334
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
for t ≥ t 0 . The result follows by taking the limit as t → +∞ in the previous
expression and considering that x i (t) = ϕ M,i (t) and that in the case of CS we have
k∈N i
d ik (ϕ M,k (t) − ϕ M,i (t)) → 0 as t → +∞.
Remark 8.3 Since x i (t) = ϕ M (t), it can be immediately checked that, in the case
of CS of (8.16) in the (ϕ, q)-domain, the state variables w c (t) in the (v, i)-domain
also achieve CS.
8.4.1 Numerical Simulations
Here, numerical simulations of 1D arrays with N = 4 identical MCCs are provided.
For the sake of simplicity, assume homogeneous ICs on the fluxes of memristors,
i.e., ϕ M,i (t 0 ) = ϕ 0 , and v C 1,i (t 0 ) = i L,i (t 0 ) = 0 for i = 1, . . . , 4. It follows that
condition (8.18) is satisfied and simplifies to (see also (8.14))
X c 0 = X 0,i = α(n(ϕ 0 ) + ϕ 0 )
(8.20)
for i = 1, . . . , 4. Moreover, it is assumed that ϕ 0 is such that X c 0 = 0, that is, the
values ϕ 0 ∈ {−1.5, 0, 1.5} are derived by using the expression (8.9). Under such
assumptions, each uncoupled MCC evolves on the zero-manifold M(0) on which
its nonlinear dynamics is the same as that of the classical Chua’s oscillator.
The dynamics of NMCC is described by (8.16) with ICs
x i (t 0 ) = ϕ 0
(8.21a)
y i (t 0 ) = 0
(8.21b)
z i (t 0 ) = −ϕ 0 + ¯
ϕ 0,i
(8.21c)
where 2 R = 1 and RC 2 v C 2,i (t 0 ) = ¯
ϕ 0,i . Finally, it is considered a uniform spaceinvariant weak coupling among the MCCs, i.e., d ik = d for all i = 1, . . . , 4. Then,
the dynamics of the whole NMCC takes place on the 3N-dimensional zero-manifold
M(0) and the same periodic/chaotic attractors and bifurcations occurring in locally
connected networks of Chua’s oscillators can be observed.
A summary of the nonlinear dynamics in cellular nonlinear networks of Chua’s
oscillators [26] is described by the following scenario:
• for the single uncoupled Chua’s oscillator (or equivalently for the uncoupled
MCC on M(0)) with
α = 8, β = 15
2 See also the normalization values in Table 6.1 of Chap. 6.
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
for t ≥ t 0 . The result follows by taking the limit as t → +∞ in the previous
expression and considering that x i (t) = ϕ M,i (t) and that in the case of CS we have
k∈N i
d ik (ϕ M,k (t) − ϕ M,i (t)) → 0 as t → +∞.
Remark 8.3 Since x i (t) = ϕ M (t), it can be immediately checked that, in the case
of CS of (8.16) in the (ϕ, q)-domain, the state variables w c (t) in the (v, i)-domain
also achieve CS.
8.4.1 Numerical Simulations
Here, numerical simulations of 1D arrays with N = 4 identical MCCs are provided.
For the sake of simplicity, assume homogeneous ICs on the fluxes of memristors,
i.e., ϕ M,i (t 0 ) = ϕ 0 , and v C 1,i (t 0 ) = i L,i (t 0 ) = 0 for i = 1, . . . , 4. It follows that
condition (8.18) is satisfied and simplifies to (see also (8.14))
X c 0 = X 0,i = α(n(ϕ 0 ) + ϕ 0 )
(8.20)
for i = 1, . . . , 4. Moreover, it is assumed that ϕ 0 is such that X c 0 = 0, that is, the
values ϕ 0 ∈ {−1.5, 0, 1.5} are derived by using the expression (8.9). Under such
assumptions, each uncoupled MCC evolves on the zero-manifold M(0) on which
its nonlinear dynamics is the same as that of the classical Chua’s oscillator.
The dynamics of NMCC is described by (8.16) with ICs
x i (t 0 ) = ϕ 0
(8.21a)
y i (t 0 ) = 0
(8.21b)
z i (t 0 ) = −ϕ 0 + ¯
ϕ 0,i
(8.21c)
where 2 R = 1 and RC 2 v C 2,i (t 0 ) = ¯
ϕ 0,i . Finally, it is considered a uniform spaceinvariant weak coupling among the MCCs, i.e., d ik = d for all i = 1, . . . , 4. Then,
the dynamics of the whole NMCC takes place on the 3N-dimensional zero-manifold
M(0) and the same periodic/chaotic attractors and bifurcations occurring in locally
connected networks of Chua’s oscillators can be observed.
A summary of the nonlinear dynamics in cellular nonlinear networks of Chua’s
oscillators [26] is described by the following scenario:
• for the single uncoupled Chua’s oscillator (or equivalently for the uncoupled
MCC on M(0)) with
α = 8, β = 15
2 See also the normalization values in Table 6.1 of Chap. 6.