252
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
ϕ L (t; t 0 ) = L
dq L (t; t 0 )
dt
− ϕ L 0
q M (t; t 0 ) = f (ϕ M (t; t 0 ) + ϕ M 0 ) − f (ϕ M 0 ).
These correspond to the system of 2b = 8 DAEs describing the tableau equations
for the circuit.
By substitution, it is found that the circuit dynamics is described by the secondorder SE
C
dϕ C (t; t 0 )
dt
= q L (t; t 0 ) − f (ϕ C (t; t 0 ) + ϕ M 0 ) + f (ϕ M 0 ) + q C 0
L
dq L (t; t 0 )
dt
= −Rq L (t; t 0 ) − ϕ C (t; t 0 ) + ϕ L 0 .
The change of variables x(t) = ϕ C (t; t 0 ) + ϕ M 0 and y(t) = q L (t; t 0 ) −
ϕ M 0 +ϕ L 0
R
yields the SE
C
dx
dt
= y − f (x) + Q 0
L
dy
dt
= −Ry − x
where we let
Q 0 = q C 0 +
ϕ M 0 + ϕ L 0
R
+ f (ϕ M 0 ).
This is in a form analogous to a Van der Pol oscillator with a constant forcing term
Q 0 .
By differentiation in time, the SEs in the (v, i)-domain are given by the thirdorder system
C
dv C
dt
= i L − G(ϕ M )v C
L
di L
dt
= −Ri L − v C
dϕ M
dt
= v C
where G(ϕ M ) = f (ϕ M ).
We leave to the reader the verification that function
Q(t) = Cv C (t) +
ϕ M (t) + Li L (t)
R
+ f (ϕ M (t))