298
7 Pulse Programming of Memristor Circuits
manifold M(k 1 ) = M(k 0 + k u ) for any t ≥ t 1 . The dynamics of N is governed by
the non-autonomous SEs (7.27) with
u x (t) = B 11
t
t 0
e(τ )dτ + B 12
t
t 0
a(τ )dτ
(7.44a)
u y (t) = B 21
t
t 0
e(τ )dτ + B 22
t
t 0
a(τ )dτ
(7.44b)
for any t ∈ [t 0 , t 1 ). Point (3) in Theorem 7.3 ensures that the effect of u x (t) and
u y (t) is to drive the solution w(t; t 0 , w 0 ) from manifold M(k 0 ) to M(k 1 ), i.e., for
any t ∈ [t 0 , t 1 ) the dynamics of N is continuously embedded in M(k(t, t 0 , w 0 ))—
see (7.31)—where M(k 0 ) and M(k 1 ) represent the “initial” and “final” manifolds,
respectively.
The following property summarizes the dynamics on manifolds for memristor
circuits subject to pulses with constant momentum for t ≥ t 1 .
Property 7.4 Let us consider a non-autonomous memristor circuit N satisfying
(A1)–(A4) and with constant momentum sources (7.6.2), in the (ϕ, q)-domain, for
t ≥ t 1 . Then, for any w 0 ∈ R (n M +n C +n L ) we have
w(t; t 0 , w 0 ) ∈ M(k 1 ), ∀t ≥ t 1
where k 1 = k 0 + k u , k 0 is given in (7.28) and k u in (7.43) describes the effect of
constant momentum sources.
The reduced-order dynamics on M(k 1 ) are described in the (ϕ, q)-domain by
the SEs (7.27) with k 0 replaced by k 1 , u x (t) by (B 11 ¯
e + B 12 ¯
a)Δ and u y (t) by
(B 21 ¯
e + B 22 ¯
a)Δ, for any t ≥ t 1 .
Remark 7.2 We stress that voltage and current sources with constant momentum
for t ≥ t 1 as in (7.6.2) can be obtained by means of voltage and/or current sources
with different pulse duration and shape. This means that only the area (i.e., the
momentum) of the waveforms e(t) and/or a(t) over different finite time intervals is
important to set the manifold M(k 1 ) on which the dynamics of N takes place once
all the pulses are over. This result agrees with the experimental results available in
literature that show how memristors can be programmed (almost) in the same way
by using for instance triangular or squared pulses or finite impulse trains.
7.6.3 Memristor Circuits with Time-Varying Sources
Let us consider memristor circuits N with voltage sources e(t) and/or current
sources a(t) varying over the infinite time interval [t 0 , +∞), i.e., N has sources
with “time-varying momentum” in the (ϕ, q)-domain. In this case the dynamics of
7 Pulse Programming of Memristor Circuits
manifold M(k 1 ) = M(k 0 + k u ) for any t ≥ t 1 . The dynamics of N is governed by
the non-autonomous SEs (7.27) with
u x (t) = B 11
t
t 0
e(τ )dτ + B 12
t
t 0
a(τ )dτ
(7.44a)
u y (t) = B 21
t
t 0
e(τ )dτ + B 22
t
t 0
a(τ )dτ
(7.44b)
for any t ∈ [t 0 , t 1 ). Point (3) in Theorem 7.3 ensures that the effect of u x (t) and
u y (t) is to drive the solution w(t; t 0 , w 0 ) from manifold M(k 0 ) to M(k 1 ), i.e., for
any t ∈ [t 0 , t 1 ) the dynamics of N is continuously embedded in M(k(t, t 0 , w 0 ))—
see (7.31)—where M(k 0 ) and M(k 1 ) represent the “initial” and “final” manifolds,
respectively.
The following property summarizes the dynamics on manifolds for memristor
circuits subject to pulses with constant momentum for t ≥ t 1 .
Property 7.4 Let us consider a non-autonomous memristor circuit N satisfying
(A1)–(A4) and with constant momentum sources (7.6.2), in the (ϕ, q)-domain, for
t ≥ t 1 . Then, for any w 0 ∈ R (n M +n C +n L ) we have
w(t; t 0 , w 0 ) ∈ M(k 1 ), ∀t ≥ t 1
where k 1 = k 0 + k u , k 0 is given in (7.28) and k u in (7.43) describes the effect of
constant momentum sources.
The reduced-order dynamics on M(k 1 ) are described in the (ϕ, q)-domain by
the SEs (7.27) with k 0 replaced by k 1 , u x (t) by (B 11 ¯
e + B 12 ¯
a)Δ and u y (t) by
(B 21 ¯
e + B 22 ¯
a)Δ, for any t ≥ t 1 .
Remark 7.2 We stress that voltage and current sources with constant momentum
for t ≥ t 1 as in (7.6.2) can be obtained by means of voltage and/or current sources
with different pulse duration and shape. This means that only the area (i.e., the
momentum) of the waveforms e(t) and/or a(t) over different finite time intervals is
important to set the manifold M(k 1 ) on which the dynamics of N takes place once
all the pulses are over. This result agrees with the experimental results available in
literature that show how memristors can be programmed (almost) in the same way
by using for instance triangular or squared pulses or finite impulse trains.
7.6.3 Memristor Circuits with Time-Varying Sources
Let us consider memristor circuits N with voltage sources e(t) and/or current
sources a(t) varying over the infinite time interval [t 0 , +∞), i.e., N has sources
with “time-varying momentum” in the (ϕ, q)-domain. In this case the dynamics of