45) imply that, if I ,4 0,

 

 

37

(2.4.46) if and only if %I(0,0,0) is transversal to 21 if and only if
$403)) is transversal to 21.
”1 .
Next from the fact that I = 3%“) 0) = :d—2(0 0) and (2 4 43)

72 ’ ”2 ’ ' ' ’
I at 0 implies %(0,0) is transversal to 22.
I = 0 implies Sim 0) = gig“) 0) = 0 which says 39;“) 0) is
T2 ’ ”2 3 ”2 3

tangent to 22.

This completes the proof. 0

 

 

 

 

CHAPTER 3. GENERIC 3—PARAMETER FAMILIES OF SYMMETRIC
PLANAR VECTOR FIELDS WITH NILPOTENT LINEAR
PART.

We will study in this chapter the symmetric planar vector fields—which
means the vector fields with the invariance by the rotation of an angle 7r with
respect to the origin in the plane—with nilpotent linear part. Also we will
classify the bifurcations of the generic 3—parameter families of those vector fields

to be mentioned later.
§1. Introduction.

There are two kinds of k—jet normal form of the vector field with

nilpotent linear part (see Guckenheimer and Holmes [10]).

(3.1.1)a 5t k+1)

y + O(Ix,y|

w

k1
+)

 

v =2: (aixi+bixi_1y) + O(|x,y|

1:

01'

k . '
(3.1.1)b it = y + 2 aixl + O(|x,y|k+1)
i=2

k .
i= ,2 bix‘ + 0(lx,ylk+1)

l:

 

k—jet normal forms (3.1.1) can be shown easily using the normal form

theory.

38

39

A. Nonsymmetric Case.
We assume. that a2b2' at 0 in (3.1.1). Hence terms of order 3 or higher

can be neglected, and we can write it down simply

><
ll
:4

(3.1.2) a

01‘

(3.1.2)b it = y + 3x2
y — bx2
where ab ¢ 0.

Bogdanov [2] analyzed (3.1.2)a and Takens ([17] and [18]) did (3.1.2)b
independently.
_ Recently, some of codimension three problems concerning the planar vector
fields with nilpotent linear part with degenerate singularity are appeared. For
example, in (3'1'2)a’ Dumortier et. al. [7] worked for the case a at 0 and
b = 0 (DRS-A), and Medved [13] and Dumortier et. al. [8] for the case
a = 0 and b it 0 (Medved—B, DRS—B) with following versal unfoldings:

i=y

° 2 3
y=x +61+y(€2+c3x.ix)

(DRS—A),

40

,1 = y
(Medved—B),
y=61+£2x+63x2=tx3+xy
x = y
(DRS—B).

y=‘el + 62X 1: x3 +,y(¢s3 + bx :t x2)

B. Symmetric Case

While parallel to the above, we can consider the unfolding of the
symmetric vector fields with nilpotent linear part with degenerate singularity on
the plane. In this case A, = bi = 0 for i even. So we assume that

a3b3 it 0 in (3.1.1). Then, similarly we have

(3.1.3)a

><
ll
i<

OI‘

(3.1.3)b it = y + ax
y = bx3

where ab it 0.

Carr [3] worked (3'1'3)a and again Takens [[17] worked (3.1.3)b.

For example, in (3’1°3)a’ if a'> 0 and b at 0, the phase portrait near
(0,0) is a degenerate saddle of codimension 2 and for a < 0 and b at 0,
it is a degenerate focus of codimension 2. (See Figure 6.)

However, if b = 0 (and a 3E 0), we have to consider

41

(3.1.4) x = y

y: 33:3 + fix4y

where ab 36 0. The phase portrait near (0,0) is a degenerate saddle or focus
(depending on the sign of a) of codimension 3, and mostly we are concerned

with these in this chapter.

If a = 0 and b at 0, we have to consider

(3.1.5) it = y

where ab it 0.
In this case we have some difficulties which will be discussed later.

In studying the equation (3.1.4) with the case a > 0, basically we
followed the similar ideas as Dumortier et.a.l.[7]. Since the type of the
equations and, its dynamical behaviors are similar. However, the equation
(3.1.4) with the case a < ‘0 has produced many difficult problems in proving '
the existence of limit cycles and new phenomena occur including triple limit
cycle bifurcation. ,

We will introduce the bifurcation diagram and the corresponding phase

portraits with a short explanation for the equation (3.1.5).
§2. Versal Deformation (a it 0. b = 0). '

First [we study the versal deformation of (3.1.4).

42

Lemma (3.2.1). Any symmetric perturbation of (3.1.4) with small parameter p

can be transformed into the form

(3.2.1)i it = y

4 2
3 + x y G(x,u) + y ‘I’(x,y,u),

i" = 901(40x +' ¢2(u)y + <p3(u)x2y 4 x
where ”CR3, G(x,0) = 1 and \Il(x,y,0) = 0.

Remark. We can take a transformation in the parameters 6i = (aim),

i = 1,2,3, such that (3.2.1)i becomes

(3.2.2), it = y

y = 61X + 62y + 63x2y i x3 + x4yG(x,c) + y2\Il(x,y,c)

where G(x,0) = 1 and ,\Il(x,y,0) = 0.’

Proof of. Lemma (3.2.1). Let the following equations

(3243) 5< = y + w1(x,y,u)

)7 = ax3 + bxliy + w2(x,y,u)

: ,uan

be the perturbed system of (3.1.4), where for i = 1,2, wi(x,y,0) = 0, mi are

sufficiently smooth (say, C°°) and symmetric (i.e., wi(—x,—y,p) = -wi(x,y,p)).
[Without loss of generality, we may assume that a = 4 1 in (3.2.3)

depending on the sign of the original a in (3.1.4) and b = 1 since

otherwise we can take scaling

 

 

43

81/3 62/3 ' a 2/3
x a x, y —+ , t -+ t.
mm" Ian576 b

By a change of coordinates p = x, q = y + “’1’ (3.2.3) becomes

(3.2.4) p = q
a: p3(a+%;w1‘) + p4qu+§§w1>
+ {—p4wl(1+g’y wl) + “120+;y wl) + mg; w1)}.

2
Let 3; w1(X(p,q,u), y(p,q,u),u) = h1(p,u) + qh2(p,u) + q h3(p,q,u) and
4 0 0 8
-—p w1(1+-5§ wl) + w2(1+-ay wl) + p(—a)—( ml)
2
= ‘1’1(p,u) + (Flam/t) + q ‘1’3(p,q,u)
for some hi and \Ili(i = 1,2,3).
Then hi = \Ili = 0 at p = 0 (i = 1,2,3). Hence the equation (3.2.4) is

(3.2.5) p q
. _ 3 4 3 2
q —- [p (a+h1) + ‘Pll + (ill) (1+h1) + p h2 + W2] + q (1).

where (I) = p3h3 + p4(h2 + qh3) + \II3. By the symmetry property, we can

say that

‘I’l = 901001) + :61(pal‘)93a

2 4
P3112 + W2 = $205) + $3001) + 52(1):”)13
for some (pi(i 2: 1,2,3) and fli (i = 1,2). Let F(p,p) = a + h1 + HI and
G(p,p) = 1 + h1 + )62. Hence (3.2.5) is changed into the following form

44

 

(3.2.6) 13

q
. 2
q .= @1001) + ¢2(u)q + 903(u)p2q +. p3F(p,/4) + p4qG(p,u) + q (1)-

By the Malgrange Preparation Theorem (see .Chow & Hale [4], pg. 43),
we get
901(44):) + F(p,u)p3 = [291001) + sgnF(0,0)p3l 0m),
where- sgn F(0,0) = a 4 0, 0(p,0) = 1 > 0, and F and 0 are even in p.
Hence in (3.2.6), V

. ~ ' w (u) - so (u) 4 2
q=[cpl(u)pip3+y%§mq+-g%fi,7p2q+%pq+$110-
Again, let u = p, v = q/JP. Then

 

(3.2.7) {1 = vfl}
3 + (95291.)-ié)‘; + 33v + %U4V + (Pi/QW-

i: "H .
[$14411 11 fl}

By the symmetry property, in (3.2.7)
‘P22 2) 2 4 ‘p3 2
— - — = ZI(M)u + 23(113/1’)“ , — = Z401) + z5(u,p)u ’
w w W

Note that zi = 0 at p = 0 (i = 1,...,5).

Let (920‘) = 210‘):
9030‘) = 2204) + 2404),
601.4) = 2301.2) + 2501.4) + GNP.

Then (3.2.7) is

(3.2.8)i i1 = M)
xv = with: + 22(u)v+23<u>u2v . u3 + é(u,u)u4v + ¢(u,v,u)v2]fll

45

In (3.2.8) é(u,0) 96 0. (3.2.8) is topologically equivalent to (3.2.1). (For
simplicity we denote {oi by (pi (i =.1,2,3) in (3.2%.) In (3.2.8)
¢(u,v,0) = 0 and é(u,0) = 1. a

Note that (3.2.2) 4 is versal to

(3.2.9)i 5c y

y = 61X 1: x3 + 62y + 63x2y + x4y

(For this, see Section 5 of Bogdanov [2].) Hence we will study (3.2.9)+ in §3
and (3.2.9)_ in §4 instead of (3.2.2)+ and (3.2.2)_ respectively.

§3. The Case a > 0.

For reminding (3.2.9) it is written down again:

+,

(3.2.9)+- 5c

y
y = 61X + x3 + y(62 + €3X2 + x4).

The equilibria in (3.2.9)+ is determined by the equations y = 0 and
x(cl+x2) = 0. Hence for 61 > 0 a saddle (0,0), and for ‘1 < 0 a focus
(0,0) and saddles (w c |,0) are equilibria.

Let the RHS of (3.2.9)+ be f€(x,y). Then

2 ( ) 0 .1

Df x,0 =
6 €1+3X2 c2+c3x2+x4

It is immediate that {61 = 0} is a pitchfork bifurcation surface. In

{51 > 0}, the phase portrait is topologically constant and it is a saddle at

46

(0,0). In {61 < 0}, several bifurcations occur. We analyze the equation
(3.2.9)+ by drawing the trace of the bifurcation surfaces on the hemisphere
S = {£'= (61,62,63)|0 <' |c| = 60 << 1, 6.1. '5 0}.

The bifurcation diagram of equation (3.2.9)+ is a cone based on its trace
in S. This trace on S consists of 4 curves: 4

,(1) Hf: Hopf bifurcation curve (62 = 0),

(2) H e: heteroclinic 100p bifurcation curve,

(3) C: semistable limit cycle bifurcation, curve, and

(4) P: pitchfork bifurcation curve (61 = 0).

The curve C joins a point h2 on Hf to a point c2 on He’ and
thecurve C is tangent to Hf and He at these points respectively. (See
Figure 7). ‘

The curves Hf and .He’ on S touch (‘38 at b1 and b2
transversally and in the neighborhood b1 and b2 we have the degenerate
saddle bifurcation of codimension 2 (see Figure 6(a)).

There exists a unique unstable closed orbit for the parameter in between
Hf and He in the neighborhood of b1 and a unique stable closed orbit in
between Hf and He in the neighborhood of b2. On Hf — {h2}, a Hopf
bifurcation of codimension 1 with appearance of an unstable (reSp. stable) closed
orbit by crossing the line b1h2 (reSp. b2h2) from right (reSp. left). On
He — {c2}, a symmetric heteroclinic 100p bifurcation of codimension 1 occurs.
By crossing the curve blc2 of He from left, we have a pair of heteroclinic
loops, then they are destroyed and unstable closed orbit appears. While by
Crossing the curve b2c2 of He from right, again we have a pair of
heteroclinic loops and then stable closed orbit appears. The point h2 (resp.
c2) corresponds to a H0pf (reSp. a heteroclinic) bifurcation of codimension 2.

The curves Hf and He intersect transversally at points b1, b2 and d.

 

 

47

The point. d corresponds to the simultaneous HOpf bifurcation of codimension
1 and symmetric heteroclinic 100p bifurcation of codimension 1. For parameter
values. in the curved triangle (dh2c2), there. are exactly two closed orbits.

One of them is stable inside an unstable closed orbit. These two closed orbits
coalesce in a generic way when crossing the curve C from left. On C itself

there exist a unique semistable limit cycle.

Theorem (3.3.1). Let Z = BS U Hf U He U C be a subset of S, where the
semisphere and curves Hf, He and C are described above. The bifurcation
diagram of (3.2.9)+ in the ball 1360 = {fl lcl _<_ c 0} is a cone

homeomorphic to {627554702 ,3 T1)|S€[0,€0], (u,7’or1).62} The t0pological type

of the phase portraits of equation (3.2.9)+ in a filxed neighborhood of the (0,0)
in IR2 is constant in each connected component surrounded by the bifurcation
surfaces (5 components: R1,...,R5, R1 n S = I,...,R5 n S = V), and-is
constant in each bifurcation surfaces (9 surfaces: 31,...,Sg,

31.“ S = 1,...,S9 n S = 9) and curves (5 curves: 01,...,C5,

C1 n S: b1,b2,c2,h2, and C5 n S: d). (See Figure 8).

Proof will be given at the end of this section.

The main difficult problem is the determination of the number of limit
cycles.

For this we use the‘blowing—up method as following for ‘1 < 0:

x -) sx _ 61 == 327)
(3.3.1) I y —» 32y - c2 = 8470

t -) t/s c3 = 3271

48

where. s > 0. First we will study the neighborhood of Gel—axis for ‘1 g 0.
Hence let 17 = —1 in (3.3.1).
Then the equation (3.2.9)+ has the form

(3.3.2) 5: = y

y: -—x + x3 + S3(T 02+71x +x 4.)y

Let ”0 = S370, p1 = S3T1, p2 = 33, then (3.3.2) becomes

(3.3.3) x = y

. 3 2 4
y = —x + x + (p0+pls +u2x )y

with p2 > 0.

By the change of parameters (61,62,63) —1 (s, ,T0. T1) —+ (#2410411) and the
change of variables (x,y,t) —) (sx,32y,t/s), (3.2.9)+ has the form (3.3.2) and
(3.3.3). Note that the equation (3.3.3) has the equilibria (0,0) and (41,0)
where (0,0) is a focus and (41,0) are saddles. I

_ Coming back to the equation (3.3.2), if s = 0, it becomes a Hamiltonian

system

ll

(3.3.4) x y

y=—x+x3

with the first integral

2 x2 x4
H(X,y) :2L+—_1_'

The phase portrait of (3 ..3 4) is shown in Figure 9.

