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This is to certify that the

dissertation entitled
ON THE EXISTENCE AND NONEXISTENCE OF PERIODIC

ORBITS IN A NEIGHBOURHOOD OF HOMOCLINIC
AND HETEROCLINIC ORBITS

presented by

Mohammad Riazi-Kermani

has been accepted towards fulfillment
of the requirements for

PH. D. degree in Mathematics

 

{Laxk LLJ

Major professor

 

Date ARM

MSU is an Affirmative Aclt'om Equal Opponunily Instilun'on 0- 12771

 

 

 

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ON THE EXISTENCE AND NONEXISTENCE OF PERIODIC
ORBITS IN A NEIGHBOURHOOD OF HOMOCLINIC
AND HETEROCLINIC ORBITS

BY

Mohammad Riazi-Kermani

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1984

ABSTRACT
ON THE EXISTENCE AND NONEXISTENCE OF PERIODIC

ORBITS IN A NEIGHBOURHOOD OF HOMOCLINIC
AND HETEROCLINIC ORBITS

BY

Mohammad Riazi-Kermani

We show that the eigenvalues of the linearized
system determines the existence or nonexistence of
periodic orbits in a neighbourhood of a homoclinic

3
for autonomous

or heteroclinic orbit in R2 and IR
systems. We also show the existence of periodic orbits
in the modified logistic equation corresponding to the
central difference scheme by the method of averaging.

The existence of a heteroclinic orbit is also proved

in the modified logistic equation.

TO
MY PARENTS
AND

MY WIFE

ii

ACKNOWLEDGMENTS

The writer is sincerely grateful to Professor

S.N. Chow for his patient counsel and guidance in the
preparation of this thesis. He wishes to offer special
thanks to Professors Lee M. Sonneborn. Clifford E. Weil,
Richard E. Phillips, Habib Salehi, Daniel A. Moran and
John Mallet-Paret for their teachings and encouragements
during his career as a graduate student. He is also
deeply appreciative of the moral support given him by

Professor Kyung Whan Kwun.

Finally, he also likes to thank Mrs. Cindy Smith,
Miss Tammy Hatfield, and Mrs. Cathy Friess for typing

this manuscript.

iii

TABLE OF CONTENTS

Chapter

INTRODUCTION . . . . . . . . . . . . . . . .

2

l. FLOWS ON HR . . . . . . . . . . . . . .
1.1 Neighbourhoods of homoclinic orbits

. 2
1n R O O O O I O O O O O O I O O

1.2 Neighbourhoods of heteroclinic orbits

inIRZ..............

2.FLOWSONJR3..............

2.1 Neighbourhoods of homoclinic orbits

in, IR3 . . . . . . . . . . . . . .

2.2 Neighbourhoods of heteroclinic orbits

in. IR3 . . . . . . . . . . . . . .

3. A SINGULAR PERTURBATION PROBLEM . . . . .

3.1 The existence of a periodic orbit in
modified logistic equation . . . .

3.2 The averaged system . . . . . . . .

4. ON THE EXISTENCE OF A HETEROCLINIC ORBIT

BIBLIOGRAPHY . . . . . . . . . . . . . . . .

iv

I NTRODUC TI ON

It is well known that the numerical approximation
of the solution of an initial value problem for an
ordinary differential equation requires many precautions
to be successful. One way of investigating the adequacy
of numerical schemes is the introduction of the concept

of modified equation. The logistic equation

six = _ =
dt y(l y) y(O) yO (1)
has been studied by M. Yamaguti and S. Ushiki. In

particular. these authors studied the central difference

scheme

= u (I‘ll) n: 112'00
n n
(2)

w1th u0 = y0 and 111 = yO-thyo(l‘-yo)

The results exhibit the phenomenon of the so called
"ghost" solution.

The iterative scheme (2) may be recreated again

in continuous form by inserting the taylor expansion

2
untl = u((nil)h) = u(nh) ihu'(nh) +g—l- u'(nh)

3
2 h u’”(nh) +0014)

37

If we put again u(1)(nh) = u(l)(t), the equation (2)

2
becomes %;'um(t)-ru'(t) = u(l‘-u)-+O(h3) with

u(O) = YO’ U'(O) = yo(l'-yo). u'(O) = (l -2yo)(l -yo)y0.

We call

ezu’”(t) +u'(t) = u(1 -u) (3)

the modified logistic equation.

The first two chapters of this manuscript are devoted
to the existence or nonexistence theorems in neighbourhoods

of homoclinic and heteroclinic orbits in m2 and 1R3 .

The existence of periodic orbits in (3) is shown in
Chapter 3 and Chapter 4 is devoted to the existence of

a heteroclinic orbit in (3).

Some graphs made by computer are also included.

CHAPTER 1
The existence and nonexistence of periodic orbits in
a small neighbourhood of homoclinic and

heteroclinic orbits in R2

Flows on R2

 

1.1. Neighbourhoods of homoclinic orbits in P?

Consider a system of two differential equations

where f is c2 and f(0) = 0. Assume that the

eigenvalues of the matrix A = f’fO) are -l and c where

c > O, l > O and c -x # 0.

Suppose there exists a homoclinic orbit issuing from

O and returning to it as shown in Figure l.

By the Stable Manifold Theorem, in a sufficiently small
neighbourhood of 0 there exists a stable manifold and
an unstable manifold correSponding to the negative and

positive eigenvalue of A respectively.

Since periodic orbits are topological invariants of the
space, following the ideas of the Grobman-Hartman theorem we
may assume that the system (1) is linear in a neighbourhood
V of the hyperbolic point and also we may assume the
stable and unstable manifolds are coordinate axes in that
neighbourhood.

The linearized system then has the form:

x = -XX
X > 0, c > 0 c-l # 0 . (2)
§=Gy

Without loss of generality we can assume that the
points (0.1) and (1,0) are in V. The lines x = l and
y = l are transversal to the solution curves of (2)

passing nearby (1,0) and (0,1). (See Figure 2)

 

 

Figure 2

Let us define: S0 = {(l,y) :0.g y‘g l} and
S1 = {(x,l) :0.g x‘g 1}. Starting at t = 0, from
(l,yO) 6 SO and following the trajectory, we get:
x(t) = e—)‘t
(3)
_ ct
y(t) — yoe
Therefore at t =‘%% zn (yo) the trajectory hits Sl
-lti%)zn(yo) ‘%
at the point x = e = y0 = y3, where

Y =-% s 1. We define the map T :S * S by

.L O O l
_ a
.l
Since v > 0, you tends to 0 as yO tends to 0 .
Therefore :Lf we define To(l,0) = (0,1), then T0 is
continuous on SO. Since the point B(O,l) is mapped by

the homoclinic orbit on the point All,0), from theorems
on the continuous dependence of solutions on initial value

and from the transversality of S the correSpondence map,

10
T1 is defined and is as smooth as f in some neighbourhood
r
S].1 :81 fl {(x,y) :x2+ (y-l)2 g ri] onto
r
SOZ==SO 0 ((x,y) :(x--l)2+y2 g r3}.
Consider the composite function TlTO :SS 4 SS and

let T = TITO' Then T(O) = 0.

