398
11 Nonlinear Dynamics of Circuits with Mem-Elements
Therefore, it is crucial to identify subclasses of nonlinear dynamical circuits in N e
for which the existence of an SE description is guaranteed. Such subclasses should
also be wide enough to include relevant circuits with mem-elements investigated in
the literature.
To this end, following the lines in Chap. 7, it is convenient to decompose any
nonlinear dynamical circuit in N e into a linear adynamic (resistive) multi-port
N R (containing only linear resistors, flux and charge sources) and a collection of
nonlinear dynamical two-terminal elements connected to N R . The standard hybrid
representation of a resistive multi-port N R is then used for deriving the SEs. Such
a decomposition permits to specify the topological conditions under which the twoterminal nonlinear dynamical elements connected to N R yield a subclass of N e that
admits of an SE representation (cf. Property 3.1 in Chap. 3):
1. Only charge-controlled nonlinear capacitors C q and flux-controlled nonlinear
inductors L ϕ are admissible.
In fact, let us consider a voltage-controlled nonlinear capacitor q = f C (v). From
the formulation given in Chap. 3 (cf. also Example 1 in Sect. 11.2), to write the
SEs it is actually needed that the incremental capacitance f
C (v) = 0 for each
v. It follows that the nonlinear capacitor results to be also charge-controlled,
i.e., v = f
−1
C (q) can be derived. Hence, without loss of generality, voltagecontrolled capacitors can be omitted and we will consider only nonlinear chargecontrolled capacitors C q . Similar considerations show that we can include only
flux-controlled inductors L ϕ .
2. Only σ -controlled memcapacitors MC σ and ρ-controlled meminductors ML ρ
are admissible.
This is obtained by a straightforward extension of the argument used in point 1).
3. Each flux-controlled memristor M ϕ should be in parallel to a charge-controlled
capacitor C q or a σ -controlled memcapacitor MC σ , while each charge-controlled
memristor M q should be in series with a flux-controlled inductor L ϕ or a ρcontrolled meminductor ML ρ .
Note that linear capacitors C or inductors L can be considered as special cases of
the corresponding nonlinear elements.
On the basis of the previous discussion and assumptions, the two following
subclasses of nonlinear dynamical circuits with mem-elements in N e can be
identified:
1. LME ⊂ N e is obtained interconnecting N R to the following nonlinear
dynamical two-terminal elements D:
– n γ μ flux-controlled memristors M ϕ (each having in parallel a linear capacitor
C)
– n λμ charge-controlled memristors M q (each having in series a linear inductor
L)
– n γ linear capacitors C
– n λ linear inductors L
– n σ memcapacitors MC σ
11 Nonlinear Dynamics of Circuits with Mem-Elements
Therefore, it is crucial to identify subclasses of nonlinear dynamical circuits in N e
for which the existence of an SE description is guaranteed. Such subclasses should
also be wide enough to include relevant circuits with mem-elements investigated in
the literature.
To this end, following the lines in Chap. 7, it is convenient to decompose any
nonlinear dynamical circuit in N e into a linear adynamic (resistive) multi-port
N R (containing only linear resistors, flux and charge sources) and a collection of
nonlinear dynamical two-terminal elements connected to N R . The standard hybrid
representation of a resistive multi-port N R is then used for deriving the SEs. Such
a decomposition permits to specify the topological conditions under which the twoterminal nonlinear dynamical elements connected to N R yield a subclass of N e that
admits of an SE representation (cf. Property 3.1 in Chap. 3):
1. Only charge-controlled nonlinear capacitors C q and flux-controlled nonlinear
inductors L ϕ are admissible.
In fact, let us consider a voltage-controlled nonlinear capacitor q = f C (v). From
the formulation given in Chap. 3 (cf. also Example 1 in Sect. 11.2), to write the
SEs it is actually needed that the incremental capacitance f
C (v) = 0 for each
v. It follows that the nonlinear capacitor results to be also charge-controlled,
i.e., v = f
−1
C (q) can be derived. Hence, without loss of generality, voltagecontrolled capacitors can be omitted and we will consider only nonlinear chargecontrolled capacitors C q . Similar considerations show that we can include only
flux-controlled inductors L ϕ .
2. Only σ -controlled memcapacitors MC σ and ρ-controlled meminductors ML ρ
are admissible.
This is obtained by a straightforward extension of the argument used in point 1).
3. Each flux-controlled memristor M ϕ should be in parallel to a charge-controlled
capacitor C q or a σ -controlled memcapacitor MC σ , while each charge-controlled
memristor M q should be in series with a flux-controlled inductor L ϕ or a ρcontrolled meminductor ML ρ .
Note that linear capacitors C or inductors L can be considered as special cases of
the corresponding nonlinear elements.
On the basis of the previous discussion and assumptions, the two following
subclasses of nonlinear dynamical circuits with mem-elements in N e can be
identified:
1. LME ⊂ N e is obtained interconnecting N R to the following nonlinear
dynamical two-terminal elements D:
– n γ μ flux-controlled memristors M ϕ (each having in parallel a linear capacitor
C)
– n λμ charge-controlled memristors M q (each having in series a linear inductor
L)
– n γ linear capacitors C
– n λ linear inductors L
– n σ memcapacitors MC σ