 

49

Every closed orbit surrounding (0,0) corresponds to a level curve
7b: H(x,y) = b, 0 < b < 1/4. On equilibria (41,0) and heteroclinic loops
joining these equilibria, H(x,y) = 1/4.

- Now we consider (3.3.2) for small 8 at 0. Every closed orbit of (3.3.2)
should intersect with the interval U = {(x,0)|0 S x 5 1} (hence with -U)
and enclosed the point B = (0,0) since the point B has an index 1 for
every 3,7 and T1. We define wb for bc[0,1/4] as follows: I

o
(1) wbc—UUU,'
(2) H(wb) = b.

Let )‘s = ((70,71),s) and Wu be an upper branch of the unstable manifold
of (3.3.2) at (1,0). We define a Poincare map P /\ : U -) —U (or
‘ s
—1

PA : -U -+ U) in the following way.
8

Let bc(0,1/4]. Then we can choose Be(0,1/4] such that the points wb
and WE are successive intersection points of U and —U respectively with
an orbit so that (i) P A : U —1 —U is defined and P A (wb) = wb- if

s s
—1 -1

W11 n (—U) )6 {}, or (ii) PA : —U -> U is defined and PA (W5) = wb if
s . s

W“ n (—U) = {}. (See Figure 10' (a) and (b).)
Let 7(b,)18) be the orbit of (3.3.2) which joins the points wb and
WE. Hence 7(b,)\s) is defined for bc(0,1/4]. Then we have a lemma.

Lemma (3.3.2).
(1) Every'closed orbit of (3.3.2) is expressed by the form 7(b,/\S) with

WE = -Wb.

(2) A trajectory 7 = 7(b,AS) of (3.3.2) is a periodic orbit if and only if

 

50

)dH" dt=0.

(3.3.5) t
' In particular 7 is a heteroclinic orbit if and only if (3.3.5) is
satisfied for b =(1/4.
(3) For 3 > 0, condition (3.3.5) is equivalent to

(3.3.6) F(b”\s) s £(b,AS)(TO+T1xz+X4)deO=.

(1) is obvious by the symmetry property

(2) )yw dt = H(WB) — H(wb) and
, fl%z‘n=x(1‘xz)1‘0 for x150 and |x| #1.

Hence H(WB) = H(wb) if and only if WE- : —wb for be(0,1/4).
(3) is immediate since

#Hx | dt=s3( 01242+Tx+x)y| dt
(.3 3. 2) (3.3.2)

= s (TO+ r1x2+x4)ydx. [I]

Let Ii(b) = J xiy dx,1= 0,,,24 where 7b: H(x ,y)- — b.
7b
We will consider F(b,/\S) as, a perturbation of F(b,/\O) = F(b,(ro,rl)). The

function F(b,AO) can be written explicitly by
(3.3.7) F(b,AO) = 7010(b) + 7112(b) + 14(b).

3114,23): 33(F(b,)\0) + (rams) — F(b,AO)))
= 33(F(b,AO) + o(s))
= s3 F(b,AO) + 0(33).

 

51

Hence by the Lemma (3.3.2) (3),
F(b,/\0) + c(s) = 0
where. 6(3) is a smooth function in all variables such that {(3) a 0 as

s -1 0. The limiting position of the_closed orbits is given by the solution of
(3.3.8) F(b,AO) = 0 for s e 0.

From now on, we denote F(b,/\0) = F(b,0,(7'0,1'1)) = F(b,(7'0,7'1)) simply by
F(b) if there is no confusion.

For a moment we study. a Hopf bifurcation curve and a symmetric
heteroclinic loop bifurcation curve in (To,rl)—plane from (3.3.7).
Lemma (3.3.3). The point (0,0) of (3.3.2) is stable (resp. unstable) if
To < 0 (resp. To > 0). It has a Hopf bifurcation of order 1 along the line
Hf = {(TO,T1)]T0 = 0} except the point 112 = (0,0) at which a Hopf
bifurcation of order 2 occurs. Moreover, there are two limit cycles at (70,71)
with r > 0, T1 < 0 around the point (x,y) = (0,0).

0

Proof. Direct calculation of formulas of the Liapunov's focal values for (3.3.2).

 

(For formulas, see Andronov et al [1], Medved [13].) 0

Next, we want to change symmetric heteroclinic loops to a homoclinic loop by
using symmetry property as follows:
For x,yc|R2, we define x-y if and only if x = y or x = —y. Let

2

* *
R = lR2/~ with a quotient topology and let x = {x,—x}. Let us regard

our symmetric vector field

52

(3.3.2) 5:

y
—x + x3 + s3y(ro+rlx2+x4)

<<.
II

in (R2 as

*
(3.3.9) dx /dt = y . .
:1: >1: ' :1: 3 3 :1: >1: 2 :1: 4
dy/dt=x +(x) +sy(r0+rl(x) +(x))
:1:
in the new phase space [R2 . Then (3.3.9) has only one saddle point
=1: ‘ '
(1,0) ; {($1,0)} and a pair of symmetric heteroclinic loops in (3.3.2)
. . >1: >1: >1:
correspond to a homoclinic loop at (1,0) in (3.3.9). [R2 — {(0,0) } has a
. . *
2—dimensional smooth manifold structure. Note that IR2 itself is not a
manifold.
2*

Now let IR: = {(x,y)|x>0}. Then the phase space IR of (3.3.9) can

be thought of the half—plane IR: U (y—axis/-). (See Figure 11.)
Then we can apply (3.3.7) F(b,/\o) = 7010(b) + 7112(b) + 14(b) to
Joyal and Rousseau (pg. 19 of [12]) on saddle quantity. (Also see Roussarie

[14])

Lemma (3.3.4). The equation (3.3.2) has a heteroclinic loop bifurcation of
order 1 along the curve He ‘= {(70,71)er + 7-1/5 + 3/35 2 0} except the
point E2 = (1/7, —8/7) at which a symmetric heteroclinic loop bifurcation of
order 2 occurs. The curves Hf and He intersect transversally at the point
a = (0, —3/7) which corresponds to a Hopf bifurcation of order 1 with
asymmetric heteroclinic loop bifurcation of order 1 simultaneously. (See Figure

13).

53

Proof. See page 19 of Joyal and Rousseau [12], Roussarie [14], Joyal [11].
Note that the trace of the saddle point (1,0) is 1'0 + Tl + 1 = 0 and
$0462) 4 0. 1:1

Next we study a semistable limit cycle bifurcation in (To,rl)—space. Equation
(3.3.7) defines a surface in the space (s,(ro,rl)). The following lemma

eliminates'the term I4(b) ' in (3.3.7).
Lemma (3.3.5). 14(b) can be expressed in terms of 10(b) and 12(b), and
(3.3.10) ,7I4(b) = 812(b) — 4130(1)).

Proof. H(x,y) = b on 5b. Hence
ydy + (x—x3)dx = 0.

4 _ 2 2

x ydx — y(2y +2x —4b)dx
= 2y3dx + 2x2ydx — 4bydx, and

y3dX = d(xy3) - 3xy2dy

d(xy3) — 3xy(x3—x)dx

d(xy3) - 3x4ydx + 3x2ydx.

Hence

4 ._ 3 2

x ydx —— 2/7 d(xy ) + 8/7 x ydx — 4/7 bydx.
Taking integration on 7b gives (3.3.10). C]

From (3.3.10), (3.3.7) becomes

(3.3.11) F(b,(ro,rl)) = (To — 4b/7)10(b) '+ (71+s/7) 12(1)).

 

54

Note that 10(0) = 12(0) = 0, 10(b) > O for bc(0,1/4] and 12(b)/Io(b) -1 0
as b -1 0. So the degeneracy of the equation (3.3.11) at b = 0 can be

removed by changing (3.3.11) into the following:

 

(3.3.12) G(b) (instead of G(b ,(1'0 ,7'1))) = To — 4b/7_ - (71+8/7)P(b)

 

where P(b) is defined by

—I2(b)/Io(b) for be ( 0,1 /4]

(3.2.13) P(b) =
0 for b = O.

 

Lemma (3.3.6). P(b) defined by (3.3.13) satisfies the equation

2

(3.3.14) 4b(4b—1) 131(1)) = 5P + (8b+4)P + 4b.

Proof. Let {I(b), u(b)} = 7b n {y=0} with 1(b) 3 u(b), and

Jib —- H(b)i bdx
,(1—{b we)

where <p(x,b) = [2b—x2+x4/2]1/2 for xc[l(b),u(b)].
Then 11(1)) = 2Ji(b).

From the definition of Ji’
(b) Iu(b) Xi

J! = X.
1 l(b) W15]

So

(b) x. x 2
(3.3.15) Ji(b) = {(b) 1% = 2bJ](b) — J]+2(b)+ 2 J+4(b)].

Also, integration by parts gives us

 

55

. 1 .
(3.3.16) Ji(b) = 1+1 [J] +201) - J] +4(13)] for all 1 2 0._
From (3.3.15) and (3.3.16), eliminating Ji+4(b), we have
(i+3)Ji = —J]+2 + 4in for all i 2 0.

In particular,

(3.3.17) 3.] = —Jé + 4bJ(')
5.1 = -J] + 4bJé.

From (3.3.10),
(3.3.18) 7J2'1(b) = 8Jé(b) — 4J0(b) — 4bJC')(b).
Plugging (3.3.18) into (3.3.17), we have

_ 1_ 1
(3.3.19) 3.1O _. 413.10 J2

1512 = 4ng) + (12b-4)Jé.

Solving for J6 and J], in (3.3.19), we get

(3.3.20) 4b(4b—1)J(') = 4(3h-1)J0 + 5.12

4b(4b—1)Jé = 4b JO + 2013.12.

Hence

—J'J +J J' J' J'

P'(b)= 2; 2°=—JZ—T°P(h)
Jo o

 

O

56

_ 5P2+(8b+4)P+4b D
‘ 4b(4b—1) -

 

Lemma (3.3.7): P(b) also satisfies:
(1) lim P(b) = -1/5 ,
b—+1/4
(2) P'(b) < O for 0 < b < 1/4, P'(O)

— 1/2 and P'(b) -1 -00
as b —1 1/4.

Proof. P(b) is a solution of the differential equation (3.3.14) with initial

condition P(O) = 0. We can interpret (3.3.14) into the form

. _ 2

(3.3.21) p — — 5P
1‘) = 4b — 16b2.

— (8b+4) P —4b

The graph of P = P(b) is the heteroclinic orbit from the saddle (0,0) to

the stable node (1/4, - 1/5) in the (b,P) -— phase plane (See Figure 12.)
The equation

(3.3.22) 5P2 + (8b+4)P + 4b = 0

describes a locus on which the tangential direction of the vector field (3.3.21) is

horizontal. The branch of hyperbola (3.3.22) above the line P = —1/2 denoted

by 7is

(3.3.23) P = P(b) s: (—(4b+2) + (16b2+6b+4)1/2)/5.

57

The vector field (3.3.21) is transverse to 7 and directed to the right of 7.
Calculation from (3.3.14) and (3.3.23) gives

lim P'(b) = — 1/2 and lim P'(b) = —1.

b—10 b—10

Therefore, the graph of P = P(b) is entirely above 7, i.e., P'(b) < 0 for
0 5 b < 1/4.

Also we can see that P'(b) -+ --00 as b -1 1/4. 13
We also need the following lemma.

Lemma (3.3.8). (See also Dumortier et a1 [7])
P"(b) < 0 for b6[0,1/4].

Proof. From (3.3.14) we get
(32b-4)P' + 4b(4b—1)P" = 10PP' + 8P + (8b+4)P' + 4. So if we solve
for P",

(3.3.24) P"(b) = [(10P—24b+8)P' + 8P + 4]/(16b2—4b)

lim P"(b) = lim [(10P'—24)P'+(10P—24b+8)P"+8P’]/(32b—4)
b-10 b—10

= [(—5—24)(-1/2)+8 lim P"(b) + 8(—1/2)]/(—4).

_)

(Note that lim P'(b) = —1/2.)
b—10

Hence lim P"(b) = —7/8 < 0.
b—10

Now let us suppose that P' has a zero on [0,1/4) and let

bO = min {be[0,1/4)|P"(b) = 0}. Hence P"(b) < 0 for all xr[0,b0). Let

58

L be a tangent line of I‘ at (b0, p(bo)), where I‘ is the graph of P(b).
Since P"(bo) = 0, the order of contact between L and I‘ is at least 2.
Let v be a. vector orthogonal to L, and L(u) be a linear parametrization
of L. Also let

f1(b,P) = 4b - 16b2,

f2(b,P) = —5P2 — (8b+4)P — 4b, and

f = (f1,f2).
Then the function (Mu) s: < f(L(u)),v > has a zero of order at least 1 in uO
with L(uo) = (b0,p(bo)) where <,> is a Euclidean inner product on IR2.
As P"(b) < 0 on [0,b0), the corresponding arc of I‘ is situated below L.
The line L cuts the line {b = 0} at a point 110 above 070 = (0,0). f
is directed downward at no. f is directed towards the half plane above L
in the neighborhood of bO in L with b < b0. Hence (b(u) must possess
a zero at some u1(#uo) with L(u1)c[n0,m0] where 110 = {b = 0} n {L(u)},
and m0 = (b0,p(bo)) = F n {L(u)}.

However, f is quadratic. So 1/)(u) is a polynomial of degree 2 in u.

¢(u0) = ¢'(u0) = 111(u1) = 1/2'(u1) = 0 with 110 at 111
implies (b 5 0, and I‘ is a line segment. This contradicts to

P'(O) = — 7/3 < o. 11

Now we consider the problem of the semistable limit cycle bifurcation

which is given by C: G(b) = G'(b) = 0 in (70,71)-space.

Lemma (3.3.9). C is a smooth curve which connects the points H2 on Hf
and 52 on He (see Lemmas (3.3.3) and (3.3.4)), and which is tangent to

59

Hf and He at these points respectively. On C the semistable limit cycle

bifurcation of the equation (3.3.2) occurs.

m. From (3.3.2)

G(b) = (To -— 4b/7) — (71 + 8/7)P(b) on [0,1/4],

G'(b) = — 4/7 — (71+8/7)P'(b), and

G"(b) = - (71+8/7)P"(b).

