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C Œ
1:4
C Œ
2:1525
K
d
0:7875
K
d
p./ D
3
K
d
C 2:0375
2
C 1:0313 C
(3.50)
Using the Routh-Hurwitz determinants one has
1:4
1
D c
1
D
K
d
C 2:0375
(3.51)
For
1
>0it should hold K
d
< 0:6871 which is not possible since K
d
is a
positive parameter.
For the determinant of
2
to be zero, where
1:4
!
c
1
c
0
c
3
c
2
K
d
C 2:0375
1
2
D
D
(3.52)
0:7875
K
d
2:1525
K
d
C 1:0313
one obtains the relation
3:0135
K
d
5:0420
K
d
C 2:1013 D 0 whichinturngivesK
d
D
1:2722, K
d
D1:1273. Again, these values are not acceptable since K
d
>0.
Therefore, no Hopf bifurcations can appear at the fixed point .x
;
1
;x
;
2
;x
;
3
/ D
.0;0;0/, under the existing set of parameters' values.
At the same conclusion one arrives if the Routh-Hurwitz matrix is used, that is
C
3
2:1525
K
d
j 1
2
1:4
0:7875
K
d
j
K
d
C 2:0375
(3.53)
1
3:0135
K
d
5:0420
K
d
j
2:1013 0
0
0:7875
K
d
j
For the first coefficient of the second row of the matrix it holds
1:4
K
d
C2:0375 > 0.To
obtain zero coefficients at the line associated with
1
(so as imaginary eigenvalues
to appear in the characteristic polynomial of the preceding line) one has again the
relation
3:0135
K
d
5:0420
K
d
C 2:1013 D 0 whichinturngivesK
d
D1:2722, K
d
D
1:1273. As mentioned, these values are not acceptable since K
d
>0. Therefore,
no Hopf bifurcations can appear at the fixed point .x
;
1
;x
;
2
;x
;
3
/ D .0;0;0/,
under the existing set of parameters' values.
C
3.6
Feedback Control of Bifurcations
Next, it is considered that input K
s
is a feedback control term of the form K
s
D
c
2
x
2
. Then the dynamics of the circadian oscillator becomes
K
i
x
1
K
m
Cx
1
x
1
D
v
s
K
i
Cx
3
v
m
x
2
(3.54)
x
2
D
v
d
K
d
Cx
2
K
1
x
2
C K
2
x
3
C c
2
x
1
x
3
x
3
D K
1
x
2
K
2
x
3