10.2 Simple Circuit with C and an Extended Memristor
383
that of a memristor-circuit with a capacitor in parallel to an ideal memristor given
by f − (ϕ M ) = −(a + G 2 )ϕ M + bϕ 3
M .
Let us focus on the nonlinear dynamics in R + . FCAM can be applied to
reduce (10.12) and (10.13) to the first-order ODE in the (ϕ, q)-domain (∀t ≥ t 0 )
C
dϕ M (t; t 0 )
dt
= −f + (ϕ M (t; t 0 ) + ϕ 0 ) + Q
+
0
(10.14)
where ϕ M (t; t 0 ) = ϕ M (t) − ϕ 0 is the incremental flux across the memristor and
Q
+
0 = Cv 0 + f + (ϕ 0 ) is a constant quantity depending on the initial conditions
(ϕ 0 , v 0 ) T ∈ R + for the state variables in the (v, i)-domain.
It can be easily seen that (10.14) has a unique solution ϕ M (t; t 0 ) which is
bounded and hence defined for any t ≥ t 0 . In addition, since (10.14) is an
autonomous first-order ODE, ϕ M (t; t 0 ) → ¯
ϕ and its time derivative ˙
ϕ M (t; t 0 ) → 0
as t → +∞ (Chap. 2). According to FCAM, the solution of (10.12) and (10.13) is
obtained as (v C (t), ϕ M (t)) = ( ˙
ϕ M (t), ϕ M (t) + ϕ 0 ) and then any solution of (10.12)
and (10.13) is bounded and defined for t ≥ t 0 . Moreover (v C (t), ϕ M (t)) → (0, ¯
ϕ)
as t → +∞.
To wrap up, the following result has been proved.
Proposition 10.1 Any solution (v C , ϕ M ) of the SEs describing the eM–C circuit in
the (v, i)-domain is defined and bounded for any t ≥ t 0 . Furthermore, the solution
converges to an EP (0, ¯
ϕ) as t → +∞.
Such a property is analogous to that of the circuit with a capacitor and an ideal
memristor in Chap. 6. The investigation of invariant manifolds for the eM–C circuit
can follow a discussion analogous to that in Example 5.15 of Chap. 5. In brief, it
can be observed that when v 0 > 0, then Q
+
0 is an invariant of motion for (10.12)
and (10.13). Therefore, we can define infinitely many positively invariant manifolds
for (10.12) and (10.13) given by M(Q
+
0 ) = {(v C , ϕ M ) T ∈ R 2 : v C > 0, Cv C +
f + (ϕ M ) = Q
+
0 }, where Q
+
0 ∈ R, and on each manifold the dynamics is described in
the (ϕ, q)-domain by the first-order ODE (10.14). The manifolds are nonintersecting
and span the whole semi-plane v C > 0 of the state-space in the (v, i)-domain.
Moreover, we can prove the existence of bifurcations without parameters of EPs,
i.e., bifurcations due to a change in the initial conditions and Q
+
0 for a fixed set of
parameters, for the reduced-order dynamic equation in the (ϕ, q)-domain.
383
that of a memristor-circuit with a capacitor in parallel to an ideal memristor given
by f − (ϕ M ) = −(a + G 2 )ϕ M + bϕ 3
M .
Let us focus on the nonlinear dynamics in R + . FCAM can be applied to
reduce (10.12) and (10.13) to the first-order ODE in the (ϕ, q)-domain (∀t ≥ t 0 )
C
dϕ M (t; t 0 )
dt
= −f + (ϕ M (t; t 0 ) + ϕ 0 ) + Q
+
0
(10.14)
where ϕ M (t; t 0 ) = ϕ M (t) − ϕ 0 is the incremental flux across the memristor and
Q
+
0 = Cv 0 + f + (ϕ 0 ) is a constant quantity depending on the initial conditions
(ϕ 0 , v 0 ) T ∈ R + for the state variables in the (v, i)-domain.
It can be easily seen that (10.14) has a unique solution ϕ M (t; t 0 ) which is
bounded and hence defined for any t ≥ t 0 . In addition, since (10.14) is an
autonomous first-order ODE, ϕ M (t; t 0 ) → ¯
ϕ and its time derivative ˙
ϕ M (t; t 0 ) → 0
as t → +∞ (Chap. 2). According to FCAM, the solution of (10.12) and (10.13) is
obtained as (v C (t), ϕ M (t)) = ( ˙
ϕ M (t), ϕ M (t) + ϕ 0 ) and then any solution of (10.12)
and (10.13) is bounded and defined for t ≥ t 0 . Moreover (v C (t), ϕ M (t)) → (0, ¯
ϕ)
as t → +∞.
To wrap up, the following result has been proved.
Proposition 10.1 Any solution (v C , ϕ M ) of the SEs describing the eM–C circuit in
the (v, i)-domain is defined and bounded for any t ≥ t 0 . Furthermore, the solution
converges to an EP (0, ¯
ϕ) as t → +∞.
Such a property is analogous to that of the circuit with a capacitor and an ideal
memristor in Chap. 6. The investigation of invariant manifolds for the eM–C circuit
can follow a discussion analogous to that in Example 5.15 of Chap. 5. In brief, it
can be observed that when v 0 > 0, then Q
+
0 is an invariant of motion for (10.12)
and (10.13). Therefore, we can define infinitely many positively invariant manifolds
for (10.12) and (10.13) given by M(Q
+
0 ) = {(v C , ϕ M ) T ∈ R 2 : v C > 0, Cv C +
f + (ϕ M ) = Q
+
0 }, where Q
+
0 ∈ R, and on each manifold the dynamics is described in
the (ϕ, q)-domain by the first-order ODE (10.14). The manifolds are nonintersecting
and span the whole semi-plane v C > 0 of the state-space in the (v, i)-domain.
Moreover, we can prove the existence of bifurcations without parameters of EPs,
i.e., bifurcations due to a change in the initial conditions and Q
+
0 for a fixed set of
parameters, for the reduced-order dynamic equation in the (ϕ, q)-domain.