Suppose Tllx) = le4-Bzx24-h.o.t. Then T(y) =

2Y

- 2 _ y
T1T0(y) - BlTo(y)-+Bz(To(y)) -+h.o.t. — 81y -+62y -+h.o.t.

p

O

1 _ Y-l 2y-l .
dy(T(y)) - Blyy 4-2y82y +-h.o.t. Bl is the

derivative of the correspondence function. Therefore Bl # 0.

Since y # l, T’(y) tends to zero in case
y > 1 and goes to infinity in case y < 1. Therefore
T cannot have any fixed point near zero and different

from zero. (See Figure 3) 117)

  

 

 

Figure 3-a: (O < Y < l)
x

‘1’")

Figure 3-b: (y > 1)

 

 

The following theorem is the direct result of the above

discussions.

Theorem 1.1. If the sum of the eigenvalues of the
linearized system at the hyperbolic point is different
from zero, then there exists a neighbourhood of the

homoclinic orbit without any periodic orbit.

1.2. Neighbourhoods of heteroclinic orbits in 3&2

Systems of differential equation in R2 with hetero-

clinic orbits joining two critical points appear naturally

in science. For example the pendulum equation
x+sinx= 0
has its solutions on the level curves of
G(x,y) =-% y2-cos x = K

where y = x. Therefore in the (x,y) plane for k = 1 the
trajectories are hetroclinic orbits joining the critical
points l-v,0) and (v.0). In the above example the
eigenvalues of the linearized system at pl(-w,0) and
p2(w,0) are $1, and every trajectory starting at a

point inside the region bounded by the hetroclinic orbit

is a periodic orbit. The existence of periodic orbits in

a neighbourhood of the hetroclinic orbit depends on the

ratio of the products of positive and negative eigenvalues,

namely:

Theorem 1.2. Let

 

)2: f(X) (1)

be a 2-dimensional ordinary differential equation with
P1 and P2 hyperbolic critical points, and let xi < 0,

Pi > 0 be the eigenvalues of the linearized system at. Pi’

X k
If y = 1 2 # 1, then there exists a neighbourhood

9192

of the heteroclinic orbit without any periodic orbits.

 

Proof: We use the method of point transformation,
assuming that the system is linear in.a small neighbourhood

of Pl'

      

__JTq’/’

Figure 4

Without loss of generality we assume that P1 = 0 and

the stable and unstable manifolds are the coordinate axes.
Let s1 = {I-1,y): Ogyg 1],

82 = {(x,l) : -l'g X‘s 0} and T1 :81 4 82 be defined

as follows.

Since S1 and 82 are transversal to the trajectories
of (1), starting at {-l,yo) the trajeptory will hit
1
S at a point (x ,l), where x = y p1 . Define
2 O O 0
X1
BI

Tlt-l.y) = (x.l) = (Y

 

5;

A~

 

S

\

Figure 5

Similarly we define the map T3 :83 * S4 in a neighbour-
hood of 92. X

 

 

 

 

Figure 6

Applying the Hartaman-Grobman theorem in a neighbourhood
of P2 and parametrizing the stable and unstable manifold
such that P2 = (0,0), and the stable manifold is

yl-axis and the unstable manifold the xl-axis, in that

neighbourhood of P2.

If 83 = {(xl,l) 30.3 Xl-g 1}, and if
is

84 = {(l,yi) :013 yi'g 1}, then the map T3 :83 + S4

10

defined and a similar computation shows that
l

_2
P2)
T3(xl,l) — (l,xl .
Since the branch ofheteroclinicorbit joining P1
to P2 takes the 0 of S2 on the zero point on S3,
S2 and 83 are transversal to the trajectories, the

correspondence map

is defined for a small neighbourhood of 0, and it is

analytic. Therefore we can assume

2
Tzlx) — clx4-c2x 4-h.o.t. (al # 0)
Similarly we define the map T4 :84 4 S1 to be the
correSpondence map and assume
2
T4(yl) = Bly1+-Bzy24-h.o.t. . (Bl # 0)

The map T = T4 0T3 oTzcaTl :S1 4 81 is defined and

continuous on a small neighbourhood of 0 on that side
of S1 which is the interior of the region bounded by
the hetroclinic orbits.

We are interested in the right-hand derivative of

T at 0. Applying the chain rule. we get

oT’wr’mr’

I_I
T‘T4 3 2 1

11

Therefore near zero, T’(y) z ay where Yl — _P_
l
_)‘2 ,
and v2 =-—B; . Since we assume y = YlYZ ; l, we get

a if v < 1

In either case there is no fixed point of T’y) near y = 0,
there is no periodic orbit in a neighbourhood of the hetro-

clinic orbit.

In case y > 1, the hetroclinic orbit is attractive
from inside. Therefore starting at a point near the hetro-
clinic orbit the trajectory will intersect a transversal
to the hetroclinic orbit infinitely many times making a

sequence which is convergent to the point of intersection.

When the distance between two consecutive terms of the
above Cauchy-sequence is less than the round off error of
the computer, then solving the differential equation by
computational methods results in the so—called ghost

solutions. The one dimensional picture looks like the

\ / ‘\ .
\ / \_

following:

 

 

 

 

Figure 7

12

The flat parts of the graph in Figure 7 corresponds to
the long time which the trajectory remains in the small
neighbourhoods of the critical points and the steep parts
correspond to the short time which the trajectory takes to

pass nearby the remainder of the heteroclinic orbits.

CHAPTER 2
The existence and nonexistence of periodic orbits in
a small neighbourhood of homoclinic and

hetroclinic orbits in R3

Flows on R3

 

In the case of three-dimensional systems of
autonomous differential equations it has been shown by
P.L. Silnikov that if the eigenvalues of the linearized
system at the hyperbolic point satisfy y > -k > 0 where
eigenvalues are: y > 0 and k:tiw, the existence of
a homoclinic orbit issuing from O and returning to it
results in the existence of a denumerable set of periodic

orbits in any neighbourhood of the homoclinic orbit.

