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ly
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m
i
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is
tic
m
ath
em
atica
l
m
o
d
els
[
1
]
–
[
3
]
.
O
n
e
o
f
th
ei
r
m
o
s
t
d
is
tin
ctiv
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p
r
o
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lik
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p
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latio
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y
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am
ic
s
[
4
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,
b
r
ain
ac
tiv
ity
[
5
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,
h
ea
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t
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h
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th
m
s
[
6
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,
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d
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is
m
s
[
7
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.
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[
8
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m
ias
[
9
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.
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k
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liter
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with
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eh
av
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[
1
0
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.
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[
1
1
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.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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I
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[
1
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1
4
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T
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On
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u
e
to
i
ts
r
o
b
u
s
tn
ess
an
d
ea
s
e
o
f
im
p
lem
en
tatio
n
[
2
0
]
,
[
2
1
]
.
Fu
r
th
e
r
m
o
r
e,
its
s
tab
ilit
y
an
aly
s
is
is
s
tr
aig
h
tf
o
r
war
d
,
e
n
s
u
r
in
g
th
at
th
e
clo
s
ed
-
lo
o
p
s
y
s
tem
co
n
v
er
g
es
q
u
ic
k
ly
to
th
e
d
esire
d
r
ef
er
en
ce
d
esp
ite
d
is
tu
r
b
an
ce
s
an
d
u
n
c
er
tain
ties
.
T
h
e
s
p
ec
ialized
liter
atu
r
e
p
r
o
p
o
s
es
s
ev
er
al
v
ar
i
atio
n
s
th
at
im
p
r
o
v
e
co
n
tr
o
l p
e
r
f
o
r
m
an
ce
a
n
d
s
ig
n
a
l q
u
ality
b
y
r
ed
u
ci
n
g
ch
atter
.
Am
o
n
g
th
ese
tech
n
iq
u
es,
we
ca
n
m
en
tio
n
:
ad
a
p
tiv
e
n
e
u
r
o
-
f
u
zz
y
in
f
er
e
n
ce
s
y
s
tem
-
b
ased
SMC
[
2
2
]
,
o
p
tim
ized
b
ac
k
s
tep
p
in
g
f
u
zz
y
SMC
[
2
3
]
,
[
2
4
]
,
a
d
ap
tiv
e
f
u
zz
y
SMC
[
2
5
]
,
[
2
6
]
,
f
u
zz
y
SMC
[
2
7
]
,
s
u
p
er
twis
tin
g
SM
C
[
2
8
]
,
ad
ap
tiv
e
f
u
zz
y
s
u
p
er
-
twis
tin
g
SMC
[
2
9
]
,
r
o
b
u
s
t
f
r
ac
tio
n
al
o
r
d
er
c
o
n
tr
o
ller
[
3
0
]
,
ter
m
in
al
f
r
ac
tio
n
al
-
o
r
d
er
SMC
(
FOSM
C
)
[
3
1
]
,
a
d
a
p
tiv
e
p
ar
ticle
s
war
m
o
p
tim
izatio
n
(
PSO
)
b
ased
g
ain
o
p
tim
izatio
n
o
f
s
lid
in
g
m
o
d
e
c
o
n
tr
o
l
[
3
2
]
,
an
d
r
o
b
u
s
t PSO tu
n
ed
FOSMC
[
3
3
]
.
I
n
th
is
wo
r
k
,
we
will
p
r
o
p
o
s
e
a
f
r
ac
tio
n
al
-
o
r
d
e
r
m
ath
em
a
tical
m
o
d
el
to
m
o
d
el
th
e
b
eh
av
io
r
o
f
a
g
iv
en
en
z
y
m
e
b
y
d
em
o
n
s
tr
atin
g
th
at
th
is
m
o
d
el
ex
h
ib
its
ch
a
o
tic
b
eh
av
io
r
f
o
r
ce
r
tain
v
alu
e
s
o
f
its
p
ar
am
eter
s
.
W
e
wil
l
th
en
p
r
o
p
o
s
e
a
s
lid
in
g
-
m
o
d
e
co
n
tr
o
l
law
to
s
y
n
ch
r
o
n
ize
an
d
co
n
tr
o
l
th
is
s
y
s
tem
,
d
em
o
n
s
tr
atin
g
its
co
n
v
er
g
en
ce
an
d
s
tab
ilit
y
u
s
in
g
L
y
ap
u
n
o
v
'
s
th
eo
r
em
ex
ten
d
e
d
to
f
r
ac
tio
n
al
-
o
r
d
er
s
y
s
tem
s
.
T
h
e
co
n
ten
t
o
f
th
is
ar
ticle
will
b
e
d
ev
el
o
p
ed
as:
Sectio
n
2
in
clu
d
es
s
o
m
e
b
asic
co
n
ce
p
ts
o
f
f
r
ac
tio
n
al
ca
lcu
lu
s
.
