IEEE Circuits and Systems Magazine - Q2 2018 - 40

The most natural fingerprint of the memristor as a fundamental
element is the unconditional preservation of its constitutive
relation in any mode of operation.

Whereas fingerprints left by human beings unmistakably point to their originator, the memristor fingerprints
are a different kettle of fish: Complying with the fingerprint is a necessary but not sufficient condition for the
analyzed element to be a memristor. This fact makes the
identification of the element more difficult. That is why,
according to today's practice, the memristor is identified if more fingerprints are held simultaneously [19].
However, a proof justifying the above procedure is not
available. This uncertainty initiated some technical disputes [20].
The objective of this paper is to refer to L. Chua's key
contribution in the area of new memory principles. For
this purpose, the memristor fingerprints appear as a
useful topic. In 1971, L. Chua outlined ideas whose revolutionariness was appreciated after as much as 37 years,
when the HP memristor came into being. The fingerprint
was one of the important ideas to which he returned after 2008 with the aim to identify the memristance as a
ubiquitous but still disregarded natural principle. He did
so in a way that is typical of him: He built solid foundations, constructed a carcass, and left others to put the
finishing touches to the building according to the current needs of the branch. The so-called Homothety fingerprint is one of such new small stones into this building. It allows demonstrating the pros and cons of the
paradox that the best known fingerprints are looked for
via nonpredictive models of the memristor in voltagecurrent coordinates. The only logical conclusion is to
humbly return to L. Chua's key idea about predictive
modeling [7], [10]: The most natural fingerprint of the
memristor as a fundamental element is the unconditional preservation of its constitutive relation in any mode
of operation. It is demonstrated how such a fingerprint
can be put into practice.
II. Homothety Fingerprint
The fingerprints of type [19] describe a phenomenon
(hysteresis in v - i coordinates) or a tendency (transforming the hysteresis loops into a single-valued function if
frequency increases to infinity), but without specifying
any details (shape of the loop, concrete regularity of diminishing the hysteresis). Some details are then specified via other fingerprints which hold for special types
of memristor, particularly ideal memristors, and driven
by concrete signals (for example the odd symmetry fin40

IEEE CIrCuItS aND SyStEmS magazINE

gerprint [12], the fingerprint of the regularity of diminishing the loop area for memristors with special types of
the constitutive relation [16], etc.).
The present fingerprints are proved only "in one direction", i.e. according to the mechanism "the element
is a memristor" => "the element must exhibit a concrete
fingerprint". This only implies that if the element does
not exhibit a fingerprint, then it surely is not a memristor. Considering any existing fingerprint, the current literature does not provide a proof of whether the element
exhibiting the given fingerprint must necessarily be
a memristor.
Such an analysis will be done below for the homothety fingerprint, published in 2015 [18]. It holds for the original memristors introduced by L. Chua in 1971 as well as
for their equivalents (siblings), i.e. ideal generic memristors, and even for some extended memristors and other
elements. The homothety fingerprint uniquely complements the existing fingerprints due to its ability to specify details. Its beauty consists in an exact prediction of
the results of the experiment, which is performed via a
series of testing signals based on an arbitrary pattern.
One version of the homothety fingerprint is as follows:
If the memristor is driven via a charge/flux of arbitrary waveform, and if this driving is repeated under
the same initial conditions via n-times accelerated
original signal, n ! R +, then in the latter case the
v - i pinched hysteresis loop will be a homothetic entity with respect to the first loop, with the homothetic
center at the v - i origin and the ratio of homothety n.
The original driving signal can be selected arbitrarily, but preferably such that it results in a set of closed
loops in v - i coordinates. Consider the driving waveform
as a flux { = { (t) . This pattern is the basis of other testing signals { n (t ) = { (nt), n ! R +. It is obvious from the
example in Fig. 3 (a), (b) that the original excitation { (t )
corresponds to the voltage excitation v = v (t ) which is
its derivative with respect to time. Similarly, the voltage
excitations v n (t) = n $ v (nt ) correspond to other testing
signals { n (t ). The homothety fingerprint states that an
n-times accelerated and n-times amplified current/voltage excitation causes an n-times accelerated and n-times
amplified voltage/current response of the memristor. It
results in a set of homothetic v - i pinched hysteresis
SECOND quartEr 2018

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q2 2018