We use the Hartman-Grobman theorem to give a more
geometrical proof of Silnikov's theorem, showing also the
existence of a neighbourhood of the homoclinic orbit
without any periodic orbit under the condition 0 < Y < -l
Where the eigenvalues of the linearized system are Y and
l:tiw.

We suppose the system is linear in a small neighbour-
hood of the hyperbolic point 0, and assume the box

[-l,l]3 is contained in that neighbourhood.

14

The motion is governed by the system:

 

i = lX-wy

jy=wx+xy (1)
z: 2

L Y

. . . 2 2
I =
Starting at a pOint 'XO'YO’ZO) on the cylinder xO-t-yO l

the solution curve would be according to (l)

x(t) = ext-'(x cos wt-y sinmt)
O 0
it .
y(t) = e (xOSinwt-i-yocosmt) (2)
\It
( = ’
z.t) e 20

The point of intersection of this trajectory with the

plane 2 = l is found easily from (2) as follows:
z‘t) = l
t—71- in (31—)
’ o
‘ l l l ->.
T£n(-—) - -
ekt = e{ 20 = 01;)Y = 20v
20
‘L. '
r v l l . l -
= .'\— — - I— I
x 20 (xocosuY £n(zo)) yOSinhan 20)))
fl '
_ Y - . l .1. .1. ‘—
y- 20 (XOSanMY jznlzo))+yocosm(V zn(zo)))
z = l

 

15

'L
V

This point is on a circle of radius 2 therefore

0 :
circles on the cylinder x2+y2 = l, 0 < z g_l will be

mapped onto circles on the punctured disc 2 = l,

0 < x2+y2 g 1. We will call this map T

O'

 

 

 

 

   

Fi re 8

Under TO each point of the cylinder will trace a funnel

which is invariant for the system (2) in the time interval
necessary to reach the plane 2 = 1.

Since 20 = e-Yt, y > 0 where t is the time

necessary for XX to hit 2 = 1, as XX

0 = (XO'YO'ZO) 0
gets closer to the stable manifold, TO(xO,yO,zO) gets
closer to the unstable manifold in longer time. This

correSpondence is illustrated in Figure 9.

l6

 

 

 

Figure 9

Without loss of generality we may assume that the
homoclinic orbit passes through points (1,0,0) and
(0,0,1). Therefore the point (0,0,1) maps to (1,0,0)

in a finite time t.

By the continuous dependence of solutions on the
initial value, there is a neighbourhood of (0,0,1) which
will be mapped onto a neighbourhood of (1,0,0) by the

trajectories of the system x = f(x).

Since the cylinder x2+y2 = 1, [2| g l is transversal

to the solution curves of ’2), we can define the correSpond-

ence map from the points on = l, nearby (0,0,1)
2

z
and the points of cylinder x24~y = l nearby (1,0,0).
We call this map T1 and by the Implicit Function

Theorem T1 is as smooth as the function f.

We define a new coordinate system on the cylinder

as follows:

For any point (x,y,z) near (1,0,0), we define

17

. -l
s = Sln y
u = 2
Then 5 = cos-1x because x2+y2 = 1.
We define T1 to be the correspondence map from
points on 2 = 1, near (0,0,1), and define a new

coordinate system by Ax = x, Ay = y. The correspondence

map Tl can be expressed as: T1 = (fl,f2), where
fl(Ax,Ay) = ole4-o25y+-GBAXAy4-h.o.t.
f2(Ax,Ay) = Ble4-fisz4-fi3AxAy4-h.o.t.

The composite map T = TITO is defined on a small
neighbourhood of 0, and maps the point ’s,z) to the

point (fliAX.Ay). f2(AX.AY)) where:

 

(Ax = 71(T0(cos 5, sins, 2))
fl
= 2 Y (cos scos fi- Ln(%) -sins sin-‘5 zn(}l+))
Ay = 72(To(cos 5, sins, 2))
'i
L = z V (cos ssin$ Ln(%) +sinscos =3 zn(—Z:L))

 

Therefore.

18

 

 

-2~.
= Y LU. J; _ ° ' 21 (l
(112 (cosscos Y zmz) SinSSin Y in 2))
‘2:
Y - L”. .1. - i (.1.
4- c122 (COSSSin Y zn’z) +SlnSCOS Y in z))
—ZX
+ 2 Y hlfs,z)
I
f2 AX.Ay)
-l
= Y .’ fl (.1:- _ ' ' .09. (.1;
£312 cosscos Y in z) SlnSSln Y Ln 2))
‘1.
Y ' LE Ll ° LE Al
+ [322 (COSSSin Y in z)) +SlnSCOS Y tn 2)))
-21
+ 2 Y h (5,2)
2

where hl(s,z) and h2(s,z) are bounded functions

near (0,0).

We are looking for the fixed points of the map T

near (0,0) that is the solutions of the system

l9

’ -i
Y

z [(o coss+o sins)cos(-L11

l
l 2 Y £n(E)) -2x

- - 23. .1 v _
< +(o2coss-o151ns) Sln(Y Ln(z))]+z h1(s,z)_
fl

Y - a (A
z [(51 coss+13231ns)cos(Y in z)) _2)\

+ (52 coss-Blsins) sin(% zn(%))]+z Y h2(X.Z) =

 

 

or simply:

(

 

13* -
1(s,z) — s
( .
f2(s,z) — 2
K
Let gl(s,z) = f:(s,z) -s: we show that the equation
gl(s,z) = 0 could be solved for s. as a function of 2

near 2 0, for positive values of 2.

Let a > 0 be small enough that the function

T = TIT is defined for (5,2) 6 [-v.w]><[0.€1o

 

 

arir
N

K“

\\_._4r’

 

Figure 10

20

Since TIT ([-v,wj x(zi) is homeomorphic to

O
TO([-v,vj x(z}) which is a circle, identifying (-v,z)

and (v,z) we get

T(-v,z) - T(*,z)
_ . 'k *
which implies fl(-v,z) = fl(v,z).
* ‘k
If fl(—7,z) = -v or f1(7,z) = r, then we define
s(z) = -v or s(z) = +v, respectively- Otherwise gl(-v,z)

and gltw,z) have different signs. In this case by the
intermediate value theorem glls,z) = 0 for some

5 e {-7T37T).

