360
9 Memristor Cellular Neural Networks Computing in the Flux-charge Domain
for i = 1, 2, . . . , n, where v C i (t) = dϕ C i (t)/dt and we have taken into
account (9.12). Let v C (t) = (v C 1 (t), v C 2 (t), . . . , v C n (t)) T , where v C i (t) =
dϕ C i (t)/dt, i = 1, 2, . . . , n, is the vector of capacitor voltages for the circuit in
Fig. 9.8.
Then, we obtain a system of 2n SEs describing the dynamics in the (v, i)-domain
in the normalized state variables q(t) = q M (t)/k q , v(t) = v C (t)/k q (we omit
dependence on t)
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
τ
dv
dt
= −v + G ˆ
Φ (q)v
dq
dt
=
1
R v
(9.21)
for t ≥ 0. The initial conditions are v(0) = v C (0)/k q = v C 0 /k q and q(0) =
q 0 = q M 0 /k q . From the standard theory of ordinary differential equations we have
that (9.21) enjoys the properties of the existence, uniqueness with respect to initial
conditions and prolongability of solutions to +∞.
Remark 9.3 It is worth noting that the SEs (9.21) can also be derived by a direct
analysis in the (v, i)-domain. Indeed, let us consider the cell implementation in
Fig. 9.8. By applying the Kirchhoff Current Law (KCL) to the operational amplifier
at the left we obtain (omitting dependence on t)
C i
d
dt
v C i = −
v C i
R
−
n
j =1
G ij v uj .
The KCL for the operational amplifier at the right leads us to
v ui = −h
ap (q M i )
v C i
R
.
Taking into account that v M i = −v uj and v C i /R = i M i = dq M i /dt, and letting
H
ap (q M ) = (h
ap (q M 1 ), . . . , h
ap (q M n ) )) T and q M = (q M 1 , q M 2 , . . . , q M n ) T , the
above equations become
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
τ
dv C
dt
= −v C + GH
ap (q M )v C
dq M
dt
=
1
R v C .
The change of variables presented in Sect. 9.2.1 yields (9.21).
By letting dv/dt = 0 and dq/dt = 0 it is seen that (9.21) has an n-dimensional
manifold of EPs
9 Memristor Cellular Neural Networks Computing in the Flux-charge Domain
for i = 1, 2, . . . , n, where v C i (t) = dϕ C i (t)/dt and we have taken into
account (9.12). Let v C (t) = (v C 1 (t), v C 2 (t), . . . , v C n (t)) T , where v C i (t) =
dϕ C i (t)/dt, i = 1, 2, . . . , n, is the vector of capacitor voltages for the circuit in
Fig. 9.8.
Then, we obtain a system of 2n SEs describing the dynamics in the (v, i)-domain
in the normalized state variables q(t) = q M (t)/k q , v(t) = v C (t)/k q (we omit
dependence on t)
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
τ
dv
dt
= −v + G ˆ
Φ (q)v
dq
dt
=
1
R v
(9.21)
for t ≥ 0. The initial conditions are v(0) = v C (0)/k q = v C 0 /k q and q(0) =
q 0 = q M 0 /k q . From the standard theory of ordinary differential equations we have
that (9.21) enjoys the properties of the existence, uniqueness with respect to initial
conditions and prolongability of solutions to +∞.
Remark 9.3 It is worth noting that the SEs (9.21) can also be derived by a direct
analysis in the (v, i)-domain. Indeed, let us consider the cell implementation in
Fig. 9.8. By applying the Kirchhoff Current Law (KCL) to the operational amplifier
at the left we obtain (omitting dependence on t)
C i
d
dt
v C i = −
v C i
R
−
n
j =1
G ij v uj .
The KCL for the operational amplifier at the right leads us to
v ui = −h
ap (q M i )
v C i
R
.
Taking into account that v M i = −v uj and v C i /R = i M i = dq M i /dt, and letting
H
ap (q M ) = (h
ap (q M 1 ), . . . , h
ap (q M n ) )) T and q M = (q M 1 , q M 2 , . . . , q M n ) T , the
above equations become
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
τ
dv C
dt
= −v C + GH
ap (q M )v C
dq M
dt
=
1
R v C .
The change of variables presented in Sect. 9.2.1 yields (9.21).
By letting dv/dt = 0 and dq/dt = 0 it is seen that (9.21) has an n-dimensional
manifold of EPs