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ity
m
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[
1
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,
[
2
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Dig
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cr
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p
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tu
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es,
an
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[
3
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[
4
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.
T
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[
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T
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[
1
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–
[
1
3
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.
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[
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[
1
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[
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[
1
8
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e
ch
ao
tic
p
r
o
p
e
r
ties
o
f
th
e
p
r
o
p
o
s
ed
m
ap
,
w
h
er
e
we
p
r
ese
n
t
v
ar
io
u
s
an
aly
tical
tech
n
i
q
u
es
an
d
s
im
u
latio
n
s
to
d
em
o
n
s
tr
ate
its
im
p
r
o
v
ed
c
h
ao
tic
b
eh
a
v
io
r
c
o
m
p
ar
e
d
to
t
h
e
tr
ad
itio
n
al
s
in
e
m
ap
an
d
o
t
h
er
r
elate
d
w
o
r
k
s
.
Fin
ally
,
s
ec
tio
n
4
p
r
esen
ts
th
e
co
n
clu
d
in
g
r
em
a
r
k
s
,
s
u
m
m
ar
izin
g
th
e
k
e
y
c
o
n
tr
ib
u
tio
n
s
o
f
o
u
r
wo
r
k
an
d
h
ig
h
lig
h
tin
g
p
o
te
n
tial f
u
tu
r
e
r
esear
ch
d
ir
ec
tio
n
s
.
2.
P
RO
P
O
SE
D
M
E
T
H
O
D
AN
D
I
T
S
ST
A
B
I
L
I
T
Y
AN
AL
Y
SI
S
T
h
is
s
ec
tio
n
p
r
esen
ts
a
n
ew
o
n
e
-
d
im
en
s
io
n
al
ch
a
o
tic
m
ap
.
T
h
e
m
ap
is
m
ath
em
atica
lly
d
ef
i
n
ed
as:
+
1
=
s
in
(
(
2
−
)
)
(
1
)
wh
er
e
0
is
th
e
in
itial st
ate
an
d
(
β,
α
)
ar
e
th
e
co
n
tr
o
l p
ar
am
ete
r
s
.
+
1
=
=
∗
=
(
(
2
−
∗
)
)
(
2
)
W
h
en
th
e
o
u
tp
u
t
o
f
th
e
m
a
p
i
n
th
e
n
ex
t
iter
atio
n
d
o
es
n
o
t
c
h
an
g
e
an
d
is
th
e
s
am
e
as
th
e
o
u
tp
u
t
it
is
cu
r
r
en
tly
p
r
o
d
u
cin
g
,
th
is
is
r
ef
er
r
e
d
to
as
a
f
ix
ed
p
o
in
t.
Un
d
er
th
e
ass
u
m
p
tio
n
th
at
x*
is
th
e
f
ix
ed
p
o
in
t,
th
e
(
2
)
is
o
b
tain
ed
b
y
s
u
b
s
titu
tin
g
xn
a
n
d
xn
+1
with
it.
I
t
is
n
o
t
p
o
s
s
ib
le
to
s
o
lv
e
th
is
eq
u
atio
n
an
aly
tically
u
s
in
g
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
2
,
Ap
r
il
20
25
:
2
1
2
8
-
2
1
3
7
2130
elem
en
tar
y
f
u
n
ctio
n
s
.
Nev
er
th
eless
,
we
ca
n
attem
p
t
to
id
en
ti
f
y
ap
p
r
o
x
im
ate
an
s
wer
s
b
y
e
m
p
lo
y
in
g
n
u
m
e
r
ical
tech
n
iq
u
es
s
u
ch
as
th
e
in
ter
v
al
b
is
ec
tio
n
m
eth
o
d
,
th
e
iter
ativ
e
f
ix
ed
-
p
o
in
ts
m
eth
o
d
,
a
n
d
t
h
e
New
to
n
-
R
ap
h
s
o
n
m
eth
o
d
.
I
n
th
is
s
tu
d
y
,
th
e
i
n
ter
v
al
b
is
ec
tio
n
m
eth
o
d
will
b
e
u
s
ed
[
2
2
]
.
Alg
o
r
ith
m
1
illu
s
tr
ates
all
th
e
s
tep
s
r
eq
u
ir
ed
to
d
ete
r
m
in
e
f
ix
ed
p
o
in
ts
an
d
in
d
icate
s
wh
eth
er
it
is
u
n
s
tab
le.
A
f
ix
ed
p
o
in
t
is
co
n
s
id
er
ed
u
n
s
tab
le
if
th
e
ab
s
o
lu
te
v
alu
e
o
f
t
h
e
d
er
iv
ativ
e
o
f
th
e
f
u
n
ctio
n
at
th
e
f
ix
ed
p
o
in
t
is
m
o
r
e
th
an
1
(
|d
f
(
x)
|
>
1
)
.
I
n
in
s
tan
ce
s
o
f
th
is
n
atu
r
e,
n
eig
h
b
o
r
in
g
p
o
in
ts
ten
d
to
d
ev
iate
f
r
o
m
t
h
e
f
ix
ed
p
o
in
t
r
ath
e
r
th
an
c
o
n
v
er
g
e
to
war
d
s
it,
s
u
g
g
esti
n
g
th
e
p
r
esen
ce
o
f
p
o
t
en
tially
ch
ao
tic
b
eh
a
v
io
r
.