Now we show the uniqueness of 5. Suppose f:(sl,z) =
*

s1 and fl(52’2) = 52, Then gl(sl,z) = 0 and
91(52'2) = 0, Therefore by the mean value theorem
_—_§§———'= 0 for some 51 < s < 52. ut
* -
~ (3 z) 5f1(3'2) A
233§4——-= -——S§———-l = 2 Y [B(s,z)]-—l where B(s,z) is

bounded in s and 2. Therefore for 2 small enough,
‘1.

|z Y [B(s,z)]1 <-%. Hence 5 = 5(2) for some
*
121 g y. Then substituting in f2(s,z) = 2, we get
‘2;

Y ( ‘ L”. 1.];
z [Bl-coss+13251ns)cos(Y zn.z))

+ (Bzcoss-Blsins) sin %‘-(Ln(%))]
—2x ‘
+-z Y h2(s,z) = z . (4)

21

Substituting 2 = e-Yt in (4) we get
it .
e [(Blcoss+stins)cos mt
+ (132 coss -Slsins)sin wt] +e2)‘t h2(s,t) = e-Yt.
We also need the continuity of the function 5 = 512) in
t. since 2 = e-xt is continuous in t, it suffices to
show that s(z) is continuous in 2. '

Let 2n 4 2. we show that s(2n) 4 5(2) by proving

that the only limit point of s(zn) is 3(2).

Suppose s(2 ) = s 4 G. Since by definition of
n n
k k
s , f(s .z ) = s I and since f is continuous,
“k “k “k “k
2n 4 2, sn 4 q = fls ,2n ) 4 f(q,z).
k k “k k
Therefore f(q,2) = o. Definition and uniqueness of s/z)

imply a = 5(2).

Since slzn) E [-n,n], it has at least one limit
point by Boltzano-Weirestrass theorem. The above argument

shows

 

 

Let f(t) = ext[(Blcoss+stins)cos mt+

(52 cos 3 - Bl sins)sin wt+ ext

elt

h2(s,t)]. Since 1 < 0 ,

h2(s,t) 4 0 exponentially as t 4 m. Therefore f(t)

is an oscillating function, and

 

22

k ekt Rt

2 g \f’tH g kle

for kl and k2 positive numbers.

In case y x -X the curves f(t) and e->‘t have

/

infinitely many intersection points as t 4 m, and in

case y < -l. the function eYt dominates the functions

it it
kle and kze

*
intersection points for t‘z t .

eventually and there will be no

 

 

Case 1. Infinitely many intersections

(Y > ->.)

Figure ll-a

 

\ / 31' VII ‘3./ \i/ ‘L,

 

 

Case 2. No intersection (y < -1)

Figure ll-b

23

The geometry of the transformations TO and T1

and T = TITO is illustrated in Figure 12.

 

 

   

 

Figure 12

 

In case 1, eventually the circles will be mapped so
that they hit the apprOpriate levels to create periodic orbits,
and in case 2, all circles are eventually mapped lower
than the appropriate level and therefore there is no
periodic orbit in a small neighbourhood of homoclinic
orbit. We can summarize the above results in the following

theorem.

Theorem 2.1. Under the above assumptions there are

 

infinitely many periodic solutions in any neighbourhood of
the homoclinic orbit in case y > -l and there is a
neighbourhood of the homoclinic orbit with no periodic

orbit in case v ( -l.

\

24

The next section will be devoted to the neighbourhoods
of hetroclinic orbits in IR; .
2.2. On the existence of a neighbourhood of a

cycle consisting of two hetroclinic orbits without any
3

 

periodic orbit in It

 

Consider a system of 3 differential equations
x = f(x) (1)

and assume that (1) has two hyperbolic critical points,

p1 and p2 with eigenvalues of the linearized system

xiztmi and Yi respectively where YlY2 < 0 and

1112 < O and also kl < 0, Yiki < 0 for i = 1,2.

In this case the well-known theorem on the stable and

unstable manifolds indicates the existence of a two dimensional
stable manifold, and a one-dimensional unstable manifold at

one point and a 2—dimensional unstable manifold along

with a one-dimensional stable manifold at the other point.

We also assume there exists a hetroclinic orbit
joining the two critical points from pl to p2 and

another one joining p2 to pl making a cycle.

The following illustration shows the complete

situation.

 

Figure 13

We claim the following:

Theorem 2.2. If le2 < Y2xl' then there exists a

neighbourhood of the cycle without any periodic orbit.

2329;: We use the method of point transformation,
assuming the system to be linear in a small neighbourhood
of each critical point. Define the maps T1' T2, T3, T4,
consider the composite map T4 0T3 °T2 0T1 and look

for its fixed points in a small neighbourhood of the cycle.

a) Construction of the map Tl:

Assume the system to be linear in a small neighbourhood

of p1' and also assume that the motion is governed by

the following system in some neighbourhood N of pl = 0:

26

< Y = lej-lly (2)

 

(.2 = le'

We also suppose that the cube [--l,l]3 is contained in N.

. . . 2 2 _
Starting at a pOint (x0,y0,zo) on the Circle xo+yO — l

the solution curve would be,according to (2)

 

r Xlt
x(t) = e (xocos wlt-yOSlnmlt)

Xlt

< y(t) = e (XOSlnmlt+ yO cos wlt) (3)
v t
'l

t =
kz( ) e 20

The point of intersection of this trajectory with the plane

2 = 1 is found easily from (2) as follows:
2(t) = l =
Y1 zo
kiwi, 11.1. '11.
x1“ Y1 2o 1 Y1 Y1
e - e — l—) 20
‘2o
‘Ll.
r = Y1 ._1_ i. _ - i. ,
x 20 (x0 cos ”(Y1 in (20)) YOSlnw(V’l in 20”)
‘11.
< .. *1). $1.44.... (in. (i m ))) (4)
2 = l

 

27

‘L1
This point is on the circle of radius 20 1. Therefore
circles on the cylinder x2+y2 = l, 0.3 z.g 1 will be

mapped onto circles on the punctured disc 2 = 1,

0 < x2+y2 g 1. This transformation is illustrated in

Figure 14 and we call it T

1'

 

T.

