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11 Nonlinear Dynamics of Circuits with Mem-Elements
In the paper [19], an adaptive reactive element is implemented and modeled as
memcapacitor/meminductor and it is demonstrated that such component is ideal
for creating adaptive metasurfaces for manipulating EM waves due to its ability
to hold the electromagnetic state without external bias. The use of memcapacitors
and meminductors for implementing nonlinear oscillators is increasingly explored
and widely investigated theoretically in the literature, see [20–22], and references
therein. More recently, the implementation of neural network synapses via memcapacitors has been suggested [23].
Memristors, memcapacitors, and meminductors are frequently used for modeling
nanoscale devices in combination with nonlinear inductors or capacitors. As a
relevant example, the classical circuit model for a Josephson junction consists
of a parallel connection of a linear capacitor, a linear resistor, and a nonlinear
flux-controlled inductor. However, a more rigorous quantum mechanical analysis
of the Josephson junction dynamics reveals the presence of an additional current
component due to interference among quasi-particle pairs, which can be modeled
with the current flowing into a flux-controlled memristor (cf. Example 1.9 in
Chap. 1).
These considerations show that a challenging topic is to develop ad hoc
methods for investigating the peculiar nonlinear dynamical properties of circuits
containing such emerging nanoscale elements with memory properties, a.k.a. memelements. This is believed to be a crucial step for further understanding stability,
oscillatory and synchronization properties, and more generally the computational
capabilities of neural architectures and reservoir systems with mem-elements, or
memcomputing machines.
Chapter 5 introduced FCAM as an effective tool for analyzing the nonlinear
dynamics of a class LM of nonlinear circuits containing ideal flux- or chargecontrolled memristors and ideal (linear) R, L, C. Goal of the chapter is to show that
FCAM can be naturally extended to a much larger class N e of circuits containing, in
addition to the elements of LM, also memcapacitors, meminductors, and nonlinear
capacitors and inductors. In the chapter we treat in a systematic way the extension to
memcapacitors and meminductors, while discussing via selected examples the case
of nonlinear capacitors and inductors.
The main results in the chapter are summarized as follows.
1. We obtain the CR and equivalent circuit in the (ϕ, q)-domain of each twoterminal element in N e and then we identify wide and relevant subclasses of
N e for which it is possible to write an SE representation in the (ϕ, q)-domain
and also in the traditional (v, i)-domain.
2. It is shown that there is a reduction of order for the SEs in the (ϕ, q)-domain,
with respect to the (v, i)-domain, leading to advantages in the nonlinear dynamic
analysis.
3. Via the extended FCAM it is shown, in the inputless (autonomous) case, that the
state space can be foliated in a continuum of invariant manifolds and there coexist
infinitely many different reduced-order dynamics and attractors for the same set
of circuit parameters. This is the basis to show the existence of bifurcations due
11 Nonlinear Dynamics of Circuits with Mem-Elements
In the paper [19], an adaptive reactive element is implemented and modeled as
memcapacitor/meminductor and it is demonstrated that such component is ideal
for creating adaptive metasurfaces for manipulating EM waves due to its ability
to hold the electromagnetic state without external bias. The use of memcapacitors
and meminductors for implementing nonlinear oscillators is increasingly explored
and widely investigated theoretically in the literature, see [20–22], and references
therein. More recently, the implementation of neural network synapses via memcapacitors has been suggested [23].
Memristors, memcapacitors, and meminductors are frequently used for modeling
nanoscale devices in combination with nonlinear inductors or capacitors. As a
relevant example, the classical circuit model for a Josephson junction consists
of a parallel connection of a linear capacitor, a linear resistor, and a nonlinear
flux-controlled inductor. However, a more rigorous quantum mechanical analysis
of the Josephson junction dynamics reveals the presence of an additional current
component due to interference among quasi-particle pairs, which can be modeled
with the current flowing into a flux-controlled memristor (cf. Example 1.9 in
Chap. 1).
These considerations show that a challenging topic is to develop ad hoc
methods for investigating the peculiar nonlinear dynamical properties of circuits
containing such emerging nanoscale elements with memory properties, a.k.a. memelements. This is believed to be a crucial step for further understanding stability,
oscillatory and synchronization properties, and more generally the computational
capabilities of neural architectures and reservoir systems with mem-elements, or
memcomputing machines.
Chapter 5 introduced FCAM as an effective tool for analyzing the nonlinear
dynamics of a class LM of nonlinear circuits containing ideal flux- or chargecontrolled memristors and ideal (linear) R, L, C. Goal of the chapter is to show that
FCAM can be naturally extended to a much larger class N e of circuits containing, in
addition to the elements of LM, also memcapacitors, meminductors, and nonlinear
capacitors and inductors. In the chapter we treat in a systematic way the extension to
memcapacitors and meminductors, while discussing via selected examples the case
of nonlinear capacitors and inductors.
The main results in the chapter are summarized as follows.
1. We obtain the CR and equivalent circuit in the (ϕ, q)-domain of each twoterminal element in N e and then we identify wide and relevant subclasses of
N e for which it is possible to write an SE representation in the (ϕ, q)-domain
and also in the traditional (v, i)-domain.
2. It is shown that there is a reduction of order for the SEs in the (ϕ, q)-domain,
with respect to the (v, i)-domain, leading to advantages in the nonlinear dynamic
analysis.
3. Via the extended FCAM it is shown, in the inputless (autonomous) case, that the
state space can be foliated in a continuum of invariant manifolds and there coexist
infinitely many different reduced-order dynamics and attractors for the same set
of circuit parameters. This is the basis to show the existence of bifurcations due