234
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
On one hand, we have the variation of the circuit parameters C, a, b. On the other
hand, we have the variation of Q 0 , which acts as an additional parameter in the r.h.s.
of the SE in the (ϕ, q)-domain (6.13). Note that Q 0 can be varied by varying the
initial conditions v C 0 and ϕ M 0 for the state variables in the (v, i)-domain even when
circuit parameters are held fixed.
Suppose first we fix Q 0 = ¯
Q 0 , i.e., the initial condition v C 0 and ϕ M 0 for the
state variables in the (v, i)-domain are such that v C 0 =
1
C (−f (ϕ M 0 ) + ¯
Q 0 ). Choose
for instance ¯
Q 0 = 0. In this case, we may have bifurcations of EPs of (6.13) only
if parameter a changes sign. These are the standard bifurcations due to changing a
parameter on a fixed invariant manifold. However, if Q 0 changes, due to changes in
the initial condition v C 0 and ϕ M 0 , then (6.13) can undergo bifurcations of EPs even if
the circuit parameters are kept fixed. In particular, (6.13) is seen to undergo a saddlenode bifurcation when ¯
ϕ
β
M and ¯
ϕ α
M (or else ¯
ϕ
β
M and ¯
ϕ
γ
M ) collide and annihilate each
other. The conditions ¯
ϕ
β
M = ¯
ϕ α
M and ¯
ϕ
β
M = ¯
ϕ
γ
M yield
Q 0 = ±Q
sn
0 = ±
2
3
3
√
a 2
√
3b
(6.21)
for the values of Q 0 at which such saddle-node bifurcation occurs, which is in
accordance with the results on EPs of (6.13) previously obtained.
Remark 6.9 Let us study in more detail the saddle-node bifurcations due to varying
Q 0 . Suppose once more C = 1, a = 1 and b = 1/3. Consider the critical value
Q 0 = 2/3 for which there is a bifurcating EP at ¯
ϕ M = −1. We have
dϕ M
dt
= ϕ M −
1
3
ϕ
3
M + Q 0
.
= F (ϕ M ).
By expanding F in Taylor series in a neighborhood of ¯
ϕ M = −1, we obtain
dϕ M
dt
= F (−1) + F
(−1)(ϕ M + 1) +
1
2
F
(−1)(ϕ M + 1)
2
= Q 0 −
2
3
+ (ϕ M + 1)
2
where we have neglected higher-order terms of the expansion. The situation in a
small neighborhood of ¯
ϕ M = −1 is illustrated in appare prima di Fig. 6.11 for three
values of Q 0 , i.e., Q 0 < 2/3, Q 0 = 2/3, and Q 0 > 2/3. Geometrically, the situation
is analogous to the saddle-node bifurcation of an EP studied in Sect. 4.4.1 of Chap. 4
(cf. Fig. 4.21 in Chap. 4 with Fig. 6.12). There is however a fundamental difference,
since in that case the bifurcation was due to varying a circuit parameter, while in
this case circuit parameters are kept fixed and the bifurcation is due to changing the
initial conditions (Q 0 ).
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
On one hand, we have the variation of the circuit parameters C, a, b. On the other
hand, we have the variation of Q 0 , which acts as an additional parameter in the r.h.s.
of the SE in the (ϕ, q)-domain (6.13). Note that Q 0 can be varied by varying the
initial conditions v C 0 and ϕ M 0 for the state variables in the (v, i)-domain even when
circuit parameters are held fixed.
Suppose first we fix Q 0 = ¯
Q 0 , i.e., the initial condition v C 0 and ϕ M 0 for the
state variables in the (v, i)-domain are such that v C 0 =
1
C (−f (ϕ M 0 ) + ¯
Q 0 ). Choose
for instance ¯
Q 0 = 0. In this case, we may have bifurcations of EPs of (6.13) only
if parameter a changes sign. These are the standard bifurcations due to changing a
parameter on a fixed invariant manifold. However, if Q 0 changes, due to changes in
the initial condition v C 0 and ϕ M 0 , then (6.13) can undergo bifurcations of EPs even if
the circuit parameters are kept fixed. In particular, (6.13) is seen to undergo a saddlenode bifurcation when ¯
ϕ
β
M and ¯
ϕ α
M (or else ¯
ϕ
β
M and ¯
ϕ
γ
M ) collide and annihilate each
other. The conditions ¯
ϕ
β
M = ¯
ϕ α
M and ¯
ϕ
β
M = ¯
ϕ
γ
M yield
Q 0 = ±Q
sn
0 = ±
2
3
3
√
a 2
√
3b
(6.21)
for the values of Q 0 at which such saddle-node bifurcation occurs, which is in
accordance with the results on EPs of (6.13) previously obtained.
Remark 6.9 Let us study in more detail the saddle-node bifurcations due to varying
Q 0 . Suppose once more C = 1, a = 1 and b = 1/3. Consider the critical value
Q 0 = 2/3 for which there is a bifurcating EP at ¯
ϕ M = −1. We have
dϕ M
dt
= ϕ M −
1
3
ϕ
3
M + Q 0
.
= F (ϕ M ).
By expanding F in Taylor series in a neighborhood of ¯
ϕ M = −1, we obtain
dϕ M
dt
= F (−1) + F
(−1)(ϕ M + 1) +
1
2
F
(−1)(ϕ M + 1)
2
= Q 0 −
2
3
+ (ϕ M + 1)
2
where we have neglected higher-order terms of the expansion. The situation in a
small neighborhood of ¯
ϕ M = −1 is illustrated in appare prima di Fig. 6.11 for three
values of Q 0 , i.e., Q 0 < 2/3, Q 0 = 2/3, and Q 0 > 2/3. Geometrically, the situation
is analogous to the saddle-node bifurcation of an EP studied in Sect. 4.4.1 of Chap. 4
(cf. Fig. 4.21 in Chap. 4 with Fig. 6.12). There is however a fundamental difference,
since in that case the bifurcation was due to varying a circuit parameter, while in
this case circuit parameters are kept fixed and the bifurcation is due to changing the
initial conditions (Q 0 ).