‘ —'-

 

 

 

 

Figure 14

b) Construction of the map T3:

We assume that the system is linear in a small
neighbourhood of P2, and since we have a one dimensional
stable manifold along with a two dimensional unstable
manifold, every trajectory starting at a point on the disk
1 i4—yi g 1 ‘will oscillate and hit the

2

cylinder xi4—y1 = 1, z g_l in a point (x

z = l, 0 < x

lO'YIO'le)
where the following relation could be easily verified

 

 

28

 

 

( ._2
x1 = zloY2 (x10 cosuu(-1— zn(—l—))
Y2 210
y sinu)( l znfz )V
< 10 2 y2 10 ’
‘12.
Y
2 . 1 l
y = z (x smwt- 2n( ))
l 10 10 {2 210
k + y COSUJ(—L'Ln(z 1)). (5)
10 2 y2 10
Therefore
".23.;
Y
2 2 _ 2
X1*‘Y1 ‘ 210
or
21
_ 2 2 2
21o ' (X14'Y1) -
SolVing the above system for (xlo,ylo), we get
( 1 1 . 1 1
x cos w(-— zn(-—-)) -y Slnw(— LIN—1)
1° Y2 210 10 Y2 210
'32
X
_ 2
“ X1210
< (6)
. l l l l
X Slnw(— zn( )) +y cosw(— zn(——))
10 Y2 21o 10 Y2 z10
‘32
_ 2 12
V ‘ y1 10

or

l 1
10 10 (x1 cos My; Lug-5))

 

)1)

+ yl sin w(71_ 1n<zl

i 2 10

4:2— (7)

"2
1o

1 1
(y cos r1;(-- Ln(—))
1 Y2 zo

y10= z

 

 

H)

x - xlsinuM3L-zn(zl
‘2 10

Therefore in fact the map T3 :(xl,yl) 4 (x10,y10,zlo)

with the following formula maps the punctured disk

_ 2 2 .
2l - l 0 < xl+yl g l to the cylinder 0 < 210 g l,
2 2 _ . .
xlO-i-ylO — l. The map T3 is not well-defined at
the point (0,0) but we define T3(0,0) = M2 where M2

is the point of intersection of the hetroclinic orbit with

the cylinder 2.

 

 

 

 

:12
2).
_ 2 2 2
Y ‘210 ‘ (X1+Y1)
3.—
. .. 2 _1- 1
x10 — zlo (xlc05112(Y2 zn(zlo))
. 1 1
< + Y Slnw (‘— zni— H)
l 2 2 210
_,,2 (a)
T
_ 2 _l_ ;
Y10 ' 210 ”1°05”sz “”210”
. 1
x - xlsmw2(:— 1n( )))

30

The map T is illustrated in Figure 15.

3

 

cylinder 1

k/ x

 

 

 

Figure 15

c) Construction of T2 and T4:

The map T2 is defined along the hetroclinic orbit
from the disc 2 = l to 21 = 1. It is defined by point
transformation by following the trajectory and since both

. 2 2 2 2
discs 2 = l, 0.3 x 4-y .g l and 21 = 1, O'g xli'Yl-S l

are transversal to the trajectories of the system, the map

T2 could be expressed as

x
ll

1 allx4-a12y4-h.o.t.
T - (9)

yl = a21x+ a22y+ h.o.t.

and the map T4 is defined similarly but from a

neighbourhood of M on cylinder (2) to a neighbourhood

l

of M2 on cylinder (1)

31

where x and y2 are new local coordinates of points

2

in a neighbourhood of M on cylinder (1) with

2
_ (
M 0,0).

To have a periodic orbit, we need a fixed point of
the composite map T = T4T3T2Tl, which means starting at

a point on a level 2 = 20 on cylinder 1, we have to come

back to the same level and the same point. We show that if

the product is greater than yle, then we can

YlXZ

never return to the starting level 2 = 2 which means

0!
no periodic orbit Could exist in a small neighbourhood

of the cycle.

From (4) it is clear that starting at the level

2 = 20 we end up at a point on a circle of radius
‘11. 1.22.1
Y Y
20 1 via Tl; that is, x2+y2 = z 1 From (9) in

the definition of T assuming the transversality of the

2'

hetroclinic in the sense that the matrix

is invertible, we get

2

2 2 2 2 2 2 2
Kl(x +y)gxl+yng2(X +y)

where (x,y) and (xl,yl) are in small neighbourhoods

of (0,0).

32

Therefore the composite map T transforms the

2T1
points of level 2 = 20 to the points on the circle of

radius Y1 where

 

 

'11 '11
VI '3 2 2
K120 .g rl'g K22O , K1+K2 > 0
T3 takes the points on the circle of radius r to the
—\{2
2 212
points of level 210 = (r ) . Therefore the composite
map T3T2Tl takes level 2 = 20 to the level 210 = r2
where
X1Y2 *1Y2
Y1*2 Y1‘2 2 2
L120 ‘; rz‘g L22O , L1+L2 > 0
b11 b12
Assuming that the matrix B == in the
b21 b22
definition of T is invertible,we get

4

2 2 2 2 2 2 2 2
Ml(XO4-yo) g X24'y2-S M2(XO4-yo) , M1+M2

\/
0

Therefore T4T3T2Tl takes the level 2 = 20 to p01nts

on circle with radius R, where

Y1‘2 Y1X2 2 2
N120 g R S N220 , Nl+ N2 > O ,

 

 

 

A Y
Since by our assumption Yllz > 1 and the equation
1 2
x112
Y1‘2
N2O = 20

does not have any nonzero solution in a small neighbourhood
of zero, the system cannot have any periodic orbit in

a small neighbourhood of the cycle.

CHAPTER 3
A singular perturbation problem on the
existence of periodic orbits for

€2u+u==u(l-u)

3.1. The differential equation:

azu+a= u(l-u) (1)

appears naturally as a modified equation for the logistic

equation u = u(l-u) solved by some numerical methods.

We show that even if the logistic equation does not
have any nontrivial,periodic solutions, the modified
equation (1) has nontrivial periodic orbits for e

sufficiently small.

The third order differential equation (1) is

equivalent to the 3-dimensional system:

 

f .
u = v
< v = w (2)
2.
evv=-v+uU-u)
K

Due to the rapid growth of w and v relative to u,
it is appropriate to change the variables and look for
the periodic solutions of the new system. The prOper

change of variable would be

34

 

U = u

V = ev
1 ‘W = e w
KT=%t

which changes the system (3) into

r

 

i1=v
(<7 = w
k1117 = -V+eU(l-U)
. U 0
or X = AX+—ef(X) where X = V , f(x) = 0
W U(1-U)
0 l
and A = O l
0 -l

The eigenvalues of A are 0, :i, with eigenvectors
l l
0 , i
0 1

Since w

5

1
(-i) respectively.

