A New Class of Chaotic Systems
with Memristor-Like Elements
Professor Milan Stork
Department of Applied Electronics and Telecommunications
Faculty of Electrical Engineering
University of West Bohemia
P.O. Box 314
30614 Plzen, Czech Republic
Abstract: We have learned at school that there are three fundamental
two-terminal elements used for electrical circuits building: resistors,
capacitors and inductors. These are known as passive elements, capable of
dissipating or storing energy - but not, as active elements are, of
generating it. The behavior of each of these elements is described by a
simple linear relationship between two of the four basic physical state
variables describing a circuit: current, voltage, charge and magnetic flux.
As the electrical engineer Leon Chua pointed out in 1971, for the sake of
the logical completeness of circuit theory, a fourth passive element should
in fact be added to the list. He named this hypothetical element, linking
flux and charge, the memristor.
The point is that such an element always appears instantanously as a
nonlinear resistor. However, in fact it represents a new passive element,
which may relate some state variable to flux without storing a magnetic
field. This contrasts strongly with behavior of an inductor, for which a
magnetic field stores all the energy (originating in the potential across
its terminals), later releasing it (as an electromotive force) within the
It is just the inability to duplicate the properties of the memistor with a
combination of the other three (classic) passive circuits elements, what
makes the existence question of the memristor fundamental.
There are six independent permutations of two objects from the bank of four
passive circuit elements above. Obviously, six binary relations might be
constructed to connect pairs of the four fundamental circuit variables
(current i; voltage v; charge q; magnetic flux φ). Five of them are well
known. Two arise from the definitions of two of the physical state variables
concerned: charge and magnetic flux are the time integrals of current and
voltage (dq = i dt and dφ = v dt), respectively.
The other three relations lead to the axiomaticaly given constituent
properties of three classic passive circuit elements: resistance R, is the
rate of change of voltage with current; capacitance C, is the rate of change
of charge with voltage; and similarly inductance L, is that of flux with
The sixth relation leads to a fourth basic circuit element, which had been
missing, and has now been found: the memristor, with memristance M, defined
as the rate of change of flux with charge.
Whether physically realized or not, since memristance was first proposed the
memristor has been successfully used as a conceptual tool for analyzing
signals, and for successful modelling, for instance, nonlinear semiconductor
devices. Beyond its fundamental interest, the excitement lies in the fact
that now - almost 40 years later, Strukov et al. present both a simple
physical model system in which a “generalized resistance” called memristance
should arise and could markedly extend how we can make electronic circuits
work. Even so, the concept has not been widely adopted, possibly because in
normal microscale chips the memristance is negligible. But everything
changes on the nanoscale, because the size of memristance effects increases
as the inverse square of device size. That could make memristors useful for
ultra-dense, non-volatile memory devices. For memristor memory devices to
become reality, and to be readily scaled downwards, the efficient and
reliable design and fabrication of electrode contacts, interconnects and the
active region of the memristor must be assured.
But even to consider an alternative to the transistor is anathema to many
device engineers, and the memristor concept will have a steep slope to climb
towards acceptance. Some will undoubtedly trivialize the realization of this
ubiquitous nanoscale concept, whereas others will embrace it only after the
demonstration of a well-functioning, large-scale array of these densely
packed devices. When that happens, the race towards smaller devices will
proceed at full steam.
In all these instances, a deeper understanding of the memristor’s dynamic
nature, as well as the non-linear energy dissipation effects is necessary.
It is often the simple ideas that stand the test of time.
Recall that essentially the memristor acts as a nonlinear resistor the
dissipation power of which depends on the history of one of the circuit
state variables, e.g. the voltage across it. Its name, a contraction of
“memory resistor”, reflects just that property.
It is easy to deduce that memristance can simply be seen as a
“charge-dependent resistance”. It means that nonzero current implies
instantanously varying charge. However, if no current is applied, the
memristance is constant, and consequently memristor reduces to a static
circuit element – ordinary linear resistor. On the other hand, it implies
that if the memristance increases rapidly, current and power consumption
will quickly stop. This is the essence of the memory effect.
At this point it seems to be evident that the memristance is a special case
of a significantly more general property, occurring in a class of nonlinear
dynamical systems, including e.g. chaos generating systems. From this point
of view a ”generalized memristor” can be seen as an abstract power
dissipation element, (or a subsystem), of a nonlinear system the dissipation
rate of which depends on the history of some system state variables.
Let us now make a further substantial step. It has been mentioned that the
most recognizable signature of the memristor is that when a sinusoidal
voltage is applied to the device, the current – voltage plot takes the form
of a Lissajous curve. A typical example can be formed by combining two
orthogonal harmonic signals i.e. harmonic oscillations that are
perpendicular to each other. It is obviously easy to generate them by a
second order linear dynamical system.
In this lecture instead of harmonic oscillations a relatively broad class of
chaotic oscillations generated by a new class of finite dimensional causal
system representations with memristor-like nonlinear dissipation subsystems
will be investigated.
Brief Biography of the Speaker:
Milan Stork received the M.Sc. degree in electrical engineering from the
Technical University of Plzen, Czech Republic at the department of Applied
electronics in 1974 and Ph.D. degree in automatic control systems at the
Czech Technical University in Prague in 1985. In 1997, he became as
Associate Professor at the Department of Applied Electronics and
Telecommunication, faculty of electrical engineering on University of West
Bohemia in Plzen, Czech Republic. He became the full professor in 2007. He
has numerous journal and conference publications. He is member of editorial
board magazine "Physician and Technology". His research interest includes
analog/digital linear and nonlinear systems, control systems, signal
processing and biomedical engineering, especially cardiopulmonary stress