8.4 Synchronization Phenomena in the NMCC
335
there are:
– two stable asymmetric limit cycles A + and A − surrounding the unstable
equilibria (+1.5, 0, −1.5) and (−1.5, 0, +1.5), respectively
– one stable symmetric limit cycle S s surrounding the unstable EP (0, 0, 0)
– one unstable symmetric limit cycle S u
• for a network composed by an arbitrary number (N) of Chua’s oscillators and
small coupling d it is seen that (see [26, p. 953]):
(a) there are 2 N stable asymmetric limit cycles with all the phase shifts equal to
π ;
(b) the other 2 N × (2 N −1 − 1) asymmetric limit cycles are unstable
(c) there is only one stable symmetric limit cycle with all the phase shifts equal
to zero
(d) the other (2 N − 1) symmetric limit cycles are unstable.
Cases (a) and (c), referred to as phase-locking in oscillatory arrays, are reported
in the next subsection, whereas the final subsection shows how the four MCC arrays
split into periodic and chaotic clusters.
8.4.1.1 Phase-Locking in Oscillatory NMCC
Consider a simple array made of four MCCs (N = 4) with uniform space-invariant
weak coupling d = 0.05 and ICs as in (8.21) with ϕ 0 = 0, ¯
ϕ 0,1 = 10 and ¯
ϕ 0,2 =
¯
ϕ 0,3 = ¯
ϕ 0,4 = −10. We have X c 0 = 0, x i (t 0 ) = 0 and y i (t 0 ) = 0 for all i =
1, . . . , 4, whereas −z 1 (t 0 ) = z 2 (t 0 ) = z 3 (t 0 ) = z 4 (t 0 ) = 10.
The previous analysis permits to conclude that for each MCC parameter Q 0 =
Q(w(t 0 )) = 0 and then Q 0 = 0 for the NMCC in (8.16). Then, the nonlinear
dynamics of the NMCC takes place on the manifold M c (0) for t ≥ t 0 . In addition,
the necessary condition for CS (in-phase synchronization), i.e., X c 0,i are equal for all
i (cf. (8.18)), is satisfied. The numerical simulation shown in Fig. 8.7 confirms the
analysis and the scenario reported in (c), that is each MCC oscillates according to the
symmetric limit cycle (only the variables x i (t) are reported in Fig. 8.7). In particular,
Fig. 8.8 presents the waveforms x i (t) over the time intervals [0, 25] and [60, 100].
Note that, after a transient lasting almost until t = 60, an in-phase oscillatory
synchronized state emerges due to the weak coupling.
We may change ICs in such a way that each uncoupled MCC would evolve on an
asymmetric limit cycle. In particular the ICs in (8.21) with ϕ 0 = 0, ¯
ϕ 0,1 = 1.1, and
¯
ϕ 0,2 = ¯
ϕ 0,3 = ¯
ϕ 0,4 = −1 have been used (all the other parameters are the same as
in the previous case) in the numerical simulation reported in Fig. 8.9. In such a case,
referred to as scenario (a) in the summary above, the weak coupling d = 0.005 gives
rise to a global periodic oscillation in the NMCC such that there exist π phase shifts
among the asymmetric limit cycles of each MCC (note the oscillations of x i (t), for
i = 1, . . . , 4, around −1.5). Such anti-phase synchronized state is shown in the
335
there are:
– two stable asymmetric limit cycles A + and A − surrounding the unstable
equilibria (+1.5, 0, −1.5) and (−1.5, 0, +1.5), respectively
– one stable symmetric limit cycle S s surrounding the unstable EP (0, 0, 0)
– one unstable symmetric limit cycle S u
• for a network composed by an arbitrary number (N) of Chua’s oscillators and
small coupling d it is seen that (see [26, p. 953]):
(a) there are 2 N stable asymmetric limit cycles with all the phase shifts equal to
π ;
(b) the other 2 N × (2 N −1 − 1) asymmetric limit cycles are unstable
(c) there is only one stable symmetric limit cycle with all the phase shifts equal
to zero
(d) the other (2 N − 1) symmetric limit cycles are unstable.
Cases (a) and (c), referred to as phase-locking in oscillatory arrays, are reported
in the next subsection, whereas the final subsection shows how the four MCC arrays
split into periodic and chaotic clusters.
8.4.1.1 Phase-Locking in Oscillatory NMCC
Consider a simple array made of four MCCs (N = 4) with uniform space-invariant
weak coupling d = 0.05 and ICs as in (8.21) with ϕ 0 = 0, ¯
ϕ 0,1 = 10 and ¯
ϕ 0,2 =
¯
ϕ 0,3 = ¯
ϕ 0,4 = −10. We have X c 0 = 0, x i (t 0 ) = 0 and y i (t 0 ) = 0 for all i =
1, . . . , 4, whereas −z 1 (t 0 ) = z 2 (t 0 ) = z 3 (t 0 ) = z 4 (t 0 ) = 10.
The previous analysis permits to conclude that for each MCC parameter Q 0 =
Q(w(t 0 )) = 0 and then Q 0 = 0 for the NMCC in (8.16). Then, the nonlinear
dynamics of the NMCC takes place on the manifold M c (0) for t ≥ t 0 . In addition,
the necessary condition for CS (in-phase synchronization), i.e., X c 0,i are equal for all
i (cf. (8.18)), is satisfied. The numerical simulation shown in Fig. 8.7 confirms the
analysis and the scenario reported in (c), that is each MCC oscillates according to the
symmetric limit cycle (only the variables x i (t) are reported in Fig. 8.7). In particular,
Fig. 8.8 presents the waveforms x i (t) over the time intervals [0, 25] and [60, 100].
Note that, after a transient lasting almost until t = 60, an in-phase oscillatory
synchronized state emerges due to the weak coupling.
We may change ICs in such a way that each uncoupled MCC would evolve on an
asymmetric limit cycle. In particular the ICs in (8.21) with ϕ 0 = 0, ¯
ϕ 0,1 = 1.1, and
¯
ϕ 0,2 = ¯
ϕ 0,3 = ¯
ϕ 0,4 = −1 have been used (all the other parameters are the same as
in the previous case) in the numerical simulation reported in Fig. 8.9. In such a case,
referred to as scenario (a) in the summary above, the weak coupling d = 0.005 gives
rise to a global periodic oscillation in the NMCC such that there exist π phase shifts
among the asymmetric limit cycles of each MCC (note the oscillations of x i (t), for
i = 1, . . . , 4, around −1.5). Such anti-phase synchronized state is shown in the