Sectio
n
3
is
d
ed
icate
d
to
th
e
f
r
ac
tio
n
al
-
o
r
d
er
m
o
d
el
in
g
o
f
th
e
b
io
lo
g
ical
en
zy
m
e
s
y
s
tem
with
ch
ao
tic
b
eh
av
io
r
.
Sectio
n
4
d
ef
i
n
es
th
e
p
r
o
p
o
s
ed
f
r
ac
tio
n
al
-
o
r
d
er
SMC
co
n
tr
o
ller
to
s
tab
ilize
t
h
e
b
io
lo
g
ical
ch
a
o
tic
s
y
s
tem
b
ased
o
n
t
h
e
s
em
i
-
f
r
ac
tio
n
al
-
o
r
d
e
r
m
o
d
el.
Simu
latio
n
r
esu
lts
ar
e
p
r
esen
ted
an
d
d
is
cu
s
s
ed
in
s
ec
tio
n
5
to
v
alid
ate
th
e
p
r
o
p
o
s
ed
co
n
tr
o
ller
.
Fin
ally
,
co
n
cl
u
d
in
g
co
m
m
en
ts
ar
e
g
iv
en
in
s
ec
tio
n
6
.
2.
E
L
E
M
E
N
T
S O
F
F
RAC
T
I
O
NAL CA
L
CU
L
US
E
x
p
er
im
en
tal
an
al
y
s
is
o
f
s
ev
er
al
n
atu
r
al
p
h
en
o
m
e
n
a
h
as
s
h
o
wn
th
e
ex
is
ten
ce
o
f
n
atu
r
al
f
r
ac
tio
n
al
-
o
r
d
er
p
atter
n
s
.
T
h
is
f
ac
t
h
as
b
ee
n
co
n
f
ir
m
e
d
f
o
r
d
ielec
tr
ic
p
o
lar
izatio
n
im
p
ed
an
ce
,
tr
a
n
s
m
is
s
io
n
lin
es,
ca
r
d
iac
r
h
y
th
m
,
i
n
ter
f
ac
es,
s
p
ec
tr
al
d
en
s
ity
o
f
p
h
y
s
ical
wav
e
[
2
0
]
.
T
h
ese
s
y
s
tem
s
ca
n
b
e
r
ep
r
esen
ted
b
y
d
i
f
f
er
en
tial
eq
u
atio
n
s
o
f
n
o
n
-
in
teg
e
r
o
r
d
e
r
.
Fra
ctio
n
al
in
teg
r
als
an
d
d
er
iv
ativ
es
h
av
e
b
ee
n
o
f
g
r
ea
t
in
ter
est
in
th
e
p
ast
to
illu
s
tr
io
u
s
m
ath
em
atician
s
wh
o
h
av
e
le
f
t
u
s
v
ar
io
u
s
d
ef
in
itio
n
s
f
o
r
th
ese
o
p
er
ato
r
s
,
t
h
e
m
o
s
t
p
o
p
u
lar
o
f
w
h
ich
ar
e
th
o
s
e
o
f
R
iem
an
n
-
L
io
u
v
ille,
C
ap
u
to
an
d
Gr
ü
n
wald
-
L
etn
i
k
o
v
[
2
1
]
.
2
.
1
.
B
a
s
ic
co
ncept
s
T
h
e
R
iem
an
n
-
L
io
u
v
ille’
s
d
ef
i
n
itio
n
o
f
f
r
ac
tio
n
al
o
r
d
e
r
in
te
g
r
al
(
R
L
)
o
f
o
r
d
e
r
>
0
f
o
r
a
f
u
n
c
tio
n
(
)
is
,
0
(
)
=
1
Γ
(
)
∫
(
−
)
−
1
(
)
0
(
1
)
wh
ile
th
e
R
L
f
r
ac
tio
n
al
-
o
r
d
er
d
er
iv
ativ
e
o
f
o
r
d
er
>
0
is
ex
p
r
ess
ed
as
(
2
)
,
0
(
)
=
−
(
)
(
2
)
h
er
e
Γ
(
.
)
r
ep
r
esen
ts
th
e
E
u
ler
’
s
g
a
m
m
a
f
u
n
ctio
n
,
with
(
n
−1
< η
< n
,
n
∈
N)
.
T
h
e
C
ap
u
to
’
s
f
r
ac
tio
n
al
-
o
r
d
er
(
C
)
d
er
iv
ativ
e
o
f
o
r
d
er
>
0
is
g
iv
en
b
y
(
3
)
,
0
(
)
=
1
Γ
(
−
)
∫
(
−
)
−
−
1
(
)
(
)
0
(
3
)
wh
er
e
μ
is
a
r
ea
l n
u
m
b
e
r
s
u
ch
th
at:
n
− 1
< η
< n
.
