330
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
Note that M c (Q 0 ) is a 3N-dimensional manifold in R 4N that coincides with the
Q 0 -level set of function Q(·) = (Q 1 (·), Q 2 (·), . . . , Q n (·)) T .
Property 8.2 The state space R 4N of the NMCC in the (v, i)-domain can be foliated
in ∞ N 3N-dimensional manifolds M c (Q 0 ) by varying Q 0 ∈ R N . Manifolds
are nonintersecting and they span the whole state space R 4N . Each manifold is
positively invariant for the dynamics of NMCC in the (v, i)-domain, i.e., if the ICs
w(t 0 ) ∈ M c (Q 0 ), where Q 0 = Q(w c (t 0 )), then the solution of the IVP (8.11)
belongs to M c (Q 0 ) for any t ≥ t 0 . On each manifold M c (Q 0 ) the dynamics
of the MCC is described in the (ϕ, q)-domain by the reduced-order system of
ODEs (8.10), whose order is 3N.
Proof The first part of the proof is analogous to that of Property 8.1 and is omitted.
To show that each manifold is positively invariant, note that we have from KqL
applied to the ith MCC
q M,i (t; t 0 ) + q C 1,i (t; t 0 ) + q C 2,i (t; t 0 ) + q L,i (t; t 0 )
−
k∈N i
1
R ik
(ϕ M,k (t; t 0 ) − ϕ M,i (t; t 0 )) = 0
for any t ≥ t 0 . Arguing as in the proof of Property 8.1, KqL at node B and KϕL at
loop Γ of the ith MCC yield
q L,i (t; t 0 ) =
ϕ M,i (t; t 0 ) − ϕ L,i (t; t 0 )
R
− q C 2,i (t; t 0 )
and substituting in the first equation we obtain
q M,i (t; t 0 ) + q C 1,i (t; t 0 ) +
ϕ M,i (t; t 0 ) − ϕ L,i (t; t 0 )
R
−
k∈N i
1
R ik
(ϕ M,k (t; t 0 ) − ϕ M,i (t; t 0 )) = 0.
Then, we obtain
f (ϕ M,i (t)) +
1
R
ϕ M,i (t) + C 1 v C 1,i (t)
−
L
R
i L,i (t) +
k∈N i
1
R ik
(ϕ M,k (t) − ϕ M,i (t))
= f (ϕ M,i (t 0 )) +
1
R
ϕ M,i (t 0 ) + C 1 v C 1,i (t 0 )
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
Note that M c (Q 0 ) is a 3N-dimensional manifold in R 4N that coincides with the
Q 0 -level set of function Q(·) = (Q 1 (·), Q 2 (·), . . . , Q n (·)) T .
Property 8.2 The state space R 4N of the NMCC in the (v, i)-domain can be foliated
in ∞ N 3N-dimensional manifolds M c (Q 0 ) by varying Q 0 ∈ R N . Manifolds
are nonintersecting and they span the whole state space R 4N . Each manifold is
positively invariant for the dynamics of NMCC in the (v, i)-domain, i.e., if the ICs
w(t 0 ) ∈ M c (Q 0 ), where Q 0 = Q(w c (t 0 )), then the solution of the IVP (8.11)
belongs to M c (Q 0 ) for any t ≥ t 0 . On each manifold M c (Q 0 ) the dynamics
of the MCC is described in the (ϕ, q)-domain by the reduced-order system of
ODEs (8.10), whose order is 3N.
Proof The first part of the proof is analogous to that of Property 8.1 and is omitted.
To show that each manifold is positively invariant, note that we have from KqL
applied to the ith MCC
q M,i (t; t 0 ) + q C 1,i (t; t 0 ) + q C 2,i (t; t 0 ) + q L,i (t; t 0 )
−
k∈N i
1
R ik
(ϕ M,k (t; t 0 ) − ϕ M,i (t; t 0 )) = 0
for any t ≥ t 0 . Arguing as in the proof of Property 8.1, KqL at node B and KϕL at
loop Γ of the ith MCC yield
q L,i (t; t 0 ) =
ϕ M,i (t; t 0 ) − ϕ L,i (t; t 0 )
R
− q C 2,i (t; t 0 )
and substituting in the first equation we obtain
q M,i (t; t 0 ) + q C 1,i (t; t 0 ) +
ϕ M,i (t; t 0 ) − ϕ L,i (t; t 0 )
R
−
k∈N i
1
R ik
(ϕ M,k (t; t 0 ) − ϕ M,i (t; t 0 )) = 0.
Then, we obtain
f (ϕ M,i (t)) +
1
R
ϕ M,i (t) + C 1 v C 1,i (t)
−
L
R
i L,i (t) +
k∈N i
1
R ik
(ϕ M,k (t) − ϕ M,i (t))
= f (ϕ M,i (t 0 )) +
1
R
ϕ M,i (t 0 ) + C 1 v C 1,i (t 0 )