254
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
f (ϕM )
qM 0
A
B
Γ
ϕM 0
C1
qC 10
L
ϕL 0
C2
qC 20
R
r
qL(t; t0)
qC 2 (t; t0)
qC 1 (t; t0)
qM (t; t0)
ϕM (t; t0)
ϕC 1 (t; t0)
ϕC 2 (t; t0)
ϕL(t; t0)
Fig. 6.26 Equivalent circuit of the MCC in the (ϕ, q)-domain
q C 1 (t 0 ) = q C 1 0 = C 1 v C 1 0 , q C 2 (t 0 ) = q C 2 0 = C 2 v C 2 0 , and ϕ L (t 0 ) = ϕ L 0 = Li L 0 .
The corresponding circuit in the (ϕ, q)-domain is reported in Fig. 6.26.
It can be easily checked that MCC satisfies the conditions in Chap. 5 for the
existence of the SE representation. Analysis of the MCC in Fig. 6.26 permits to
write, by means of the KϕL at the loop Γ and the KqLs at the nodes A and B ,
the following SEs in the (ϕ, q)-domain in terms of the state variables ϕ C 1 (t; t 0 ),
ϕ C 2 (t; t 0 ) and q L (t; t 0 ). We have for any t ≥ t 0
C 1
dϕ C 1 (t; t 0 )
dt
=
1
R
(ϕ C 2 (t; t 0 ) − ϕ C 1 (t; t 0 ))
− f (ϕ C 1 (t; t 0 ) + ϕ M 0 ) + f (ϕ M 0 ) + q C 1 0
(6.37a)
C 2
dϕ C 2 (t; t 0 )
dt
= −
1
R
(ϕ C 2 (t; t 0 ) − ϕ C 1 (t; t 0 ))
+ q L (t; t 0 ) + q C 2 0
(6.37b)
L
dq L (t; t 0 )
dt
= −rq L (t; t 0 ) − ϕ C 2 (t; t 0 ) + ϕ L 0
(6.37c)
ϕ C 1 (t 0 ; t 0 ) = 0
(6.37d)
ϕ C 2 (t 0 ; t 0 ) = 0
(6.37e)
q L (t 0 ; t 0 ) = 0.
(6.37f)
By following a procedure analogous to that in Sect. 4.3.1 of Chap. 4, the first
three equations can be rewritten in the adimensional form
dϕ C 1 (τ ; t 0 )
dτ
= α
(ϕ C 2 (τ ; t 0 ) − ϕ C 1 (τ ; t 0 ))
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
f (ϕM )
qM 0
A
B
Γ
ϕM 0
C1
qC 10
L
ϕL 0
C2
qC 20
R
r
qL(t; t0)
qC 2 (t; t0)
qC 1 (t; t0)
qM (t; t0)
ϕM (t; t0)
ϕC 1 (t; t0)
ϕC 2 (t; t0)
ϕL(t; t0)
Fig. 6.26 Equivalent circuit of the MCC in the (ϕ, q)-domain
q C 1 (t 0 ) = q C 1 0 = C 1 v C 1 0 , q C 2 (t 0 ) = q C 2 0 = C 2 v C 2 0 , and ϕ L (t 0 ) = ϕ L 0 = Li L 0 .
The corresponding circuit in the (ϕ, q)-domain is reported in Fig. 6.26.
It can be easily checked that MCC satisfies the conditions in Chap. 5 for the
existence of the SE representation. Analysis of the MCC in Fig. 6.26 permits to
write, by means of the KϕL at the loop Γ and the KqLs at the nodes A and B ,
the following SEs in the (ϕ, q)-domain in terms of the state variables ϕ C 1 (t; t 0 ),
ϕ C 2 (t; t 0 ) and q L (t; t 0 ). We have for any t ≥ t 0
C 1
dϕ C 1 (t; t 0 )
dt
=
1
R
(ϕ C 2 (t; t 0 ) − ϕ C 1 (t; t 0 ))
− f (ϕ C 1 (t; t 0 ) + ϕ M 0 ) + f (ϕ M 0 ) + q C 1 0
(6.37a)
C 2
dϕ C 2 (t; t 0 )
dt
= −
1
R
(ϕ C 2 (t; t 0 ) − ϕ C 1 (t; t 0 ))
+ q L (t; t 0 ) + q C 2 0
(6.37b)
L
dq L (t; t 0 )
dt
= −rq L (t; t 0 ) − ϕ C 2 (t; t 0 ) + ϕ L 0
(6.37c)
ϕ C 1 (t 0 ; t 0 ) = 0
(6.37d)
ϕ C 2 (t 0 ; t 0 ) = 0
(6.37e)
q L (t 0 ; t 0 ) = 0.
(6.37f)
By following a procedure analogous to that in Sect. 4.3.1 of Chap. 4, the first
three equations can be rewritten in the adimensional form
dϕ C 1 (τ ; t 0 )
dτ
= α
(ϕ C 2 (τ ; t 0 ) − ϕ C 1 (τ ; t 0 ))