-1
1 {o
u4-iv, where u = 0 , v = l ,
1 \0

II
A
H P' H
v
N

l l
the matrix P = 0 0 1 has the prOperty that,
0 -l
O 0 0
B = P-lAP = 0 0 l is the Jordan Canonical form of
0 -l

A and the transformation y = P_lx changes the linear

system X = AX into y = By.

36

x
Let y=<y> . The transformation y = P-lx
2

has the form:

x = U4-W
y: —W (4)
z = V

and the system (3) becomes:

 

r

x=efix+yHl-x-y)
<y==z-dx+yHl-x-y) w)
ké=-y.

To show the existence of a periodic solution in (S) we use

another change of variable:

 

(y'=~{sin 9
4 z = y cos 9 (6)
x = p
k
. 1 . . , . . . 2
Y = ¥(yy+zz) = -c-; 3111 e(p+ysine)+631ne(P+Y51n9)

= —g sine(p+ysine)(l-p-Ysin6)

tan a =‘E = é(14—tanze) =-JE(yz-y2)
2

 

 

'— iz‘yé -12‘Yé -XZ—zfi:
e _ 2(1+tan2 ) - 2 2 - 224- 2
z 9 2 (“15) Y

Z

_ 22--ez(x+y)+e2(x+y)2+y2 = l-c 2(X+Y)(1-X-Y)
" 2 2 ' 2 2 '
y 4-2 Y 4-2

37

Therefore
é-l—ccos Q(p+ysine)(1-p-ysine)
_ ‘ Y

and

P: )2 = €(X+Y) -e:(x+y)2

g(p+ysin9)(l-p-y sin e)

The new system in (y,p,e) is

r
p = e(p+ysin8)(l-p-Ysine)

A
-<o
II

-esin9(p+ysine)(l-p—ysina) (7)

 

e = 1-. -%cose(p+ysin9) (1-p-ysine).

The existence of a periodic solution for (7) is equivalent

to the existence of a 2n periodic solution of

 

 

 

r

39 _ eUL+YsineH1 -P—1_sin 9)

de y-ecose(p+ysine)(l-p-Ysine)

( (a)
sly. = _€ YSinG(Pj'LSinBHl-p-ysine)

\de y-ecos8(p+ysine)(l-p-Ysine)

for y in a compact set.

Consider the Zw-periodic change of variable
2
Y()

= + e + e + ---
(l) (2) fl
(9)

Y Y
taking ( ) into ( O) and try to determine Y1(Y0'p0'a)
P PO

2 .
y(2)(YO.PO.B).....pl(YO.POa6). p‘ )(Yo'Po'e) and functions

38

l (2)

R (yo,po), R (yo,po),... so that the differential

Y

p0) is autonomous and given by
O

equation for (

d Yo _ 1 2 (2)
E§'< > - 6 R (YOIPO)*'€ R (YO'pO)Y' . (10)

If such a transformation can be found, then the 27
periodic solutions of (8) coincide with the equilibrium

points of (10).

We can find functions Rl(yo,p0), R(2)(yo,po),...

(2) (l)

(l
) (YO.PO.6).---oP (YOoPOoe):

Y (YOpPOoB): Y

9(2)(YO.PO.6),... inductively by the Method of Averaging,

using the requirement of 22-periodicity of y(l)(yo.po.e)

and p(i)(yo.po.8). i = 1,2,... . Since
(1)
d Y d YO d Y (Yorpooe)
.__ =.__ + e ——- + --' we get
C” (p) d9 (90) d9 (p‘l)(vo.po.e>

(leYIPIE)
8

> = e R1(YO.pO)4-ez R2(YO.PO)+-°°°+
92(Y.P.e)

(11),

2
e .— + e — + 00.

d6 (1) d8 (2)
p (Yolpooe) P (Yoopooe)

where

-Y(P+Y3in8)(l-p-ysine)sine
Y-ecose(p+ysine)(1-p-ysine)

91(Y.P.e)

y(p+ysine)(l-p-ysine)
r-ecose(p+ysin9Xl-p-ysine)

92(Y.p.e)

39

From (11) we see that the following equation must be
satisfied:

1
R (YOIPO) = + '55 1

or

1
d Y (Yon/3019) _ l 91(Y00p000)
d9 1 ‘ R (YO’pO) ‘ ° (12)

(12) is equivalent to

271' g (Y op 00)
alw ,p ) =-1—=" (1 0 0 de (12.1)
0 0 2w J

Y(l) (“{OoPO:8)>
(9(1)(Y0o90:8)

g (Y '9 ,0) \
=f R(1)(Yo,po) -< 1 ° 0 >)de, (12.2)
92(Y00p000)

1 2" 1 27’ . .
EEO 91(Y0.p0,0)de =EI0 -Sln 9(po+y051ne)

. _‘1
(1-po- {Os-:Ln3)de— 2 Y0(2 pO-l)

2W 2F

.1. .. .1. r - _ _ .

O O

_ 2 l 2
‘ po“’o"2 Y0'

2 l 2

l
po‘po'i Yo

Therefore Rl(yo,po):=(

40

Y(l)(YO:PO:9) = 01' % Y0(2 pO-l)+sin 8(po+vosin9)
. 2 1 .
(l-pO-v031n8)d8 = -(pO-po)cos 9+2}- YO(2 pO-l)Sin29-

1
2( 3

Y0? cos e-cos e)

(l 2 l 2 .

P )(YOoPO.8) = 01' pO-p0-2 Y0- (po+y051n9)
- .. 1

(l-pO-y051n9)de-yo(1-2 po)cos 9-

4 YCZ) sin 28 . T herefore

1

d Yo 210(29‘1) 2 2

56( )=€( 2 (12 +8 R(10"30H'”
po po‘po'i' Yo

I
‘Y (2p -l)

e (2 02 C]? 2)+€R(2)(YOoPO)+”° (13)
p0"90'2'Yo

€ G(Yo,poo€) .

