11.1 Nonlinear Circuits with Mem-Elements
389
to changing initial conditions (ICs) for fixed parameters, i.e., bifurcations without
parameters.
4. In the non-autonomous case, formula for designing independent pulse current
or voltage sources to drive trajectories through manifolds with different regimes
and dynamics are obtained. These can be used to effectively design circuits with
mem-elements that behave as controllable sources of complex dynamics to be
used in future neuromorphic systems.
11.1 Nonlinear Circuits with Mem-Elements
For the reader’s convenience, we first summarize some facts from Chap. 1 which are
needed in this chapter. Let v(t) and i(t) be the voltage and current of a two-terminal
circuit element D. Define the higher-order derivatives and integrals of v(t), denoted
by v (α) (t) and i (β) (t), respectively, where α, β are integers (i.e., 0, ±1, ±2, . . .).
While v (0) (t) = v(t) and i (0) (t) = i(t), the following notation and nomenclature
are also used:
v
(−1) (t) = ϕ(t) =
t
−∞
v(τ )dτ
is the flux (or voltage momentum) and
i
(−1) (t) = q(t) =
t
−∞
i(τ )dτ
is the charge (or current momentum). Note that, from a physical viewpoint, ϕ(t)
and q(t) do not necessarily correspond to the flux of a magnetic field or the charge
accumulated in D. Let us also introduce the electrical variables
v
(−2) (t) = ρ(t) =
t
−∞
τ
−∞
v(τ 1 )dτ 1
dτ =
t
−∞
ϕ(τ )dτ
i.e., the time integral of flux (or flux momentum) and
i
(−2) (t) = σ (t) =
t
−∞
τ
−∞
i(τ 1 )dτ 1
dτ =
t
−∞
q(τ )dτ
i.e., the time integral of charge (or charge momentum).
According to the axiomatic approach in Chap. 1, a two-terminal element D is said
to be algebraic if and only if its CR can be expressed by an algebraic relationship that
involves at most two independent electric variables v (α) (t) and i (β) (t). An algebraic
element is also referred to as an (α, β)-element.
389
to changing initial conditions (ICs) for fixed parameters, i.e., bifurcations without
parameters.
4. In the non-autonomous case, formula for designing independent pulse current
or voltage sources to drive trajectories through manifolds with different regimes
and dynamics are obtained. These can be used to effectively design circuits with
mem-elements that behave as controllable sources of complex dynamics to be
used in future neuromorphic systems.
11.1 Nonlinear Circuits with Mem-Elements
For the reader’s convenience, we first summarize some facts from Chap. 1 which are
needed in this chapter. Let v(t) and i(t) be the voltage and current of a two-terminal
circuit element D. Define the higher-order derivatives and integrals of v(t), denoted
by v (α) (t) and i (β) (t), respectively, where α, β are integers (i.e., 0, ±1, ±2, . . .).
While v (0) (t) = v(t) and i (0) (t) = i(t), the following notation and nomenclature
are also used:
v
(−1) (t) = ϕ(t) =
t
−∞
v(τ )dτ
is the flux (or voltage momentum) and
i
(−1) (t) = q(t) =
t
−∞
i(τ )dτ
is the charge (or current momentum). Note that, from a physical viewpoint, ϕ(t)
and q(t) do not necessarily correspond to the flux of a magnetic field or the charge
accumulated in D. Let us also introduce the electrical variables
v
(−2) (t) = ρ(t) =
t
−∞
τ
−∞
v(τ 1 )dτ 1
dτ =
t
−∞
ϕ(τ )dτ
i.e., the time integral of flux (or flux momentum) and
i
(−2) (t) = σ (t) =
t
−∞
τ
−∞
i(τ 1 )dτ 1
dτ =
t
−∞
q(τ )dτ
i.e., the time integral of charge (or charge momentum).
According to the axiomatic approach in Chap. 1, a two-terminal element D is said
to be algebraic if and only if its CR can be expressed by an algebraic relationship that
involves at most two independent electric variables v (α) (t) and i (β) (t). An algebraic
element is also referred to as an (α, β)-element.