6 Synchronisation and Non-autonomicity
103
This stabilising effect of the external modulation can be compared to that of
bounded noise illustrated in Fig. 6.5. The net effect averaged over long time is similar,
such that curves showing the parameter-dependence of the ALE can be made to look
similar between the case of bounded noise and the case of deterministic slow driving
frequency modulation. The same is also true for the parameter-dependence of ψ ,
which likewise represents an average over infinite time. However, even if the effects
of bounded noise and of slow deterministic modulation are similar to each other on
average over long times, the actual dynamics observed in time is very different in
nature. This is discussed in more detail in [35].
Now that we have described in detail the three dynamical regimes over short and
long times, we generalise this to networks of oscillators in the next section.
6.4.4 Network: Time-Varying Frequency Driving
Now, results from the previous section are extended to networks. Instead of considering the single driven oscillator of system (6.9), we now present an arbitrary
(undirected) network of N identical such oscillators, driven by the external phase
θ 0 (t), studied in [34]
˙
θ i = ω + D
N
j=1
A i j sin(θ i − θ j ) + γ sin[θ i − θ 0 (t)],
(6.24)
with frequency ω, coupling constant D, symmetric adjacency matrix A. Entries A i j
are set to 1 for connected oscillators, and 0 otherwise. All oscillators in the networks
are driven with strength γ by the external oscillator θ 0 (t) evolving according to
Eq. (6.20) as in the previous section. As mentioned earlier, system (6.24) is a direct
generalisation of Eq. (6.9) to networks.
Without external driving, γ = 0, the network converges to a fully synchronous
solution θ 1 (t) = θ 2 (t) = · · · = θ N (t) = c + ωt for some c, provided that the network
does not have disconnected components, and that couplings are attractive, D < 0,
see [58–60].
With external driving, γ > 0, three regimes are possible for the fully synchronous
solution: it can synchronise to the external driving, or not synchronise to the external
driving, or intermittently synchronise to the external driving. These regimes correspond to the three regimes of the previous section, Sect. 6.4.3, for a single driven
oscillator. In fact, when all nodes in the network are in a synchronous state, they can
be seen as one single node driven by the external phase. The region in parameter
space for each of the regimes is also identical to the single oscillator case illustrated
in Figs. 6.6 and 6.8 of Sect. 6.4.3.
For convenience, system (6.24) can be rewritten in the rotating reference frame
of the driving oscillator, ψ i = θ i − θ 0 (t),
103
This stabilising effect of the external modulation can be compared to that of
bounded noise illustrated in Fig. 6.5. The net effect averaged over long time is similar,
such that curves showing the parameter-dependence of the ALE can be made to look
similar between the case of bounded noise and the case of deterministic slow driving
frequency modulation. The same is also true for the parameter-dependence of ψ ,
which likewise represents an average over infinite time. However, even if the effects
of bounded noise and of slow deterministic modulation are similar to each other on
average over long times, the actual dynamics observed in time is very different in
nature. This is discussed in more detail in [35].
Now that we have described in detail the three dynamical regimes over short and
long times, we generalise this to networks of oscillators in the next section.
6.4.4 Network: Time-Varying Frequency Driving
Now, results from the previous section are extended to networks. Instead of considering the single driven oscillator of system (6.9), we now present an arbitrary
(undirected) network of N identical such oscillators, driven by the external phase
θ 0 (t), studied in [34]
˙
θ i = ω + D
N
j=1
A i j sin(θ i − θ j ) + γ sin[θ i − θ 0 (t)],
(6.24)
with frequency ω, coupling constant D, symmetric adjacency matrix A. Entries A i j
are set to 1 for connected oscillators, and 0 otherwise. All oscillators in the networks
are driven with strength γ by the external oscillator θ 0 (t) evolving according to
Eq. (6.20) as in the previous section. As mentioned earlier, system (6.24) is a direct
generalisation of Eq. (6.9) to networks.
Without external driving, γ = 0, the network converges to a fully synchronous
solution θ 1 (t) = θ 2 (t) = · · · = θ N (t) = c + ωt for some c, provided that the network
does not have disconnected components, and that couplings are attractive, D < 0,
see [58–60].
With external driving, γ > 0, three regimes are possible for the fully synchronous
solution: it can synchronise to the external driving, or not synchronise to the external
driving, or intermittently synchronise to the external driving. These regimes correspond to the three regimes of the previous section, Sect. 6.4.3, for a single driven
oscillator. In fact, when all nodes in the network are in a synchronous state, they can
be seen as one single node driven by the external phase. The region in parameter
space for each of the regimes is also identical to the single oscillator case illustrated
in Figs. 6.6 and 6.8 of Sect. 6.4.3.
For convenience, system (6.24) can be rewritten in the rotating reference frame
of the driving oscillator, ψ i = θ i − θ 0 (t),