Since

1 2

1
- Y (2 P -1)
90‘90‘2 Yo

the equation G(yo.po.0) = 0 has solutions
_ .12 1
(YOIPO) " ( 2 :2): (000): (Oil)

We are not interested in the trivial periodic solutions

(0,0) and (0,1). Therefore we use The Implicit

Function Theorem at («(0.90) = (3% '2') to show the

existence of an equilibrium point near (E l

2'2)'

41

 

 

l
- (2 p 1-1) Y
BG(YO,pO,O) _ 2 0 0
BTY .p )
O 0
-YO l«-2 p0
Therefore
aG(YO.pO.O) _ 1
C181: 5 (Y ) _ — i 5! O
0'90 “/2 _1_)
2 '2

which gives (YO(€) .po(€) . G(YO(€) .po(€) .e)) = o for

small enough 5 and a 2v-periodic solution for (8).

We have proved

Theorem 3.1. The modified equation
2." . _
e 11-Fu - u(l -u)

has a periodic solution for sufficiently small 6 > 0.

3.2. The Averaged System

The averaged system

(
dr _ 1
_g_r(p_§)

( .(14)
§§=p-92-§_r2 '

is conservative with the energy function

 

K

H(r,p) = r2(4p-4p2-r2) .

because

42

SE =2‘((4p -4p2 -Y2) -2Y3 = 8Y(p -p2 .0:— Y2)
Y
8H _ 2 _ 1
5-5- Y (4-8p) - -8Y(YP-2 Y).
aH . aH ._ . .
Therefore, SY'Y + 33 p — 0, which shows the solution

curves of the averaged system (14) are on the level curves

of H(Y,p) = c.

For c = 0, the level curves correspond to Y = 0
and Y = zjfiTITTET' which are the heteroclinic orbits
whose existence was shown in previous chapters by
studying the qualitative behavior of the averaged

system.

 

 

 

 

 

 

For c # 0, H(Y,p) = c implies 4p -4p2--Y2 = 5%, (Y510).
Y

The positive values of c correspond to the periodic

orbits inside the heteroclinic. The maximum value of

43

H is i- and it is attained at Y = 1%Z, p =‘%
which is a critical point of the averaged system
corresponding to the periodic orbit of the original
3-dimesnional system. For negative values of c,

the level curves are unbounded as shown in the graph.
We are only interested in the positive values of c,
corresponding to the periodic orbits and we would like
to investigate the existence of periodic solutions of (8).
Since it takes infinite time to trace the heteroclinic

orbitpthe period map goes to infinity as c tends to

0*.
11¢)

 

 

I'a- n

44

5.2 The existence of periodic orbits in (8)

Since (2) remains unchanged under the transformation

,
Y"Y

(921-9 (15)
84-9:

 

[/—
starting at e = 0. p(0) = %-, 0 < y(O) < 1%-. the
trajectory T+ corresponding to the positive values
of e traces a curve in (Y,p) plane symmetric t0

the trace of the trajectory F- corresponding to the

 

 

negative values of 9, with respect to the line p = %.
(=19

I -1

4)
?= ?o 1"

1

‘ —Ju——— —-——-— —-—.— ———.—

(”=2 ‘1’

 

Since the solutions to the system (8) remains
close to the solutions of the averaged system for finite
time, and due to the 2V periodicity of (8),the stable
and unstable manifolds at the hyperbolic points are

Zw periodic, and as a point approaches the stable

45

manifold on the line p = p0, the solution curve
passing through that point hits the line Y = YO at

a point approaching the unstable manifold, and from

the line Y = Y0 it takes finite time to hit p = % ,
at a point close to (l,%). We conclude that starting
at e = 0, p(0) = %, y(O) > 0 small, r+ would hit

p = % at some y(el) near 1. For 91 = nw we get
n distinct periodic solutions of period 2nw in (8)

because (8) is 22 periodic and

F 9 9

p(-81) p(91)

 

K

We have proved

Theorem 3.2. For every 6 > 0 there is

 

N = N(e) > 0 such that for every n, 0; 113 N,
there are n distinct periodic orbits with period

~ 2nv in (8). Furthermore N(€) 4 a as e 4 0.

CHAPTER 4

On the Existence of a heteroclinic orbit

4.1. The differential equation

€2u+u==u(l-u) (1)

has been studied in the previous chapter and the existence

of periodic orbits for small 5 was proved.

The existence of a heteroclinic orbit from (1,0,0)
to (0,0,0) has been observed in computer experiments

using the Shampine-Gordon Method to solve the associated

 

system
f .
u = v
3 v = w (2)
2.
c w = -v4-u(l-u) ,
k

We prove the existence of a spiral heteroclinic orbit
using a prOper change of variable and the variation of

constant formula.

The change of variable which suits our purpose would be:

2

U = u, V = av, W = e w, T = t (3)

(oh-4

which changes the system into

46

47

v
w (4)
W = -V+ e U(l—U)

<°cr
II II

System (4) could be written in the form X = AX+F(e,X)

where:
,_
0 l 0 0
A = O l p F(€'X) = 6 O
o -1 0 u(1 -U)
U
and X = V
W
. . . erA
A Simple computation Will show that e = I, and the

solution to X = AX is:

u‘(t) = uo+vO Si~nt+w0(l-cos t)
V1. (t) = v0 cos t+w0 Slnt (5)
w‘ (t) = -vO Slnt+WO cos t 0

These solutions are 22-periodic, and satisfy

v2(t) +w2(t) = v2(0) +w2(0)
tEIR (6)
u(t) +w(t) = u(O) +w(0) .

Solutions to the purturbed system (3) satisfy the variation

of constant formula:
t
At _
x(t) = e X0+i eA(t S)F(x(s),e)ds.
O

48

For t = 2V.
Y- e(l-coss)u(s)(l—u(s))q
X(2*r) = x(o) +j -€ (sin s)u(s) (l-u(s)) dS (7)
O ecoss u(s)(1‘u(S))
L ._

 

 

Since the solutions of X = AX+F(e,X) depend smoothly

on e, and for e 0, the solutions satisfy (5), we

get

u(t) = uL(t) + eh
where h is bounded and uz(t) = uO+vO sin t +
wO (1 - cos t) . Therefore,

u(t) (l-u(t)) = uL(t) (l-u£(t)) +eB ,

Substituting the above expression for u(s) (l -u(s)) in
(7) we get

211‘

of (l-cos S)U(s) (l-u(S)) ds
0

Au

27

e I (l-cos s) (uz(s)) (l-uz(s)) + eZBlds.
0

Computing the integral part in Au, we get

_ _ 2_ 2__ 2 2
Au—Tr(2uo+3w0 2uO 5wO V )6+ 3B

O - 6quO 1 .

