Chapter 6
Memristor Circuits: Invariant Manifolds,
Coexisting Attractors, Extreme
Multistability, and Bifurcations Without
Parameters
Chapter 5 has developed the flux-charge analysis method (FCAM) for the analysis of
a class LM of memristor circuits containing memristors, linear resistors, inductors,
capacitors, and independent voltage and current sources. The formulation of circuit
equations (DAEs and SEs) has been provided in the (ϕ, q)-domain and in the
(v, i)-domain and relationships between the analysis in the two domains have
been discussed. A fundamental property is that the SE description in the (ϕ, q)domain of a circuit in LM has a lower order with respect to the corresponding
description in the (v, i)-domain, so that it is expected that the dynamic analysis
in the former domain is simpler. The goal of this chapter is to highlight some
main advantages of FCAM by studying the dynamics of a number of low-order
autonomous memristor circuits in the (ϕ, q)-domain, namely, circuits described by
a first-order SE in the (ϕ, q)-domain, oscillatory circuits described by a secondorder SE in the (ϕ, q)-domain, and chaotic circuits described by a third-order SE in
the (ϕ, q)-domain. Such a study permits to show that memristor circuits display a
number of fundamental peculiar dynamic features. First of all, the state space in the
(v, i)-domain can be foliated in a continuum of invariant manifolds, thus implying
the coexistence of infinitely many different reduced-order dynamics and attractors.
Moreover, such circuits display bifurcations due to changing the initial conditions
for a fixed set of circuit parameters that are named bifurcations without parameters.
6.1 First-Order Memristor Circuits
For didactic purposes, we find it useful to first illustrate the idea of the invariant
of motions, invariant manifolds, and coexisting dynamics using a simple linear
circuit. Bifurcations without parameters are instead displayed only when the circuit
is nonlinear, as in the case where the circuit contains also memristors.
© Springer Nature Switzerland AG 2021
F. Corinto et al., Nonlinear Circuits and Systems with Memristors,
https://doi.org/10.1007/978-3-030-55651-8_6
219
Memristor Circuits: Invariant Manifolds,
Coexisting Attractors, Extreme
Multistability, and Bifurcations Without
Parameters
Chapter 5 has developed the flux-charge analysis method (FCAM) for the analysis of
a class LM of memristor circuits containing memristors, linear resistors, inductors,
capacitors, and independent voltage and current sources. The formulation of circuit
equations (DAEs and SEs) has been provided in the (ϕ, q)-domain and in the
(v, i)-domain and relationships between the analysis in the two domains have
been discussed. A fundamental property is that the SE description in the (ϕ, q)domain of a circuit in LM has a lower order with respect to the corresponding
description in the (v, i)-domain, so that it is expected that the dynamic analysis
in the former domain is simpler. The goal of this chapter is to highlight some
main advantages of FCAM by studying the dynamics of a number of low-order
autonomous memristor circuits in the (ϕ, q)-domain, namely, circuits described by
a first-order SE in the (ϕ, q)-domain, oscillatory circuits described by a secondorder SE in the (ϕ, q)-domain, and chaotic circuits described by a third-order SE in
the (ϕ, q)-domain. Such a study permits to show that memristor circuits display a
number of fundamental peculiar dynamic features. First of all, the state space in the
(v, i)-domain can be foliated in a continuum of invariant manifolds, thus implying
the coexistence of infinitely many different reduced-order dynamics and attractors.
Moreover, such circuits display bifurcations due to changing the initial conditions
for a fixed set of circuit parameters that are named bifurcations without parameters.
6.1 First-Order Memristor Circuits
For didactic purposes, we find it useful to first illustrate the idea of the invariant
of motions, invariant manifolds, and coexisting dynamics using a simple linear
circuit. Bifurcations without parameters are instead displayed only when the circuit
is nonlinear, as in the case where the circuit contains also memristors.
© Springer Nature Switzerland AG 2021
F. Corinto et al., Nonlinear Circuits and Systems with Memristors,
https://doi.org/10.1007/978-3-030-55651-8_6
219