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Since there are no changes of sign in the coefficients of the first column of the Routh
table it can be concluded that polynomial p
2
./ is also stable. The associated Routh
table for Kharitonov polynomials p
3
./ and p
4
./ are
3
j 1
70:39
2
j 31:91 23:27
(3.70)
1
j 69:66 0
0
j 23:27
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
3
./ is a stable one.
Finally, the associated Routh table for the fourth Kharitonov polynomial p
4
./ is
3
j 1
68:33
2
j 31:31 23:67
(3.71)
1
j 67:57 0
0
j 23:67
Since there are no changes of sign in the coefficients of the first column of the
Routh tables it can be concluded that polynomial p
4
./ is stable. From the previous
analysis it can be seen that all Kharitonov polynomials have stable roots therefore
the fixed point .x
;
1
;x
;
2
;x
;
3
/ of the circadian oscillator under feedback gain
c
2
2Œ60:2; 50:2 is a stable one.
Next, the case that the feedback gain c
2
2 Œ270:2; 260:2 is examined. In such
a case the variation ranges for the coefficients of the characteristic polynomial are
c
f
2
2 Œ58:52; 55:12, c
f
1
2Œ175:03; 163:82 and c
f
0
2 Œ88:17; 84:13.The
four Kharitonov polynomials of the system are:
p
1
./ D84:13 163:82 58:52
2
C
3
p
2
./ D88:17 175:03 55:12
2
C
3
(3.72)
p
3
./ D88:17 163:82 55:12
2
C
3
p
4
./ D84:13 175:03 58:12
2
C
3
The associated Routh table for the first Kharitonov polynomial p
1
./ is
3
j 1
163:82
2
j58:52 84:14
(3.73)
1
j165:25 0
0
j84:14
Since there is 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 an unstable one. The
associated Routh table for Kharitonov polynomial p
1
./ and p
2
./ are
