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a
b
25
14
12
20
10
8
15
6
10
4
2
5
0
0
−2
0.4
0.5
0.6
0.7
0.8
0.9
1
−100
−80
−60
−40
−20
0
20
Kd
c2
2
(
blue line
)andx
;
2
(
red line
) of the circadian
oscillators under variation of the Michaelis parameter K
d
,(
b
) Bifurcation of fixed point x
;
2
(
blue
line
)andx
;
2
(
red line
) of the circadian oscillators under state feedback with gain c
2
(
a
) Bifurcation of fixed point x
;2
Fig. 3.1
The control input of the circadian oscillator is taken to be zero, that is K
s
D 0.
It is assumed that the bifurcating parameter is K
d
(Michaelis constant). By substi-
tuting the numerical values of the parameters of the model into the characteristic
polynomial one obtains
x
2
f0:3750.k
d
1/x
2
C .0:2913K
d
C 0:7587/x
2
C .0:0039K
d
0:8196/gD0
(3.34)
A first fixed point is located at
.x
;
1
;x
;
2
;x
;
3
/ D .0;0;0/
(3.35)
For the binomial appearing in the above equation the determinant is D
0:0791K
d
C 1:6773K
d
0:6536 which is positive for K
d
0:3891. In such a case
there are two more fixed points at
q
0:0791K
d
C 1:6773K
d
0:6538
0:75.K
d
1/
.0:2913K
d
C 0:7587/ C
x
2
D
(3.36)
q
0:0791K
d
C 1:6773K
d
0:6538
0:75.K
d
1/
.0:2913K
d
C 0:7587/
x
2
D
(3.37)
The bifurcation diagram of the fixed points considering as bifurcation parameter
the Michaelis constant K
d
is given in Fig.
3.1
a. Next, the Jacobian matrix of the
dynamic model of the system is computed, which is given by
