52
2 Fundamental Properties of Mem-Elements
Example 2.14 (POP in Memristive Devices) Let us consider a voltage-controlled
memristive device (2.16) with a scalar state variable x defined as
i = G(x, v)v = x
2 v
where
dx
dt
= g(x, v) = −x + α (|x + 1| − |x − 1|) + v.
(2.19)
The POP is given by
dx
dt
= g(x, 0) = −x + α (|x + 1| − |x − 1|) .
(2.20)
Parameter α allows us to change the memory and volatility properties of such
memristive device. In particular, the following cases are considered:
• α = 1 gives the POP in Fig. 2.20a. Note that there are two points x A = x B
where the POP intersects the x-axis with negative slope and one point x 0 where
the intersection has positive slope. It follows that there exist two (isolated)
asymptotically stable EPs x A and x B , and an unstable EP x 0 . Arrows on the
dynamic route denote that x increases (decreases) when ˙
x > 0 ( ˙
x < 0). It
follows that when t → ∞ then x(t) → x A (x(t) → x B ) for any initial condition
x(0) > 0 (x(0) < 0) and so the memristive device can encode two memory
states into the memductance values G(x A , 0) = x 2
A and G(x B , 0) = x 2
B . Hence
the memristive device is nonvolatile and it acts as a discrete memory memristor.
It is worth to note that such discrete (nonvolatile) memory states occur under zero
input voltage, thus this is an inherent property of the memristive device. On the
contrary, an ideal memristor can present discrete memory states only if they are
induced by suitable voltage pulses (cf. Example 2.11).
• α = 0.5 gives the POP in Fig. 2.20b, that has an interval [x B , x A ] with zero slope.
Then, we have a continuum of (non-isolated) stable EPs corresponding to any
value of x in the interval [x B , x A ]. In such a case the memristive device operates
as a continuum memory memristor. The memory states are all the memductance
values G(x, 0) = x 2 with x ∈ [x B , x A ].
• α = 0.25 gives the POP in Fig. 2.20c that intersects the x-axis with negative
slope only in x A = 0, thus there exists a unique asymptotically stable EP. In this
case the memristive device serves as volatile memory memristor, because when
the input is turned off the state variable x(t) converges to x A for t → ∞ and
the memristive device has a unique memductance value G(x A , 0) = x 2
A = 0 as
memory state.
If we exclude situations where the POP has intervals with zero slope (i.e., there
are non-isolated EPs for the memristor), we can summarize some of the previous
considerations with the following result.
2 Fundamental Properties of Mem-Elements
Example 2.14 (POP in Memristive Devices) Let us consider a voltage-controlled
memristive device (2.16) with a scalar state variable x defined as
i = G(x, v)v = x
2 v
where
dx
dt
= g(x, v) = −x + α (|x + 1| − |x − 1|) + v.
(2.19)
The POP is given by
dx
dt
= g(x, 0) = −x + α (|x + 1| − |x − 1|) .
(2.20)
Parameter α allows us to change the memory and volatility properties of such
memristive device. In particular, the following cases are considered:
• α = 1 gives the POP in Fig. 2.20a. Note that there are two points x A = x B
where the POP intersects the x-axis with negative slope and one point x 0 where
the intersection has positive slope. It follows that there exist two (isolated)
asymptotically stable EPs x A and x B , and an unstable EP x 0 . Arrows on the
dynamic route denote that x increases (decreases) when ˙
x > 0 ( ˙
x < 0). It
follows that when t → ∞ then x(t) → x A (x(t) → x B ) for any initial condition
x(0) > 0 (x(0) < 0) and so the memristive device can encode two memory
states into the memductance values G(x A , 0) = x 2
A and G(x B , 0) = x 2
B . Hence
the memristive device is nonvolatile and it acts as a discrete memory memristor.
It is worth to note that such discrete (nonvolatile) memory states occur under zero
input voltage, thus this is an inherent property of the memristive device. On the
contrary, an ideal memristor can present discrete memory states only if they are
induced by suitable voltage pulses (cf. Example 2.11).
• α = 0.5 gives the POP in Fig. 2.20b, that has an interval [x B , x A ] with zero slope.
Then, we have a continuum of (non-isolated) stable EPs corresponding to any
value of x in the interval [x B , x A ]. In such a case the memristive device operates
as a continuum memory memristor. The memory states are all the memductance
values G(x, 0) = x 2 with x ∈ [x B , x A ].
• α = 0.25 gives the POP in Fig. 2.20c that intersects the x-axis with negative
slope only in x A = 0, thus there exists a unique asymptotically stable EP. In this
case the memristive device serves as volatile memory memristor, because when
the input is turned off the state variable x(t) converges to x A for t → ∞ and
the memristive device has a unique memductance value G(x A , 0) = x 2
A = 0 as
memory state.
If we exclude situations where the POP has intervals with zero slope (i.e., there
are non-isolated EPs for the memristor), we can summarize some of the previous
considerations with the following result.