If G(b) = G'(b) = 0, then 71 + 8/7 76 0. Hence by the Lemma
(3.3.8), G"(b) )6 0. By the implicit function theorem, there exists b = b(rl)
such that G'(b(rl)) = 0, and so 7'O = T0(b,1'1) = 70(b(7'1),71) from

G(b) = 0. Hence To = O(1'1) is smooth and the semistable limit cycle

bifurcation occurs on C. From G(b) = G'(b) = 0, we get

(3.3.25) 7'0 = 4b/7 — %

Ti=‘53/7‘71ri713).

Note that as b -+ 0, P(b) -1 0, P'(b) —1 — 1/2, and (70(b),71(b)) -1 (0,0) = h2.
Also as b -1 1/4, P(b) —1 — 1/5, P'(b) —1 — co, and

d7
(70(b), 71(b)) a (1/7, — 8/7) = c2. From (3.3.25), 37% = P(b) along the
curve C: To = 70(71). This implies that C is tangent to Hf and He at
52 and 62 respectively. :1

Given (70,71), the number of limit cycles of equation (3.3.2) is

determined by the number of roots of equation G(b) = 0 for 0 < b < 1/4.

If 1'1 + 8/7 = 0, then G(b) = 0 if and only if T0 = 4b/7. So

60

706(0,1/7) and 1'1 = —8/7 at which G(b) = 0 has a unique solution
(note that G'(b) at 0).

We suppose that 1'1 + 8/7 96 0. Then

G(b) = (r1+8/7)1A(b)—P(b))1
where A(b) = (70—4/7b)/(71+8/7) which is linear in b. The roots of
G(b) = 0 is the intersection of the straight line P = A(b) and the curve
P = P(b) on the (b,P)—plane. Since P"(b) < 0, the graph of P = P(b) is
concave downward. P(b) is independent of r

o
P(O) = 0, P(1/4) = — 1/5. A(b) depends on To and 71, and for

and TI, and

T1 + 8/7 > 0,
(a) A(O) = 0 (reSp. > 0 or < 0)
+——1 (T0,Tl)£Hf(l‘eSp. is on the RHS, or LHS of Hf).
(b) A(1/4) = —- 1/5(resp. > — 1/5, or < — 1/5)
1—1 (7'0,1'1)6He (reSp. is above, or below He)‘
(c) The straight line P = A(b) _is tangent to the curve P = P(b)
1—1 (70,71)6C (i.e., G(b) = G'(b) = 0 for some bc(0,1/4)).

The bifurcation diagram of equation (3.3.2) is as in Figure 13 and the
relative positions between the straight line P = A(b) and the curve
P = P(b) is as in Figure 14.

Lemmas (3.3.3.), (3.3.4) and the implicit function theorem provide us the

following extended results from s = O to s > 0. (Also see Dumortier et. al.

[7].)

61

Lemma (3.3.10). Let K be a compact neighborhood of

{H(x,y) 5 1/4} 11 {|x|$1} in the (x,y)—plane, and D be a compact
neighborhood of the curved region h2dc2 in (To,rl)—plane. Then there exists
a(D) > 0 such that the bifurcation diagram of the equation (3.3.2) consists of
three surfaces and three curves in C(D) = (0,a(D)) x D which is as follows

up to a diffeomorphism of C(D) equal to the identity at s = 0:
(1

V

SHf = (0,a(D)) x (Hf — {H2}) is a surface of Hopf bifurcation of

codimension 1,

(2

V

SHe = (0,a(D)) x (He — {E2}) is a surface of heteroclinic loop

bifurcation of codimension 1.
(3) SC = (0,a(D)) x C is a surface of semistable limit cycle bifurcation

of codimension 1,

A
A
V

(0,a(D)) x {H2} and (0,a(D)) x {62} are curves of Hopf and

heteroclinic loop bifurcation of codimension 2 respectively.

A
Us
V

(0,a(D)) x {d} = SHf n SHe is a curve of simultaneous Hopf

bifurcation and heteroclinic loop bifurcation.
Outside these bifurcation sets, the topological type of the phase portraits

of the equation (3.3.2) is constant in K.

For §c(0,a(D)), we denote the intersection of the bifurcation diagram of
equation (3.3.2) with the plane {(s,(1'0,71))|s = E} by Wg. Then W-S— has
a cone structure (see Figure 15).

The bifurcation diagram for the equation (3.2.9)+ with ‘1 < 0 can be
constructed from Lemma (3.3.10). The blowing—up (3.3.1) with 17 = —1

gives a transformation ¢:(s,(70,71)) —) (51,52,53), and

 

62

(33-25) ¢((0,0(D)) x D) = “-52 184 T 0,8 2T1)|86(0 0(1)» (T0 T1)€D}o

Let E6 (D) be the RHS of (3.3.26). Then E]E (D) is a cone in
1 1

(61,62,63)—space around the axis 061 for 61 < 0 (see Figure 15).
The bifurcation diagram of (3.2.9)+ in E]E (D) is the image of those
1
sets described in Lemma (3.3.10) by the transformation and thus homeomorphic
to cones based on H ,He,C,Hz,52, and d with curves 3 -1 (3217,3470 ,827'1)
with 17 = —1, or equivalently 61 -1 (cl,c%ro,(—el)rl) for 61 < 0.

Now we will study the behavior of (3.2.9) in a sector around 063—axis

+
for 61 _<_ 0. In the blowing—up (3.3.1), we take 71 = :1: 1 instead of
17 = —1. Since both cases 71 = 1 and 71 = —1 are similar, we will
consider the case 71 = 1. By (3.3.1) with T1 = 1, equation (3.2,9)+

becomes

(3.3.27) x = y
3 3( 2 4)

y=7]x+x +s To+x+x y.

Let 31 > 0 be fixed. Then for each se(0,sl] we take a blowing—up again:

x-+rx 7)=—r2
y-1r2y 7' =r2?
o o

t-1t/r

Then (3.3.27) becomes

 

63

(3.3.28) x = y

y = -x + x3 + s3(r(?0+x2)y + o(r3))

which is a perturbation of the Hamiltonian system (r = 0)

(3.3.29) x = y

y -x+x

with first integral
2 2 4
H(x,y) = g— + g— — 31—.
As in Lemma (3.3.2), we have
H(wE) — H(wb) = s3r(F(b,?O) + o(r))
where 'F(b,?0) = ?OIO(b) + 12(b). Finally it leads to study F(b),?0) = 0,

and get to the following conclusion:

Lemma (3.3.11). In the halfplane {17,70,71)|1) S 0, T1 = 1} there is a fixed
compact subset B+, diffeomorphic to a disk having a contact of order 1 with
axis 070 at (17,70,71) = (0,0,1), and such that for equation (3.3.27) the
results of Carr—Takens (see page 54—81 and, in particular, Figure 3 on page 59
of Carr[3], also Guckenheimer and Holmes [10], Takens [17] and [18]) are valid
for any (17,70) 6 13+ and any sc(0,sl]. (See Figure 16.) (3.3.1) with

71 = 1 gives a mapping (3,17,70) -1 (61,6263) which maps (0,31] x B+ to

E-l-

_ 2 4 2 + + . .
£3 — {(3 11,8 70,3 )|sc(0,sl], (17,70)cBO}. E63 1s a cone in (51,62,63)—space

around 063—axis for 63 > 0 based on B+. The bifurcation diagram of

(3.3.2) in E: consists of cones based on Hf, He and {1—51} with
3

64

generating curves 3 -1 (8211,841'0 ,s 2.) (See Figure 17). For 1'1 = —1, we can

get a cone E around 06 —axis for < 0 based on H , H and
63 3 3 f e

{'62} Similarly.
Proof. See Carr [3]. 1:1

Proof of Theorem (3.3.1). Let E: and E; be the two cones from B+
3 3

and ‘B— respectively as above. We can choose a compact set D in the
(70 ,1'1) —-p1ane to use Lemma (3. 3. 10) in such a way that (see Figure 18.)

(1) E6 (D) U E: U E; contains a cone C(M) based on a disc
1 3 3

M in the hemiSphere S,
(2) 8M is tangent to BS at the point b1 = (0,0,(50) and
b2 = (0,0,-co),

(3) M contains the curves SHf n S, SHe n S, and SC 11 S where

SHf’ SHe and SC are defined in Lemma (3.3.10).

Condition (3) is possible because the curve of Hopf bifurcation and the

curve of heteroclinic loop bifurcation in M n E: are connected with the
3

curves Hf = SHf n S and He = SHe n S respectively.

To show this, we consider the equations of curves Hf and He' From

Lemmas (3.3.3) and (3.3.4), we have

(3.3.30) Hf: To = 0 and He: To =—1/57'1 — 3/35
2 2 2 _ 2 2 2 2 _ 2
€1+€2+€3—60 €1+€2+€3—€0.

65

From (3.3.1)
3 = (—r1)1/2
(3.3.31) To = (52/(—1€1)2
7'1 = 53/(—€1)a

where 61 < 0. Substituting (3.3.31) into (3.3.30) we obtain
6 e 2

2 2 2 2 1 3 2
Hfzcl+c3=60 and H:61+(-5—- )

2 2 .
e + 63 = 60 respectively.

__1_

35
Let (1 -> 0. Then 63 —1 :1: 60.
This implies that if 61 —1 0, then the curve Hf and He tend to the

points b1 and b2 in as, hence that Hf and He are connected with the

Hopf bifurcation curve and the heteroclinic loop bifurcation curve in M n E: .
3

Thus we can choose D and M satisfying the condition (1), (2) and
(3). The conclusion of the Theorem, for 6 near (0,0,ieo) with 61 < 0,
follows from Lemma (3.3.10) and Lemma (3.3.11), and for 61 > 0 or £1 = 0

but 6 ,4 (0,0,i60) are obvious. n
§4. The Case a < 0
(3.2.9)_ is as follows:

(3.2.9)_ 5r = y

y = 61X — x3 + y(1€2+63x2 + x4).

66

The equilibria in (3.2.9)_ is determined by the equation y = 0 and
x(cl—x2) = 0. Hence for ‘1 < 0 (0,0) is a focus, for 61 > 0 (0,0) is a
saddle and (an/51,0) are foci.

Let the RHS of (3.2.9)_ be f€(x,y). Then
0 , 1

61-382 , 62+ 63X2+X4 .

Df€(x,0) =

It is immediate that {61 = 0} is a pitchfork bifurcation surface. We
will consider two cases (cl > 0 and ‘1 < 0) and by the same way as in
the previous section, we will use a hemisphere section to be easy to understand
the bifurcation diagram.

Next, we combine the above two results to get our complete bifurcation
diagram on'the sphere.

First let us consider the case 61 > 0.
Let

S = {(61,62,63) = (ID < |c| = 60 << 1}, 5+ = S r1{e1 2 0}.
The bifurcation diagram of (3.2.9)_ is a cone based on its trace with S.
This trace on S consists of 4 curves:

(1) Hf: Hopf bifurcation curve,

(2) HO:

(3) C = C1 U C2 U C3 U C 4: semistable limit cycle bifurcation curve,

(symmetric) homoclinic loop bifurcation curve.

and

(4) P: pitchfork bifurcation curve (61 = 0).

The curve C1 joins a point a on Hf to a point e on Ho’ and it
is tangent to Hf and H0 at these points respectively. Also the curve C2

joints a point b1 on P to a point e on H and it is tangent to H0 at

0,
e, however, intersects with P transversally at b1. The curve C3 (resp. C 4)

joins a point f to a point b2(resp. b3) on P. C3(resp. C4) intersects with

67

P transversally at b4 (reSp. b3) and C3 meets C4 tangentially at f.
(See Figure 19.) In the neighborhood of b1 and b2 we have the degenerate
focus bifurcation of codimension 2. (See Figure 6 (b).)

The point a (reSp. e, f) corresponds to a Hopf (reSp. homoclinic 100p,
triple limit cycle) bifurcation of codimension 2. At points b3,c,d,g and h

two corresponding bifurcations occur simultaneously.

Theorem (3.4.1). Let 2 = (S n {61 = 0}) U Hf U HO U C be a subset of
8+. The bifurcation diagram of (3.2.9)_ in the ball
+ ,
B = {c = (61,62,63)] [c] 5 60,61 2 0}
is a cone hemeomorphic to
2 4 2
{(8 17,3 70,8 T1)|86[0,6Ol, (11.70.7063)
The topological type of the phase portraits of equation (3.2.9)_ in a
neighborhood of (0,0) in R2 is constant in each connected component
surrounded by the bifurcation surfaces (10 components:
R1,...,R10, R1 n 8"” = 1,...,R10 1) S+ = X) and is constant in each
bifurcation surfaces (19 surfaces.
81,...,Sg, S1 r1 S+ = aboo,...,S19 n SJr = eg) and curves (10 curves:

C1,...,CIO, 01 n 3+ = {bl},...,c10 11 3+ = {b3}). (See Figure 19 (3)—((1).)

Proof will be given later.

Next'let us consider the case 61 < 0. Let S— = S n {61 g 0}.

As before the trace of bifurcations on 8’ consists of 3 curves:

68

(1) Hf: Hopf bifurcation curve,

(2) C: semistable limit cycle bifurcation curve, and

(3) P: pitchfork bifurcation curve (61 = 0).

In this case (61 < 0), the HOpf bifurcation curve is described by
(,2 = 0 and connects b1 and b2 at which Carr—Takens bifurcation of
codimension 2 occur. C is tangent to Hf at k and connects k on Hf
and b3 on P. Note that b1,b2 and b3 in Figure 20 (a) are same
points as those in Figure 19 (a) respectively.

Theorem (3.4.2). Let E = (S n {61 = 0}) U Hf U C be a subset of S.
The bifurcatiOn diagram of (3.2.9)_ in the ball
B- = {c = (61,62,63)] If] S 60’ 61 S 0} is a cone homeomorphic to

2

{(8277, 8470, s T1)]SE[0,€O], (17,70,706) The t0pological type of the phase

2 is constant in

portraits of equation (3.2.9)_ in a neighborhood of (0,0) in IR
each connected component surrounded by the bifurcation surfaces (3 components:
R1,R2,R3, ’RlflS = I, R2118 = 11, R308 = III) and is constant in each
bifurcation surfaces (6 surfaces: S1,...,86, 81118 = blk, S608 = b1b3) and
curves (4 curves: C1,...,C4, 01118 = {k},...,C4nS = {b3}). (See Figure 20

(EU-((1))-
For the proof of this Theorem, see the proof of Theorem (3.4.1).
A. 61 > 0.
We use the blowing—up technique (3.3.1) for 61 > 0. First we investigate

the behavior of (3.2.9)_ in a neighborhood of Del—axis for ‘1 2 0. Hence let
1) = 1m (3.3.1).