Gr
ü
n
wald
-
L
etn
ik
o
v
(
GL
)
d
ef
i
n
itio
n
is
as
(
4
)
:
0
(
)
=
l
im
ℎ
→
0
∑
(
)
(
ℎ
−
ℎ
)
=
0
(
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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t J E
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&
C
o
m
p
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n
g
I
SS
N:
2088
-
8
7
0
8
F
r
a
ctio
n
a
l
-
o
r
d
er c
h
a
o
s
mo
d
e
liz
a
tio
n
a
n
d
s
lid
in
g
mo
d
e
c
o
n
t
r
o
l
in
a
b
io
lo
g
ica
l
...
(
S
a
ki
n
a
B
en
r
a
b
a
h
)
731
wh
er
e
ℎ
is
th
e
s
am
p
lin
g
tim
e,
an
d
th
e
c
o
ef
f
icien
ts
(
)
ar
e
co
m
p
u
ted
as
(
5
)
:
(
)
=
(
−
1
)
Γ
(
+
1
)
Γ
(
+
1
)
Γ
(
−
+
1
)
, j
=
0
,
1
,
.
.
.
k
(
5
)
On
e
ca
n
r
em
ar
k
th
at
th
is
las
t
d
ef
in
itio
n
s
er
v
es
also
as
n
u
m
er
ical
ap
p
r
o
x
im
atio
n
f
o
r
t
h
e
f
r
ac
tio
n
al
o
r
d
er
d
if
f
er
en
tiatio
n
b
y
f
ix
in
g
t
h
e
d
i
s
cr
ete
tim
e
=
ℎ
.
3.
P
RO
P
O
SE
D
F
RAC
T
I
O
NA
L
-
O
RDER
B
I
O
L
O
G
I
CA
L
M
O
DE
L
I
n
th
is
p
ap
er
,
a
n
ew
f
r
ac
tio
n
al
-
o
r
d
er
b
io
l
o
g
ical
m
o
d
el,
b
ase
d
o
n
th
e
in
teg
e
r
m
o
d
el
p
r
o
p
o
s
ed
in
[
2
2
]
wh
ich
co
n
s
is
ts
o
f
en
z
y
m
e
s
u
b
s
tr
ate
r
ea
ctio
n
in
b
o
th
au
to
n
o
m
o
u
s
an
d
n
o
n
-
au
t
o
n
o
m
o
u
s
ca
s
e.
T
h
is
s
y
s
tem
h
as
b
ee
n
ex
ten
s
iv
ely
a
n
aly
ze
d
in
[
1
1
]
.
T
h
e
ac
tiv
ated
e
n
zy
m
e
m
o
lecu
les
wer
e
o
r
ig
in
ally
e
x
p
r
ess
ed
u
s
in
g
th
e
f
o
llo
win
g
s
ec
o
n
d
o
r
d
er
m
o
d
el,
{
x
̇
=
y
̇
=
(
1
−
x
2
+
α
x
4
−
β
x
6
)
y
−
x
+
E
c
os
(
t
)
)
(
6
)
wh
er
e
th
e
v
ar
iab
le
is
th
e
s
y
s
tem
’
s
s
tate
co
r
r
esp
o
n
d
in
g
t
o
th
e
en
zy
m
e
c
o
n
ce
n
tr
atio
n
,
µ
is
an
in
d
ex
o
f
n
o
n
lin
ea
r
ity
.
is
a
m
ea
s
u
r
e
o
f
th
e
ex
ter
n
al
ex
citatio
n
am
p
litu
d
e,
an
d
is
th
e
elec
tr
o
m
ag
n
etic
ex
citatio
n
f
r
eq
u
e
n
cy
.
an
d
ar
e
two
p
o
s
itiv
e
p
ar
am
eter
s
.
T
h
is
m
o
d
el
ex
h
ib
its
ch
ao
tic
b
eh
av
io
r
f
o
r
th
e
f
o
llo
win
g
p
ar
am
eter
s
v
alu
es:
=
2
.
55
;
=
1
.
7
;
=
3
.
465
;
=
2
.
001
;
=
8
.
27.
T
h
e
p
ar
am
eter
s
µ
an
d
wer
e
d
eter
m
in
ed
b
ased
o
n
b
if
u
r
ca
tio
n
d
iag
r
am
s
[
1
1
]
.
3
.
1
.
G
l
o
ba
l f
ra
c
t
io
na
l
-
o
rder
m
o
del
I
n
th
is
s
tu
d
y
,
we
in
tr
o
d
u
ce
a
f
r
ac
tio
n
al
-
o
r
d
er
m
at
h
em
atica
l
m
o
d
el
to
r
e
p
r
esen
t
th
e
ch
a
o
tic
b
eh
av
io
r
o
f
th
e
en
zy
m
e
.
