232
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
and
¯
ϕ
γ
M =
a
3b ˆ
σ
+ ˆ
σ .
4. Similarly, if Q 0 = −Q sn
0 , there are two EPs given by
¯
ϕ
α
M = −
a
3b ˆ
σ
− ˆ
σ
and
¯
ϕ
βγ
M =
a
3b
.
5. Finally, suppose |Q 0 | < Q sn
0 . In this case we have Δ < 0 and so we need to
consider
√
Δ in the complex sense. It can be shown via suitable manipulations
that there are three EPs
¯
ϕ
γ
M =
a
3b ˆ
σ
+ ˆ
σ
(6.18)
¯
ϕ
α
M =
1
2
− ¯
ϕ
γ
M + j
√
3(− ¯
ϕ
γ
M + 2 ˆ
σ )
(6.19)
¯
ϕ
β
M =
1
2
− ¯
ϕ
γ
M − j
√
3(− ¯
ϕ
γ
M + 2 ˆ
σ )
(6.20)
and that all three EPs turn out to be real numbers.
For instance, when Q 0 = 0 the three EPs are simply given by
ˆ
ϕ
α
M = −
a
b
, ˆ
ϕ
β
M = 0; ˆ
ϕ
γ
M =
a
b
.
Let us now study the global dynamics of (6.13). The following holds.
Property 6.2 Consider the M − C circuit with the cubic nonlinearity (6.15). Any
solution of the SE (6.13) describing the circuit in the (ϕ, q)-domain is bounded and
hence defined for t ≥ t 0 ; moreover, it converges to an EP as t → +∞.
Proof First, let us verify that any solution of (6.13) is defined and bounded for any
t ≥ t 0 . Consider the vector field −f (ϕ M ) + Q 0 defining (6.13). There exists ˜
ϕ > 0
and > 0 such that −f (ϕ M ) + Q 0 < − < 0 for ϕ M ≥ ˜
ϕ and −f (ϕ M ) + Q 0 >
> 0 for ϕ M ≤ − ˜
ϕ. Then, the set S = [− ˜
ϕ, ˜
ϕ] is positively invariant for (6.13);
moreover, any solution with initial condition ϕ M (t 0 ) /
∈ S reaches S in finite time
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
and
¯
ϕ
γ
M =
a
3b ˆ
σ
+ ˆ
σ .
4. Similarly, if Q 0 = −Q sn
0 , there are two EPs given by
¯
ϕ
α
M = −
a
3b ˆ
σ
− ˆ
σ
and
¯
ϕ
βγ
M =
a
3b
.
5. Finally, suppose |Q 0 | < Q sn
0 . In this case we have Δ < 0 and so we need to
consider
√
Δ in the complex sense. It can be shown via suitable manipulations
that there are three EPs
¯
ϕ
γ
M =
a
3b ˆ
σ
+ ˆ
σ
(6.18)
¯
ϕ
α
M =
1
2
− ¯
ϕ
γ
M + j
√
3(− ¯
ϕ
γ
M + 2 ˆ
σ )
(6.19)
¯
ϕ
β
M =
1
2
− ¯
ϕ
γ
M − j
√
3(− ¯
ϕ
γ
M + 2 ˆ
σ )
(6.20)
and that all three EPs turn out to be real numbers.
For instance, when Q 0 = 0 the three EPs are simply given by
ˆ
ϕ
α
M = −
a
b
, ˆ
ϕ
β
M = 0; ˆ
ϕ
γ
M =
a
b
.
Let us now study the global dynamics of (6.13). The following holds.
Property 6.2 Consider the M − C circuit with the cubic nonlinearity (6.15). Any
solution of the SE (6.13) describing the circuit in the (ϕ, q)-domain is bounded and
hence defined for t ≥ t 0 ; moreover, it converges to an EP as t → +∞.
Proof First, let us verify that any solution of (6.13) is defined and bounded for any
t ≥ t 0 . Consider the vector field −f (ϕ M ) + Q 0 defining (6.13). There exists ˜
ϕ > 0
and > 0 such that −f (ϕ M ) + Q 0 < − < 0 for ϕ M ≥ ˜
ϕ and −f (ϕ M ) + Q 0 >
> 0 for ϕ M ≤ − ˜
ϕ. Then, the set S = [− ˜
ϕ, ˜
ϕ] is positively invariant for (6.13);
moreover, any solution with initial condition ϕ M (t 0 ) /
∈ S reaches S in finite time