   

69

Then the equation (3.2.9)_has the form

(3.4.1) 5: = y
3" = X — X3 + S3(TO+Tlx2+x4)y.
Equation (3.4.1) has the equilibria (0,0) and (41,0) where (0,0) is a saddle and

(41,0) are foci. If s = 0, (3.4.1) becomes a Hamiltonian system

(3.4.2) x = y

2 2 4
with the first integral H(x,y) = %l— — g— + 21(— .

The phase portrait of (3.4.2) is shown in Figure 21.

Closed orbits surrounding A = (—1,0) or C = (1,0) (type 1) correspond to
level curves 7b: H(x,y) = b, -—1/4 < b < 0 and those surrounding
A,B = (0,0) and C at the same time (type 2) correspond to level curves
7b: H(x,y) = b, 0 < b < oo. 7_1/4 = {A,C}. 70 corresponds to a level
curve of a figure-eight homoclinic orbit.

Now we consider (3.4.1) for small 3 it 0. Every closed orbit of (3.4.1)
should intersect with the interval U = {(x,0)|x21} and/or —U. Since closed
orbits of type 1 enclosing A is a 1 — 1 correspondence to those of type 1
enclosing C, we only consider the latter.

We define wb for b6[-1/4,oo) as follows:

(a) wch, (b) H(wb) = b.

Let As = ((70,71),s) and WS(resp. Wu) be a stable (resp. unstable)

manifold of (3.4.1) at B in the right half—plane. We define a Poincare map

70

P A : U —1 U in the following way:
3

Let {wbS} = WS n U and {wbu} = W11 n U. For b6[—1/4,oo), we can
choose Bc[—1/4,oo) such that the point wb and WE are successive
intersection points of U with an orbit so that

PAS: U -1 U is defined and PAS(wb) 2 WE, PAS(wbs) = wbu if

H(wbs) < H(wbu)’ and
—1 . —1 —1 .
PAS: U -1 U 13 defined and PAS(WB) = wb, PAS(wbu) = wbs 1f
H(wbu) < H(wbs)' (See Figure 22 (a) and (b).)
Let 7(b,)rs) be the orbit of (3.4.1) which joins the points wb and WE.
Hence 7(b,AS) is defined for b6(-1/4,oo). Then we have the lemma.

Lemma (3.4.3).
(1) Every closed orbit of (3.4.1) is expressed by the form 7(b,/\S) with

WE : who
(2) A trajectory 7 = 7(b,)\S) of (3.4.1) is a periodic orbit if and only if

(3.4.3)

In particular 7 is a homoclinic orbit if and only if (3.4.3) is satisfied for
b = 0.
(3) For s > 0, condition (3.4.3) is equivalent to

(3.4.4) F(b,/\S) (r0+r1x2+x4) ydx = 0.

isms)

70

P /\ : U —1 U in the following way:
8

Let {wbs} = WS (1 U and {wbu} = Wu (1 U. For b€[-1/4,oo), we can
choose Bc[-1/4,oo) such that the point wb and WE are successive
intersection points of U with an orbit so that

PAS: U -+ U is defined and PAS(wb) = W5, PAS(wbs) = wbu if

H(wbs) < H(Wbu)’ and
—1 . —1 —1 .
PAS: U -1 U 18 defined and PAS(WB) = wb, PAS(wbu) 2 W133 1f
H(wbu) < H(wbs). (See Figure 22 (a) and (b).)

Let 7(b,AS) be the orbit of (3.4.1) which joins the points wb and wb-.

Hence 7(b,/\S) is defined for b€(—1/4,oo). Then we have the lemma.

Lemma (3.4.3).

(1) Every closed orbit of (3.4.1) is expressed by the form 7(b,)\S) with

WE = Wb.
(2) A trajectory 7 = 7(b,)\s) of (3.4.1) is a periodic orbit if and only if

Ide

t t=0.

(3.4.3)

In particular 7 is a homoclinic orbit if and only if (3.4.3) is satisfied for
b = 0.
(3) For s > 0, condition (3.4.3) is equivalent to
2 4
)

.4.4 Fb,)\ E: T 7x x
(3 ) < ,) 17mg) 0+1 +

ydx : 0.

71

Proof. See the proof of Lemma (3.3.2). 1:1

Let Ii(b) = I xiydx, i. = 0,2,4, where 7 : H(x,y) = b.
. 7b b

We will consider F(b,/\S) as a perturbation of F(b,)r0). The function
F(b,/\O) can be written explicitly by

(3.4.5) F(b,AO) = 7010(b) + 7112(b) + 14(1)).

s3F(b,/\S) = s3(F(b,AO) + (F(b,AS) — F(b,AO)))
= s3 (F(b,AO) + O(s))
= s3 F(b,/\O) + 0(33).
Hence by (3) of the Lemma (3.4.3),
F(b,)\o) + c(s) = 0
where c(s) is a smooth function in all variables such that c(s) -+ 0 as s —1

0. The limiting position of the closed orbits is given by the solution of
(3.4.6) F(b,)\0) = 0 for s —1 0.

From now on, we denote F(b,(7'0,71)) E: F(b,/\O) = F(b,(7'0,71),0). (Also F(b)
instead F(b,(ro,71)) if no confusion.)
Equation (3.4.6) defines a surface in the (s,(70,71))—space. The following

lemma eliminates the term I 4(b) in (3.4.5).

E_mma_(3_w‘e4).
(3.4.7) 714(b) = 812(b) +' 4bIO(b).

72

Proof. See the proof of Lemma (3.3.5). 1:1

From (3.4.7), (3.4.5) becomes

(3.4.8) F(b,('ro,71)) = (70+4b/7)Io(b) + (71+8/7)I2(b).

Note that IO(—l/4) = I2(—1/4) = 0, Io(b) > 0 for b > —1/4 and
12(b)/Io(b) —+ 1 as b —1 —1/4. 33 we can change (3.4.8) into the following:

(3.4.9) G(b) (instead G(b,(70,71))) = 70 + 4b/7 + (71+8/7)P(b)

where P(b) is defined by

12(b)/Io(b) for 7 b > —1/4

(3.4.10) P(b) = 1 for b = _1/4 .

Lemma (3.4.5). P(b) defined in (3.4.10) satisfies
(3.4.11) 4b(4b+1)P'(b) = 5P2 + (8b—4)P — 4b.
Proof. See the proof of Lemma (3.3.6). B

Lemma. (3.4.6). lim P(b) = 00.

b-mo

Proof. (Carr [3]) Let Ji(fl) = I xiy dx where 6 > 0 is a maximum of
0

the solutions to 64 - 232 = 4b and y = (2b+x2—x4/2)1/2. Then

73

P(b) = J2(fl)/Jo(fl). Let x =_hfl.
Then

1 .
1,01) = F? g (410‘ 14111411. 1 = 0,2,

where

g(h) = 1530—1141 + 012-1111”.
Since g(h) S g(l/fl) for 0 S h g l, JO()6) 3 C133 for some constant

C1 > 0. Also we have J2Ifl) Z 0255 for some constant C2 > 0. Hence
lim P(b) = kiln J2(fl)/J0(fl) = 00. D

(D (D

Lemma (3.4.7 ). P(b) also has the following properties:
(1) lim P(b) = 4/5,
b—10

(2) There exists b* > 0 such that
P'(b) < 0 for bc[—1/4,b*) — {0} and
P'(b) > 0 for b > b*.
Also b* satisfies: P(b*) > 1/2, P'l'(b*) > 0.

Proof. Rewrite (3.4.11) into the form

(3.4.12) 6 = 4b(4b+1)
P = 5P2 + (8b—4)P — 4b

_ (1
Where "I — Hf . '
Since P(-1/4) = 1, the graph of P = P(b) for bc[-—1/4,0] is the heteroclinic
orbit between the saddle (-1/4,1) and the node (0,4/5) in the (b,P)—plane.
(See Figure 23.) Hence lim P(b) = 4/5. We denote two branches of the

_’

74

 

hyperbola .5P2 + (8b—4)P — 4b = 0 by P(b) (upper) and P(b) (lower).

b = 0, b = —1/4, P = P(b) and P = P(b) divide (b,P)-plane into 9

regions in each of which g; has a constant sign. We denote 4 regions

among them (which are interested in) by A,B, C, and D as in Figure 23.
First one Can show that

nmnm=4mlmnm=m,
b—1-1/4 b30

lim P'(b) = —1, and lim P'(b) = —3/5.
b+1/4 . b—10

Hence the graph of P = P(b) must stay in region A for —1/4 < b < 0

 

and enter into region D for . 0 < b << 1. In regions A and D, g]; < 0.
But P(b)-100 as b-+oo and P(b)-11/2- as b—tm.

Thus there exists b* > 0 such that P(b*) = P(b*) (hence P'(b*) = 0)
and P'(b) > 0 for b > b*. Since P'(b) < 0 and P(b) .. 1/2
as b .. e, F(b*) = P(b*) > 1/2. By (3.4.11),
4b(4b+1)P" = (10P—24b—8) P' + 8(P—1/2). So we have
4b*(4b*+1)F"(b*) = 3(F(b*)—1/2) > 0 which implies P"(b*) > 0. 1:1

Lemma (3.4.8). P(b) ~ CA? as b -1 00 for some c > 0.

Proof. From (3.4.11) P'(b) = (5P2+(8b—4)P—4b)/(4b(4b+1)), we easily see that

lb

it can not be that P(b) .. ce as b -1 00 for any c,chR—{0}. Hence let

P(b) ~ cbr (c¢0) as b -1 00 for some rclR. Also let
A = 5P2 4 (8b—4)P — 4b, . B = 4b(4b+1).
ME = {5(P/b)2 + (8—4/b)(P/b)—4/b}/{4(4+1/b)}.

P'(b) ~ crbr—1 as b —1 oo. .
502b2r-2
(1) _Ifr>1,A/B~—m———. as b—1m.

 

75

2 2r—2
r—l _ 5c b
. Hence crb —— —1-6——. -
This implies r— 1 = 2r — 2 and thus r = 1 which contradicts to r > 1.

2 . .
(2) If. r.=1,A/B..5c'+8c as b—>oo.’

1 .
2 .
Hence c = éE—Gj—SE. Since c 9% 0, c = 8/5, i.e., P(b) ~ 8b/5 as b -1 oo.

‘ (a) P'(b) .. 3/5+ as b a .0.

Since P'(b*) = 0, there. must be h > b* such that P"(b) = 0 and
P'(b) > .8/5 but P“(b) s 0. (See Figure 24 (3).) Note that P'(b) > 0
for. b > b* by Lemma (3.4.7). P' = A/B (implies P'B + P"B' = A' and
so P"'B + 2P"B' + P'B" = A". Hence at b = b, BP'” = (10P'—16)P'.

Since P'(b) >~ 8/5, P"'(b) >' 0, which gives a contradiction to P“(b) S 0.

*
Thus P'(b) S 8/5 for any b ((1) ,oo), and there is no local maximum since, if

~

. _ ‘ , *
there is, then there must be a local minimum point b C(b ,oo), so

~ ~ ~

0 _< P'(b) < 3/5, P”(b) = 0, P"'(b) 2 0, however,

-

BP'” = (10P'416)P' < 0 at b = b so that P"'(b) < 0 which is 3

~

contradiction to P"'(b) > 0.
(b) P'(b) -1 8/5- as b —1 00.

Since there is no local maximum of y = P'(b) on (b*,oo), y = P'(b) is
monotonically increasing. Also we can get a contradiction easily if we assume
that there is an, inflection point in the graph'of y = P'(b) on (b*,oo). So
P"(b) > 0 on (b*,e). (See Figure 24 (b).)

BP" = (10P—24b-8)P' + 8P—4

BQP"

(3.4.13) —2—— : (5P—12b—4) (5P2+(8b—4)P—4b) + (4P—2) (16b2+4b).

76

Let f(b,P) be the RHS of (3.4.13) and let x = P/b. Then
$23 = (5x—12—4/b)(5x2+(8—4/b)x-4/b) + (4x—2/b)(16x2+4/b)

= x(5x+4)(5x—8) — (4/b)(18x2—3x—12) + (8/b2)(2x+1).
Since for b >> 1, x(b) = 2‘82) is less than 8/5 but near 8/5, we have
x(5x+4)(5x—8) < 0, and —(4/b)(18x2—3x—12) + (8/b2)(2x+1) < 0. Thus
f(b,P) < 0 for b >> 1.3 From (3.4.13) this implies that P" < 0 for
b >> 1, a contradiction. ,
(c) P'(b) = 8/5, b 2 b6 for some bO >> 1.
Then P(n)(bo) = 0 I for n = 2,3,.... (See Figure 24 (c).)
From BP" = (10P—24b—3)P"+3P—4, at b = b0
0 = (10P—24b—8)(8/5) + 8P—4
which gives P .Q—.. 8b/5 + 7/10.
While from'uP'(b) 2: A/B, we have

2 + 20(2b—1)P —4b(32b+13) = 0.

25P
If we solve the following system
P = (8b/5) + 7/10
2 + 20(2b—1)P — 4b(32b+13) = 0,
then the solution is (b0,P(bO)) = (—1/4, 3/10), which also gives a

*
contradiction because bO > b > 0.

25P

By (1) and (2), r < 1.

In this case, A/B ~ 213—8 bf"1 as b —1 00.
Hence £13% br"1 2 crbr_1, and so

r = 1/2. 1:1

b—mo

Cor llar 3.4.9 . lim P'(b) = 0.

77

Proof. Easy frOm Lemma (3.4.8). 1]

Lemma (3.4.10). If P"(b) = 0 for b > b*, then P"'(b) it 0.

M. P"(b) = 0 gives BP"'(b) = 2(5P'(b)—8)P'(b). Suppose
P"(b) = 0 = P"'(b) for some b > b*. Then P'(b) = 3/5.
From BP" = (10P—24b—8)P' +_8P—4, we have
P = 8b/5 + 7/10.
From P' = A/B,
25P2 + 20(2b—1)P - 4b(32b+13) = 0.

