126
J. M. I. Newman et al.
7.5 Slow-Fast Finite-Time Dynamical Systems
In view of the conclusions at the end of Sects. 7.3 and 7.4, a natural question is
whether there might be other approaches to formally defining concepts of stability
and neutral stability other than the long-time-asymptotic approaches introduced by
Aleksandr Lyapunov.
For any physical process in the real world, time is a finite parameter just like
all other parameters, and we are interested in the behaviour on finite timescales.
The philosophy behind traditional long-time-asymptotic dynamics analysis is that
for many physical processes, what is practically observed on such finite timescales
of interest matches the theoretical limiting behaviour of a suitable mathematical
model of the process as t → ∞. However, we propose that for the kinds of slowly
time-dependent systems that we have been studying in this chapter, an appropriate
approach is not to treat time as the special parameter that can be approximated by an
infinite limit, but instead to treat the timescale separation as the special parameter that
can be approximated by an infinite limit. We do anticipate (as evidenced numerically
in [27]) that the timescale separation will often not need to be very great in order for
the infinite-limit approximation to give a suitable description of dynamics.
For one-dimensional phase-oscillator models, our proposed approach means that
we consider finite-time differential equations of the form
˙
θ(t) = F(θ (t), εt),
t ∈ [0,
1
ε
]
(7.9)
for some function F : R × [0, 1] → R that is 2π -periodic in its first input, and we
consider the limiting behaviour as ε → 0. For example, non-autonomous Adler equation models could take the form
˙
θ(t) = −a sin(θ ) + ˜
G(εt)
for some ˜
G : [0, 1] → R, and one can consider the limiting behaviour as ε → 0.
The study of limiting dynamical behaviour as timescale separation tends to infinity
is the subject of slow-fast dynamical systems [19]. Here, we are combining this slowfast approach with the philosophy of FTDS theory, by restricting the slow time εt
to the unit interval. In the paper [27], we use this “slow-fast FTDS” approach to
define notions of stability and neutral stability that are analogous to those seen in
scenarios (A) and (B) of Proposition 1 but from an “ε → 0 perspective” in place of a
“t → ∞ perspective”. We use these definitions to address rigorously the stabilisation
phenomenon that we have discussed in this chapter.
This chapter has focused on stabilisation of one-dimensional phase dynamics.
But as seen in [22, 24] and also numerically evidenced in [23], similar dynamical
phenomena are likely to be prevalent in higher-dimensional multiple-timescale nonautonomous systems as well.
J. M. I. Newman et al.
7.5 Slow-Fast Finite-Time Dynamical Systems
In view of the conclusions at the end of Sects. 7.3 and 7.4, a natural question is
whether there might be other approaches to formally defining concepts of stability
and neutral stability other than the long-time-asymptotic approaches introduced by
Aleksandr Lyapunov.
For any physical process in the real world, time is a finite parameter just like
all other parameters, and we are interested in the behaviour on finite timescales.
The philosophy behind traditional long-time-asymptotic dynamics analysis is that
for many physical processes, what is practically observed on such finite timescales
of interest matches the theoretical limiting behaviour of a suitable mathematical
model of the process as t → ∞. However, we propose that for the kinds of slowly
time-dependent systems that we have been studying in this chapter, an appropriate
approach is not to treat time as the special parameter that can be approximated by an
infinite limit, but instead to treat the timescale separation as the special parameter that
can be approximated by an infinite limit. We do anticipate (as evidenced numerically
in [27]) that the timescale separation will often not need to be very great in order for
the infinite-limit approximation to give a suitable description of dynamics.
For one-dimensional phase-oscillator models, our proposed approach means that
we consider finite-time differential equations of the form
˙
θ(t) = F(θ (t), εt),
t ∈ [0,
1
ε
]
(7.9)
for some function F : R × [0, 1] → R that is 2π -periodic in its first input, and we
consider the limiting behaviour as ε → 0. For example, non-autonomous Adler equation models could take the form
˙
θ(t) = −a sin(θ ) + ˜
G(εt)
for some ˜
G : [0, 1] → R, and one can consider the limiting behaviour as ε → 0.
The study of limiting dynamical behaviour as timescale separation tends to infinity
is the subject of slow-fast dynamical systems [19]. Here, we are combining this slowfast approach with the philosophy of FTDS theory, by restricting the slow time εt
to the unit interval. In the paper [27], we use this “slow-fast FTDS” approach to
define notions of stability and neutral stability that are analogous to those seen in
scenarios (A) and (B) of Proposition 1 but from an “ε → 0 perspective” in place of a
“t → ∞ perspective”. We use these definitions to address rigorously the stabilisation
phenomenon that we have discussed in this chapter.
This chapter has focused on stabilisation of one-dimensional phase dynamics.
But as seen in [22, 24] and also numerically evidenced in [23], similar dynamical
phenomena are likely to be prevalent in higher-dimensional multiple-timescale nonautonomous systems as well.