W
e
s
elec
t
1
0
0
v
alu
es
o
f
β
b
etwe
en
[
0
,
1
]
an
d
α
=
3
,
5
0
0
.
T
h
e
n
u
m
b
er
o
f
u
n
s
tab
le
f
i
x
ed
p
o
in
ts
ca
lcu
lated
f
o
r
ea
ch
v
alu
e
o
f
β
is
d
is
p
lay
ed
in
Fig
u
r
e
1
;
f
o
r
o
u
r
m
ap
an
d
th
e
MST
en
t
m
ap
[
1
6
]
,
th
e
to
tal
n
u
m
b
er
f
o
r
all
β
v
alu
es
is
2
,
0
7
9
an
d
1
,
9
4
0
,
r
esp
ec
tiv
ely
.
C
o
m
p
ar
in
g
th
ese
to
tals
,
o
u
r
m
a
p
ap
p
ea
r
s
to
ex
h
ib
it
m
o
r
e
ch
a
o
tic
b
eh
av
i
o
r
s
in
ce
it
h
as a
h
ig
h
e
r
n
u
m
b
er
o
f
u
n
s
tab
le
f
ix
ed
p
o
in
ts
.
Alg
o
r
ith
m
1
.
Fin
d
f
ix
e
d
p
o
i
n
t
P
rocedure findFixedPoint(f (x), α, β, [−1, 1], N , ϵ)
In
pu
t:
Fu
nc
ti
on
f(
x)
,
pa
ra
me
te
rs
α
an
d
β,
in
it
i
al
in
te
rv
al
[−
1,
1]
,
ma
xi
mu
m
it
er
at
io
ns
N=1,000, tolerance ϵ=1e−10.
Output: Return the value of fixed points.
Procedure:
Divide the interval [−1, 1] into 100 small intervals [a, b].
for each small interval [a, b] do
if f (a, α, β)=a then
return a.
end if
if f (b, α, β) = b then
return b.
end if
Initialize iteration counter i=0.
while i<N do
Increment i.
if (f(a, α, β)
-
a)*(f(b, α, β)
-
b)>0 then
Set atemp=a.
Set a=a+b
if f (a, α, β)=a then
return a.
end if
if (f(a, α, β)
-
a)*(f(b, α, β)
-
b)>0 then
Set b = a.
Set a = atemp.
end if
else
break.
end if
end while
end for
end procedure
Fig
u
r
e
1
.
C
o
m
p
a
r
in
g
t
h
e
n
u
m
b
er
o
f
u
n
s
tab
le
f
ix
e
d
p
o
in
ts
b
e
twee
n
o
u
r
m
a
p
an
d
th
e
MST
e
n
t
m
ap
[
1
6
]
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
A
n
o
ve
l o
n
e
-
d
imen
s
io
n
a
l c
h
a
o
tic
ma
p
w
ith
imp
r
o
ve
d
s
in
e
ma
p
d
yn
a
mics
…
(
Mo
h
a
med
Hti
ti
)
2131
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
Her
e,
we
ev
alu
ate
th
e
p
er
f
o
r
m
an
ce
an
d
c
h
ao
tic
b
eh
av
i
o
r
o
f
th
e
p
r
o
p
o
s
ed
m
ap
.
T
o
ass
es
s
th
e
d
is
o
r
d
er
ly
b
eh
av
io
r
o
f
o
u
r
s
y
s
tem
,
we
u
s
e
estab
lis
h
ed
ass
es
s
m
en
t
tech
n
iq
u
es
lik
e
th
e
b
if
u
r
ca
tio
n
d
iag
r
am
,
th
e
L
y
ap
u
n
o
v
ex
p
o
n
en
t,
th
e
s
en
s
itiv
ity
to
war
d
s
t
h
e
in
itial
co
n
d
itio
n
s
,
th
e
p
ar
am
eter
s
o
f
th
e
ch
ao
tic
m
ap
,
en
tr
o
p
y
an
aly
s
is
,
an
d
th
e
r
esu
lts
o
f
th
e
0
-
1
test
.
T
o
s
h
o
wca
s
e
th
e
b
etter
ch
ao
tic
q
u
alities
o
f
t
h
e
p
r
o
p
o
s
ed
m
ap
,
we
co
m
p
ar
e
its
ass
ess
ed
r
esu
lts
w
ith
th
o
s
e
o
f
o
th
er
ex
is
tin
g
c
h
a
o
tic
m
ap
s
.
3
.
1
.
B
if
urca
t
io
n dia
g
r
a
m
Fig
u
r
e
2
d
is
p
lay
s
th
e
b
if
u
r
ca
tio
n
d
iag
r
am
o
f
t
h
e
s
u
g
g
ested
m
ap
,
with
a
f
ix
e
d
v
al
u
e
o
f
al
p
h
a
s
et
at
3
,
5
0
0
.