*
There is no solution on (b ,oo) which satisfies both above. Hence if

P"(b) I: 0 for some b > b*, then P”'(b) # 0. El

Lemma (3.4.11). There exists b**€(b*,oo) such that P"(b) > 0 for
** ** **
bc(0,b ) and P"(b) < 0 for bc(b ,oo), but P"'(b ) 7t 0.

3):
Proof. Existence follows from Lemmas (3.4.7) and (3.4.8) since P"(b ) > 0

 

and P"(b) < 0 for b >> 1. Note that from
BP" = (10P—24b—8)P' + 8P—4
= (10(P—4/5) — 24b)P' + 8(P—1/2),
on (0,b*), since 1/2 < P < 4/5 and P' < 0, we have P" > 0.
This says, inflection points exist in (b*,oo). To complete the proof, we will
show the uniqueness: i.e., P"(bl) = P"(b2) = 0, b1, b2€(b*,oo) implies
b1 = b2.
If P"(b) = 0 at more than one point, then there must exist at least

*
three points (and an odd number of points) since P"(b ) > 0 and P"(b) < 0

78

for b >> 1. and P"'(b) at 0 if P"(b) = 0 (Lemma (3.4.10)).
Suppose P"(bi) = 0, i = 1,2,3 with b3 > b2 > b1 > b* and
P"(b) 4 0 for all be(0,b3) — {b1,b2}. Then P"'(bl) < 0,
P"'(b3) < 0 and P"'(b2) > 0 (see Figure 25).
At b = b2,
0 < BP"'(b2) = (10P'(b2) — 16)P'(b2)
which implies that P'(b2) > 8/5. On the other hand, at b = bi(i=1,3),
0 < P'(bi) < 8/5.
But this contradicts because y = P'(b) has local maxima at b = b1 and
b3 and a local minimum at b = b2 so that P'(bl), P'(b3) > P'(b2).

Hence b1 2 b2 = b3. D

*
Corollary (3.4.12). 0 < P'(b) < 8/5 on (b ,oo).

**
Proof. Lemma (3.4.10) and Lemma (3.4.11) P"'(b ) < 0, so
0 < P'(b**) < 8/5.
** *
y = P'(b) has a miximum at b = b and P'(b) > 0 on (b ,oo).

*
Hence 0 < P'(b) < 8/5 for all b > b . 1:1

Lemma (3.4.13). P"(b) < 0 for bc(—1/4,0).

Proof. See the proof of Lemma (3.3.8). 13

Now it is time to investigate bifurcation curves in (70,71)—plane. The
Hopf bifurcatidn curve Hf of order 1 in (3.4.1) is given by the equation
G(—1/4) = 0 and G'(—l/4) at 0, that is, Hf: To + T1 + l = 0 except

(70,71) = (—1,0) = 3: where the HOpf bifurcation of order 2 occurs since

79

G(—1/4) = G'(-1/4) = 0 but G"(—1/4) # 0 at E. (Andronov et.a1. [1]).
Also the symmetric homoclinic 100p bifurcation curve HO of order 1 in (3.4.1)

is HO: To + 471/5 + 32/35 = 0 except (70,71) = (0,—8/7) = e where the
symmetric homoclinic 100p bifurcation of order 2 occurs. (Roussarie [14], Joyal
et.a1. [11]).

Clearly Hf and H0 intersect transversally at the point
E = (—4/7,—3/7). (See Figure 26).

The semistable limit cycle. bifurcation curve C is given by the equation
G(b) = G'(b) = 0 for b6(—1/4,00) - {0,b*,b**}. C1 is a smooth curve
which connects the points 5 on Hf and E on H0, and which is tangent

to I-l'f and H0 at these points respectively.

4
C = U C1
i=1
where U1: G(b) = G'(b) = 0, bc(—l/4,0),
c2: C(b) = G'(b) = 0, be(0,b*),
C3: G(b) = G'(b) = 0, br(b*,b**),
and C4: G(b) = G'(b) = 0, bc(b**,oo).

The behavior of H, H0 and C1 are same as H, He and C in §3 (U1
corresponds to C in the previous section.) So we concentrate our attention
to Ci(i=2,3,4). The (To,rl)—plane is divided into 10 regions by curves H,
HO, Ci(i=1,2,3,4) (see Figure 26). Let

Cl(Hf) 001ml) = {a},c1(nf) nCl(HO) = {E},Cl(Hf) 001(c2) = {a},

01(H0) 001ml) = {é},Cl(C3) nCl(C4) = {f},Cl(HO) 001(134) = {g},

C1(Hf) nCl(C4)‘= {H}.

80

Lemma (3.4.14).

.The curves Ci(i=2,3,4) satisfy the following properties (see Figure 26).

(1)
' (2)
(3)

(11)

Proof.

' C

Ci(i=2,3,4) are smooth curves.

2 ‘ is tangent to H0 at 5 = (0,—8/7).

*
On (0,b ), the s10pe of C2 is decreasing monotonically and tends

:1: =1:_
to —1/P(b), as b-1b .

Hf . and C2 intersect transversally at d.

** **
There is an 1' = (To ,71 ) in (ro,rl)—plane such that

' ** ** **
G(b ) = G'(b ) = G"(b ) = 0 at (70,71) = f.
** **
T1 < —8/7 and To
* **
On (b ,b ), the s10pe of C3 is increasing monotonically and tends

* *+
to —1/P(b) as b-1b.

**
+ 471 -/5 + 32/35 < 0.

**
_ On (b ,oo), the SIOpe of C4 is increasing monotonically and tends

to 0 as b—1oo.

C3 and C4 are tangent at f.

C; intersects transversally with Hf at g and with H0 at H
respectively.
C3 r1 H0 = {}.

For (1) see the proof of Lemma (3.3.9).
Next, from G(b) = G'(b) = 0, we have

T 2

7'1:

— 7% — 3/7.

 

 

81

d7'1 1

By (3.4.14), we'have properties (2), (3), (4), (7), (3) and (9). (5) is immediate
from Lemma (3.4.11).
**
**
(6) r0 = — f; b + £19.41
7P'(b )

._ 4 _
‘ 7P'(b ) 8.”

** **-
Hence 71 <—8/7 since P'(b )> 0.

q
)_.1
II

** 4 ** . -

Now 70 + 5— .71 + 32/35 < 0 1S equivalent to

** ** **

b P'(b )—P(b )+4/5>0.
Let f(b) = bP'(b) — P(b) + 4/5.

f(b*) = 4/5 — P(b*) > 0

- *
f'(b) = bP"(b) > 0 on (b ,b**) by Lemma (3.4.11).

Hence f(b**) > 0. In fact,
* **
(3.4.15) f(b) > 0 on [b ,b ].

(10) is clear from (6) and (8). (Note that C4 is concave upward.)
(11) Suppose C3 11 H0 at {} and let (7 1' 13)cC 11 HO.

 

.0,
Then 70 + 371+32/35 = 0 since (70,71)6H0.
Also 7'0:— 7b+man 71:- -—8/7
7P‘ (b) ~ ~ 7P'(b)
for some bc(b* ,b* *) since (70,71) (C3.

Hence C3 11 Ho # {I is equivalent to
bP'(b)— P(b)+ 4/5: 0 for some bc(b*,b**).

However, from (3.4.15),

 

 

82

' ' f(b) = bP'(b) — P(b) + 4/5 > 0 on (b*,b**).
Hence C3 11 HO = {}. 13

Now we consider the number of limit cycles. For a given (70,71), the
number of limit cycles of equation (3.4.1) is determined by the number of roots
of equation G(b) = 0 for b > -—1/4. The roots of G(b) = 0 for
bc(—1/4,0) correspond to the limit cycles of type 1 (two limit cycles for each
root; one surrounds the point 'A = (—1,0) and the other surrounds the point
C = (1,0), see Figure 21) and those for b€(0,oo) correspond to the limit cycles
of type 2 (one for each root which surrounds A, B = (0,0) and C
simultaneously).

If 1'1
for 706(m,1/7) -— {0} (i.e. b6(-—1/4,oo)- {0}) and 71 = —8/7, G(b) = 0 has

+ 8/7 = 0, then G(b) = 0 if and only if To = — 4b/7. Hence

aunique root since G'(b) = 4/7 at 0 and for T06(1/7,oo) and 71 = — 8/7,
G(b) = 0 has no root.

We suppose that 71 + 8/7 )6 0, and rewrite
G(b) = To + $- b + (71+8/7)P(b) into the form

G(b) = (T1+8/7)IP(b)-A(b)l
where A(b) = —(ro+4b/7)/(rl_+8/7).
For given (70,71), A is linear in b. Again the root of G(b) = 0 is the
intersection of the straight line P =_A(b) and the curve P = P(b) on the
(b,P)—plane. The curve P = P(b) is concave downward (P"(b) < 0) on
(-—1/4,0) U(b**,oo) and concave upward (P”(b) > 0) on (0,b**). Thus
points (0,4/5) and (b**,P(b**)) in (b,P)-planeare inflection points.
P'(b) < 0 on (—1/4,b*) and P'(b) > 0 on (b*,oo) so that the point
(b*,P(b*)) is an extreme point. A(b) depends on 70 and 71 while P(b)
does not. Recall that P(-1/4) = 1, P(O) = 4/5.

83

 

A(b).;has the following properties;
(1) A(—1/4)_ =1 (resp. A(—1/4) > 1, A(—1/4) < 1) if and only if
_ (70,71)er (resp. is below Hf, is above Hf).
(2) A(O) = 4/5 (reSp. A(O) > 4/5, A(O) < 4/5) if and only if
(TO,T1')€HO (reSp. is below H0, is above Ho)'
(3) The straight line ' P = A(b) is tangent to the curve P = P(b) for

4
bc(-—1/4,oo) — {0,b*,b**} if and only if (r0,r1)c U Ci.
. i=1

The relative positions between the straight line P = A(b) and the curve

\

 

P = P(b) are as inFigure 27 (a) — (h). In the strict sense, I to X in
Figure ‘26 are different from those in Figure 19,. however, we will use the same
notations for convenience.
For g and H in (70,71)_—plane, we have associated straight lines, say,

g H P = A(b). and H 4..) P =3 H(b). .Let b (reSp. B) be the solution of
the system

A(b) = P(b) reSp. K(b) = P(b)

F(b) = P'(b) ' F(b) = P'(b).
It is easy to see that b* < b** < B < B < 00 (see Figure 27 (g) and (h)).

Lemma (3.4.15).

Let K be a compact neighborhoOd of ' {H(x,y) S b") and let D be a
compact neighborhood of the curved region 5 5 g H d E and a curve 1' g in
(76,rl)—plane. Then there exists a(D) > 0 such that the bifurcation diagram
of the equation'(3.4.1) can be described in C(D) = (0,a(D)) x D as follows
(up to a diffeomorphism of C(D) equal. to the identity of s = 0):

(1) SHf = (0,a(D)) x. Hf is a surface of H0pf bifurcation of

codimension 1.

84

(2) SH = (0,a(D)) x H0 is a surface of homoclinic 100p bifurcation of
- o

codimension 1. I

(3) SC. = (0,'a(D)) x Ci (i=1,2,3,4) is a surface of semistable limit cycle
1

bifurcation of codimension 1.
(4) (0,003)) X {5}, ‘(0,a(D)) X {5}, and (0,0(D)) x {T} are curves of
H0pf, homoclinic 100p, and triple limit cycle bifurcations of

codimension 2 respectively.
(5) (0.4(D)>x161 = (21(st) 1‘1 cusHO)
(0,a(D)) x {El} 2 Cl(SHf) n Cl(SC2)
(0.44m) x {a} =‘ 01(SH ) n Cl<sC )
_ o 4
(0,a(D)) x {H} = C1(SHf) n Cl(S4)
are curves of the corresponding two simultaneous bifurcations.
Outside these bifurcation sets in C(D), the t0pological type of the phase

portraits of the equation (3.4.1) is constant in K.

Proof. The paragraph after Lemma (3.4.13) plus the implicit function theorem
applied to F(b,AS) = s3(rOIO(b) +' 7112(b) + I4(b)) + 0(s3) give the proof.

For details, see Dumortier et a1 [7]. 1:1

The blowing—up (3.3.1) with 17 = 1 gives a transformation

(1): (S,(7'0,7'1)) -1 (61,62,63),, and
(3.4.16) ¢((0,a(D)) x n) = {(82,84TO,S271)[SC(0,01(D)), (r0,r1)tn}.

Let Ef (D) be the RHS of (3.4.16). The bifurcation diagram of (3.2.9)_ in
1

85

E C (D) is the image of those‘described in Lemma (3.4.15) by the
1 .

transformation and thus homeomorphic to cones based on H , HO, O, and
3: - h with curves 3 -1 (S217, 841'0, szrl) with 17 = 1, or equivalently

2
61 -1 (61, 617'0, 617’1) for 61 > 0.

Now we will consider the behavior of (3.2.9)_ in a sector around 063-axis.

For this we take 7 = *1 instead of 17 = 1 in (3.3.1). Since both cases

1
T1 = 1 and 71 = —1 are similar, we will consider only the case T1 = 1

like in §3.3. By (3.3.1) with 71 = 1, equation (3.2.9)_ becomes

(3.4.17): x = y

y = 17x — x3 + S3(TO+X2+X4) y.

Let s1 > 0 be fixed. Then for each se(0,sl] we take a blowing—up again:

2

(3.4.18) 17 = r x -1 rx
' and y —1 r2y

2_
TO = T To t —1 t/r.

Then (3.4.17) becomes

(3.4.19) it = y

y = x - x + s3(r(?o+x2)y + 0(r3))

which is a perturbation of the Hamiltonian system (r = 0)

86

(3.4.20) 5r

II
<<

4
x

with the first integral H(x,y) = 5—2— — g— + 1— . As in Lemma (3.4.3), we
have I .

. H(wB) — H(wb) = s3r(F(b,?o) + o(r))
where F(b,'r'0) = 7010(b) + 12(b).

Lemma (3.4.16).

In the plane {(17,70,'r1)|71 = 1} there is a fixed compact neighborhood
B+ of (17,70,71) = (0,0,1) such that for equation (3.4.19) the results of

Carr—Takens are valid for any (1),T0)6B+ and any sc(0,s1]. (See Figure 28.)