Mo
d
elin
g
t
h
e
b
io
lo
g
ical
s
y
s
tem
u
s
in
g
f
r
ac
tio
n
al
-
o
r
d
e
r
d
if
f
er
en
tial
e
q
u
at
io
n
s
allo
ws
f
o
r
th
e
in
clu
s
io
n
o
f
v
e
r
y
i
n
ter
esti
n
g
p
h
y
s
ical
p
r
o
p
er
ties
(
o
f
te
n
n
eg
lecte
d
in
in
teg
e
r
-
o
r
d
er
m
o
d
els)
s
u
ch
as
th
e
m
em
o
r
y
ef
f
ec
t,
f
r
ac
tal
p
r
o
p
e
r
t
ies,
tis
s
u
e
h
eter
o
g
en
eity
a
n
d
n
o
n
-
lo
ca
l
b
eh
a
v
io
r
.
On
e
o
f
th
e
m
ain
m
o
tiv
atio
n
s
f
o
r
th
is
m
eth
o
d
o
f
d
escr
ib
in
g
r
ea
l
s
y
s
tem
s
is
th
e
p
o
s
s
ib
ilit
y
o
f
n
atu
r
ally
in
tr
o
d
u
cin
g
f
r
ac
t
io
n
al
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o
r
d
er
co
n
tr
o
l
in
to
a
m
o
d
el
th
at
is
alr
ea
d
y
f
r
ac
tio
n
al.
T
h
is
allo
ws
u
s
to
b
e
n
ef
it
f
r
o
m
t
h
e
r
o
b
u
s
t
n
ess
an
d
im
p
r
o
v
e
d
p
er
f
o
r
m
an
ce
p
r
o
p
er
ties
o
f
th
e
s
e
m
o
d
els.
I
n
s
p
ir
ed
b
y
th
e
in
t
eg
er
o
r
d
er
m
o
d
el
(
4
)
th
e
p
r
o
p
o
s
ed
r
ep
r
esen
tatio
n
o
f
th
is
p
h
y
s
ical
p
h
en
o
m
e
n
o
n
is
g
iv
en
as,
{
x
q
1
=
y
2
=
(
1
−
x
2
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x
4
−
β
x
6
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y
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os
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g
th
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ar
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eter
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d
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55
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3
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465
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2
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001
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h
u
s
,
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wer
e
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le
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o
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tain
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h
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o
r
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al
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er
s
(
1
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2
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(
0
.
9
8
,
0
.
9
9
)
.
Fig
u
r
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1
illu
s
tr
ates
th
e
c
h
ao
tic
b
eh
av
io
r
o
f
th
e
f
r
ac
tio
n
a
l
-
o
r
d
er
en
zy
m
e
m
o
d
el
f
o
r
th
e
ex
ter
n
al
ex
citatio
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p
a
r
am
eter
v
alu
e
8
.
2
7
.
F
ig
u
r
e
1
(
a)
r
e
p
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th
e
p
h
ase
p
lan
e
b
e
h
av
io
r
an
d
F
ig
u
r
e
1
(
b
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g
iv
es
th
e
tim
e
s
er
ies
p
lo
t
o
f
th
e
b
io
l
o
g
ical
s
y
s
tem
.
Fig
u
r
e
2
s
h
o
ws
th
e
s
y
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tem
r
esp
o
n
s
e
f
o
r
eq
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1
1
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4
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F
ig
u
r
e
2
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e
p
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th
e
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e
p
lan
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d
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r
e
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i
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e
tim
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ies
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t
o
f
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h
e
b
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l
o
g
ical
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y
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tem
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Fig
u
r
e
3
p
r
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t
h
e
ch
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tic
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esp
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n
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e
o
f
th
e
f
r
ac
tio
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al
-
o
r
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er
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n
zy
m
e
m
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d
el
f
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r
th
e
e
x
ter
n
al
ex
citatio
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p
ar
am
ete
r
v
alu
e
1
7
.
5
0
.
F
ig
u
r
e
3
(
a)
r
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th
e
p
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ase
p
lan
e
b
eh
a
v
io
r
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d
F
ig
u
r
e
3
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b
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illu
s
tr
ates th
e
tim
e
s
er
ies p
lo
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o
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th
e
b
i
o
lo
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ical
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y
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tem
.
3
.
2
.
Se
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i
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r
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(
1
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lead
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to
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r
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al
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o
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m
o
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{
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̇
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y
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1
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2
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x
4
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x
6
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y
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wh
er
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ter
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citatio
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ated
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u
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4
f
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s
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1
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1
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u
r
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4
(
a
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p
r
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th
e
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ase
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lan
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b
eh
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io
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d
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u
r
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4
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b
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th
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tim
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ies
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th
e
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i
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lo
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ical
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y
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tem
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h
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d
if
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er
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t
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o
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d
ch
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av
io
r
s
m
a
k
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it
p
o
s
s
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le
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a
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io
r
s
o
f
th
e
b
io
lo
g
ical
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zy
m
e
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d
m
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k
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s
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n
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in
th
e
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ea
lity
o
f
th
e
b
io
lo
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ical
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o
d
y
.
(
a)
(
b
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u
r
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1
.
Fra
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al
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o
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ch
a
o
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b
eh
av
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r
with
(
1
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2
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=
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0
.
