IEEE Circuits and Systems Magazine - Q2 2018 - 73

Exploiting their peculiar nonlinear behaviour, strongly dependent upon
their material composition, it is possible to devise new dynamic
circuits and systems computing data in unconventional forms.

subject to. A positive DC current of modulus I m = 1 A is
inserted through the device and its state response is plotted over time for a number of initial conditions, specifically x 0 ! {- 5, - 3.5, - 2.5, - 1.5, - 0.5, 0, .25, 0.75, 2, 4}. As
expected, all state solutions with initial conditions to the
left (right) of the unstable equilibrium Xr 0, 1A = 0 approach
the leftmost (rightmost) equilibrium Xr -,1A ^ Xr +,1A h as time
goes by. For the null initial condition, the state will keep unchanged at all times, since the origin is an equilibrium. In
fact it is called separatrix, because it separates the basins of
attraction of the two locally stable equilibria. The memory
state exhibits a similar bistable dynamic behaviour under
any other DC value for the input current. All in all, the memristor features local fading memory at DC, i.e. it experiences
memory loss in the basin of attraction of each of the two
locally-stable outer equilibria its state equation admits for
all constant input current values.
Interestingly, bi-stability is responsible for the emergence of local fading memory in the memristor under
focus also under AC periodic excitation. As clearly illustrated in Fig. 35, the time waveform of the memory state
in response to a sine wave input current i m = itm sin (2r ft)
with amplitude itm = 10 A and frequency f = 25 Hz approaches a negative (positive)-valued oscillatory solution for all initial conditions to the left (right) of the
origin, which, similarly to the DC case, represents the
separatrix between the basins of attraction of the two

x0 = 10 x = 5
0

x0 = 2.5
x0 = 1

x0 = -1
x0 = -2.5
x0 = -5
x0 = -10

-5

x0 = 0 (Separatrix)

-10
0

0.05

0.1
t /s

0.15

0.2

Figure 35. state over time under ac periodic excitation of
the memristor modelled by the daE set (51)-(52) with an
input current of the form i m = itm sin (2r f t) with amplitude
itm = 10 A and frequency f = 25 Hz for each initial condition
within the set x 0 ! " - 10, - 5, - 2.5, - 1, 0, 1, 2.5, 5, 10 , .

sEcOnd quartEr 2018

locally-stable attractors. For more details on the AC bistability of the memristor modelled by the DAE set (52)(53) the interested reader is invited to consult [47].
In conclusion, it is worthy to mention that a memristor circuit emulator with bistable response to specific
sets of DC and AC stimuli has been already designed by
combining a limited number of known electrical devices. All details may be found in [47].
VI. Conclusions
The most popular applications of memristors lie in the
design of extremely-dense, ultra-fast, and very low-power
non-volatile memories and in the development of novel neuromorphic systems operating according to biological principles. But this is only one side of the landscape of opportunities memristors may offer in the electronics of the future.
Exploiting their peculiar nonlinear behaviour, strongly dependent upon their material composition, it is possible to
devise new dynamic circuits and systems computing data
in unconventional forms. Furthermore, the unique combined capability of memristors with non-volatile memory
capability [40] to store the information embedded in their
state for a very long time after the power is switched off,
and to compute data through their particular dynamic behaviour under input application may allow the development
and circuit implementation of mem-computing paradigms
[37], [38], and, eventually, guide towards the conception of
beyond-Von Neumann computing architectures, overcoming the performance bottleneck associated to the physical
separation between the Central Processing Unit (CPU) and
the memory, and due to the inability of the first to run at
its maximum possible speed due to the frequent inactive
periods it is forced to while waiting for data to be fetched
from/written to the storage units and/or transferred via the
limited bandwidth data communication bus. Last but not
least, the extreme sensitivity of the electrical characteristics of real-world memristors to tiny changes in their initial
condition and/or inputs ([33], [66]-[67]) may be harnessed
to develop novel biological signal sensors [68], while the
intrinsic stochastic variability in the switching kinetics of
certain physical realizations may be leveraged for cryptographic applications. In view of all these promising applications, it is essential to investigate the nonlinear dynamics
of memristors thoroughly ([32], [50]) to allow their more
conscious use in future circuit design. A nonlinear system
is said to be endowed with fading memory capability, a
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