322
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
f (ϕM )
qM 0
A
B
Γ
ϕM 0
C1
qC 10
L
ϕL 0
C2
qC 20
R
ϕR(t; t0)
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. 8.2 Equivalent circuit of MCC in the (ϕ, q)-domain. We have let q C 1 0 = q C 1 (t 0 ), q C 2 0 =
q C 2 (t 0 ), ϕ L 0 = ϕ L (t 0 ) and ϕ M 0 = ϕ M (t 0 )
Kirchhoff charge law (KqL) for incremental charges and fluxes. The reduction of
order for the associated circuit follows from the fact that in the (ϕ, q)-domain the
ideal memristor results to be a memoryless nonlinear element.
By using KϕLs and KqLs, the following SEs describing the MCC in the (ϕ, q)domain are obtained:
C 1
dϕ C 1 (t; t 0 )
dt
= −
1
R
(ϕ C 1 (t; t 0 ) − ϕ C 2 (t; t 0 )) + q C 1 (t 0 )
− f (ϕ C 1 (t; t 0 ) + ϕ M (t 0 )) + f (ϕ M (t 0 ))
(8.1a)
C 2
dϕ C 2 (t; t 0 )
dt
= −
1
R
(ϕ C 2 (t; t 0 ) − ϕ C 1 (t; t 0 )) + q C 2 (t 0 )
− q L (t; t 0 )
(8.1b)
L
dq L (t; t 0 )
dt
=ϕ C 2 (t; t 0 ) + ϕ L (t 0 )
(8.1c)
for all t ≥ t 0 , where we have taken into account that ϕ C 1 (t; t 0 ) = ϕ M (t; t 0 ). The
initial conditions are ϕ C 1 (t 0 ; t 0 ) = 0, ϕ C 2 (t 0 ; t 0 ) = 0 and q L (t 0 ; t 0 ) = 0. Note
that (8.1) is an initial value problem (IVP) for a third-order system of ODEs in
the three state variables (ϕ C 1 (t; t 0 ), ϕ C 2 (t; t 0 ), q L (t; t 0 )) in the (ϕ, q)-domain. Also
note that the right-hand side of (8.1) contains constant terms depending on the ICs
for the state variables in the (v, i)-domain.
From FCAM, time differentiation of (8.1) yields the SEs of the MCC in the
(v, i)-domain
C 1
dv C 1 (t)
dt
= −
1
R
(v C 1 (t) − v C 2 (t)) − f
(ϕ M (t))v C 1 (t)
(8.2a)
C 2
dv C 2 (t)
dt
= −
1
R
(v C 2 (t) − v C 1 (t)) − i L (t)
(8.2b)
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
f (ϕM )
qM 0
A
B
Γ
ϕM 0
C1
qC 10
L
ϕL 0
C2
qC 20
R
ϕR(t; t0)
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. 8.2 Equivalent circuit of MCC in the (ϕ, q)-domain. We have let q C 1 0 = q C 1 (t 0 ), q C 2 0 =
q C 2 (t 0 ), ϕ L 0 = ϕ L (t 0 ) and ϕ M 0 = ϕ M (t 0 )
Kirchhoff charge law (KqL) for incremental charges and fluxes. The reduction of
order for the associated circuit follows from the fact that in the (ϕ, q)-domain the
ideal memristor results to be a memoryless nonlinear element.
By using KϕLs and KqLs, the following SEs describing the MCC in the (ϕ, q)domain are obtained:
C 1
dϕ C 1 (t; t 0 )
dt
= −
1
R
(ϕ C 1 (t; t 0 ) − ϕ C 2 (t; t 0 )) + q C 1 (t 0 )
− f (ϕ C 1 (t; t 0 ) + ϕ M (t 0 )) + f (ϕ M (t 0 ))
(8.1a)
C 2
dϕ C 2 (t; t 0 )
dt
= −
1
R
(ϕ C 2 (t; t 0 ) − ϕ C 1 (t; t 0 )) + q C 2 (t 0 )
− q L (t; t 0 )
(8.1b)
L
dq L (t; t 0 )
dt
=ϕ C 2 (t; t 0 ) + ϕ L (t 0 )
(8.1c)
for all t ≥ t 0 , where we have taken into account that ϕ C 1 (t; t 0 ) = ϕ M (t; t 0 ). The
initial conditions are ϕ C 1 (t 0 ; t 0 ) = 0, ϕ C 2 (t 0 ; t 0 ) = 0 and q L (t 0 ; t 0 ) = 0. Note
that (8.1) is an initial value problem (IVP) for a third-order system of ODEs in
the three state variables (ϕ C 1 (t; t 0 ), ϕ C 2 (t; t 0 ), q L (t; t 0 )) in the (ϕ, q)-domain. Also
note that the right-hand side of (8.1) contains constant terms depending on the ICs
for the state variables in the (v, i)-domain.
From FCAM, time differentiation of (8.1) yields the SEs of the MCC in the
(v, i)-domain
C 1
dv C 1 (t)
dt
= −
1
R
(v C 1 (t) − v C 2 (t)) − f
(ϕ M (t))v C 1 (t)
(8.2a)
C 2
dv C 2 (t)
dt
= −
1
R
(v C 2 (t) − v C 1 (t)) − i L (t)
(8.2b)