9
8
,
0
.
9
9
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d
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e
p
ar
am
eter
E
=
8
:2
7
:
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a)
p
h
ase
p
la
n
e
b
eh
a
v
io
r
a
n
d
(
b
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im
e
s
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lo
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t
h
e
b
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lo
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ical
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(
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b
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Fig
u
r
e
2
.
Fra
ctio
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o
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ch
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b
eh
av
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r
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2
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9
8
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0
.
9
9
)
an
d
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h
e
p
ar
a
m
eter
E
=
1
1
.
4
0
:
(
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ase
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la
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e
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v
io
r
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n
d
(
b
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e
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er
ies p
lo
t o
f
t
h
e
b
io
lo
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ical
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(
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Fig
u
r
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3
.
Fra
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ar
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eter
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1
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ase
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io
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er
ies p
lo
t o
f
t
h
e
b
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lo
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ical
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tem
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r
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ar
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2
7
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ase
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la
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io
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a
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b
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t
im
e
s
er
ies p
lo
t o
f
t
h
e
b
io
lo
g
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s
y
s
tem
4.
F
RACTI
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N
AL
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O
RD
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S
M
C
CO
NT
RO
L
D
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e
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ed
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ield
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e
s
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s
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ch
ch
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h
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n
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en
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s
o
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te
n
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ity
,
h
en
ce
th
e
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ter
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in
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is
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.
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h
e
r
ea
s
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n
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at
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g
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ch
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alter
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th
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n
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r
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n
with
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r
ain
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ea
[
2
3
]
,
[
2
4
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.
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h
is
m
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ates
th
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p
r
esen
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r
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r
k
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ich
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ilize
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o
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eh
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o
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in
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ty
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e
o
f
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io
lo
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ical
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zy
m
e
cr
u
cial
f
o
r
th
e
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in
g
b
o
d
y
.
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h
e
c
o
n
tr
o
l
a
ctio
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is
p
er
f
o
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m
e
d
u
s
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g
ap
p
r
o
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r
iate
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o
lecu
les th
at
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n
b
lo
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k
o
r
p
r
o
m
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en
z
y
m
e
f
u
n
ctio
n
.
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h
e
au
g
m
e
n
ted
f
r
ac
tio
n
al
-
o
r
d
er
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zy
m
e
m
o
d
el
(
8
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is
ex
p
r
ess
ed
as
(
9
)
:
{
x
q
1
=
y
2
=
(
1
−
x
2
+
α
x
4
−
β
x
6
)
y
−
x
+
E
c
os
(
t
)
)
+
u
(
9
)
wh
er
e
q
1
=1
an
d
q
2
=q
is
th
e
f
r
ac
tio
n
al
d
er
iv
ativ
e
o
r
d
e
r
,
an
d
t
h
e
s
y
s
tem
ca
n
b
e
r
ew
r
itten
as
(
1
0
)
:
{
x
(
1
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=
y
=
(
1
−
x
2
+
α
x
4
−
β
x
6
)
y
−
x
+
E
c
os
(
t
)
)
+
u
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1
0
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Ou
r
o
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jectiv
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is
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esig
n
th
e
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n
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o
l
(
)
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u
ch
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at
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s
ta
tes
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d
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n
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n
v
e
r
g
e
to
th
e
o
r
ig
i
n
.
No
w,
we
n
ee
d
to
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esig
n
a
SMC
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(
)
to
ac
h
iev
e
th
e
asy
m
p
to
tic
s
tab
ilit
y
o
f
th
e
f
r
ac
tio
n
al
-
o
r
d
er
s
y
s
tem
d
y
n
am
ics
(
10
)
[
2
5
]
.
L
et
th
e
f
r
ac
tio
n
al
-
o
r
d
er
s
lid
in
g
s
u
r
f
ac
e
(
)
b
e
d
ef
in
ed
as
(
1
1
)
:
(
)
=
−
(
)
+
(
)
(
1
1
)
wh
er
e
>
0
.
T
h
e
eq
u
iv
ale
n
t sli
d
in
g
m
o
d
e
c
o
n
tr
o
l is o
b
tain
e
d
b
y
tak
in
g
th
e
f
r
ac
tio
n
al
o
r
d
er
d
er
iv
ativ
e
o
f
(
1
0
)
as
:
(
)
=
−
+
(
)
+
(
)
=
0
(
1
2
)
t
h
u
s
,
(
)
+
(
1
−
x
2
+
α
x
4
−
β
x
6
)
y
−
x
+
E
c
os
(
t
)
)
+
u
=
0
(
1
3
)
(
)
=
−
[
1
−
2
+
4
−
6
]
+
(
1
−
)
−
c
os
(
)
(
1
4
)
c
h
o
o
s
in
g
th
e
f
o
llo
win
g
s
witch
co
n
tr
o
l la
w
=
−
−
1
ℎ
(
)
(
1
5
)
w
e
o
b
tain
,
(
)
=
(
)
+
(
)
(
1
6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
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lec
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C
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,
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l.
