94
M. Lucas et al.
Numerical integration of the systems is performed with a Runge-Kutta 4 scheme
with a timestep of 0.01 s. The time-frequency representations are computed according
to the Morlet wavelet transform ( p = 1) from the PyMODA package [7].
6.4.1 Single Oscillator: Periodic Driving
We start with the following driven phase oscillator system
˙
θ 1 = ω + γ sin[θ 1 − θ 0 (t)],
(6.9)
where the driving has strength γ , phase θ 0 (t), and a constant frequency
˙
θ 0 = ω 0 .
(6.10)
System (6.9)–(6.10) is well known in the literature, see for example [49], and it
is schematically shown in Fig. 6.2a. We nonetheless present the parts of its analysis
that are relevant for the remainder of the present chapter.
System (6.9) with its driving given by Eq. (6.10) is more conveniently studied
in the reference frame of the driving. The phase difference ψ = θ 1 − ω 0 t evolves
according to what is known as the Adler equation
˙
ψ = + γ sin ψ,
(6.11)
where = ω − ω 0 is called the frequency mismatch. The frequency mismatch
and the driving strength determine the dynamics of the system, and in particular the
stability of the driven oscillator. There are two scenarios: either Eq. (6.11) has a
Fig. 6.2 Synchronisation in
coupled oscillators with
time-varying parameters:
hierarchy of systems
presented, in increasing
complexity. a Single
oscillator with periodic
driving. b Single oscillator
with noisy driving. c Single
oscillator driven at a
time-varying frequency. The
last of these is generalised to
d a driven network and e a
driven network with
time-varying driving strength
in place of time-varying
driving frequency
Single oscillator
Network
ω
ω 0
ω
ω 0 (t)
Time-varying frequency
ω
ω 0 -ξ(t)
Noisy frequency
ω
ω
ω
ω
ω
ω 0 (t)
Time-varying frequency
ω
ω
ω
ω
ω
ω 0
γ(t)
Time-varying coupling
Periodic driving
(a)
(b)
(c)
(e)
(d)
M. Lucas et al.
Numerical integration of the systems is performed with a Runge-Kutta 4 scheme
with a timestep of 0.01 s. The time-frequency representations are computed according
to the Morlet wavelet transform ( p = 1) from the PyMODA package [7].
6.4.1 Single Oscillator: Periodic Driving
We start with the following driven phase oscillator system
˙
θ 1 = ω + γ sin[θ 1 − θ 0 (t)],
(6.9)
where the driving has strength γ , phase θ 0 (t), and a constant frequency
˙
θ 0 = ω 0 .
(6.10)
System (6.9)–(6.10) is well known in the literature, see for example [49], and it
is schematically shown in Fig. 6.2a. We nonetheless present the parts of its analysis
that are relevant for the remainder of the present chapter.
System (6.9) with its driving given by Eq. (6.10) is more conveniently studied
in the reference frame of the driving. The phase difference ψ = θ 1 − ω 0 t evolves
according to what is known as the Adler equation
˙
ψ = + γ sin ψ,
(6.11)
where = ω − ω 0 is called the frequency mismatch. The frequency mismatch
and the driving strength determine the dynamics of the system, and in particular the
stability of the driven oscillator. There are two scenarios: either Eq. (6.11) has a
Fig. 6.2 Synchronisation in
coupled oscillators with
time-varying parameters:
hierarchy of systems
presented, in increasing
complexity. a Single
oscillator with periodic
driving. b Single oscillator
with noisy driving. c Single
oscillator driven at a
time-varying frequency. The
last of these is generalised to
d a driven network and e a
driven network with
time-varying driving strength
in place of time-varying
driving frequency
Single oscillator
Network
ω
ω 0
ω
ω 0 (t)
Time-varying frequency
ω
ω 0 -ξ(t)
Noisy frequency
ω
ω
ω
ω
ω
ω 0 (t)
Time-varying frequency
ω
ω
ω
ω
ω
ω 0
γ(t)
Time-varying coupling
Periodic driving
(a)
(b)
(c)
(e)
(d)