246
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
q M (t 0 ) = q M 0
(6.32f)
where h (q M (t)) = −a + 3bq 2
M (t). In particular, it can be easily checked that if
ϕ C (t; t 0 ), q L (t; t 0 ) is the unique solution of the IVP given in (6.29), then v C (t) =
dϕ C (t; t 0 )/dt, i L (t) = dq L (t; t 0 )/dt, q M (t) = q L (t; t 0 ) + q M 0 , is the unique
solution of the IVP (6.32) for t ≥ t 0 . Conversely, if v C (t), q L (t), ϕ M (t) is the
solution of the IVP (6.32) for t ≥ t 0 , then ϕ C (t; t 0 ) =
t
t 0
v C (τ )dτ , q L (t; t 0 ) =
t
t 0
i L (τ )dτ is the solution of the IVP (6.29) for t ≥ t 0 .
Note that (6.32) is an IVP for a third-order system in the (v, i)-domain
whereas (6.29) is an IVP for a second-order system in the (ϕ, q)-domain, i.e.,
FCAM leads to formulate the circuit equations by means of a reduced-order SE
in the (ϕ, q)-domain. It is also worth noting that the third-order oscillator (6.32)
would be difficult to analyze directly in the (v, i)-domain since its equations do not
resemble those of any typical oscillator.
6.2.3 Invariant Manifolds
The KqLs (6.28b) and (6.28c) imply the law of conservation of incremental charge
q C (t; t 0 ) + q M (t; t 0 ) = 0, i.e., we have
Q(t)
.
= q C (t) + q M (t) = q C 0 + q M 0 = Q 0
for any t ≥ t 0 , where Q 0 is given in (6.31). This is equivalent to saying that Q(t)
is an invariant of motion for the SEs (6.32) and so we can define the positively
invariant manifolds
M(Q 0 ) = {(v C (t), i L (t), q M (t))
T
∈ R
3
: Cv C (t) + q M (t) = Q 0 }
(6.33)
for the circuit dynamics in the (v, i)-domain. Note that each manifold is simply
a plane in the state-space (v C (t), i L (t), q M (t)) T ∈ R 3 in the (v, i)-domain. Since
each manifold is identified by Q 0 ∈ R, there are ∞ 1 of such manifolds. Moreover,
it can be easily checked that they are non-intersecting and that the whole state-space
in the (v, i)-domain is covered by such manifolds by varying Q 0 in R. In this way
we obtained a foliation of the phase-space (v C (t), i L (t), q M (t)) in ∞ 1 invariant
manifolds M(Q 0 ) and on each manifold the dynamics is described in the (ϕ, q)domain by the second-order system (6.30).
Note that the vector field defining the SEs (6.30) depends on the circuit
parameters (L, C, a, b) and on Q 0 as well, where Q 0 in turn depends on the initial
conditions v C 0 , i L 0 and q M 0 for the state variables in the (v, i)-domain.
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
q M (t 0 ) = q M 0
(6.32f)
where h (q M (t)) = −a + 3bq 2
M (t). In particular, it can be easily checked that if
ϕ C (t; t 0 ), q L (t; t 0 ) is the unique solution of the IVP given in (6.29), then v C (t) =
dϕ C (t; t 0 )/dt, i L (t) = dq L (t; t 0 )/dt, q M (t) = q L (t; t 0 ) + q M 0 , is the unique
solution of the IVP (6.32) for t ≥ t 0 . Conversely, if v C (t), q L (t), ϕ M (t) is the
solution of the IVP (6.32) for t ≥ t 0 , then ϕ C (t; t 0 ) =
t
t 0
v C (τ )dτ , q L (t; t 0 ) =
t
t 0
i L (τ )dτ is the solution of the IVP (6.29) for t ≥ t 0 .
Note that (6.32) is an IVP for a third-order system in the (v, i)-domain
whereas (6.29) is an IVP for a second-order system in the (ϕ, q)-domain, i.e.,
FCAM leads to formulate the circuit equations by means of a reduced-order SE
in the (ϕ, q)-domain. It is also worth noting that the third-order oscillator (6.32)
would be difficult to analyze directly in the (v, i)-domain since its equations do not
resemble those of any typical oscillator.
6.2.3 Invariant Manifolds
The KqLs (6.28b) and (6.28c) imply the law of conservation of incremental charge
q C (t; t 0 ) + q M (t; t 0 ) = 0, i.e., we have
Q(t)
.
= q C (t) + q M (t) = q C 0 + q M 0 = Q 0
for any t ≥ t 0 , where Q 0 is given in (6.31). This is equivalent to saying that Q(t)
is an invariant of motion for the SEs (6.32) and so we can define the positively
invariant manifolds
M(Q 0 ) = {(v C (t), i L (t), q M (t))
T
∈ R
3
: Cv C (t) + q M (t) = Q 0 }
(6.33)
for the circuit dynamics in the (v, i)-domain. Note that each manifold is simply
a plane in the state-space (v C (t), i L (t), q M (t)) T ∈ R 3 in the (v, i)-domain. Since
each manifold is identified by Q 0 ∈ R, there are ∞ 1 of such manifolds. Moreover,
it can be easily checked that they are non-intersecting and that the whole state-space
in the (v, i)-domain is covered by such manifolds by varying Q 0 in R. In this way
we obtained a foliation of the phase-space (v C (t), i L (t), q M (t)) in ∞ 1 invariant
manifolds M(Q 0 ) and on each manifold the dynamics is described in the (ϕ, q)domain by the second-order system (6.30).
Note that the vector field defining the SEs (6.30) depends on the circuit
parameters (L, C, a, b) and on Q 0 as well, where Q 0 in turn depends on the initial
conditions v C 0 , i L 0 and q M 0 for the state variables in the (v, i)-domain.