7 Non-asymptotic-time Dynamics
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well-defined on infinite time; but in the setting of autonomous dynamical systems,
this is not generally a problem. As above, long-time-asymptotic properties include
traditionally defined notions of stability and neutral stability, as well as asymptotic
Lyapunov exponents.
The classical theory of autonomous dynamical systems has been applied throughout the sciences to describe the qualitative behaviour of systems given a quantitative
model, and inversely to inform the inference of underlying physics from observational data [28, 34].
7.1.2 The Limitations of Long-Time-Asymptotic Analysis
The aim of the present chapter is to explore some of the ways in which long-timeasymptotic formalisms of stability can be inadequate or unsuitable when studying
open systems and the parameter-dependence of their stability.
As we have seen in Chap. 6 [24], in order to model the time-evolution of a process
subject to time-variable external influences (i.e. a process that does not possess the
kind of “timelessness” described above for Newtonian celestial mechanics), it would
not be suitable to use an autonomous dynamical system defined on the space of states
of the process. Instead, non-autonomous [18] and noise-perturbed [1] dynamical
systems can be used. (It is well-known that one can convert a non-autonomous system
on the state space into an autonomous system on the higher-dimensional state-time
space. But this does not make traditional autonomous dynamical systems theory, as
developed for isolated systems, applicable to analysis of open systems: all solutions
of this higher-dimensional system simply blow up to infinity as their time-coordinate
blows up to infinity [18, Remark 2.5].)
Nonetheless, merely carrying over traditional long-time-asymptotic analysis methods from the classical autonomous setting to the non-autonomous setting still inherently limits the extent to which the free time-variability of open systems can be
suitably treated. The main points that we will highlight in this chapter (see also [15,
Sects. 1.1, 6]) are:
(1) that the behaviour of an open system on the finite timescales of interest need not
follow (even approximately) any particular deterministic or statistical rule that
can be extended indefinitely in time;
(2) that a finite-time dynamical model which admits no natural extension to infinite
time may still clearly exhibit important dynamical phenomena;
(3) that even for an indefinite-time non-autonomous model, simply pursuing a longtime-asymptotic analysis of dynamics may hinder the recognition of physically
significant dynamical phenomena.
Chapter 6 [24] highlighted how non-autonomous driving can induce stability,
through the example of a phase oscillator governed by the Adler equation being
driven by an external phase oscillator with time-dependent frequency, where the
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well-defined on infinite time; but in the setting of autonomous dynamical systems,
this is not generally a problem. As above, long-time-asymptotic properties include
traditionally defined notions of stability and neutral stability, as well as asymptotic
Lyapunov exponents.
The classical theory of autonomous dynamical systems has been applied throughout the sciences to describe the qualitative behaviour of systems given a quantitative
model, and inversely to inform the inference of underlying physics from observational data [28, 34].
7.1.2 The Limitations of Long-Time-Asymptotic Analysis
The aim of the present chapter is to explore some of the ways in which long-timeasymptotic formalisms of stability can be inadequate or unsuitable when studying
open systems and the parameter-dependence of their stability.
As we have seen in Chap. 6 [24], in order to model the time-evolution of a process
subject to time-variable external influences (i.e. a process that does not possess the
kind of “timelessness” described above for Newtonian celestial mechanics), it would
not be suitable to use an autonomous dynamical system defined on the space of states
of the process. Instead, non-autonomous [18] and noise-perturbed [1] dynamical
systems can be used. (It is well-known that one can convert a non-autonomous system
on the state space into an autonomous system on the higher-dimensional state-time
space. But this does not make traditional autonomous dynamical systems theory, as
developed for isolated systems, applicable to analysis of open systems: all solutions
of this higher-dimensional system simply blow up to infinity as their time-coordinate
blows up to infinity [18, Remark 2.5].)
Nonetheless, merely carrying over traditional long-time-asymptotic analysis methods from the classical autonomous setting to the non-autonomous setting still inherently limits the extent to which the free time-variability of open systems can be
suitably treated. The main points that we will highlight in this chapter (see also [15,
Sects. 1.1, 6]) are:
(1) that the behaviour of an open system on the finite timescales of interest need not
follow (even approximately) any particular deterministic or statistical rule that
can be extended indefinitely in time;
(2) that a finite-time dynamical model which admits no natural extension to infinite
time may still clearly exhibit important dynamical phenomena;
(3) that even for an indefinite-time non-autonomous model, simply pursuing a longtime-asymptotic analysis of dynamics may hinder the recognition of physically
significant dynamical phenomena.
Chapter 6 [24] highlighted how non-autonomous driving can induce stability,
through the example of a phase oscillator governed by the Adler equation being
driven by an external phase oscillator with time-dependent frequency, where the