Information Technology Reference
In-Depth Information
5.5
Robust Synchronization of Coupled Circadian
Oscillators Using Differential Geometry Methods
The nonlinear state-space model of the coupled circadian oscillators is given by
0
1
0
1
0
1
K
i
x
1
K
M
Cx
1
v
s
K
i
Cx
3
v
m
x
1
x
2
x
3
x
4
x
5
x
6
00
x
1
0
00
00
0x
4
00
@
A
@
A
@
A
x
2
v
d
K
d
Cx
2
K
1
x
2
C K
2
x
3
C K
c
.x
2
x
5
/
K
1
x
2
K
2
x
3
u
1
u
2
D
C
K
i
K
i
Cx
6
v
m
x
4
K
M
Cx
4
v
s
x
5
v
d
K
d
Cx
5
K
1
x
5
C K
2
x
6
C K
c
.x
5
x
2
/
K
1
x
5
K
2
x
6
(5.33)
The following two functions are defined:
z
1
D h
1
.x/ D x
1
and
z
4
D h
2
.x/ D x
4
.It
holds that
K
i
x
1
K
M
C x
1
z
2
D L
f
h
1
.x/ D
v
3
K
i
C x
3
v
m
(5.34)
z
3
D L
f
h
1
.x/ D
v
3
K
i
nx
n
1
.K
i
Cx
3
/
2
.K
1
x
2
K
2
x
3
/
3
.K
M
Cx
1
/
2
v
s
K
m
Cx
1
(5.35)
K
i
K
m
x
1
v
m
K
i
Cx
3
v
m
Moreover, it holds that
z
3
D L
f
h
1
.x/ C L
g
1
L
f
h
1
.x/
u
1
C L
g
2
L
f
h
1
.x/
u
2
(5.36)
where
n
v
m
K
m
2.K
m
C
x
1
/
.K
m
Cx
1
/
4
h
v
s
K
m
Cx
1
i
K
i
x
1
L
f
h
1
.x/ D
K
i
Cx
3
v
m
.K
m
Cx
1
/
2
oh
v
s
K
m
Cx
1
i
K
i
K
m
K
m
x
1
C
v
m
.K
m
Cx
1
/
2
v
m
K
i
Cx
3
v
m
n
v
s
K
i
nx
n
1
.K
i
Cx
3
/
2
K
1
oh
K
d
Cx
2
K
1
x
2
C K
2
x
3
C K
c
.x
2
x
5
/
i
x
2
v
d
3
C
n
v
m
(5.37)
.K
m
Cx
1
/
2
v
s
K
i
nx
n
1
K
m
3
.K
i
Cx
3
/
2
v
s
K
i
n.n
1/x
n
3
.K
i
C
x
3
/
K
i
nx
n
3
2.K
i
C
x
3
/
.K
1
x
2
K
2
x
3
/
.K
i
Cx
3
/
4
.K
i
Cx
3
/
2
K
2
o
ŒK
1
x
2
K
2
x
3
C
v
s
K
i
nx
n
1
3
L
g
1
L
f
h
1
.x/ D
v
s
K
i
nx
.n1/
3
.K
i
C x
3
/
2
K
1
x
1
(5.38)
L
g
2
L
f
h
2
.x/ D 0
(5.39)