274
7 Pulse Programming of Memristor Circuits
a(t) =
⎧
⎨
⎩
0, t ∈ [t 0 , t 1 )
I 0 , t ∈ [t 1 , t 2 ]
0, t > t 2 .
To see the effect of the current source note that the KCL yields, for t ≥ t 0 , i M (t) +
i C (t) = a(t). Since i M (t) = dq M (t)/dt, we obtain
d(f (ϕ M (t)))
dt
+
d(Cv C (t))
dt
=
dQ(t)
dt
= a(t).
By integrating between t 2 and t 1 we obtain
Q(t 2 ) − Q(t 0 ) =
t 2
t 1
a(τ )dτ = I 0 (t 2 − t 1 ).
Hence, if we choose I 0 = ( ¯
Q − Q 0 )/(t 2 − t 1 ), we obtain Q(t 2 ) = ¯
Q. Since
a(t) = 0 for t ≥ t 2 , it follows that the dynamics of the M − C circuit evolves on
manifold M( ¯
Q) for any t ≥ t 2 . Finally note that, as required, the dynamics evolves
on M(Q 0 ) in [t 0 , t 1 ] since a(t) = 0, t ∈ [t 0 , t 1 ), implies Q(t) = Q 0 , t ∈ [t 0 , t 1 ].
It is worth noting that a single current source and a single pulse is able to switch
the dynamics of the M − C circuit between manifolds M(Q 0 ) and M( ¯
Q). It is also
clear that what counts is the area of the pulse impressed by the current source, not
the shape of a(t). In other words, any current pulse with area
t 2
t 1
a(τ )dτ = ¯
Q − Q 0
would result in the M − C circuit evolving on M( ¯
Q) for t ≥ t 2 .
Switching the dynamics of the M − C circuit between different manifolds is
of potential practical interest since, as we have seen in Chap. 6, the dynamics on
different manifolds may be qualitatively different. In fact, in the M − C circuit in
Example 7.1 there coexist both mono-stable dynamics and bistable dynamics on
different manifolds for a cubic memristor nonlinearity q = −ϕ +
1
3 ϕ 3 and so via a
pulse we can easily switch between these two different types of dynamics.
The idea in this simple example will be greatly generalized in this chapter by
showing how we can programme the dynamics of a large class of memristor circuits
via the application of suitable finite-duration impulsive current or voltage sources.
7.3 Structure of Memristor Network and Differential
Algebraic Equation Description
Consider as in Chap. 5 the class LM of memristor circuits constituted by n C ideal
capacitors, n L ideal inductors, n R ideal resistors, n E ideal independent voltage
sources, n A ideal independent current sources, and n M memristors that are either
flux-controlled or charge-controlled. Assume that any flux-controlled memristor is
described by a CR of the type q(t) = f (ϕ(t)), where ϕ(t) and q(t) are the flux
7 Pulse Programming of Memristor Circuits
a(t) =
⎧
⎨
⎩
0, t ∈ [t 0 , t 1 )
I 0 , t ∈ [t 1 , t 2 ]
0, t > t 2 .
To see the effect of the current source note that the KCL yields, for t ≥ t 0 , i M (t) +
i C (t) = a(t). Since i M (t) = dq M (t)/dt, we obtain
d(f (ϕ M (t)))
dt
+
d(Cv C (t))
dt
=
dQ(t)
dt
= a(t).
By integrating between t 2 and t 1 we obtain
Q(t 2 ) − Q(t 0 ) =
t 2
t 1
a(τ )dτ = I 0 (t 2 − t 1 ).
Hence, if we choose I 0 = ( ¯
Q − Q 0 )/(t 2 − t 1 ), we obtain Q(t 2 ) = ¯
Q. Since
a(t) = 0 for t ≥ t 2 , it follows that the dynamics of the M − C circuit evolves on
manifold M( ¯
Q) for any t ≥ t 2 . Finally note that, as required, the dynamics evolves
on M(Q 0 ) in [t 0 , t 1 ] since a(t) = 0, t ∈ [t 0 , t 1 ), implies Q(t) = Q 0 , t ∈ [t 0 , t 1 ].
It is worth noting that a single current source and a single pulse is able to switch
the dynamics of the M − C circuit between manifolds M(Q 0 ) and M( ¯
Q). It is also
clear that what counts is the area of the pulse impressed by the current source, not
the shape of a(t). In other words, any current pulse with area
t 2
t 1
a(τ )dτ = ¯
Q − Q 0
would result in the M − C circuit evolving on M( ¯
Q) for t ≥ t 2 .
Switching the dynamics of the M − C circuit between different manifolds is
of potential practical interest since, as we have seen in Chap. 6, the dynamics on
different manifolds may be qualitatively different. In fact, in the M − C circuit in
Example 7.1 there coexist both mono-stable dynamics and bistable dynamics on
different manifolds for a cubic memristor nonlinearity q = −ϕ +
1
3 ϕ 3 and so via a
pulse we can easily switch between these two different types of dynamics.
The idea in this simple example will be greatly generalized in this chapter by
showing how we can programme the dynamics of a large class of memristor circuits
via the application of suitable finite-duration impulsive current or voltage sources.
7.3 Structure of Memristor Network and Differential
Algebraic Equation Description
Consider as in Chap. 5 the class LM of memristor circuits constituted by n C ideal
capacitors, n L ideal inductors, n R ideal resistors, n E ideal independent voltage
sources, n A ideal independent current sources, and n M memristors that are either
flux-controlled or charge-controlled. Assume that any flux-controlled memristor is
described by a CR of the type q(t) = f (ϕ(t)), where ϕ(t) and q(t) are the flux