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k
2
C
K
d
C x
2
2
K
1
c
2
x
1
v
m
C
K
m
C x
1
2
v
d
k
d
K
m
c
f
2
D
C
(3.65)
The associated Routh matrix is
3
j
1
c
f
1
2
j
c
f
2
c
f
0
(3.66)
1
j c
f
0
c
f
1
c
f
2
0
0
j
c
f
0
The condition for obtaining Hopf bifurcation is c
f
2
>0and c
f
0
c
f
1
c
f
2
D 0. It can
be confirmed that for the fixed point .x
;
1
;x
;
2
;x
;
3
/ and for the previously used set
of values of the parameters of the circadian oscillator model, solutions c
2
obtained
from the condition c
f
0
c
f
1
c
f
2
D 0 result in c
f
2
<0, therefore Hopf bifurcations
do not appear.
The bifurcation diagram of Fig.
3.1
b shows the limit values of c
2
beyond which
x
;
2
remains positive. Such a value is c
2
< 45:2. It is examined next if for
specific range of variations of parameter c
2
the fixed point .x
;
1
;x
;
2
;x
;
3
/ is a
stable or an unstable one. To this end Kharitonov's theorem will be used again. For
c
2
2Œ60:2; 50:2 it holds that c
f
2
2Œ31:31;31:91, while c
f
1
2Œ68:33;70:39 and
c
f
0
2Œ23:27;23:67. The associated four Kharitonov polynomials take the values
p
1
./ D 23:67 C 70:39 C 31:31
2
C
3
p
2
./ D 23:27 C 68:33 C 31:91
2
C
3
(3.67)
p
3
./ D 23:27 C 70:39 C 31:91
2
C
3
p
4
./ D 23:67 C 68:33 C 31:31
2
C
3
The Routh table for Kharitonov polynomial p
1
./ is
3
j 1
70:39
2
j 31:31 23:67
(3.68)
1
j 69:63 0
0
j 23:67
Since there is no change of sign of the coefficients of the first column of the Routh
table it can be concluded that the characteristic polynomial p
1
./ is a stable one.
The associated Routh table for the second Kharitonov polynomial p
2
./ is
3
j 1
68:33
2
j 31:91 23:27
(3.69)
1
j 67:59 0
0
j 23:27
