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8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
and from KϕL at the loop Γ (see again Fig. 8.2)
ϕ M (t; t 0 ) = ϕ R (t; t 0 ) + ϕ L (t; t 0 ).
These two last equations yield
q L (t; t 0 ) =
ϕ M (t; t 0 ) − ϕ L (t; t 0 )
R
− q C 2 (t; t 0 )
and substituting in the first equation we obtain
q M (t; t 0 ) + q C 1 (t; t 0 ) +
ϕ M (t; t 0 ) − ϕ L (t; t 0 )
R
= 0.
Since q M (t) = f (ϕ M (t)), we have
f (ϕ M ) +
1
R
ϕ M + C 1 v C 1 −
L
R
i L = f (ϕ M (t 0 )) +
1
R
ϕ M (t 0 )
+C 1 v C 1 (t 0 ) −
L
R
i L (t 0 )
for any t ≥ t 0 .
Remark 8.1 The technique here used for proving invariance of M(Q 0 ) in Property 8.1 is based on KϕLs and KqLs of the associated circuit and, as such, it differs
from that given in Chap. 6, that is instead based on algebraic manipulations of SEs in
the (ϕ, q)-domain. This new proof, in addition to being simpler and based on circuit
theoretic ideas, has also the advantage of lending itself to an extension to arrays of
coupled MCCs (cf. Property 8.2 in Sect. 8.3). Of course, the invariance of manifold
M(Q 0 ) is equivalent to saying that Q(·) as in (8.3) is an invariant of motion for the
dynamics of the MCC in the (v, i)-domain.
8.2.1 Complex Dynamics in the Single MCC
To analyze the dynamics of the MCC, it is convenient to recast the SEs (8.1) in a
more compact normal form. Let us define parameters (cf. Table 6.1 in Chap. 6)
α =
C 1
C 2
, β =
R 2 C 2
L
(8.5)
introduce the normalized time t → t/(RC 2 ) and consider the change of variables
x(t) = ϕ C 1 (t; t 0 ) + ϕ M (t 0 ) = ϕ M (t)
(8.6a)