Proof. Carr [3]. Cl

 

Blowing—up (3.3.1) with 71 = l (i.e., £1 =' 7782, 62 = 7034, 63 = 32)

gives a mapping (s, 17,70) -1 (61,62,63) which maps (0,31] x B+ to
+ _ 2 4 , 2 +

E+

‘ 3
3+.

is a cone in (51,62,63)—space around 063—3X1S (63 > 0) based on

The bifurcation diagram of (3.4.1) in E: consists of cones based on

3
Hf, HO, C2, P, Hf and {51} with generating curves s —1 (S277, 3470,32).

For 1'1 = —1, we can get a cone E-
c3.
‘11,, 1‘10, 63, P, Hf and {'62} similarly.

around 063—axis (£3 < 0) based on

87

Now we consider the case ‘1 = 0. In (3.2.9)_, if ‘1 = 0, we have the
following equation

it = y

y =‘ —x3 -I- (62+e3x2+x4)y.
After blowing—up ((3.3.1) with 7) = 0), we get

(3.4.21) it

y

y = —x3 + s3(ro+7'1x2+x4)y.

If s = 0 in (3.4.21), it becomes a Hamiltonian system

x=y

i=—x3

with the first integral

2 4
__ X
(3.4.22) H(x,y) _ g— + 4— .

As before, we get a similar lemma of Lemma (3.4.3) and lead to solve
(3.4.23) F(b,/\O) = 7010(b) + 7112(b) + 14(b) = 0.

where Ii(b) = ) xiydx, 7b; H(x,y) =' b in (3.4.22).
7b
(All the same notations will be used again.)

Lemma (3.4.17).

(3.4.24) 714(b) = 4b 10(b).

 

88

Proof. Omit. :1
Hence by (3.4.24) we get a new form in F(b,AO) from (3.4.23).
(34 25) F(bA ) = (r +4b) 1 (b) + r1 (b)

' ° ’ o o 7— o 12 '

Let

12(b)/Io(b) for b > 0
(3.4.26) P(b) = .
I 0 for b = 0

and from (3.4.25) and (3.4.26) we define

. 1 . . .

4b
= TO + T— + T1P(b)
(simply write G(b) instead G(b,)t0)).

Lemma (3.4.18). P(b) defined in (3.4.26) satisfies

(3.4.28) P'(b) = %B .

Proof. Immediate from 310(b) = 4b IO'(b)

51 b) = 4b Ié(b). n

2(

By (3.4.28),

89

(3.4.29) P(b) = Z 75, b 2 0, l is a constant independent of b.

If b = 1/4, 4: P(1/4) = J x2ydx/J ydx
' ' 71 71

= I1X2(1*X4)1/2 dx/I1 (1-x4)1/2 dx.
_1 _1

For the semistable limit cycle bifurcation, if we solve
G(b) = To + 1% + T1P(b) = 0
G'(b) = f; + 71P'(b) = 0,

then we get
_ 4
T0 — 7' b
. _ - 846
1 _ W’
Hence
62 r2
3 . 1 16
(3.4.30) — = ——- = —
62 To 732

We restrict (3.4.30) to a Sphere ~S (actually S n {51 = 0} = {€2+€3=60})
so that the solution is exactly (0,072,a3)
8 8 2 2 1/2
- —- + (I ) + 6 )
7132 fig 0
We let b3 = (0,07 ,a3).

2_16
and 3—7702(073<0).

where 072 = 7

The HOpf bifurcation of order 1 occurs at b1 2 (0,0,60) and
= (0,0,—eo). (G(O) = 0 # G'(0) —1 To = 0 —1 62
so 3 0 {c1 = 0} = (b1,b2) u (b2,b3) u (b3,b1) u {b1,b2,b3}.

b = 0).

2

In (3.4.27), G(b) = 0 has only one generic solution if 62 < 0. So on

 

 

 

90

(b1,b2), there is one limit cycle. If 6% < fig (2 or (62"3) is in the first

quadrant, G(b) = 0 has no solution. So on (b1,b3), there is no limit cycle.

If 0 < $3 ‘62 < 5% and 63 < 0, G(b) = 0 has two generic solutions. So
7!

on . (b2,b3) there are two limit cycles.

Those limit cycles are generic on (b1,b2) U (b2,b3) U (b1,b3) so that
they still persist if 61 is a nonzero small number such that c = (61,62,63)
is in S.

Let R C S be a compact neighborhood of b3 diffeomorphic to a closed
disk and let D0 (resp. D1,D2) C S be a compact neighborhood of part of
(bl,b3) (resp. (b1,b2), (b2,b3)) around the circle {(1 = 0} n s in such a way

that (1) limit cycle(s) persist on _Do (resp. D1, D2), and (2)

2 ,
(u D.) 11 13+ 0 E" u R 3 {61: 0} n s as in Figure 29. We choose a
i=0 1 63 63
compact set D in Lemma (3.4.15) such that union of E6 (D) n S, D0, D1,
1
D2, E: n S, E; n S and R covers hemisphere {cl 2 0} 11 S. (See
- 1 3 '
Figure 29.)

Then we will show that the curves of Hopf bifurcation, homoclinic loop

bifurcation and semistable limit cycle bifurcation (C2 and C3) in

E6 (D) n S are connected with those in E: n S (and thus Hf, H0 and
l 3

C2 with b2).

with b1) and in E23 11 S (and thus Hf, H0 and C3

We have
:Hf: T+T1+1=0

and

 

 

91

H: To + 41'1/5 + 32/35 = 0
2 2 2 _ 2
_ “€1+62+€3—€0,61>0
by the transformation (3.3.1)
. _ _ 2 _
. s — E, To — 62/61, 1'1 — 63/61, 61> 0.
Let ‘1 -+ 0 on Hf and H0' Then 62 -1 0. Hence (1 —1 0 implies
(61,62,63) —'1 (0,0,ieo) = b1 or b2.
Now let C2: 71 = f(ro).

. * d71
, . . _ 1
As b varies from 0 to, .b (i.e., 7'0 varies from 0 to —oo), 37—0 — f (To)

- *
varies from —5/4 to —1/P(b ) and f'(70) is monotonically decreasing on

(0,b*).

7 .
0 .
——-T< f(r) <—5r/4—8/7.
P(b ) ° 0- .
7
Since on the straight lines 71 = — —O—,,.— and 71 = — 27 - 8/7, if 61
P(b) ' 0
tends to 0, then 62 tends to 0 and To -1 — co and thus 63 —1 60. Hence

by squeezing, c2 -1 0 and 63 -1 60 as 61 -1 0, i.e. as 61 -+ 0, (61,62,63) on
C2 tends to b1. Similarly, as ‘1 -1 0, (61,62,63) on C3 tends to b2.
Next we will show that C4 in IE6' (D) n S is connected with b3 in
1

R.
Note that U~(r r‘) — (— 4b +4P — 4 —8/7) for bc(b** )
. -, 4611'- 7 7P“ 715T 1°°-
2_ _ 4 4P ‘3_ _ 4
From 62/61—T0—-7b+7p‘r and q—T1—-7P-r-8/7,

as b -1 oo P'(b) -+ 0+, 7'1-1 — co and 63/61"1 — co,
and '

[as b-160 70-100,62/6¥7160 and 6962-10

since To =.—i§-b+%];-4b/7 as b-loo.

 

92

 

in: #10:.-.
o ‘2 ‘2 ‘2 ‘2 2
and
Ti (‘ 7f“ ' $2 16
—= 4 4P 47%b4m where c>0isaconstantin
. o (— 7b + 7F) _7c
Lemma (3.4.8).

. Hence 61 -i 0 but 62 and, £3 -/-+ 0 as b -» oo. (61,62,63) on C4
does not tend to b1 or b2 as b -i 00 since 52 —/—» 0. The only other
point at_which the number of limit cycle change on {61 = 0} n S is b3.
Thus (61,62,63) on C4 must tend to b3 as- ‘1 —) 0 and so it must be
that c = Z.

We have proved Theorem (3.4.1).

B. 61 < 0.

As in the case 61 > 0, we use (3.3.1) with 77 = —1 to investigate the
behavior of (3.2.9)_ in a neighborhood of Del—axis for 51 S 0. Hence the
equation (3.4.31)_ has the form

(3.4.31) ii = y
y = -—x - x3 + 83(TO+T1X2+X4)y.
Equation (3.4.31) has the equilibrium point (0,0) which is a focus. If

s = 0, (3.4.31) becomes a Hamiltonian system

(3.4.32) )1 = y

y=-——x——x

 

93

withthe first integral
'H(x,y)=g3+)§&+i:

and the phase portrait of (3.4.32) is shown in Figure 30.

Closed orbits surrounding (0,0) correspond to level curves
(7b: H(x,y) = b, b€(0,oo). Now. we consider (3.4.31) for small 3 1: 0. Every
closed orbit of (3.4.31) should intersect with the interval U = {(x,0)|x 2 0}.
We define Wb for b€[0,oo) as follows:

(a) wch (b) H(wb) = b.
Let As = ((70,71),s) and we define a Poincare map PAS: U -+ U by a

successive intersection points of U with an orbit in an obvious way. (See
Figure 31.)
Let 7(b,/\S) be the orbit of (3.4.31) which joins the point wb and WE.

Hence 7(b,)\s) is defined for b€(0,oo). Then we have the lemma.

Lemma (3.4.19).

(1) Every closed orbit of (3.4.31) is expressed by the form 7(b,AS) with
WB = Wb. V

(2) A trajectory 7 = 7(b,)\s) of (3.4.31) is a periodic orbit if and only
if
(3.4.33) 1 dH if (it = 0.

(3) For x > 0, condition (3.4.33) is equivalent to

94'

 

(3.4.34) F(b,A ) ’2: J (7' +7 x2+x4)y dx = 0.
s o 1
Proof. See the proof of Lemma (3.3.2). a

Following the same procedure as Subsection A, it reduced to solve (see the

 

paragraphs after Lemma (3.4.2).)

(3.4.35) F(b,)\o) = 7010(1)) + 7112(3) + 14(3) = 0

 

where Ii(b) = J xiy dx, 7b; H(x,y) = b.
7b

Lemma (3.4.20).
(3.4.36) 714(b) = 4bIO(b) — 812(b).

Proof. Similar calculation of ’the Lemma (3.3.5). D
From (3.4.36), (3.4.35) becomes

(3.4.37) F(b,(ro,rl)) = (70+ 4b/7) 10(3) + (7'1 — 8/7)I2(b).

Note that 10(0) = 12(0) = 0, 10(1)) > 0 for b > 0 and 12(b)/Io(b) .. 0

as b -+ 0. So we can change (3.4.37) as follows:

(3.4.38) G(b,(TO,T1)) = TO + 4b/7 + (71—8/7)P(b)

(We will denote G(b,(TO,T1)) simply by G(b)) where

95

I2(b)/Io(b) for b > 0

(3.4.39) P(b) _=
. 0 for b =

Lemma (3.4.21). P(b) defined-in (3.4.39) satisfies
4b(4b+1) P'(b) = —5P2 + (8b—4)P + 4b

 

Proof.- See the proof of Lemma (3.3.6). 0

Lemma (3.4.22). P(b) also satisfies:

(1) lim P'(b) = 1/2 and P'(b) > 0 for b > 0
b—»0 ' '
(2) tlyi’m P"(b) = —7/8 and P"(b) < 0 for b > 0
0

hence the graph of P = P(b) is concase downward.

(3) P(b) ... mJb’ as b —i 00 for some positive constant m.

 

So ,
lim P(b) = 00 and lim P'(b) = 0.
b—ioo ' b-ioo
Proof. (1) and (2) are easy calculations from the system of equations
13 = -5P2 + (8b—4)P + 4b
13 = 4b(4b+1). . D

which is gotten by Lemma (3.4.21).
See Lemma (3.3.7) and Lemma (3.3.8) for details.
For (3), see Lemma (3.4.8). o

 

 

96

Now we will compute the bifurcation curves of the equation (3.4.31) in
To,rl)—plane. The H0pf bifurcation curve Hf of order 1 in (3.4.31) is
Hf: .To

where the Hopf bifurcation of order 2 occurs. (Compute the Liapunov's focal

:0 except (70,11) = (0,—16/7) = IE

values in (3.4.31).) The semistable limit cycle bifurcation curve C is given
by the equation G(b) =-G-?_(b) = 0 for b€(0,oo) and is
4 4P 4
CI (TO’TI) = (- 7b _.7PT’ - TPT + 8/7), b€(0,oo).

(See'Figure 32.) ‘

Lemma 3.4.23 .

(1) The curve C issmooth.
1 (2) C is tangent to Fl'f at E: (0,—16/7).

' ' ~ d7
(3) As b -+ 00, the SIOpe (a—Tl) of U is increasing monotonically and
o .

tends to 0.
Proof. See Lemma (3.4.14). ~ ' n

It works through on the semistable curve almost the same as the case
‘1 > 0. For the number of limit cycles, we consider the relative positions of
the curve P = P(b) and the straight line P = A(b) shown in Figure 33,
where A(b) is such that '

G(b) =,(r1—8/7)(P(b)—A(b))

SO
To + 4b/7

43:77.7...

97

We can get a similar version of Lemma (3.3.15) and, after secondary
blowing—up, Lemma (3.3.16), then using the results for 61 = 0 ((3.4.21) —
(3.4.30)) and the same idea as before (61 > 0), we can prove that (61,62,63)
on C tends to b3 as 61 4 0 and m = t. We omit the details because

the steps are routine repeated arguments of Subsection A.
§5: The Case: b at 0.
We can get the versal deformation of (3.1.5) as follows:

Lemma (3.5.1). Any symmetric perturbation of (3.1.5) with small parameter p

can be transformed into the form

(3.5.1)i x =- y

9: (41(4) + 423(4))? .4 x4)x + (42(4) + Gm)? + y¢(x,y,u))y,

where ”(R3, G(x,0) = [all/2 and ¢(x,y,0) = 0.

Proof. By an appropriate scaling, we can change (3.1.5) into the following

form:

(3.5.2) x = y.

ly=ax5+flx2y

Where a = 8311(5), fl = (AH/2.
Using (3.5.2) to follow the same steps as in the proof of Lemma (3.2.1), we get
(3.5.1) 4' D

98

Remark. We can take a transformation (a = ((p1,(p2,(p3): p «—» e in the

parameter space so that (3.5.1) i becomes

(3‘5°3)¢ x = y
- _ 2 4 . 2
u —- (61+63X ix )x + (62+G(x,p)x +y¢(x,y,e))y
_ i 1/2 ._ _
where G(x,0) - |a| (.=c), ¢(x,y,0) — 0.
(3.5.3) 4 is versal to

 

(3.5.4)i x = y :2 f1(x,y,c~)

. 2 4 2 _
y = (€1+€3X =sz )x + (62+CX )y :: f2(x,y,c).

