7 Non-asymptotic-time Dynamics
125
seen to occur for the model in Sect. 7.3.2. Furthermore, for A > k − a the FTLEs
are indistinguishable from the negative value < 0 given in Eq. (7.8). However,
Corollary 1 says that whatever the value of A, it is possible for ω to be arbitrarily
small and still for (7.7) to have neutrally stable dynamics where the ALE of every
trajectory is 0.
So the question arises of how to reconcile these two seemingly conflicting descriptions of the dynamics of (7.7). Now it turns out that when A > k − a, the width of the
ω-intervals of neutral stability is extremely small compared to the separation between
them; this is proved rigorously in [7] for A = a and k ∈ (a, 2a). So, in other words,
while this neutral stability can occur for all A-values as in Corollary 1, nonetheless it is an extremely fine-tuned phenomenon in the case that A > k − a. In these
narrow ω-intervals of neutral stability for A > k − a, the mechanism for the breakdown of the conceptual analysis in Sect. 7.2.2 is a kind of canard-like phenomenon:
at some of the times when the forcing G(t) := k + A cos(ωt) starts to re-enter the
interval (−a, a), the previously formed cluster of mutually synchronised trajectories
happens to spend too much time tracking the motion of the slowly moving source
z(t) := π − arc sin
G(t)
a
and is thus de-synchronised.
However, the mathematical tools needed to obtain Proposition 1 and Corollary 1
neither reveal the narrowness of the ω-intervals of neutral stability when A > k −
a, nor hint at the existence of an adiabatic mechanism of stability for A > k − a
(namely as described in Sect. 7.2.2) which needs some other peculiar mechanism of
destabilisation to undo.
So in conclusion:
• We have seen in Fig. 7.3 a clear and physically important qualitative change in the
behaviour of the model (7.7) as A crosses from below k − a to above k − a, and
we have explained this in terms of very simple non-asymptotic-time reasoning, of
exactly the same kind as was applied to the model in Sect. 7.3.2.
• But simply pursuing a classical approach to stability analysis risks positively hindering the recognition of this important qualitative change.
Fig. 7.3 Dynamics of (7.7) with varying A, for ω = 10 −3 rad/s, from time 0 s up to time T = 2π ×
10 5 s =
200π
ω . Other parameters are a =
1
3 rad/s and k = 1 rad/s. In (a) and (b), for each A-value,
results for the evolution θ(t) of 50 equally spaced initial conditions θ(0) =
2πi
50 , i = 0, . . . , 49, are
shown: a shows the finite-time Lyapunov exponents λ T , as defined by (7.6), for these trajectories,
and also shows (defined in (7.8)) in grey; (b) shows the positions θ(T ) of these trajectories at time
T . In (c), for each A-value, the positions of θ(0) for the 50 trajectories of (7.7) with θ(T ) =
2πi
50 ,
i = 0, . . . , 49, are shown. The value k − a =
2
3 rad/s is marked by the black dashed line
125
seen to occur for the model in Sect. 7.3.2. Furthermore, for A > k − a the FTLEs
are indistinguishable from the negative value < 0 given in Eq. (7.8). However,
Corollary 1 says that whatever the value of A, it is possible for ω to be arbitrarily
small and still for (7.7) to have neutrally stable dynamics where the ALE of every
trajectory is 0.
So the question arises of how to reconcile these two seemingly conflicting descriptions of the dynamics of (7.7). Now it turns out that when A > k − a, the width of the
ω-intervals of neutral stability is extremely small compared to the separation between
them; this is proved rigorously in [7] for A = a and k ∈ (a, 2a). So, in other words,
while this neutral stability can occur for all A-values as in Corollary 1, nonetheless it is an extremely fine-tuned phenomenon in the case that A > k − a. In these
narrow ω-intervals of neutral stability for A > k − a, the mechanism for the breakdown of the conceptual analysis in Sect. 7.2.2 is a kind of canard-like phenomenon:
at some of the times when the forcing G(t) := k + A cos(ωt) starts to re-enter the
interval (−a, a), the previously formed cluster of mutually synchronised trajectories
happens to spend too much time tracking the motion of the slowly moving source
z(t) := π − arc sin
G(t)
a
and is thus de-synchronised.
However, the mathematical tools needed to obtain Proposition 1 and Corollary 1
neither reveal the narrowness of the ω-intervals of neutral stability when A > k −
a, nor hint at the existence of an adiabatic mechanism of stability for A > k − a
(namely as described in Sect. 7.2.2) which needs some other peculiar mechanism of
destabilisation to undo.
So in conclusion:
• We have seen in Fig. 7.3 a clear and physically important qualitative change in the
behaviour of the model (7.7) as A crosses from below k − a to above k − a, and
we have explained this in terms of very simple non-asymptotic-time reasoning, of
exactly the same kind as was applied to the model in Sect. 7.3.2.
• But simply pursuing a classical approach to stability analysis risks positively hindering the recognition of this important qualitative change.
Fig. 7.3 Dynamics of (7.7) with varying A, for ω = 10 −3 rad/s, from time 0 s up to time T = 2π ×
10 5 s =
200π
ω . Other parameters are a =
1
3 rad/s and k = 1 rad/s. In (a) and (b), for each A-value,
results for the evolution θ(t) of 50 equally spaced initial conditions θ(0) =
2πi
50 , i = 0, . . . , 49, are
shown: a shows the finite-time Lyapunov exponents λ T , as defined by (7.6), for these trajectories,
and also shows (defined in (7.8)) in grey; (b) shows the positions θ(T ) of these trajectories at time
T . In (c), for each A-value, the positions of θ(0) for the 50 trajectories of (7.7) with θ(T ) =
2πi
50 ,
i = 0, . . . , 49, are shown. The value k − a =
2
3 rad/s is marked by the black dashed line