6 Synchronisation and Non-autonomicity
93
6.3.2.2 Wavelet Transform
Just as the ALE only gives an average measure of stability without time resolution,
the DFT only shows a static picture of the frequency content of a time series. To
overcome this limitation, one can use time-frequency representation which gives
how the frequency content of a trajectory evolves over time.
The continuous wavelet transform (CWT) is a time-frequency representation that
was developed to obtain a good time-frequency resolution simultaneously across
a broad range of timescales. The idea is to perform frequency analysis in a timewindow whose width is scaled in proportion with the timescale being investigated
The CWT of a signal x(t) is given by
W (s, t) =
1
s p
∞
−∞
u − t
s
x(u) du,
(6.7)
where s is a scaling factor, p is number typically chosen as either 1 or
1
2
, and is
the “wavelet function” which represents mathematically the time-localisation used
for the frequency analysis. Different types of wavelet function can be used. In this
chapter, the Morlet wavelet is used, which has good time-frequency resolution [10].
Taking the scaling factor s as reciprocally proportional to frequency ω, we can obtain
the power in a frequency interval [ω −
δω
2
, ω +
δω
2
] around each frequency ω at time
t as
P W (ω, t) =
ω+
δω
2
ω−
δω
2
|W (ω, t)|
2 dω
(6.8)
for p =
1
2
. A more detailed presentation of the CWT can be found in [10] and
references therein. The wavelet transform has been used e.g. in [10, 29, 30].
6.4 Systems Analysis
In this section, we present a succession of coupled oscillators models, in increasing
order of complexity. Figure 6.2 illustrates the different systems considered and the
relationship between them. In Sect. 6.4.1, we start with the simplest case: a periodically driven phase oscillator. Then, we present two models in which the driving is
made aperiodic: noisy, in Sect. 6.4.2, and deterministic with a time-varying driving
frequency, in Sect. 6.4.3. Finally, we present results of the former case generalised
to networks, in Sect. 6.4.4, for which we also present the case of a deterministic
time-varying coupling strength. For each model, we apply the methods described in
the previous section, and report results about the stability of the system.
93
6.3.2.2 Wavelet Transform
Just as the ALE only gives an average measure of stability without time resolution,
the DFT only shows a static picture of the frequency content of a time series. To
overcome this limitation, one can use time-frequency representation which gives
how the frequency content of a trajectory evolves over time.
The continuous wavelet transform (CWT) is a time-frequency representation that
was developed to obtain a good time-frequency resolution simultaneously across
a broad range of timescales. The idea is to perform frequency analysis in a timewindow whose width is scaled in proportion with the timescale being investigated
The CWT of a signal x(t) is given by
W (s, t) =
1
s p
∞
−∞
u − t
s
x(u) du,
(6.7)
where s is a scaling factor, p is number typically chosen as either 1 or
1
2
, and is
the “wavelet function” which represents mathematically the time-localisation used
for the frequency analysis. Different types of wavelet function can be used. In this
chapter, the Morlet wavelet is used, which has good time-frequency resolution [10].
Taking the scaling factor s as reciprocally proportional to frequency ω, we can obtain
the power in a frequency interval [ω −
δω
2
, ω +
δω
2
] around each frequency ω at time
t as
P W (ω, t) =
ω+
δω
2
ω−
δω
2
|W (ω, t)|
2 dω
(6.8)
for p =
1
2
. A more detailed presentation of the CWT can be found in [10] and
references therein. The wavelet transform has been used e.g. in [10, 29, 30].
6.4 Systems Analysis
In this section, we present a succession of coupled oscillators models, in increasing
order of complexity. Figure 6.2 illustrates the different systems considered and the
relationship between them. In Sect. 6.4.1, we start with the simplest case: a periodically driven phase oscillator. Then, we present two models in which the driving is
made aperiodic: noisy, in Sect. 6.4.2, and deterministic with a time-varying driving
frequency, in Sect. 6.4.3. Finally, we present results of the former case generalised
to networks, in Sect. 6.4.4, for which we also present the case of a deterministic
time-varying coupling strength. For each model, we apply the methods described in
the previous section, and report results about the stability of the system.