122
J. M. I. Newman et al.
7.4 The Adler Equation with Sinusoidal Driving
In order to illustrate point (3) of the three main points in Sect. 7.1.2, we now consider
the Adler equation with slow-timescale sinusoidal driving (just as was used for the
numerical illustrations in the chapter [24]). The strict periodicity of the driving means
that the theoretical analysis in Sect. 7.2, as well as numerics of the kind presented in
Sect. 7.3, can now be compared with a long-time-asymptotic analysis of stability.
We consider the system (7.3) with g(t) = cos(ωt), i.e.
˙
θ(t) = −a sin(θ (t)) + k + A cos(ωt),
(7.7)
with a, k, ω > 0 and A ≥ 0. We assume slow time-dependence of the driving, specifically in the sense that both ω and the product Aω are very small. We will also mostly
focus on the case that k > a, so that the picture in Sect. 7.2.4 is applicable.
Let us first present the basic results of a classical stability analysis.
7.4.1 An Initial Long-time-asymptotic Analysis of Dynamics
The case that A = 0 is the autonomous case for which a classical stability analysis
was carried out in Sect. 7.2.1 (with k = G), as well as in Sect. 6.4.1 of the chapter
[24]. We now address the general case from the same classical point of view.
The ALE of a trajectory θ(t) of (7.7) is precisely the limit as T → ∞ of the FTLE
λ T given by (7.6). Now let be the “mean frequency” ˙
θ of trajectories of (7.7),
namely
= lim
T →∞
1
T
T
0
˙
θ(t) dt,
which does not depend on the initial condition θ(0) of the trajectory θ(t), but does
depend on the system parameters a, k, A and ω. For the autonomous case A = 0, as in
[24, Sect. 6.4.1] we have = 0 in the stable scenario (k < a), but =
√
k 2 − a 2 >
0 in the neutrally stable scenario (k > a).
Proposition 1 For any given parameter values a, k, A, ω in Eq. (7.7), exactly one
of the following three statements holds.
(A) Equation (7.7) exhibits neutral stability in the following sense:
– There is a finite bound M > 0 such that for any two distinct solutions θ 1 (t)
and θ 2 (t) of (7.7), for all time t ∈ R,
1
M
≤
distance between θ 1 (t) and θ 2 (t)
distance between θ 1 (0) and θ 2 (0)
≤ M.
– The ALE of every trajectory is 0.
J. M. I. Newman et al.
7.4 The Adler Equation with Sinusoidal Driving
In order to illustrate point (3) of the three main points in Sect. 7.1.2, we now consider
the Adler equation with slow-timescale sinusoidal driving (just as was used for the
numerical illustrations in the chapter [24]). The strict periodicity of the driving means
that the theoretical analysis in Sect. 7.2, as well as numerics of the kind presented in
Sect. 7.3, can now be compared with a long-time-asymptotic analysis of stability.
We consider the system (7.3) with g(t) = cos(ωt), i.e.
˙
θ(t) = −a sin(θ (t)) + k + A cos(ωt),
(7.7)
with a, k, ω > 0 and A ≥ 0. We assume slow time-dependence of the driving, specifically in the sense that both ω and the product Aω are very small. We will also mostly
focus on the case that k > a, so that the picture in Sect. 7.2.4 is applicable.
Let us first present the basic results of a classical stability analysis.
7.4.1 An Initial Long-time-asymptotic Analysis of Dynamics
The case that A = 0 is the autonomous case for which a classical stability analysis
was carried out in Sect. 7.2.1 (with k = G), as well as in Sect. 6.4.1 of the chapter
[24]. We now address the general case from the same classical point of view.
The ALE of a trajectory θ(t) of (7.7) is precisely the limit as T → ∞ of the FTLE
λ T given by (7.6). Now let be the “mean frequency” ˙
θ of trajectories of (7.7),
namely
= lim
T →∞
1
T
T
0
˙
θ(t) dt,
which does not depend on the initial condition θ(0) of the trajectory θ(t), but does
depend on the system parameters a, k, A and ω. For the autonomous case A = 0, as in
[24, Sect. 6.4.1] we have = 0 in the stable scenario (k < a), but =
√
k 2 − a 2 >
0 in the neutrally stable scenario (k > a).
Proposition 1 For any given parameter values a, k, A, ω in Eq. (7.7), exactly one
of the following three statements holds.
(A) Equation (7.7) exhibits neutral stability in the following sense:
– There is a finite bound M > 0 such that for any two distinct solutions θ 1 (t)
and θ 2 (t) of (7.7), for all time t ∈ R,
1
M
≤
distance between θ 1 (t) and θ 2 (t)
distance between θ 1 (0) and θ 2 (0)
≤ M.
– The ALE of every trajectory is 0.