98
M. Lucas et al.
where ξ(t) is a noise process. This system is schematically shown in Fig. 6.2b. The
phase difference ψ = θ 1 − ω 0 t now evolves as
˙
ψ = ω + γ sin ψ + ξ(t).
(6.19)
Analysis of this equation can be found in [49], and we present here only what is
relevant to the comparison with the other systems of this chapter.
When γ > |ω|, the effect of the noise term in Eq. (6.19) is to push the trajectory
away from the stable fixed point ψ S . If at any time this noisy perturbation becomes
too strong, it can push the trajectory ψ all the way to the other side of the unstable
fixed point, inducing a phase slip where ψ makes a quick full 2π revolution before
returning to the vicinity of the stable fixed point. We now consider separately the
case of unbounded noise and bounded noise.
If ξ is an unbounded noise process such as stationary Gaussian white noise, then
phase slips will always take place, and so overall we do not have phase-locking
between the driven and driving oscillators, regardless of the values of γ and ω.
Defining the mean frequency difference ψ according to Eq. (6.17), the phase slips
induced by unbounded noise will completely destroy the ψ = 0 plateau that was
observed in Fig. 6.3c to correspond to γ > |ω| in the case of strictly periodic
driving. However, the λ < 0 region which also corresponded to γ > |ω| as seen
in Fig. 6.3b is not destroyed; instead, the extreme opposite happens, namely that
the λ < 0 region becomes the entire (γ , ,ω)-parameter space (see the analytical
derivation in [49]). In other words, regardless of the parameter values the trajectories
ψ of Eq. (6.19) have negative ALE. Physically, this indicates that different trajectories
of the driven oscillator θ 1 evolving under a common noise realisation of ξ(t) but
starting at different initial phases θ 1 (0) will mutually converge towards each other.
Synchronisation of oscillators by common noise is a well-studied phenomenon, see
for example [3, 37, 39, 47].
Now suppose ξ is a bounded noise process (e.g. dichotomous Markov noise [23]),
meaning that ξ(t) can only take values in a bounded interval. Then the ψ = 0 plateau
is decreased in width compared to the scenario without noise, as shown in Fig. 6.5b,
and the λ < 0 region is widened compared to the scenario without noise, as shown in
Fig. 6.5a. However, if the bound on the noise strength |ξ(t)| is less than the coupling
strength γ , then phase slips cannot occur when |ω| is small enough, and so the
ψ = 0 plateau is not completely destroyed.
So we have seen that in contrast to the deterministic case of strictly periodic driving
considered in Sect. 6.4.1, now for both bounded and unbounded noise, having ψ = 0
is not equivalent to having λ < 0. Indeed, they have different physical interpretations:
having ψ = 0 indicates that the frequency of the driven oscillator is entrained to that
of the driving; but having a negative ALE λ < 0 indicates that the driven oscillator
loses memory of its initial phase (i.e. solutions starting at different initial phases
mutually converge) due to the influence of the driver.
We will see in the next Sect. 6.4.3 how the bounded noise case relates to that of
a deterministic time-varying driving frequency.
M. Lucas et al.
where ξ(t) is a noise process. This system is schematically shown in Fig. 6.2b. The
phase difference ψ = θ 1 − ω 0 t now evolves as
˙
ψ = ω + γ sin ψ + ξ(t).
(6.19)
Analysis of this equation can be found in [49], and we present here only what is
relevant to the comparison with the other systems of this chapter.
When γ > |ω|, the effect of the noise term in Eq. (6.19) is to push the trajectory
away from the stable fixed point ψ S . If at any time this noisy perturbation becomes
too strong, it can push the trajectory ψ all the way to the other side of the unstable
fixed point, inducing a phase slip where ψ makes a quick full 2π revolution before
returning to the vicinity of the stable fixed point. We now consider separately the
case of unbounded noise and bounded noise.
If ξ is an unbounded noise process such as stationary Gaussian white noise, then
phase slips will always take place, and so overall we do not have phase-locking
between the driven and driving oscillators, regardless of the values of γ and ω.
Defining the mean frequency difference ψ according to Eq. (6.17), the phase slips
induced by unbounded noise will completely destroy the ψ = 0 plateau that was
observed in Fig. 6.3c to correspond to γ > |ω| in the case of strictly periodic
driving. However, the λ < 0 region which also corresponded to γ > |ω| as seen
in Fig. 6.3b is not destroyed; instead, the extreme opposite happens, namely that
the λ < 0 region becomes the entire (γ , ,ω)-parameter space (see the analytical
derivation in [49]). In other words, regardless of the parameter values the trajectories
ψ of Eq. (6.19) have negative ALE. Physically, this indicates that different trajectories
of the driven oscillator θ 1 evolving under a common noise realisation of ξ(t) but
starting at different initial phases θ 1 (0) will mutually converge towards each other.
Synchronisation of oscillators by common noise is a well-studied phenomenon, see
for example [3, 37, 39, 47].
Now suppose ξ is a bounded noise process (e.g. dichotomous Markov noise [23]),
meaning that ξ(t) can only take values in a bounded interval. Then the ψ = 0 plateau
is decreased in width compared to the scenario without noise, as shown in Fig. 6.5b,
and the λ < 0 region is widened compared to the scenario without noise, as shown in
Fig. 6.5a. However, if the bound on the noise strength |ξ(t)| is less than the coupling
strength γ , then phase slips cannot occur when |ω| is small enough, and so the
ψ = 0 plateau is not completely destroyed.
So we have seen that in contrast to the deterministic case of strictly periodic driving
considered in Sect. 6.4.1, now for both bounded and unbounded noise, having ψ = 0
is not equivalent to having λ < 0. Indeed, they have different physical interpretations:
having ψ = 0 indicates that the frequency of the driven oscillator is entrained to that
of the driving; but having a negative ALE λ < 0 indicates that the driven oscillator
loses memory of its initial phase (i.e. solutions starting at different initial phases
mutually converge) due to the influence of the driver.
We will see in the next Sect. 6.4.3 how the bounded noise case relates to that of
a deterministic time-varying driving frequency.