T
h
e
m
ap
'
s
o
u
tp
u
t
v
alu
es
ar
e
co
n
f
in
ed
to
th
e
r
an
g
e
o
f
[
-
1
,
+1
]
.
T
h
e
r
an
g
e
o
f
b
eta
v
alu
es
h
as
b
ee
n
ex
ten
d
ed
f
r
o
m
[
0
,
1
]
to
m
o
r
e
th
an
[
-
5
0
,
5
0
]
,
ex
h
i
b
itin
g
a
g
lo
b
al
ch
ao
tic
b
eh
av
i
o
r
in
co
m
p
ar
is
o
n
with
th
e
b
if
u
r
ca
tio
n
d
iag
r
am
o
f
t
h
e
s
in
e
m
ap
p
lo
tted
in
Fig
u
r
e
3
.
Fig
u
r
e
2
.
B
if
u
r
ca
tio
n
d
iag
r
am
o
f
o
u
r
m
ap
,
with
α
=3
,
500
Fig
u
r
e
3
.
Sin
e
m
a
p
b
if
u
r
ca
tio
n
d
iag
r
am
Ad
d
itio
n
ally
,
th
e
b
if
u
r
ca
tio
n
d
iag
r
am
is
p
lo
tted
f
o
r
eig
h
t
d
if
f
er
en
t
v
alu
es
o
f
α
in
Fig
u
r
e
4
.
I
t
d
em
o
n
s
tr
ates
th
at
wh
en
th
is
v
alu
e
is
g
r
ea
te
r
th
a
n
3
0
,
0
0
0
,
o
u
r
m
ap
ex
h
ib
its
to
tal
c
h
ao
tic
b
e
h
av
io
r
.
As
a
r
esu
lt,
f
o
r
t
h
e
r
est
o
f
th
is
s
tu
d
y
,
w
e
f
ix
ed
α
to
3
5
,
0
0
0
an
d
β
b
etwe
en
[
-
5
0
,
5
0
]
.
Fo
r
an
y
PR
NG
o
r
en
cr
y
p
tio
n
ap
p
licatio
n
,
a
s
m
all
k
e
y
s
p
ac
e
is
u
n
d
esira
b
le.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
2
,
Ap
r
il
20
25
:
2
1
2
8
-
2
1
3
7
2132
Fig
u
r
e
4
.
B
if
u
r
ca
tio
n
d
iag
r
am
with
d
if
f
er
en
t
v
alu
es o
f
α
f
o
r
o
u
r
p
r
o
p
o
s
ed
m
ap
3
.
2
.
L
y
a
pu
no
v
e
x
po
nent
T
h
e
L
y
ap
u
n
o
v
ex
p
o
n
e
n
t
(
L
E
)
is
an
ess
en
tia
l
to
o
l
in
co
m
p
u
ter
s
cien
ce
f
o
r
ev
alu
atin
g
m
o
d
els
an
d
tech
n
iq
u
es
th
at
s
im
u
late
o
r
d
escr
ib
e
d
y
n
am
ical
s
y
s
tem
s
[
2
3
]
.
R
esear
ch
er
s
ca
n
ass
e
s
s
a
s
y
s
tem
's
s
tab
ilit
y
an
d
co
n
f
ir
m
if
ch
a
o
tic
d
y
n
am
ics
ar
e
p
r
esen
t
b
y
co
m
p
u
tin
g
th
e
L
E
.
Sin
ce
it
im
p
lies
th
at
clo
s
e
t
r
ajec
to
r
ies
d
iv
er
g
e
ex
p
o
n
e
n
tially
,
a
p
o
s
itiv
e
L
E
is
u
s
u
ally
s
ee
n
as
a
s
tr
o
n
g
d
e
g
r
ee
o
f
ch
ao
s
,
r
esu
ltin
g
in
an
u
n
p
r
ed
ictab
le
an
d
ex
tr
em
ely
s
en
s
itiv
e
s
y
s
tem
.
T
h
e
L
E
ca
n
b
e
m
ath
em
atica
lly
d
ef
in
ed
u
s
in
g
(
3
)
,
wh
er
e
F
(
xi)
r
ep
r
esen
ts
th
e
s
tate
o
f
o
u
r
s
y
s
tem
at
iter
atio
n
i
.
=
→
∞
1
∑
|
′
(
)
|
−
1
=
0
(
3
)
Fro
m
th
e
cu
r
v
e
i
n
Fig
u
r
e
5
,
we
ca
n
o
b
s
er
v
e
th
at
o
u
r
p
r
o
p
o
s
ed
m
ap
ex
h
i
b
its
a
L
y
ap
u
n
o
v
ex
p
o
n
en
t
th
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I
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e
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2
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]
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ar
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34
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in
Fig
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.
I
t
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m
ap
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
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I
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I
n
t J E
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&
C
o
m
p
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g
I
SS
N:
2088
-
8
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0
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n
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p
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Mo
h
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med
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2135
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o
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ased
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B
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titi
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fr
o
m
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M
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h
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m
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ll
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Un
iv
e
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in
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p
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r
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h
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Ay
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Evaluation Warning : The document was created with Spire.PDF for Python.