Bfl 6f2 ' 2
Note that div(f) = y(xq) + W(x,y) = 62 + cx .

Here we are not able to control cx2, hence (3.5.4) 4 can't be regarded as a
perturbation of the Hamiltonian system. This implies that it's likely hard to
apply to abelian integrals and Picard—Fuchs equations. So probably we have to
approach from other directions.

There is a recent paper by Dangelmayr et.al [6] which introduces an
equation for a laser with saturable absorber. After using a center manifold
theorem and (a normal form theorem, they reduced it to the form (3.1.5) with a
leading fifth order term. It described also the bifurcation diagrams and phase
portraits of (3.5.4)_ without a detailed analysis (Figure 1 (a), (b) and Figure 2
in [6]). I '

However, the conjecture of the bifurcation diagrams has a couple of

mistakes, one is a saddlehnode homoclinic bifurcation (Sechecter [15]) and the

99

other is a‘pitchfork homoclinic orbit in the symmetric vector field (Chapter 2
of this thesis). I

We will describe about (3.5.4)_ briefly. For simplicity we assume that
c = .—1. First, equilibria of (3.5.4)_ is determined by the equations y = 0,
(614-5382 — x4)x = 0. Hence. {cl = 0} is a pitchfork bifurcation surface and
{£3 + .461 = 0, 63 > 0} is a saddle—node bifurcation surface. Also {62 = 0,
£1 < 0} is a Hopf bifurcation of order 1. The number of equilibrium points
are as follows. (See Figure 34.)

(1) 61 '> 0: 3

 

(2) —£§/4<€1<0 and c3>021

2
c
3
(3) (£1<0 and c3<0)U(61<—1— and €3>0)25.

Bifurcation diagram and phase portraits (Figure 35) are based on numerical
results, Chapter 2 of this thesis, and Schecter [15]. (Also, of course referred
Dangelmayr et.a.l. [6].)

We divide three cases depending on the sign of 63;

(i) Case 1: 63 > 0
(ii) Case 2: g, = 0
' : (iii) Case 3: e3 < 0,

and codimension two bifurcations are explained in each case.
(i) Case 1: 63 > 0.

For a small but fixed 63 > 0, we can change (3.5.4)_ into the following

form

 

100

(3.5.5) 5: = y

. , 3
y=u1x+u2y+X(

1—x2) + x2y
by an appropriate scaling, where 441 = 61/63, p2 = 62/63.

(3.5.5) gives us a local behavior near (p1,;r2) = (0,0) (i.e.,

063—axis (63 > 0); (5) + in Figure 35). Also we can see that Takens—Bogdanov
bifurcation occiirs at (111,142) 2 (—1/4,1/2) ’

(i.e.,.e = (—c§/4,e3/2,c3); (1) in Figure 35), H0pf—saddle node bifurcation at

(141,442) = (—1/4,0) (i.e., c = (- €3/4,0,€3); (9) in Figure 35), Hopf—pitchfork

 

bifurcation at 011,432) = (0,1) (i.e., c = (0,63,63),63>0; (7) in Figure 35).
(ii) Case 2: 63 < 0.

As above, we have the following:-
x = y

- 3 2 2
y=u1x+u2y-X(1+X)-Xy,

‘ 2
On 063—axis (63 < 0), we have a Takens—Carr bifurcation ((5)_ in Figure 35)
which is only a codimension two bifurcation.
(iii) Case 3: c3 = 0.
Symmetric version of cusp bifurcation occurs at (0,62,0) ((6) in Figure 35).
In this case (63 = 0), HOpf, homoclinic and semistable bifurcation curves are

tangent to the pitchfork bifurCation curve.

 

101

There are nine different codimension two bifurcation occurring in (3.5.4)_.

(see. Figure 35).

(1') Takens—Bogdanov bifurcation.

This— bifurcation has an equilibrium point whose linearization is nilpotent.
The equations defining the locus of Takens—Bogdanov bifurcations for a family
5: = f(x,A), de2 are f(xO,A) = o,
trace Dxf(xo”\) = det Dxf(xo”\) = 0.

A normal form for the Takens—Bogdanov bifurcation is

 

x = y
_ . 2
y—61+£2six +xy.
(See Bogdanov [2] and Figure 36.)

(2) Saddle—node homoclinic ' bifurcation.

The vector field of this bifurcation has a homoclinic orbit at an equilibrium
point whose linear part has a simple zero eigenvalue and one nonzero
eigenvalue, and the center direction is a saddle—node codimension one
bifurcation. A homoclinic bifurcation meets the saddle—node bifurcation with

quadratic tangency. (See Schecter [15] and Figure 37.)

(3) Homoclinic bifurcation of order 2. g

This is a homoclinic'bifurcation at a saddle point where the Jacobian
derivative has a trace 0 and a certain prOperty is satisfied.

The semistable bifurcation curve approaches the homoclinic bifurcation
curve tangentially with infinite order contact. (See Roussarie [14], Joyal [11] and

Figure 38.)

 

102

(4) Pitchfork homoclinic bifurcation. .

(see Chapter 2 of this thesis and Figure 39.)

(5) a: Takens—Carr bifurcation.

If a symmetric vector field has a nilpotent linear part at 0, then it occurs
with two distinct qualitative behaviors (depending on the sign of coefficient in
x3 in the nbrmal form) whose: normal form is as follows:

it = y
y: clx+ c2yix3+x2y
(See Carr [3] and Figure 40 (a), (b).)

 

(6) Pitchfork saddle—node bifurcation.
Ifa pitchfork bifurcatiOn occurs at (0,0) and a saddle—node bifurcation
occurs at an equilibrium point Splitted from (0,0) in the symmetric vector

field, we have a pitchfork saddle—node bifurcation. The normal form is as

follows:
x = 61X + €2X3 — x5
r = cry ((#0)-

(See Chapter 6 of Golubitsky and Schaeffer [9] and Figure 41.)

The following are codimension two bifurcation curves where two

codimension one bifurcation curves meet transversally each other:

(7) Hopf pitchfork bifurcation I
(8) Pitchfork semistable bifurcation, and

(9) Hopf saddle—node bifurcation.

APPENDIX

Theorem. Let OeIR2 be a pitchfork of 11:33‘ :(IR2), r _>_ 3. Then there is a
neighborhood B of f in 42‘ $0112), N a neighborhood of 0 in R2 and
a: B —i R, Cr"1 function such that

(1) ch has a pitchfork as a unique equilibrium point in N if and
only if d(g) = 0. If d(g) < 0, g has three equilibrium points, origin is
node, other two, both generic, are saddles. If d(g) > 0, g has one saddle
point, origin in N.

(2) d(f) = 0 and dozf 9f 0.

Before proving it, first we introduce some notations and definitions. Let

M be a 2—dimensional smooth manifold. Let fee? 1(M) such that

2
f(p) = 0 for some pcM, and f = 2 fi 3%. Then define
i=1 i

. 2 _, = =
Dfp. DpM Dp2M by Dfp(v) [g,f](p) where g(p) v
for some g = 2 gi 3:3— 63 1.(M), and [,o] is a Lie bracket.

i=1 i

Also define

A(f,p) = Det(Dfp)

a(f,p) = Trace (Dfp).
Let r 2 3 and Spec(Dfp) = {A1 = 0, A2} and Ti’ a corresponding
eigenspace of Dfp and iri: DpM -» Ti projection (i = 1,2).
For chI, v i 0, we define A1(f,p,v)v by 7rl[g,[g,f])(p) = A1(f,p,v)v, and
likewise

A2(f,p,V) by 7rllg,ls,[3,flll(p) = A2(f4>,v,V)V-

Then A1(i=1,2) does not depend on g. P is called a saddle—node of
f if A1(f,p,v) at 0 for some v ¢ 0.

103

 

104

If fee?)r :(IR2) and A2(f,0,v,v) ,4 0 for some v ,4 0, 0 is called a pitchfork
of f.
i

2 such that 7r, v = uv and fl,ui,v

Let u be the vector on DpM
components of f,u,v respectively with reSpect to a coordinate system (x1,x2)

around p. Then

(1) 41333) = ulg,[g,fll(p) 131‘” 62915 mv’vk u,

(2) A, does not depend on g. i.e., ulg,[g,fll(p) = uisisimp)
for every ges%r(M) with g(p) = g(p) =

(3) Let fc$;(lR2), ge$r(lR2). Then

A2(f,p,v V)— _ 11[g [g [g {1le ):

63fi jk 1
2 p) v v v u.
i,i,k,1 afiak‘f‘l ‘

(4) Likewise, A2 does not depend on g.

Proof of Theorem. Let x = (x1,x2) be a coordinate system around 0

2 .
such that x(0) = 0 and 33—(0)6Ti(i=1,2). Then f: 2 fl 3:— satisfies
i i=1 i
1 1 2 2
f f f f
"35(0) 1' £50» 2 331(0) - 0, 3;“) 0) = 0(fa0),
azfi (0)_ _ 0 f 2
W or every i,j,k— — 1,2 since fee? S(IR ),
33413 a
—3-(0)= A2670, (0), (0))-
6x1 33‘? 3‘71—
That is,
1
f (x1,x2) = A2x:13 + bx%x2 + cxlx?2 + dxg + M1(x1,x2)

2 3
f(x1,x2) = 0x2 + axl

2
where Mi(x1,x2) = 0(le ).

2 2 3
+ flxlx2 + 7x1x2 + 6x2 + M2(x1,x2)

 

105

0 0
Assume a(f,0) < 0 and A2(f,0,3x—(0), 571(0)) > 0
Let N and B be neighborhoods of 0 and f respectively such that

o
for g— - 2g i 3;ch the following hold on NO:
=1g i

(1) 42.

HO

0,
2
6‘“ 'k1
(2) A(0,,vv)=2 vvaug>0
2g i,j,k,1 j k 1gggl
where
v1=1v2 = -(§§3)'1(§3)
g ’g 2 1
2 2 1
—2 a —-1
ui=<1+<31L1 3% >
2 1 2
1 2
g_- -1. s
“2‘ 3,3,) 1,,
1 2
(3) a(g)=%+%§<0

Existence of such N 0 and B 0 can be guaranteed because of the

continuity since f satisfies (1), (2), and (3) at 0.

[\D

2 . 2
Tk =2 ‘6 =02 ia,ad 8=2u5d.
3.8 Vg i=1Vga;,WgH:1Wg3x—l D 11 i=1 1 X1
where

2 1
1 __ a -1 0 2 _
wg .. (Big) £3 wg ._ 1.
If OcN0 is an equilibrium point of g and so A(g,0) = 0, then Vg(0) is
an eigenvector corresponding to A: 0, Wg(0) is an eigenvector corresponding

to A = a(g,0), u g(0) is the covector such that ug(vg) = 1 and

Hence by (2) and (3), non—generic point OeNO of chO is such that
o(g,0) < 0 and A2(g,0,vg,vg) > 0, i.e., 0 is a pitchfork.

 

106

Define F. B0 xNO'-11R by F(g ,x( ,x2))= g2(x1,x2). F is Cr
since g2 is and it is an evaluation map. F(f (0 0)): 0, gF—(f (00

2
m 0,0) = a(f,0) < 0. By implicit function theorem, there exist B1 x I1
2

a neighborhood of (f,0) and I2 a neighborhood of 0, Ii C IR (i=1,2)

and F1: B.1 x I1 -» 12. a unique Cr—function such that

(*1) F1(f,—0) — 0, F(g,x(1F(g,x1))) E 0 for every (g,x1)1€B1 x 11.

Define F2: B1 x I1 —+ IR by

(*2) _ F2331) = 631313311).

From (*1) and (*2), we have

2 = g —1
(6) 3x7— (111) [A2(810,Vgivg) > 0-

The computations of (4), (5), and (6) will be later.

Define a: B o -+ [R by
0F 1
2 0
d(g) = 571(310) = 3%(0 F (3,0))
r—l 6F2
Then a is C and d(g) is the minimum of ax—(g,x1) at
1

0F
x1 = 0 611 by (5) and (6). Hx—i(g,x1) is even in x1.

 

107

(i). Case 1: d(g) > 0.

There is a unique equilibrium point (0,0)6N which is saddle since

0f
-a(g) > 0 if and only if BX—2(g,0) > 0 if and only if
1

A(g)(x1) < 0 at x1 = 0 by (4) and (1).

(ii) ' Case 2: d(g) = 0.
1) f f 6F2( 1 f f
ag =‘ 0 i and onlyi g,0 = 0 i and onlyi
3Y1

A(g)(x1) = 0 at x - 0. Hence (0,0) is a pitchfork by (6) since g

1 _
is symmetric.

(iii) Case 3: d(g) < 0

BF
d(g) < 0 if and only if gx—2(g,0) < 0 if and only if
, 1

A(g)(x1) = 0 at x1

be x1 = 0, 7(3), and -7(g)-
That is,

= 0. Hence (0,0) is node. Let zeros of F2(g,x1)

(0,0) and (47(3), F1(3,i7(3))) are zeros of 3

BF 2
3x—1(g,i7(g)) > 0 if and only if A(g)(*7(3)) < 0-

Hence (i7(g), F1(g,i7(g))) are saddle.

