Preface
xv
Aims and Scope
The main goal of this book is to develop systematic theoretical methodologies to
analyze nonlinear circuits including memristors as nonlinear dynamic elements.
Such dynamical nonlinear circuits are referred to as memristor circuits. These
methods are essential for understanding the peculiar dynamic properties and
computational capabilities of memristors/mem-elements and exploit their dynamics
to implement future unconventional computing systems.
It is known from circuit theory that it is natural and effective to analyze a
nonlinear RLC circuit, i.e., a circuit with linear/nonlinear resistors (R), inductors
(L), and capacitors (C) (but without memristors), in the traditional voltage-current
(v, i)-domain, i.e., using constitutive relations of circuit elements and Kirchhoff
laws expressed in terms of voltages v and currents i. However, since the definition
of a memristor involves a link between flux ϕ (the integral of voltage or voltage
momentum) and charge q (the integral of current or current momentum), it is natural
to ask the following:
1. Is there a more effective domain, with respect to the traditional (v, i)-domain, to
analyze the nonlinear dynamics of memristor circuits?
2. Do we expect to observe new and peculiar nonlinear dynamic phenomena when
adding one (or more) memristors to a classical RLC circuit?
An extended literature is available on the analysis of dynamic memristor circuits.
Several papers point out via experimental or numerical means that, generally
speaking, including a memristor in an RLC circuit greatly enriches the dynamics. In
particular, it appears that due to the memristor there can coexist different dynamics
and regimes (e.g., convergent, oscillatory, complex dynamics) for the same set of
circuit parameters. Several basic aspects of the observed behaviors are however
elusive and remain unclear and in large part unexplained. It is not even clear from
a mathematical viewpoint whether the order of the dynamics of an RLC circuit
increases or not when a memristor is added to it.
The chief aim of the book is to answer these fundamental questions and provide
an analytic treatment and clear explanation of dynamic phenomena reported in
the literature via numerical or experimental means. The treatment is rigorous and
based on tools and techniques with foundation in nonlinear circuit theory. Whenever
possible, we try however to keep mathematics at the minimum indispensable level
for an accurate description.
In this book, we identify and select progressively relevant classes of memristor
circuits, widely investigated in the literature and used in the applications, and
describe a new method for their dynamic analysis. The new analysis method
introduced in this book is based on suitable forms of Kirchhoff laws and constitutive
relations of circuit elements expressed in the flux-charge (ϕ, q)-domain rather than
in the traditional (v, i)-domain. Thus the name Flux-Charge Analysis Method, or
FCAM, in short. FCAM has been mainly developed in a series of recent articles
[14–20].
xv
Aims and Scope
The main goal of this book is to develop systematic theoretical methodologies to
analyze nonlinear circuits including memristors as nonlinear dynamic elements.
Such dynamical nonlinear circuits are referred to as memristor circuits. These
methods are essential for understanding the peculiar dynamic properties and
computational capabilities of memristors/mem-elements and exploit their dynamics
to implement future unconventional computing systems.
It is known from circuit theory that it is natural and effective to analyze a
nonlinear RLC circuit, i.e., a circuit with linear/nonlinear resistors (R), inductors
(L), and capacitors (C) (but without memristors), in the traditional voltage-current
(v, i)-domain, i.e., using constitutive relations of circuit elements and Kirchhoff
laws expressed in terms of voltages v and currents i. However, since the definition
of a memristor involves a link between flux ϕ (the integral of voltage or voltage
momentum) and charge q (the integral of current or current momentum), it is natural
to ask the following:
1. Is there a more effective domain, with respect to the traditional (v, i)-domain, to
analyze the nonlinear dynamics of memristor circuits?
2. Do we expect to observe new and peculiar nonlinear dynamic phenomena when
adding one (or more) memristors to a classical RLC circuit?
An extended literature is available on the analysis of dynamic memristor circuits.
Several papers point out via experimental or numerical means that, generally
speaking, including a memristor in an RLC circuit greatly enriches the dynamics. In
particular, it appears that due to the memristor there can coexist different dynamics
and regimes (e.g., convergent, oscillatory, complex dynamics) for the same set of
circuit parameters. Several basic aspects of the observed behaviors are however
elusive and remain unclear and in large part unexplained. It is not even clear from
a mathematical viewpoint whether the order of the dynamics of an RLC circuit
increases or not when a memristor is added to it.
The chief aim of the book is to answer these fundamental questions and provide
an analytic treatment and clear explanation of dynamic phenomena reported in
the literature via numerical or experimental means. The treatment is rigorous and
based on tools and techniques with foundation in nonlinear circuit theory. Whenever
possible, we try however to keep mathematics at the minimum indispensable level
for an accurate description.
In this book, we identify and select progressively relevant classes of memristor
circuits, widely investigated in the literature and used in the applications, and
describe a new method for their dynamic analysis. The new analysis method
introduced in this book is based on suitable forms of Kirchhoff laws and constitutive
relations of circuit elements expressed in the flux-charge (ϕ, q)-domain rather than
in the traditional (v, i)-domain. Thus the name Flux-Charge Analysis Method, or
FCAM, in short. FCAM has been mainly developed in a series of recent articles
[14–20].