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not be forgotten but nonetheless it will remain small. This corresponds physically
to neutral stability. In this case, the ALE of every trajectory is 0.
So in short, if G ∈ (−a, a) then the system (7.1) describes a stable process, but
if G /
∈ [−a, a] then the system (7.1) describes a neutrally stable process.
Finally, let us briefly mention the boundary between stability and neutral stability,
which occurs when |G| is exactly equal to a. In this case, there is a unique fixed point,
which is attracting from one direction but is unstable due to being repelling in the
other direction, and all trajectories converge to this unstable fixed point and have an
ALE of 0. For most of this chapter, we will leave out the analysis of such degenerate
cases lying on the boundary between two scenarios.
7.2.2 Stability and Neutral Stability in the Non-autonomous
Case
In view of the above statemets for the autonomous case, now under the assumption
that G(t) varies very slowly with time t, a “conceptual-level” analysis of (7.1) gives
us the following:
• While G(t) ∈ (−a, a), the trajectories move away from the vicinity of the slowly
moving source π − arc sin
G(t)
a
, and cluster together into an increasingly tight
cluster near the slowly moving sink y(t) := arc sin
G(t)
a
. In this clustering, the
separation between different trajectories decays approximately according to an
exponential decay of exponent −a cos(y(t)) = −
a 2 − G(t) 2 .
• While G(t) /
∈ [−a, a], trajectories move approximately periodically round the
circle with common approximate period
2π
√
G(t) 2 −a 2
.
• Thus overall, over a given time-interval [0, T ], we have the following:
(a) If there are subintervals during which G(t) ∈ (−a, a), then the solutions starting
at different initial conditions end up clustered extremely close to each other
(apart from those solutions that start extremely near the maximally repulsive
solution which follows the slowly moving source while G(t) ∈ (−a, a)). So
the effect of a perturbation to a solution’s initial condition will eventually be
forgotten over time.
(b) If there are no times at which G(t) ∈ [−a, a] then the system does not exhibit
significant separation or attraction between the trajectories of different initial
conditions. So the effect of a small perturbation to a solution’s initial condition
will not be forgotten but on the other hand will remain small.
Once again, the loss of memory of initial condition in case (a) corresponds physically to stability, and the lack of significant mutual attraction or separation of
solutions in case (b) corresponds physically to neutral stability.
J. M. I. Newman et al.
not be forgotten but nonetheless it will remain small. This corresponds physically
to neutral stability. In this case, the ALE of every trajectory is 0.
So in short, if G ∈ (−a, a) then the system (7.1) describes a stable process, but
if G /
∈ [−a, a] then the system (7.1) describes a neutrally stable process.
Finally, let us briefly mention the boundary between stability and neutral stability,
which occurs when |G| is exactly equal to a. In this case, there is a unique fixed point,
which is attracting from one direction but is unstable due to being repelling in the
other direction, and all trajectories converge to this unstable fixed point and have an
ALE of 0. For most of this chapter, we will leave out the analysis of such degenerate
cases lying on the boundary between two scenarios.
7.2.2 Stability and Neutral Stability in the Non-autonomous
Case
In view of the above statemets for the autonomous case, now under the assumption
that G(t) varies very slowly with time t, a “conceptual-level” analysis of (7.1) gives
us the following:
• While G(t) ∈ (−a, a), the trajectories move away from the vicinity of the slowly
moving source π − arc sin
G(t)
a
, and cluster together into an increasingly tight
cluster near the slowly moving sink y(t) := arc sin
G(t)
a
. In this clustering, the
separation between different trajectories decays approximately according to an
exponential decay of exponent −a cos(y(t)) = −
a 2 − G(t) 2 .
• While G(t) /
∈ [−a, a], trajectories move approximately periodically round the
circle with common approximate period
2π
√
G(t) 2 −a 2
.
• Thus overall, over a given time-interval [0, T ], we have the following:
(a) If there are subintervals during which G(t) ∈ (−a, a), then the solutions starting
at different initial conditions end up clustered extremely close to each other
(apart from those solutions that start extremely near the maximally repulsive
solution which follows the slowly moving source while G(t) ∈ (−a, a)). So
the effect of a perturbation to a solution’s initial condition will eventually be
forgotten over time.
(b) If there are no times at which G(t) ∈ [−a, a] then the system does not exhibit
significant separation or attraction between the trajectories of different initial
conditions. So the effect of a small perturbation to a solution’s initial condition
will not be forgotten but on the other hand will remain small.
Once again, the loss of memory of initial condition in case (a) corresponds physically to stability, and the lack of significant mutual attraction or separation of
solutions in case (b) corresponds physically to neutral stability.