Next,
0F2 Bil Bfl
“(fl = ax—lfiiol = Emil-’10,”) = 3,750.0) = 0
2

__ 0 r2
For h_i:1hi3’q€£3(m)’

108

 

(101(1)) = lim a(f+ch)—a(f)
6
6+0
= lim (l/c) a(f+ch)
£40
1
. f+eh
= 11m (1 e 0,F (f+fh,0))]
= lim (1/€)-g)f(—1(0,F1(f+eh,0)) + lim 6111 0,F1(f+ch,0))
6-10 61 6-10 1
6241 F1 0111
= W010 f+ch,0) + (0,0)
1 2 ) Bfi |c=0 H1
1 .
= 12512409).
. 1
Hence dof 76 0. 1:1
Computations of (4), (5), and [6).
For (g,xl) (B1 x 11, g2(x1,F1(g,x1)) E 0.
s a 2 2 (Fl (1 h
o + = 0 an ten
W1 % in _
2 2
1_ 0 -1
“in“ 31‘??? '
6F 1 1 6F
2 8 1
(4) = + 35x;
3? 1 3’71
1 1 2 2
- -1
= 3% + 3%, <— 34 (ii)
1 1 2
2 .
—1
= (3,55) A(3)
()02F2 621 82101“, (321 3218F1)(8F1) g£1021?
5 ——§ = —5— + 37%),— + ax—éx— + +
0x1 0x? 123’? 12 6x23371551— 20x2
2 31 62F

 

109

since

i .
fl = 0 (iajrk = 112),

.1

3223233111 32232 1121
52+axlaxzexl+(ar%r+g§—ax—laq+%g2—=Oiand

1 1.2

 

031 6381 0F1 8F1

 

 

 

 

 

 

 

(QT 3+ 2 8x+2( + 23x)6x
6x1 0x1 3x16x2 1 6x16x2 6X10x2 1 1
62g1 0F1 62F1 a3 1 a3 1 (Fl 6F12

+ 2 . + ( '1' ——§_ ) ( )
3x 6x Bx 2 "(J—x a?
1 2 1 0x1 6x16x2 6x2 1 1
. 2 63
32g16F162F1 321 a216F1 61F1 01 Fl
+ 2 + (37%; + g ) + 5%; —
8x3 3X1 0:512 1 2 (9x3 3X1 (9x? 0x?
_a3§1+3 8381 0F1+963g1 (6F1)2+.6;38_1_(g§3)3
— 6x ax2ax 3X1 ”ax‘ 2 3X1 3 1
1 1 2 13x2 ax2
1 63F1
+ 3%; T
6‘1
8F 2 2
. 1 g -1 a
Slnce — - and
6i1 ( 2) 1
03F 2 2 2 2 2
1 a —1 a3 a3 3i —1 3
———-(£—) [—§-+—-5—(-3( ) 3,55)
(2613 2 0x1 6x?8x2 332 1
632 32—13122 032953—33
+2x—Lag) (31,—) -—i( 1321
2 3 2x ’
0x16x2 2 1 6x2 2

we have

63F

2 g _
—3'ax1 (111) - A2(gavgavg) _ 0'

 

 

FIGURES

Figure 1

Figure 2

110

++

+g+

+3—

"+.—-(-
Q
-—>-~<— A

 

Figure 3 (a)

(b) Codimension 0:

(c) Codimension 1:

0‘

<2)
93
\ \%
95

8 Ggégz:: 10 (jééié/{/fl

 

 

 

 

 

(d) Codimension 2:

Figure 3 .

/C/@
\l \1/

 

 

Figure 4

112

 

 

 

 

 

 

 

 

)M
A
‘f’itw .412)
‘(t )
Q) 9U13U2
11=u1>0 :- saddle 7. L
i
i
I
S
/,.J.P(t903112)
/ 95(t309U2)
u1=0 ’ pitchfor /"/“" ~\
ll : \x L
l \
/ I \
/ 1 \
I, : \

    

 

 

YégfsstZ) //

Y§(t3U13U2)

5(t9'69 )
(F “2 g g
u =-62>0 ‘* >: < :Xi/{/

' 1

 

 

 

f

 

 

P('59U ) O = (090) P(59U2)
saddlg node Saddl 6
Figure 5

113

 

y

1 1
/

Efl
\

(a) degenerate saddle (b) degenerate focus

/ .

 

/

 

 

 

Figure 6

 

“"‘1 /// ‘1 "——

 

 

 

 

 

(a) )The parameter space (b) The trace of the
e=(el,cz,c3) on S bifurcation diagram
on S (e “I <0)
Figure 7

114

 

 

 

‘ WM‘ i/‘1 i”/\/
w) >® >@/

>Q< .61 M

Figure 8: Phase portraits of codimension 0

115

 

1 and 9 2
Pitchfork M heteroc] inic ‘\//@\\ /
z”'*"\\ ‘//\‘\\\\‘_’////‘/\\

4

\T/.\/ \m/
/®Q/\\ /'®\

5 6
heteroci iniC\./_@—;\o/ semistable \ 0%
)5 v X N \'\

7 \W 8 \/\J
”/Q/x @9/

 

Figure 8: Phase portraits of codimension 1 (c)
b1 and b2: d:
M \,/a\’\/
fo:::&::::5\ //\k\\\£;::////”h\
degenerate saddle of codim 2 heterociinic - H0pf
C2: h2:
\ C; / \W
Wang/AK @\
heterociinic loop of codim 3 Hopf of codim 2 (d)

Figure 8: Phase portraits of codimension 2

116

 

\/«j—\/ A = mm

B = (0,0)
fl\ / c=<m>

 

 

’/’_‘\\“-______;?_,.//”’fl—_\‘ Figure 9

 

Level curves of (3.3.4)

 

y
45
L-\/ \/ WU
sflé\/ x _séfix
é/st "52m A/\"E B wb/xg
(a) PAS(wb)=w- (‘.r/‘r~(-u)#g}) (b) P;1(w5)=wb(wuo(-U)={}).
S
Figure 10
y
’P

 

f_’///,zér’_“‘\ Figure 11

117

 

 

 

 

> (1/4.-1/5)

 

l\

 

Figure 12: Phase portrait of (3.3.21) in
. (b,P)-plane

 

    
 

T
/\1
IV
n 5% )
e h = 0,0
3 > To
v
III I
d-(0,-3/7)
. 'EZ=(1/7.-8/7)
---------- - - -'“'r +8/7=o
II 1

 

Figure 13: The bifurcation diagram of equation (3.3.2)
in (to,rl)-plane for small s>0

118

Illll

o 0

Nice: 3 3:. s E 3:. e E 2:3: 3
s 3 33:25.0: E

_HHoAHP.opv Auv >HuAHP.o

 

 

 

 

O.

119

 

 

------"

on~\m+~u Loo .oAN\m+HP goev Aav<ua m=__ ¢;m_atum as»
vcm Aavaum m>g=o mg» $o m:o_u_moa m>wum—mg ash "ea usamwm

 

mm zo_an mmaAHP.OPV on
@215 E Nears: E
m 322 €223 E: 3

mm a>onm emafiso.oav A_v

m 22:. were: E

mire: E

 

 

Axv III/III:

A5 ltlllllllllile

:v $F§€Ill. llllll

on -.u..

:vI/Iz

A: ll 3:&

no / \

\iiilx \\ 85V

Lt lliil.

 

“I|

“- ‘

 

120

 

 

(D)

\\

E
/ E1

Figure 15

 

 

 

 

 

 

Figure 16

 

----\ E

 

Figure 17

121

 

 

 

 

Figure 18

122

 

 

 

 

 

II, IV (2,2)

1 (4,1)

Figure 19

123

 

 

y ' y

K

, X

/
\

o :

 

 

III (2,0)
I :X

F“ @/ x
\j /\

Pv
o

 

 

VI (0,1)

@Y-

J

o

 

V

 

r—\
VIII, X (0,2) IX (0,0)
(m,n) m = no. of inner limit cycles.0,2,4
n = no. of outer limit cycles,0,1,2,3
Figure 19: Phase portraits of codimension O (b)

124

 

 

 

 

 

 

 

 

 

 

 

Figure 19: Phase portraits of codimension 1

 

 

 

 

 

 

 

 

 

 

 

125

 

   

h
"2%
Q/

 

Takens-Carr

 

Hopf-semistable

 

Homoclinic-semistable

Figure 19:

a 1
fl

_/

 

H0pf of order 2

 

Homoclinic of order 2

-
I“ ~

 

 

Takens-Carr

 

Hopf-Homoclinic

 

Triple limit cycle

Pitchfork-semistable

Phase portraits of codimension 2

126

 

 

 

(b Phase portraits of codimension O

(I'lllllllllllll’l/ III

)Phase portraits of codimension 1

Q

b k

”0

kb.

127

fl,-\\
I \
\ ' '

\ /

kb

 

 

For ble’ b2b3 and b1b3, See Figure 19 (c)

(d) phase portraits of codimension 2

///1E§§:> Hopf of order 2

For b1, b2 and b3, see Figure 19 (d)

Figure 20

128

 

A
p...

t—IOI
U
CO“
V

OED)
II II II
a—-.av-~

 

 

 

Figure 21: Level curves of (3-4-2)
X

\ fl. 7 7 X
\AJ/ \3 C wbu ]

 

 

 

k\__'/
(a) H(wbu) < H(wbs)
Y
e __g Ybsfi: x
A B C wbu
w
(b) H(Wbs) < H(Wbu)
Figure 22

129

 

 

 

   

 

 

”F
A A
H/4,1)
\
x L(-i/4,1/2)(0,1/fi) we ‘ b“ > b
\1>\(-1/4,1/5)
‘ \ o

 

Figure 23: The raph of P(b)
(mm = puibu) = 0)

130

 

 

 

 

(a) P'(b) + 8/5+

 

 

 

I
..... _--.-_-__--__------__--------- y = 8/5
y=P'(b)
2— %> b
b*
(b) P'(b) + 8/5’
9(
y = 8/5
2% b

 

 

 

(c) P'(b) a 8/5 for b >> 1

Figure 24

131

 

 

 

 

 

Figure 25

 

 

Figure 26

132

 

 

 

 

 

VI
'5
VII

 

III IIIIIIIIIII.

11111111:

(0,4/5)”\

PllIl

__ b**

b*

-1/4

_a

(a)

 

 

133

 

   

(“l/4:1)I

 

 

 

   

 

. V
-<I-l"r : VI
. VII __
; : b]: 2
l 5 IX
-1/4 0 11* 3“ >b

 

1 (b) infinite limit
I aS b + b*(Bi=Bz)

 

134

 

 

 

 

 

 

 

1- 113--17-;

 

3-1/4

135

\

 

 

 

 

136

IIIIIIIIIIIIIIIIIII

 

,

5* \ 'w

0

-1/4

II

III

VI

 

1

(-1/4,1)

 

(e)

137

VI

 

b'**

r

5*

-1/4

 

('1/491) ‘

 

138

 

b**

6*

 

//

-1/4

 

1-1/4.1P

 

139

 

 

 

 

 

 

>b

 

 

O
01--------
* 1
1
1
I
1
0".
:1-
a.

 

-1/4

..-___..- _.._._ 4’____

 

Figure 27: The relative positions of the curve P=P(b)

and the straight line P=A(b)
(See Figure 26 for a to‘fi and I to X)

140

 

 

 

 

 

A
//’”’fi fi
f =
H
a
13 ' i 1"
b1
”r-
(a) rl=1 (3“)

 

 

     

 

 

 

Figure 28
+
m’ n S
,/ \ QC*
3 \ I \\ 8x.
1/ \ \ \\‘ \
x. ’3 \' \"\ \ x
\\\f _ \
\‘1 ‘
\ Es (0)05 \\‘ 3
1 1 *\:

E!

Moog;
6,”
7x3

EénS

Figure 29

141

 

 

as
Qfl 3-

 

 

 

 

 

V
w
Ab w__> x
\ b
Y\\\\\\\~—_
PAWD) = WE, Y=Y(b9>‘s)
Figure 31
T
131
F
........ Wf---_---- =
T1 8/7
> To
1 ‘k'

 

Figure 32

0|

142

p blk A,k kbz II kb3
‘3 ' 111

/// 1/\/>1>=1>(b)
/ 1
’7
/ .

Figure 33

The relative positions between the
curve P=P(b) and the straight line
P=A(b) (See also Figure 20)

\\

‘

 

 

 

 

 

E:3
2 -
€3+4el-0
SN
\ 81
P
Figure 34

l
I
1
143

 

 

 

 

 

®.~m\
W
X / II\/x . 3V F’s/P) 3
3; /@@UI.@ g/@ A .@\W\
(“JV
AW a: 3% /
% 2ch§ 1.
W.®JJ§ 1r v
e . .a \ Q

 

 

 

 

144

 

   

 

 

 

 

 

 

 

 

=0

3

Figure 35 (b) e

145

 

 

 

PF

 

 

 

Figure 35 (c) 53 < O

146

f
@@ @
° -éf.>§g3/
/\"”

C
<9

 

 

 

 

 

 

Ho
SN
Figure 36
C
Ho
Figure 38
H
Hf I
P 1
P
+ -
(a) (5) (b) (5)
Figure 40

147

 

5N

 

Figure 37

0

 

031

P

 

Figure 39

 

Figure 41

- .3 Asymptotic to a. Saddle—Saddle Equilibrium. Preprint.

10.

11.
12.

13.

'Chow, S.-N. and Hale, J.K., Methods of Bifurcation Theory.
Springer—Verla‘g, New York, Heidelberg, Berlin, 1982.

BIBLIOGRAPHY

Andronov, A.A., Leontovich, E.A.,‘Gordon, 1.1. and Maier, A.G., Theory
of Bifurcations of Dynamical Systems on a Plane, Wiley, 1973. '

Bogdanov, R.I., Versal Deformation of a' Singular Point on the Plane in
the Case of Zero Eigenvalues, Seminar Petrovskii, (1976), (in Russian),
Sel. Math. Sov. 1, (1981), pp. 388—421 (in English).

Carr, J ., Applications of Centre Manifold Theory, Applied Math Sciences,
35, Springer—Verlag, New York, 1981.

Chow, S.—N. and Lin, X.-B., Bifurcations ‘of Homoclinic Orbit

 

Dangelmayr, G., Armbruster, D., and Neveling, N., A Codimension
Three Bifurcation for the Laser with Saturable Absorber. Z. Phys.
B - Codensed Matter 59, 365—370, 1985:

Dumortier, F., Roussarie, R. and Sotomayor, J., Generic 3—parameter
Families of Vector Fields on. the Plane, Unfolding a Singularity with
Nilpotent Linear Part. The Cusp Case of Codimension 3. Ergod.
Theory and Dynam. Systems 7, pp. 375—413, 1987.

, Generic 3—parameter Families of Planar Vector Fields, Unfolding
of Saddle, Focus and Elliptic Singularities with Nilpotent Linear Part.
Part 1: Presentation of the Results and Normalization. Preprint.

 

‘ Golubitsky, M. and Schaeffer, D.G., Singularities and Groups in

Bifurcation Theory, Vol 1. Applied Math. Sciences, 51, Springer-Verlag,
New York, 1985. ‘ .

Guckenheimer, J. and Holmes, P.,‘ Nonlinear Oscillations, Dynamical
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