49

Similarly:
Av = -wa(v -2u v -2v w )4-52B
0 0 0 0 0 2
Aw = ve(-w04-2uow04-2wo)+-€2B3.
For small 6
F All-1 ‘4 7(2uo+ 3wO-2ué-5wé-vg-6uowo)
< AlV % -v(vO-2uOvO-2vowo)
Alw z Tr(-w0+ 2uowo+ 2wC2)) .

 

Since w = 0 is transversal to the trajectories for

v0 # eu0(1-u ), the Poincare's map is defined on w = 0,

O

a compact set disjoint from a neighbourhood of

 

v0 = auo(l-uo). Therefore for T = T’s) z 27
F _ 2 2
Alu — 7r(2uO--2uO--VO)+€Bll
4 Alv = -Tr(vO--2uovo)+eB2l
A W = O
K 1
We divide the (uO,vO) plane into 4 regions, namely
Alu > 0 Alu > 0
R1 : , R2 :
AlV > 0 Alv < 0
Alu < 0 Alu < 0
R3 : , R4 :
Alv ) O AlV < O .

50

fl

 

0 y\ U” “‘1’: o)

 

we

 

Figure 22

For 6 = 0, starting at a point (uO,vO), wb = 0, the
trajectory will be periodic in the (u,v) plane. It will

trace the circle

2 2 _ 2 _ z
(u no) -+v — R0 R0 - gv

returning to (u0,vo) at t = 2v. For a > 0, t = 7(a)
the trajectory will be nearby. (u0,vb) and according to
the position of (uO,vO) above, below, to the right or

to the left of (uo,vo). For instance,starting near (1,0)
in the (uO,vO) plane in R3, the trajectory will reach a

point (u -+Av) which is located to the left and

O 0

above (uo,vb). (See Figure 21)

+Au, V

51

 

 

V
\L '4." 0")
D l
t"°°) (“5
+0
t,1r
4. I \“‘;10)
\“ofld Q3
11
' W U
0 1
Figure 23

While starting at a point in R2. the trajectory will reach

a point to the right and below (uO,vO). Therefore starting
in R3, the trajectory starts to oscillate and move to the
left and the radius of oscillation will grow until it reaches

u0 = 1/2.

After entering R4 the trajectory starts to trace
smaller and smaller rings until it tends to zero, or escapes

to the region u < 0. (See Figure 24.)

 

 

 

Uo

 

 

 

Figure 24

We are looking for a point (uo,vo) such that the trajectory
starting at (u0,Vb) moves to the left and tends to (0,0)

in positive time and tends to (1,0) in negative time.

Since the 2-dimensional unstable manifold at the critical
point (1,0,0) is almost perpendicular to the u-axis, the
intersectionof that manifold with the plane w = 0 is a
curve passing through (1,0) almost perpendicular to the

u-axis at u = 1. (Figure 25)

53

- (no
The intersection '1 a“
(,1. moniiv“

K) “(\5‘3 w - o
0“", (x 0,0) Ni“ ‘
1, 13 f/ I

K U

 

 

Figure 25

We pick a point (uo,vb) on the intersection of w = O

and the unstable manifold in R following the trajectory

30
we get another point of the intersection forward in time,

(t7! 211’) say (u1,vl) 6R3. Since (ul,vl) 6R3, starting
at (ul,vl) the trajectory will reach a point (u2,v2) 6R3

forward in time after At F‘ZF, and in this way we define

If for some k, Pk enters R from the positivity of

1'
Au in R1. the trajectory will reenter R3 ,

Therefore, let us assume the trajectory enters R4 for

some Pk where Pk-IER This simply means uk—l > 1/2

3.
and uk < 1/2.

Let Aun = u -u Then we have

n n-l'

54

k k-l
v~ Y‘

uO-r :.Aun < 1/2 and uO-t :IAUn > 1/2.
n—1 n—1

Since solutions to 01333 are continuously dependent on

the initial values (uO,vO), uk is a continuous function

of (uO,vO).
Let the unstable manifold on w = 0 be parametrized
by = $(s), v = 1(5), -sO S s S so. Then as s-40,

k-Oo.

Therefore, k depends on s and 11k is a continuous

function of s. We define a step function k = k(s) as
follows:

For 0 < s S so, let k(s) be the number of oscilla-

tions required to reach uk(s) 2 1/2 and uk(s)+1 < 1/2.

Then clearly as s-40 :k-oo. (See Figure 26.)
‘F

} 1

lit—I
(“LT
1

(I

 

 

 

’ l
l l l 1
l ' I l.__.l k(s)
- l l i E 1 5
0 55 3: 55 53 s; s‘
Figure 26

From the definition of k(s), it is clear that for

the jumping points, $1.32,..., uk(sr) = 1/2: that is,

there are infinitely many points on the unstable manifolds
such that the trajectory starting at those points hits the

plane u = l/2 after an integral number of oscillations.

Let s e (0.50] be such a value; therefore,

l/2. We claim that starting at u0 = uk(s),

the trajectory will be an oscillating

uk(s)
wO = 0, v = v0,

heteroclinic orbit.

Since the system remains unchanged under the change

of variable

U = l-u
V = v
W = -w
T=-T,
the trajectories are symmetric with reSpect to u =-% .
' ' = ( = =
The trajectory starting at 110 uk 3), wO 0, v VO

would merge to (0,0,0) as t 4 m.

The graphs on the next 3 pages were made by the
computer, showing the oscillating heteroclinic orbit

with different scalings in (uo,v0) plane.

We have proved

Theorem 4.l. There is a heteroclinic orbit in the

 

modified logistic equation

€2.11.+ u = u(1-u)

for e > 0 sufficiently small.

56

.omTuooso so ooouoao ooo w o w o

you

noduoHOm umosm mo nmmum one

 

 

57

(3+

5+

 

 

 

The oscillating heteroclinic orbit connecting two

critical points of ofii-o = u(1-u).

BIBLIOGRAPHY

[l]

[2]

[3]

[51

[6]

[71
[81

[9]

BIBLIOGRAPHY

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L.P. Silnikov, Existence of a countable set of periodic
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Nauk SSSR 172 (1967), 54-57 = Soviet Math. Dokl.
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L.P. Silnikov, 0n the generation of a periodic motion
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(119) (1969). 461-472 = Math. USSR Sb. 6 (1968)
427-438.

Hartman, P., Ordinary differential equations,
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Grobman, D., Dokl. Akad. Nauk. USSR 128, 880 (1965).
A. Arneodo, P. Coullet, and C. Tresser, Possible
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S.N. Chow and E.M. deJager, 0n the discretisation
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