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6
,
No
.
2
,
Ap
r
il
20
2
6
:
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2
9
-
738
734
an
d
,
(
)
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−
[
1
−
2
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4
−
6
]
+
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1
−
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−
c
os
(
)
−
−
1
ℎ
(
)
(
1
7
)
T
h
e
s
lid
in
g
m
o
d
e
m
o
tio
n
o
cc
u
r
r
en
ce
is
p
er
m
itted
b
y
a
n
ad
eq
u
ate
co
n
tr
o
l la
w
(
)
is
d
esig
n
ed
i
n
(
17
).
T
h
en
,
we
ca
n
s
tate
th
e
f
o
llo
wi
n
g
m
ain
r
esu
lt:
T
h
eo
r
em:
C
o
n
s
id
er
th
e
f
r
ac
tio
n
al
o
r
d
er
b
io
lo
g
ical
ch
a
o
tic
s
y
s
tem
(
9
)
.
T
h
en
,
t
h
e
s
lid
in
g
m
o
d
e
co
n
t
r
o
l
law
(
1
7
)
g
u
ar
an
ties
th
e
co
n
v
er
g
e
o
f
t
h
e
en
z
y
m
e
s
y
s
tem
tr
ajec
to
r
ies
to
th
e
s
lid
in
g
s
u
r
f
ac
e
s
(
t)
=
0
an
d
th
e
s
tate
v
ar
iab
les v
an
is
h
asy
m
p
t
o
tically
.
P
r
o
o
f o
f th
e
th
e
o
r
em:
L
et
u
s
co
n
s
id
er
th
e
f
o
llo
win
g
L
y
ap
u
n
o
v
ca
n
d
id
ate
f
u
n
ctio
n
:
=
1
2
2
(
1
8
)
I
ts
tim
e
d
er
iv
ativ
e
is
g
iv
en
as
(
1
9
)
:
̇
=
̇
=
1
−
{
}
=
1
−
{
(
−
)
+
}
=
1
−
{
+
(
1
−
x
2
+
α
x
4
−
β
x
6
)
y
−
x
+
E
c
os
(
t
)
+
u
}
(
1
9
)
w
h
er
e
(
)
=
(
)
+
(
)
(
2
0
)
T
h
e
co
n
tr
o
l
(
)
is
ch
o
s
en
as
(
2
1
)
:
(
)
=
−
(
1
−
2
+
4
−
6
)
+
(
1
−
)
−
(
)
(
2
1
)
T
h
u
s
̇
ca
n
b
e
ex
p
r
ess
ed
as:
̇
=
1
−
{
}
(
2
2
)
c
h
o
o
s
in
g
:
=
−
[
(
)
]
−
1
(
2
3
)
w
h
er
e
K
is
a
s
tr
ictly
p
o
s
itiv
e
n
u
m
b
er
.
W
e
h
a
v
e
f
r
o
m
(
2
2
)
,
̇
≤
−
1
−
{
[
(
)
]
−
1
}
≤
−
|
|
≤
0
(
2
4
)
T
h
is
co
n
clu
d
es th
e
p
r
o
o
f
.
As th
e
s
ig
n
f
u
n
ctio
n
is
d
is
co
n
tin
u
o
u
s
at
0
,
we
r
ep
lace
u
sw
by
(
2
5
)
,
=
−
−
1
ℎ
(
)
(
2
5
)
I
n
f
ac
t
,
u
sw
is
d
esig
n
ed
u
s
in
g
ta
n
h
(
s
)
as
a
co
n
tin
u
o
u
s
ap
p
r
o
x
im
atio
n
o
f
th
e
s
ig
n
f
u
n
ctio
n
to
elim
in
ate
th
e
ch
atter
in
g
p
h
en
o
m
e
n
o
n
in
in
p
u
t sig
n
als.
T
h
u
s
,
th
e
f
i
n
al
f
o
r
m
u
la
f
o
r
th
e
co
n
tr
o
l la
w
is
g
iv
en
b
y
(
2
6
)
:
(
)
=
(
)
+
(
)
=
−
(
1
−
x
2
+
α
x
4
−
β
x
6
)
y
+
(
1
−
c
)
x
−
E
c
os
(
t
)
−
K
D
q
−
1
ta
h
(
s
)
(
2
6
)
5.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
C
o
n
s
id
er
th
e
s
em
i
-
f
r
ac
tio
n
al
-
o
r
d
er
m
o
d
el
o
f
th
e
en
z
y
m
e
b
io
lo
g
ical
s
y
s
tem
g
i
v
en
b
y
(
1
0
)
with
th
e
p
ar
am
eter
s
v
alu
es:
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
F
r
a
ctio
n
a
l
-
o
r
d
er c
h
a
o
s
mo
d
e
liz
a
tio
n
a
n
d
s
lid
in
g
mo
d
e
c
o
n
t
r
o
l
in
a
b
io
lo
g
ica
l
...
