7 Non-asymptotic-time Dynamics
115
7.2 The Non-autonomous Adler Equation
As in [24], the phase difference for a unidirectionally coupled pair of phase-oscillators
with Kuramoto-type coupling can be modelled by an Adler equation with additive
forcing, i.e.
˙
θ(t) = −a sin(θ (t)) + G(t).
(7.1)
Physically, if the oscillator-pair is an isolated system then the driving frequency will
be constant (i.e. not time-dependent), and as a result the term G(t) will simply be a
constant (i.e. with no dependence on t). In this case, (7.1) is an autonomous dynamical
system on the circle. But if, instead, the driving oscillator is open to external influence,
then this is likely to mean that the driving frequency will be time-dependent, and
as a result the term G(t) will depend on t. In this case, (7.1) is a non-autonomous
dynamical system on the circle. In either case, for studying dynamics, we can without
loss of generality take a > 0.
For our consideration here, just as in the previous chapter [24], we will take the
forcing term G(t) to be slowly time-dependent, corresponding to slow frequency
modulation of the driving oscillator. In other words, the timescale of variation of
G(t) is slower than the timescale of the dynamics of θ itself. (This can be formalised
as in Sect. 7.5.).
7.2.1 Stability and Neutral Stability in the Autonomous Case
Before considering the non-autonomous case where G(t) depends slowly on t, let us
first consider the autonomous case where G is constant, in which case we can apply
a classical dynamical systems analysis to (7.1).
• If G ∈ (−a, a), all solutions θ(t) of (7.1) converge to the sink y := arc sin
G
a
,
apart from the repulsive constant solution at the source π − arc sin
G
a
. Hence in
particular, for any solution other than the repulsive solution, a small perturbation to
the initial condition will be “forgotten over time”, since eventually the solution will
settle at the sink in any case. This “loss of memory of perturbation” corresponds
physically to stability. The separation between solutions starting at different initial
conditions (other than the source) will decay at an exponential rate, with exponent
−a cos(y) = −
√
a 2 − G 2 ; this is the ALE of every trajectory of (7.1) other than
the single repulsive trajectory, and quantifies the system’s stability.
• If G /
∈ [−a, a], all the trajectories θ(t) of (7.1) move strictly periodically round
the circle with common period
2π
0
dθ
|−a sin(θ)+G|
=
2π
√
G 2 −a 2 . Hence, on the one hand,
the separation between solutions starting at two nearby conditions does not decay
as time tends to infinity, but on the other hand, the two solutions will remain nearby
forever. So the effect of a small perturbation to a solution’s initial condition will
115
7.2 The Non-autonomous Adler Equation
As in [24], the phase difference for a unidirectionally coupled pair of phase-oscillators
with Kuramoto-type coupling can be modelled by an Adler equation with additive
forcing, i.e.
˙
θ(t) = −a sin(θ (t)) + G(t).
(7.1)
Physically, if the oscillator-pair is an isolated system then the driving frequency will
be constant (i.e. not time-dependent), and as a result the term G(t) will simply be a
constant (i.e. with no dependence on t). In this case, (7.1) is an autonomous dynamical
system on the circle. But if, instead, the driving oscillator is open to external influence,
then this is likely to mean that the driving frequency will be time-dependent, and
as a result the term G(t) will depend on t. In this case, (7.1) is a non-autonomous
dynamical system on the circle. In either case, for studying dynamics, we can without
loss of generality take a > 0.
For our consideration here, just as in the previous chapter [24], we will take the
forcing term G(t) to be slowly time-dependent, corresponding to slow frequency
modulation of the driving oscillator. In other words, the timescale of variation of
G(t) is slower than the timescale of the dynamics of θ itself. (This can be formalised
as in Sect. 7.5.).
7.2.1 Stability and Neutral Stability in the Autonomous Case
Before considering the non-autonomous case where G(t) depends slowly on t, let us
first consider the autonomous case where G is constant, in which case we can apply
a classical dynamical systems analysis to (7.1).
• If G ∈ (−a, a), all solutions θ(t) of (7.1) converge to the sink y := arc sin
G
a
,
apart from the repulsive constant solution at the source π − arc sin
G
a
. Hence in
particular, for any solution other than the repulsive solution, a small perturbation to
the initial condition will be “forgotten over time”, since eventually the solution will
settle at the sink in any case. This “loss of memory of perturbation” corresponds
physically to stability. The separation between solutions starting at different initial
conditions (other than the source) will decay at an exponential rate, with exponent
−a cos(y) = −
√
a 2 − G 2 ; this is the ALE of every trajectory of (7.1) other than
the single repulsive trajectory, and quantifies the system’s stability.
• If G /
∈ [−a, a], all the trajectories θ(t) of (7.1) move strictly periodically round
the circle with common period
2π
0
dθ
|−a sin(θ)+G|
=
2π
√
G 2 −a 2 . Hence, on the one hand,
the separation between solutions starting at two nearby conditions does not decay
as time tends to infinity, but on the other hand, the two solutions will remain nearby
forever. So the effect of a small perturbation to a solution’s initial condition will