332
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
When r = 1 and N i = {i − 1, i, i + 1}, the SEs (8.13) can be written in the form
(i = 1, . . . , N)
dx i (t)
dt
= α(−x i (t) + y i (t) − n(x i (t)))
+
k∈N i
d ik (x k (t) − x i (t)) + X c 0,i
(8.16a)
dy i (t)
dt
= x i (t) − y i (t) + z i (t)
(8.16b)
dz i (t)
dt
= −βy i (t).
(8.16c)
This shows that the NMCC is analogous to an array of diffusively coupled Chua’s
oscillators [1]. A relevant difference is however due to the additional constants terms
X c 0,i at the right-hand side of (8.16), depending on the ICs for the state variables in
the (v, i)-domain, which need to be carefully taken into account in the investigation
of synchronization phenomena.
8.4 Synchronization Phenomena in the NMCC
The analytical results on invariant manifolds for the single MCC and the whole
NMCC are fundamental to analyze synchronization phenomena and to exploit the
theory of weakly connected oscillatory networks developed in the literature [36]. In
particular, in order to demonstrate the role of invariant manifolds on the occurrence
of synchronization in the NMCC, let us introduce the synchronization manifold of
the whole NMCC in the (ϕ, q)-domain
M S = {(x 1 , y 1 , z 1 , . . . , x N , y N , z N ) ∈ R
3N
: x 1 = · · · = x N ;
y 1 = · · · = y N ; z 1 = · · · = z N } ⊂ R
3N
and define the synchronization errors
e ij =
⎛
⎝
x i − x j
y i − y j
z i − z j
⎞
⎠
for i, j = 1, 2 . . . , N.
Manifold M S is positively invariant for the dynamics of (8.16) if and only if
w c (t) ∈ M S implies de ij (t)/dt = 0, t ≥ t 0 . Since for w c (t) ∈ M S
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
When r = 1 and N i = {i − 1, i, i + 1}, the SEs (8.13) can be written in the form
(i = 1, . . . , N)
dx i (t)
dt
= α(−x i (t) + y i (t) − n(x i (t)))
+
k∈N i
d ik (x k (t) − x i (t)) + X c 0,i
(8.16a)
dy i (t)
dt
= x i (t) − y i (t) + z i (t)
(8.16b)
dz i (t)
dt
= −βy i (t).
(8.16c)
This shows that the NMCC is analogous to an array of diffusively coupled Chua’s
oscillators [1]. A relevant difference is however due to the additional constants terms
X c 0,i at the right-hand side of (8.16), depending on the ICs for the state variables in
the (v, i)-domain, which need to be carefully taken into account in the investigation
of synchronization phenomena.
8.4 Synchronization Phenomena in the NMCC
The analytical results on invariant manifolds for the single MCC and the whole
NMCC are fundamental to analyze synchronization phenomena and to exploit the
theory of weakly connected oscillatory networks developed in the literature [36]. In
particular, in order to demonstrate the role of invariant manifolds on the occurrence
of synchronization in the NMCC, let us introduce the synchronization manifold of
the whole NMCC in the (ϕ, q)-domain
M S = {(x 1 , y 1 , z 1 , . . . , x N , y N , z N ) ∈ R
3N
: x 1 = · · · = x N ;
y 1 = · · · = y N ; z 1 = · · · = z N } ⊂ R
3N
and define the synchronization errors
e ij =
⎛
⎝
x i − x j
y i − y j
z i − z j
⎞
⎠
for i, j = 1, 2 . . . , N.
Manifold M S is positively invariant for the dynamics of (8.16) if and only if
w c (t) ∈ M S implies de ij (t)/dt = 0, t ≥ t 0 . Since for w c (t) ∈ M S