(
S
a
ki
n
a
B
en
r
a
b
a
h
)
735
=
2
.
55
;
=
1
.
7
;
=
3
.
465
;
=
2
.
001
,
=
8
.
27
;
=
1
.
2
.
l
ea
d
in
g
to
th
e
m
o
d
el:
{
x
(
1
)
=
y
(
1
.
2
)
=
(
1
−
x
2
+
α
x
4
−
β
x
6
)
y
−
x
+
E
c
os
(
t
)
)
+
u
(
2
7
)
Un
d
er
th
ese
co
n
d
itio
n
s
,
th
is
s
y
s
tem
ex
h
ib
its
ch
ao
tic
b
e
h
av
i
o
r
as
illu
s
tr
ated
in
Fig
u
r
e
5
.
E
n
zy
m
es
h
av
e
to
b
e
ca
r
ef
u
lly
co
n
tr
o
lled
b
ec
a
u
s
e
th
ey
r
eg
u
late
a
n
d
g
u
id
e
th
e
m
etab
o
lis
m
o
f
th
e
b
o
d
y
c
ells
.
B
y
ap
p
ly
in
g
th
e
f
r
ac
tio
n
al
-
o
r
d
er
SMC
co
n
tr
o
l
law
(
1
9
)
at
th
e
tim
e
=7
0
s
,
we
o
b
tain
th
e
n
u
m
e
r
ical
s
im
u
latio
n
r
esu
lts
illu
s
tr
ated
in
Fig
u
r
es 6
to
8
.
Fig
u
r
e
6
p
r
esen
ts
th
e
s
tates
b
eh
av
io
r
a
n
d
co
n
v
er
g
en
ce
to
th
e
o
r
ig
in
in
a
f
in
ite
tim
e.
T
h
e
s
lid
in
g
s
u
r
f
ac
e
s
is
illu
s
tr
ated
in
Fig
u
r
e
7
a
n
d
clea
r
l
y
is
s
et
to
ze
r
o
a
f
ter
th
e
ap
p
licatio
n
o
f
th
e
co
n
tr
o
l
s
ig
n
al
at
t=7
0
s
.
W
e
o
b
s
er
v
e
th
at
th
e
s
y
s
tem
s
tab
ilizes
r
ap
id
ly
af
ter
th
e
la
u
n
ch
o
f
th
e
co
n
t
r
o
l
s
ig
n
al,
wit
h
ze
r
o
s
tead
y
-
s
tate
er
r
o
r
an
d
th
e
a
b
s
en
ce
o
f
an
y
o
s
cillatio
n
.
Fig
u
r
e
8
p
r
esen
ts
th
e
co
n
tr
o
l
s
ig
n
al
a
n
d
s
h
o
ws
a
s
lig
h
t
ch
atter
in
g
p
h
en
o
m
en
o
n
af
ter
th
e
c
o
n
tr
o
l
b
eg
in
n
in
g
in
s
tan
t.
Fo
r
tu
n
ately
,
th
e
s
p
ec
ialized
liter
at
u
r
e
o
f
f
er
s
s
ev
er
al
tech
n
iq
u
es
f
o
r
r
ed
u
cin
g
,
o
r
ev
en
elim
in
atin
g
,
th
is
ch
atte
r
in
g
p
h
e
n
o
m
e
n
o
n
,
s
u
c
h
as
h
ig
h
er
-
o
r
d
e
r
SMC
,
ad
ap
tiv
e
g
ain
s
ch
ed
u
lin
g
,
s
u
p
er
-
twis
tin
g
SMC
.
I
n
o
r
d
er
to
h
ig
h
lig
h
t
th
e
lev
el
o
f
p
er
f
o
r
m
a
n
ce
ac
h
iev
ed
b
y
th
e
p
r
o
p
o
s
ed
co
n
tr
o
l sch
em
e,
we
will m
ak
e
u
s
e
o
f
th
e
i
n
teg
r
al
o
f
s
q
u
ar
ed
e
r
r
o
r
(
I
SE)
J
as,
=
√
∫
(
2
+
2
)
(
2
8
)
wh
er
e
t
c
is
th
e
tim
e
o
f
co
n
tr
o
l
ap
p
licatio
n
an
d
t
f
is
th
e
s
im
u
latio
n
tim
e
d
u
r
atio
n
.
T
a
b
le
1
illu
s
tr
ates
th
e
p
er
f
o
r
m
an
ce
o
f
th
e
p
r
o
p
o
s
ed
SMC
co
n
tr
o
l sy
s
tem
(
r
esp
o
n
s
e
tim
e
τ
r
an
d
I
SE)
.
Fig
u
r
e
5
.
Fra
ctio
n
al
-
o
r
d
er
ch
a
o
tic
attr
ac
to
r
Fig
u
r
e
6
.
C
o
n
tr
o
lled
s
tate
v
ar
i
ab
les o
f
th
e
b
i
o
lo
g
ical
s
y
s
tem
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.
