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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)
 
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