6 Synchronisation and Non-autonomicity
97
of ˜
ψ(t) gives λ = 0 as shown in Fig. 6.4g in dashed grey, and so the phase of θ 1 is
only neutrally stable.
The Lyapunov exponents tells us about the stability of the solution. We complement this information with a measure about the motion of the solution itself: the
mean frequency difference, often called beat frequency, which is defined as
ψ = 2π
2π
0
dψ
ω + γ sin ψ
−1
,
(6.16)
and is related to the other frequencies of the system by
˙
θ 1 = ω 0 + ψ .
(6.17)
In the synchronised case, the driven oscillator rotates with the frequency of the
driving, and hence ψ = 0, otherwise it is non-zero. So, here, the region of synchronisation is equivalently characterised by the plateau ψ = 0 and a negative
ALE λ < 0, as illustrated in Fig. 6.3b, c by the solid black curves. This equivalence
between a negative LE and a zero beat frequency no longer holds in more complicated cases, as we shall discuss in the next sections, and is illustrated by the other
curves in Fig. 6.5.
One can obtain further information about the frequency content of solutions by
applying a Fourier transform, for a static picture, or a wavelet transform, for a timeresolved picture. In this periodic driving case, however, both methods yield equivalent
results, as shown in by the constant content of the time-frequency representation in
Fig. 6.4d–h). In the synchronised case shown in Fig. 6.4d, we see only one component,
namely at
ω 0
2π
, indicating that the driven frequency is entrained to that of the driver.
The non-synchronised case is shown in Fig. 6.4h. In this case, we do not have
frequency entrainment; instead, the driven oscillator is quasi-periodic, containing
natural frequencies of both
ω 0
2π
(coming from the driving) and
ψ
2π
.
Now that we have presented the simplest case, that of a periodic (fixed-frequency)
driving, we consider more complicated driving scenarios.
6.4.2 Single Oscillator: Noisy Periodic Driving
Time-variability due to external perturbations is often modelled in the literature with
noisy processes, as described in the Introduction. For the purpose of comparison
with the deterministic autonomous and non-autonomous settings (the latter presented
shortly), we now describe such a case where the driving in system (6.9) has a timevariable frequency modelled as
˙
θ 0 = ω 0 − ξ(t),
(6.18)
97
of ˜
ψ(t) gives λ = 0 as shown in Fig. 6.4g in dashed grey, and so the phase of θ 1 is
only neutrally stable.
The Lyapunov exponents tells us about the stability of the solution. We complement this information with a measure about the motion of the solution itself: the
mean frequency difference, often called beat frequency, which is defined as
ψ = 2π
2π
0
dψ
ω + γ sin ψ
−1
,
(6.16)
and is related to the other frequencies of the system by
˙
θ 1 = ω 0 + ψ .
(6.17)
In the synchronised case, the driven oscillator rotates with the frequency of the
driving, and hence ψ = 0, otherwise it is non-zero. So, here, the region of synchronisation is equivalently characterised by the plateau ψ = 0 and a negative
ALE λ < 0, as illustrated in Fig. 6.3b, c by the solid black curves. This equivalence
between a negative LE and a zero beat frequency no longer holds in more complicated cases, as we shall discuss in the next sections, and is illustrated by the other
curves in Fig. 6.5.
One can obtain further information about the frequency content of solutions by
applying a Fourier transform, for a static picture, or a wavelet transform, for a timeresolved picture. In this periodic driving case, however, both methods yield equivalent
results, as shown in by the constant content of the time-frequency representation in
Fig. 6.4d–h). In the synchronised case shown in Fig. 6.4d, we see only one component,
namely at
ω 0
2π
, indicating that the driven frequency is entrained to that of the driver.
The non-synchronised case is shown in Fig. 6.4h. In this case, we do not have
frequency entrainment; instead, the driven oscillator is quasi-periodic, containing
natural frequencies of both
ω 0
2π
(coming from the driving) and
ψ
2π
.
Now that we have presented the simplest case, that of a periodic (fixed-frequency)
driving, we consider more complicated driving scenarios.
6.4.2 Single Oscillator: Noisy Periodic Driving
Time-variability due to external perturbations is often modelled in the literature with
noisy processes, as described in the Introduction. For the purpose of comparison
with the deterministic autonomous and non-autonomous settings (the latter presented
shortly), we now describe such a case where the driving in system (6.9) has a timevariable frequency modelled as
˙
θ 0 = ω 0 − ξ(t),
(6.18)