1
6
,
No
.
2
,
Ap
r
il
20
2
6
:
7
2
9
-
738
736
Fig
u
r
e
7
.
Sli
d
in
g
s
u
r
f
ac
e
S
(
t)
Fig
u
r
e
8
.
Fra
ctio
n
al
-
o
r
d
er
SMC
co
n
tr
o
l sig
n
al
T
ab
le
1
.
C
o
n
tr
o
l sy
s
tem
p
er
f
o
r
m
an
ce
t
c
:
Ti
m
e
o
f
c
o
n
tro
l
a
p
p
li
c
a
ti
o
n
(s)
t
f
:
S
imu
latio
n
t
ime
d
u
ra
ti
o
n
(s)
τ
r
:
Re
sp
o
n
se
ti
m
e
(s)
J
:
(IS
E)
70
150
5
9
.
2
T
h
e
s
im
u
latio
n
r
esu
lts
d
em
o
n
s
tr
ate
th
e
ef
f
icien
cy
o
f
th
e
p
r
o
p
o
s
ed
SMC
co
n
tr
o
l
m
eth
o
d
to
ac
h
iev
e
th
e
s
tab
ilizatio
n
o
f
th
e
f
r
ac
ti
o
n
al
-
o
r
d
er
c
h
ao
tic
b
i
o
lo
g
ical
en
zy
m
e
s
y
s
tem
.
T
h
e
p
r
o
p
o
s
ed
co
n
t
r
o
l
m
a
d
e
it
p
o
s
s
ib
le
to
d
r
asti
ca
lly
r
ed
u
ce
th
e
co
n
v
er
g
en
ce
tim
e
o
f
th
e
s
y
s
tem
co
m
p
ar
ed
to
th
e
in
te
g
e
r
o
r
d
er
m
o
d
el
an
d
SMC
co
n
tr
o
l.
Ho
wev
er
,
it
is
im
p
o
r
tan
t
to
m
en
tio
n
t
h
e
lim
itatio
n
s
o
f
th
is
s
tu
d
y
o
n
a
b
io
lo
g
ical
en
zy
m
e
d
u
e
t
o
s
en
s
itiv
ity
to
p
ar
am
eter
s
,
u
n
m
o
d
eled
d
y
n
am
ics an
d
o
th
er
b
io
lo
g
ical
in
ter
ac
tio
n
s
.
6.
CO
NCLU
SI
O
N
I
n
th
is
s
tu
d
y
,
th
e
m
o
d
eliza
tio
n
o
f
a
b
io
lo
g
ical
en
zy
m
e
s
y
s
tem
,
wh
ich
ca
n
e
x
h
ib
it
ch
a
o
tic
b
eh
av
io
r
u
s
in
g
a
f
r
ac
tio
n
al
-
o
r
d
er
m
o
d
e
l,
is
p
r
o
p
o
s
ed
an
d
a
f
r
ac
tio
n
a
l
-
o
r
d
er
s
lid
in
g
m
o
d
e
c
o
n
tr
o
l
is
d
esig
n
ed
f
o
r
its
s
tab
ilizatio
n
.
T
h
e
p
r
o
p
o
s
ed
f
r
a
ctio
n
al
-
o
r
d
e
r
m
o
d
el
is
in
s
p
ir
ed
f
r
o
m
t
h
e
o
r
ig
i
n
al
in
teg
er
o
r
d
er
m
o
d
el
p
r
esen
ted
in
th
e
liter
atu
r
e.
T
h
is
in
n
o
v
ati
o
n
h
as
m
ad
e
it
p
o
s
s
ib
le
to
in
tr
o
d
u
ce
s
ev
er
al
f
ea
tu
r
es
th
at
ar
e
m
o
r
e
r
ea
lis
tic
an
d
ad
v
an
tag
e
o
u
s
p
r
o
p
er
ties
to
t
h
e
p
r
o
p
o
s
ed
m
o
d
el,
s
u
ch
as
th
e
m
em
o
r
y
e
f
f
ec
t,
f
aster
an
d
m
o
r
e
co
m
p
lex
d
y
n
am
ics,
an
d
a
n
atu
r
al
r
o
b
u
s
tn
ess
ag
ain
s
t n
o
is
e
an
d
d
is
tu
r
b
an
ce
s
.
B
y
ap
p
ly
in
g
a
f
r
ac
tio
n
al
-
o
r
d
er
SMC
co
n
tr
o
ller
,
we
w
er
e
ab
le
t
o
g
u
ar
an
tee
th
e
asy
m
p
to
tic
s
tab
ilizatio
n
o
f
th
e
d
esig
n
ed
b
io
lo
g
ical
en
zy
m
e
m
o
d
el
u
s
in
g
th
e
L
y
a
p
u
n
o
v
th
e
o
r
y
.
Sim
u
latio
n
r
esu
lts
in
a
MA
T
L
AB
en
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