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11 Nonlinear Dynamics of Circuits with Mem-Elements
The four basic algebraic elements, i.e., the resistor, capacitor, inductor, and
memristor, are defined axiomatically by an algebraic CR involving the electric
pairs (v (0) , i (0) ) = (v, i), (v (0) , i (−1) ) = (v, q), (v (−1) , i (0) ) = (ϕ, i), and
(v (−1) , i (−1) ) = (ϕ, q), respectively. Higher-order elements as a memcapacitor or a meminductor have been introduced axiomatically in a similar way in
Chap. 1. Namely, a memcapacitor is defined by an algebraic CR involving the
electric quantities (v (−1) , i (−2) ) = (ϕ, σ ) and a meminductor by a CR involving
(v (−2) , i (−1) ) = (ρ, q). The algebraic (α, β)-elements can be systematically
organized in a “periodic table” where each element exhibits peculiar properties of
the small-signal impedance at an operating point. The elements of the periodic table
named memristors, memcapacitors, and meminductor are also designated as memelements, in contrast to the classical nonlinear resistors, capacitors, and inductors.
The next class N e of nonlinear dynamical circuits containing mem-elements is
the main subject of the chapter. Specifically, N e includes the following types of
two-terminal elements D:
• flux- or charge-controlled memristors, denoted M ϕ and M q , resp.
• nonlinear charge- or voltage-controlled capacitors C q and C v , resp.
• nonlinear flux- or current-controlled inductors L ϕ and L i , resp.
• σ - or flux-controlled memcapacitors MC σ and MC ϕ , resp.
• ρ- or charge-controlled meminductors ML ρ and ML q , resp.
• linear resistors, capacitors, and inductors R, C, and L, resp.
• time-varying current or voltage sources, a(t) and e(t), resp.
Since a generic circuit in N e is made of algebraic elements with different α and
β (i.e., the trivial case of circuits obtained interconnecting elements of one single
kind is excluded), the following question arises:
Which is the domain (α, β), if any, where the dynamics of the circuit is described in the
simplest form (e.g., the SEs contain the lowest number of differential equations)?
If a nonlinear dynamical circuit in N e contains as mem-elements only memristors, 1 then the answer to the previous question, which is provided in Chap. 5,
is that the most suitable domain is (α, β) = (−1, −1), i.e., the (ϕ, q)-domain.
The goal of this chapter is to extend FCAM to circuits with not only memristors,
but memcapacitors and meminductors as well, and to show that the analysis in
the (ϕ, q)-domain presents several advantages over the traditional (v, i)-domain. In
other words, the answer to the stated question continues to be that the most suitable
domain is the (ϕ, q)-domain also for the extended class N e .
Suppose we wish to analyze the dynamics of a circuit in N e for t ≥ t 0 , where t 0
is a finite initial instant, in the (ϕ, q)-domain. According to FCAM, the analysis has
to exploit the incremental flux and incremental charge
1 Such a class of nonlinear dynamical circuits is denoted by LM in Chap. 5. Note that LM ⊂ N e .
11 Nonlinear Dynamics of Circuits with Mem-Elements
The four basic algebraic elements, i.e., the resistor, capacitor, inductor, and
memristor, are defined axiomatically by an algebraic CR involving the electric
pairs (v (0) , i (0) ) = (v, i), (v (0) , i (−1) ) = (v, q), (v (−1) , i (0) ) = (ϕ, i), and
(v (−1) , i (−1) ) = (ϕ, q), respectively. Higher-order elements as a memcapacitor or a meminductor have been introduced axiomatically in a similar way in
Chap. 1. Namely, a memcapacitor is defined by an algebraic CR involving the
electric quantities (v (−1) , i (−2) ) = (ϕ, σ ) and a meminductor by a CR involving
(v (−2) , i (−1) ) = (ρ, q). The algebraic (α, β)-elements can be systematically
organized in a “periodic table” where each element exhibits peculiar properties of
the small-signal impedance at an operating point. The elements of the periodic table
named memristors, memcapacitors, and meminductor are also designated as memelements, in contrast to the classical nonlinear resistors, capacitors, and inductors.
The next class N e of nonlinear dynamical circuits containing mem-elements is
the main subject of the chapter. Specifically, N e includes the following types of
two-terminal elements D:
• flux- or charge-controlled memristors, denoted M ϕ and M q , resp.
• nonlinear charge- or voltage-controlled capacitors C q and C v , resp.
• nonlinear flux- or current-controlled inductors L ϕ and L i , resp.
• σ - or flux-controlled memcapacitors MC σ and MC ϕ , resp.
• ρ- or charge-controlled meminductors ML ρ and ML q , resp.
• linear resistors, capacitors, and inductors R, C, and L, resp.
• time-varying current or voltage sources, a(t) and e(t), resp.
Since a generic circuit in N e is made of algebraic elements with different α and
β (i.e., the trivial case of circuits obtained interconnecting elements of one single
kind is excluded), the following question arises:
Which is the domain (α, β), if any, where the dynamics of the circuit is described in the
simplest form (e.g., the SEs contain the lowest number of differential equations)?
If a nonlinear dynamical circuit in N e contains as mem-elements only memristors, 1 then the answer to the previous question, which is provided in Chap. 5,
is that the most suitable domain is (α, β) = (−1, −1), i.e., the (ϕ, q)-domain.
The goal of this chapter is to extend FCAM to circuits with not only memristors,
but memcapacitors and meminductors as well, and to show that the analysis in
the (ϕ, q)-domain presents several advantages over the traditional (v, i)-domain. In
other words, the answer to the stated question continues to be that the most suitable
domain is the (ϕ, q)-domain also for the extended class N e .
Suppose we wish to analyze the dynamics of a circuit in N e for t ≥ t 0 , where t 0
is a finite initial instant, in the (ϕ, q)-domain. According to FCAM, the analysis has
to exploit the incremental flux and incremental charge
1 Such a class of nonlinear dynamical circuits is denoted by LM in Chap. 5. Note that LM ⊂ N e .