7 Non-asymptotic-time Dynamics
123
(B) Equation (7.7) exhibits stability in the following sense:
– There is a
2π
ω
-periodic solution p(t) that attracts all solutions as t → ∞,
apart from a single repulsive periodic solution π − p(−t).
– The ALE of every trajectory other than the repulsive trajectory π − p(−t) is
a negative number λ.
(C) Equation (7.7) lies “on the boundary between stability and neutral stability”, in
the following sense: There is a unique
2π
ω
-periodic solution, which is attracting
from one direction but is unstable due to being repulsive in the other direction,
and all solutions converge to this solution as t → ∞ and have an ALE of 0.
Furthermore, if the mean frequency is not an integer multiple of ω, then we must
be in case (A).
This proposition is essentially Theorems 1 and 4 of [12]. So we see that the set of
possible behaviours for the periodically forced Adler equation is directly analogous
to the set of possible behaviours for the autonomous Adler equation in Sect. 7.2.1.
(This is quite a special property of Adler equations, that does not generalise to other
one-dimensional phase-oscillator models [6].)
For the case that k > a, we emphasise the following corollary of Proposition 1.
Corollary 1 Assume k > a, and fix any A ≥ 0. There are intervals of ω-values
arbitrarily close to 0 for which Eq. (7.7) exhibits neutral stability in the sense of
Proposition 1(A).
Proof For any trajectory θ(t), since
k − a + A cos(ωt) ≤ ˙
θ(t) ≤ k + a + A cos(ωt)
for all t, and since the term A cos(ωt) has an average of 0, it follows that
k − a ≤ ≤ k + a.
Now depends continuously on the parameters of Eq. (7.7) [17, Proposition 11.1.6]. Hence, since is constrained to the interval [k − a, k + a] ⊂ (0, ∞),
it follows that as ω → 0, there will always be intervals of ω-values for which is
not an integer multiple of ω.
Note that Corollary 1 applies to every A ≥ 0: whatever the value of A, it is possible for the time-dependence of the driving G(t) := k + A cos(ωt) to be arbitrarily
slow (corresponding to arbitrarily small ω) and yet for the system to exhibit neutral
stability in the sense of Proposition 1(A). In both the statements and the proofs of
Proposition 1 and Corollary 1, no distinction whatsoever appears between the situation that G(t) remains forever outside [−a, a] and the situation that there are times
t when G(t) ∈ (−a, a), as considered in Sect. 7.2.2. In particular, the statement and
proof of Corollary 1 recognise no distinction between A-values less than a critical
value A ∗ and A-values larger than a critical value A ∗ .
123
(B) Equation (7.7) exhibits stability in the following sense:
– There is a
2π
ω
-periodic solution p(t) that attracts all solutions as t → ∞,
apart from a single repulsive periodic solution π − p(−t).
– The ALE of every trajectory other than the repulsive trajectory π − p(−t) is
a negative number λ.
(C) Equation (7.7) lies “on the boundary between stability and neutral stability”, in
the following sense: There is a unique
2π
ω
-periodic solution, which is attracting
from one direction but is unstable due to being repulsive in the other direction,
and all solutions converge to this solution as t → ∞ and have an ALE of 0.
Furthermore, if the mean frequency is not an integer multiple of ω, then we must
be in case (A).
This proposition is essentially Theorems 1 and 4 of [12]. So we see that the set of
possible behaviours for the periodically forced Adler equation is directly analogous
to the set of possible behaviours for the autonomous Adler equation in Sect. 7.2.1.
(This is quite a special property of Adler equations, that does not generalise to other
one-dimensional phase-oscillator models [6].)
For the case that k > a, we emphasise the following corollary of Proposition 1.
Corollary 1 Assume k > a, and fix any A ≥ 0. There are intervals of ω-values
arbitrarily close to 0 for which Eq. (7.7) exhibits neutral stability in the sense of
Proposition 1(A).
Proof For any trajectory θ(t), since
k − a + A cos(ωt) ≤ ˙
θ(t) ≤ k + a + A cos(ωt)
for all t, and since the term A cos(ωt) has an average of 0, it follows that
k − a ≤ ≤ k + a.
Now depends continuously on the parameters of Eq. (7.7) [17, Proposition 11.1.6]. Hence, since is constrained to the interval [k − a, k + a] ⊂ (0, ∞),
it follows that as ω → 0, there will always be intervals of ω-values for which is
not an integer multiple of ω.
Note that Corollary 1 applies to every A ≥ 0: whatever the value of A, it is possible for the time-dependence of the driving G(t) := k + A cos(ωt) to be arbitrarily
slow (corresponding to arbitrarily small ω) and yet for the system to exhibit neutral
stability in the sense of Proposition 1(A). In both the statements and the proofs of
Proposition 1 and Corollary 1, no distinction whatsoever appears between the situation that G(t) remains forever outside [−a, a] and the situation that there are times
t when G(t) ∈ (−a, a), as considered in Sect. 7.2.2. In particular, the statement and
proof of Corollary 1 recognise no distinction between A-values less than a critical
value A ∗ and A-values larger than a critical value A ∗ .