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frequency modulation of the driving oscillator is such that the driven oscillator intermittently synchronises with the driving oscillator. We will use this same example to
illustrate the above three points; further details are presented in [27]. The physical
phenomenon of phase stabilisation by time-dependent external influence has previously been described within the framework of chronotaxic oscillators [33], where
the mathematical concept of pullback-attraction in deterministic nonautonomous
dynamical systems was used to describe the stability of self-sustained oscillators
with time-dependent phase dynamics. However, this requires the model of the timedependent external forcing to be well-defined indefinitely far back into the past. This
present chapter draws attention to how the same physical stabilisation effect can be
understood in terms of dynamical behaviour on bounded time-intervals even when
indefinite-time models are inherently irrelevant or unsuitable.
Now let us mention that the paper [27], and this present chapter, are far from
being the first investigation into finite-time dynamics motivated by the limitations of
long-time-asymptotic dynamics. The field of non-autonomous finite-time dynamical
systems, though relatively recent, has seen important progress being made [2, 3, 5,
9, 15, 16, 21, 31]. So far, the primary application of FTDS theory has been the study
of Lagrangian coherent structures in diverse fluid flows [10, 11, 13, 20, 26, 30,
32, 35, 36, 38]. Other examples include population dynamics [37] and activation
of biochemical signalling pathways [3]. Our present contribution to this field of
finite-time dynamics is the consideration of FTDS with slow-timescale driving (as
formalised in Sect. 7.5). Let us mention that fast-timescale driving of the Adler
equation has also been treated rigorously from a finite-time perspective in [8], in the
context of synchronisation transitions induced by time-dependence of the shape of
phase-coupling function.
7.1.3 Structure of the Chapter
In Sect. 7.2, we will present a general analysis of Adler equations with slowly
time-dependent additive forcing, following the same line of reasoning as in the
chapter [24]. In particular, in Sect. 7.2.4 we will describe the time-variability-induced
stabilisation phenomenon that forms the basis of the considerations in this chapter.
In Sect. 7.3, we will illustrate points (1) and (2) of Sect. 7.1.2 through the example
of an Adler equation driven by a low-pass-filtered sample realisation of a Brownian
bridge. In Sect. 7.4, we will illustrate point (3) of Sect. 7.1.2 through the example of
an Adler equation with slow sinusoidal driving. In Sect. 7.5, we will briefly describe
the framework of slow-fast FTDS that can be used to formalise the stabilisation phenomenon seen in both [24] and this chapter. In Sect. 7.6, we will summarise and
conclude.
J. M. I. Newman et al.
frequency modulation of the driving oscillator is such that the driven oscillator intermittently synchronises with the driving oscillator. We will use this same example to
illustrate the above three points; further details are presented in [27]. The physical
phenomenon of phase stabilisation by time-dependent external influence has previously been described within the framework of chronotaxic oscillators [33], where
the mathematical concept of pullback-attraction in deterministic nonautonomous
dynamical systems was used to describe the stability of self-sustained oscillators
with time-dependent phase dynamics. However, this requires the model of the timedependent external forcing to be well-defined indefinitely far back into the past. This
present chapter draws attention to how the same physical stabilisation effect can be
understood in terms of dynamical behaviour on bounded time-intervals even when
indefinite-time models are inherently irrelevant or unsuitable.
Now let us mention that the paper [27], and this present chapter, are far from
being the first investigation into finite-time dynamics motivated by the limitations of
long-time-asymptotic dynamics. The field of non-autonomous finite-time dynamical
systems, though relatively recent, has seen important progress being made [2, 3, 5,
9, 15, 16, 21, 31]. So far, the primary application of FTDS theory has been the study
of Lagrangian coherent structures in diverse fluid flows [10, 11, 13, 20, 26, 30,
32, 35, 36, 38]. Other examples include population dynamics [37] and activation
of biochemical signalling pathways [3]. Our present contribution to this field of
finite-time dynamics is the consideration of FTDS with slow-timescale driving (as
formalised in Sect. 7.5). Let us mention that fast-timescale driving of the Adler
equation has also been treated rigorously from a finite-time perspective in [8], in the
context of synchronisation transitions induced by time-dependence of the shape of
phase-coupling function.
7.1.3 Structure of the Chapter
In Sect. 7.2, we will present a general analysis of Adler equations with slowly
time-dependent additive forcing, following the same line of reasoning as in the
chapter [24]. In particular, in Sect. 7.2.4 we will describe the time-variability-induced
stabilisation phenomenon that forms the basis of the considerations in this chapter.
In Sect. 7.3, we will illustrate points (1) and (2) of Sect. 7.1.2 through the example
of an Adler equation driven by a low-pass-filtered sample realisation of a Brownian
bridge. In Sect. 7.4, we will illustrate point (3) of Sect. 7.1.2 through the example of
an Adler equation with slow sinusoidal driving. In Sect. 7.5, we will briefly describe
the framework of slow-fast FTDS that can be used to formalise the stabilisation phenomenon seen in both [24] and this chapter. In Sect. 7.6, we will summarise and
conclude.