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C Π1:4
C Π2:1525
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
 
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