230
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
The EPs of (6.13) are obtained by letting dϕ M (t)/dt = 0 and they correspond to
the intersection of M(Q 0 ) with v C = 0, i.e., they are solutions of
− bϕ
3
M + aϕ M + Q 0 = 0.
(6.16)
From a graphical viewpoint, to find the EPs we need to find the intersections
between the cubic memristor characteristic
f (ϕ M ) = −aϕ M + bϕ
3
M
and the horizontal straight line
(ϕ M ) = Q 0 .
From the graphic of f (·) (Fig. 6.8), we conclude the following. Let
Q
sn
0 =
2a
3
a
3b
.
(6.17)
1. If |Q 0 | < Q sn
0 , then there are three EPs ¯
ϕ α
M = ¯
ϕ
β
M = ¯
ϕ
γ
M . The dynamic route
(see Fig. 6.10 in the case a = 1, b = 1/3, hence Q sn
0 = 2/3) permits to figure
out the stability properties of the EPs. Note that, according to the arrowheads
on the dynamic route, ¯
ϕ α
M and ¯
ϕ
γ
M attract solutions starting nearby, hence they
are asymptotically stable. Instead, ¯
ϕ
β
M repels solutions starting nearby, i.e., it is
unstable (Fig. 6.10 in the case Q 0 = 0 and Q 0 = 1/3).
2. If Q 0 = Q sn
0 , then there are two EPs ¯
ϕ
αβ
M and ¯
ϕ
γ
M . According to the dynamic
route (Fig. 6.10 in the case Q 0 = 2/3), the EP ¯
ϕ
γ
M is asymptotically stable, while
¯
ϕ
αβ
M is unstable since it repels solutions starting nearby at its right.
3. If Q 0 = −Q sn
0 , then there are two EPs ¯
ϕ α
M and ¯
ϕ
βγ
M . The first one is
asymptotically stable while the latter is unstable.
4. If Q 0 > Q sn
0 , then there is a unique EP ¯
ϕ
γ
M . From the dynamic route (cf. Fig. 6.10
in the case Q 0 = 1), it follows that ¯
ϕ
γ
M is a globally attracting asymptotically
stable EP.
5. If Q 0 < −Q sn
0 , then there is a unique EP ¯
ϕ α
M which is globally attracting.
Remark 6.7 It is also possible to analytically find the roots of a third-order algebraic
equation by using some known techniques and formulas (e.g., [5]).
Let
ˆ
σ =
3
Q 0
2
+
√
Δ
where
Δ =
Q 2
0
4b 2 −
a 3
27b 3 =
Q 2
0
4b 2
1 −
Q sn
0
Q 0
2
.
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
The EPs of (6.13) are obtained by letting dϕ M (t)/dt = 0 and they correspond to
the intersection of M(Q 0 ) with v C = 0, i.e., they are solutions of
− bϕ
3
M + aϕ M + Q 0 = 0.
(6.16)
From a graphical viewpoint, to find the EPs we need to find the intersections
between the cubic memristor characteristic
f (ϕ M ) = −aϕ M + bϕ
3
M
and the horizontal straight line
(ϕ M ) = Q 0 .
From the graphic of f (·) (Fig. 6.8), we conclude the following. Let
Q
sn
0 =
2a
3
a
3b
.
(6.17)
1. If |Q 0 | < Q sn
0 , then there are three EPs ¯
ϕ α
M = ¯
ϕ
β
M = ¯
ϕ
γ
M . The dynamic route
(see Fig. 6.10 in the case a = 1, b = 1/3, hence Q sn
0 = 2/3) permits to figure
out the stability properties of the EPs. Note that, according to the arrowheads
on the dynamic route, ¯
ϕ α
M and ¯
ϕ
γ
M attract solutions starting nearby, hence they
are asymptotically stable. Instead, ¯
ϕ
β
M repels solutions starting nearby, i.e., it is
unstable (Fig. 6.10 in the case Q 0 = 0 and Q 0 = 1/3).
2. If Q 0 = Q sn
0 , then there are two EPs ¯
ϕ
αβ
M and ¯
ϕ
γ
M . According to the dynamic
route (Fig. 6.10 in the case Q 0 = 2/3), the EP ¯
ϕ
γ
M is asymptotically stable, while
¯
ϕ
αβ
M is unstable since it repels solutions starting nearby at its right.
3. If Q 0 = −Q sn
0 , then there are two EPs ¯
ϕ α
M and ¯
ϕ
βγ
M . The first one is
asymptotically stable while the latter is unstable.
4. If Q 0 > Q sn
0 , then there is a unique EP ¯
ϕ
γ
M . From the dynamic route (cf. Fig. 6.10
in the case Q 0 = 1), it follows that ¯
ϕ
γ
M is a globally attracting asymptotically
stable EP.
5. If Q 0 < −Q sn
0 , then there is a unique EP ¯
ϕ α
M which is globally attracting.
Remark 6.7 It is also possible to analytically find the roots of a third-order algebraic
equation by using some known techniques and formulas (e.g., [5]).
Let
ˆ
σ =
3
Q 0
2
+
√
Δ
where
Δ =
Q 2
0
4b 2 −
a 3
27b 3 =
Q 2
0
4b 2
1 −
Q sn
0
Q 0
2
.