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Since there is no change of sign in the first column of the Routh matrix it can be
concluded that the characteristic polynomial is a stable one. For polynomial p 3 ./
it holds
3
j 1 6:41
2
j 5:54 0:88
(3.44)
1
j 6:25 0
0
j 0:88
Since there is no change of sign in the first column of the Routh matrix it can be
concluded that the characteristic polynomial is a stable one. For polynomial p 4 ./
it holds
3
j 1 3:42
2
j 3:59 1:97
(3.45)
1
j 2:87 0
0
j 1:97
Since there is no change of sign in the first column of the Routh matrix it can be
concluded that the characteristic polynomial is a stable one. Consequently, the fixed
point .x ; 1 ;x ; 2 ;x ; 3 / D .0;0;0/is a stable one (sink) for the considered variations
of K p .
3.5.2
Method to Detect Hopf Bifurcations
in the Circadian Cells
Using the characteristic polynomial of the linearized equivalent of the biological
model, it is possible to formulate conditions for the emergence of Hopf bifurcations.
The whole procedure makes again use of the Routh-Hurwitz criterion.
Consider the characteristic polynomial computed at a fixed point
P./D c n n
C c n1 n1
C c n2 n2
(3.46)
CCc 1 C c 0
The coefficients c i ;i D 0 ;nare functions of the bifurcating parameter, which is
denoted as .
The following matrix is introduced
0
@
1
A
c n1 c n 0 0 0 0
c n3 c n2 c n1 0 0 0
c n5 c n4 c n3 c n2 0 0
c n.2j1/ c n.2j2/ c n.2j3/ c n.2j4/ c n.2jnC1/ c n.2jn/
j D
(3.47)
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