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Since there is no change of sign in the first column of the Routh matrix it can be
concluded that polynomial p
2
./ has stable roots. For polynomial p
3
./
3
j 1
9:5
2
j 7:7
8
(3.12)
1
j 8:4610 0
0
j 8
Since there is no change of sign in the first column of the Routh matrix it can be
concluded that polynomial p
3
./ has stable roots. For polynomial p
4
./
3
j 1
9:5
2
j 6:3
12
(3.13)
1
j 7:5952 0
0
j 12
Since there is no change of sign in the first column of the Routh matrix it can be
concluded that polynomial p
4
./ has stable roots.
Using that all four Kharitonov characteristic polynomials p
i
./ have stable
roots it can be concluded that the initial characteristic polynomial p./ has stable
roots when its parameters vary in the previously defined ranges. Equivalently, the
computation of the four Kharitonov characteristic polynomials can be performed as
follows:
p
1
./ D c
max
0
C c
ma
1
C c
mi
2
2
C c
mi
3
3
C c
ma
4
4
C c
ma
5
5
C c
mi
6
6
C
p
2
./ D c
min
0
C c
mi
1
C c
ma
2
2
C c
ma
3
3
C c
mi
4
4
C c
mi
5
5
C c
ma
6
6
C
p
3
./ D c
min
0
C c
ma
1
C c
ma
2
2
C c
mi
3
3
C c
mi
4
4
C
ma
5
5
C c
ma
6
6
C
p
4
./ D c
max
0
C c
mi
1
C c
mi
2
2
C c
ma
3
3
C c
ma
4
4
C c
mi
5
5
C c
mi
6
6
C
(3.14)
while an equivalent statement of Kharitonov's theorem is as follows [
162
,
192
]:
Theorem.
Each characteristic polynomial in the interval polynomials family
p./
is stable if and only if all four Kharitonov polynomials
p
i
./; i D 1; ;4
are
stable.
3.3
Stages in Bifurcations Analysis
The classification of fixed points according to their stability features has been given
in Sect.
2.4
. Here the classification of fixed points' bifurcations is overviewed. Bifur-
cation of the equilibrium point means that the locus of the equilibrium on the plane
(having as horizontal axis the bifurcation parameter and as vertical axis the fixed
point variable) changes due to variation of the system's parameters. Equilibrium
