2
P. V. E. McClintock and A. Stefanovska
the contributions that follow. These methods are not yet in widespread use, and one
purpose of the book (and of the Workshop before it) is to help to promulgate them.
Science is, of course, seamless and indivisible but, for the convenience of readers,
we have divided the book into four Parts covering: theory; model-driven and datadriven approaches; biological oscillators; and applications. These are not rigidly
separated topics but, rather, an indication of emphasis. The applications chapters, for
example, draw freely on the ideas discussed in the first three Parts.
1.1 Theory
The theory Part opens with two chapters devoted to different aspects of phase reduction, applied to autonomous oscillatory systems, an approach that will be key to most
of what follows on non-autonomous oscillators. The underlying idea is to reduce a
multi-dimensional dynamical equation describing a nonlinear limit cycle oscillator
to a one-dimensional phase equation. Many rhythmic phenomena can in practice be
considered as nonlinear limit cycle oscillators, and hence described in terms of their
phase dynamics, usually amounting to an enormous simplification. This approach
is particularly useful in relation to the analysis of synchronising oscillators. In realworld situations the oscillators are of course subject to external perturbations that
take the system away from its limit cycle temporarily, and much interest attaches
to what happens when two such systems interact with each other. Note however
that the core of the book involves consideration of the situation that arises when
one or more of the oscillators is non-autonomous, so that the frequency of the limit
cycle itself is being perturbed by external agency. Chapter 2 by Nakao makes use
of a recently-introduced extension of the classical phase reduction method that also
includes amplitude degrees of freedom. He considers phase-amplitude reduction in
a spatially-extended reaction-diffusion system exhibiting stable oscillatory patterns,
and its entrainment by optimized periodic forcing with additional stabilization by
feedback. Chapter 3 by Pietras and Daffertshofer shows how different reduction
techniques applied to a network of interacting neural oscillators can lead to different dynamics of the reduced network, thereby identifying some delicate issues in
the application of the method. They demonstrate that an accurately-derived phase
model can properly capture the collective dynamics and they discuss the effect of
biologically plausible connectivity structures on the network behaviour.
In Chap. 4, Kloeden and Yang outline relevant ideas from the mathematical
theory of non-autonomous attractors, explaining that the nature of time in a nonautonomous dynamical system is very different from that in autonomous systems.
They point out that this difference has profound consequences in terms of the interpretation of dynamical behaviour, and that many of the familiar concepts developed
for autonomous dynamical systems are either too restrictive, or invalid, in the nonautonomous context. Chapter 5 by MacKay presents a view of a non-autonomous
oscillator as a mapping from input functions of time to a circle of possible solutions
(state functions of time). The author indicates how this view encompasses chrono-
P. V. E. McClintock and A. Stefanovska
the contributions that follow. These methods are not yet in widespread use, and one
purpose of the book (and of the Workshop before it) is to help to promulgate them.
Science is, of course, seamless and indivisible but, for the convenience of readers,
we have divided the book into four Parts covering: theory; model-driven and datadriven approaches; biological oscillators; and applications. These are not rigidly
separated topics but, rather, an indication of emphasis. The applications chapters, for
example, draw freely on the ideas discussed in the first three Parts.
1.1 Theory
The theory Part opens with two chapters devoted to different aspects of phase reduction, applied to autonomous oscillatory systems, an approach that will be key to most
of what follows on non-autonomous oscillators. The underlying idea is to reduce a
multi-dimensional dynamical equation describing a nonlinear limit cycle oscillator
to a one-dimensional phase equation. Many rhythmic phenomena can in practice be
considered as nonlinear limit cycle oscillators, and hence described in terms of their
phase dynamics, usually amounting to an enormous simplification. This approach
is particularly useful in relation to the analysis of synchronising oscillators. In realworld situations the oscillators are of course subject to external perturbations that
take the system away from its limit cycle temporarily, and much interest attaches
to what happens when two such systems interact with each other. Note however
that the core of the book involves consideration of the situation that arises when
one or more of the oscillators is non-autonomous, so that the frequency of the limit
cycle itself is being perturbed by external agency. Chapter 2 by Nakao makes use
of a recently-introduced extension of the classical phase reduction method that also
includes amplitude degrees of freedom. He considers phase-amplitude reduction in
a spatially-extended reaction-diffusion system exhibiting stable oscillatory patterns,
and its entrainment by optimized periodic forcing with additional stabilization by
feedback. Chapter 3 by Pietras and Daffertshofer shows how different reduction
techniques applied to a network of interacting neural oscillators can lead to different dynamics of the reduced network, thereby identifying some delicate issues in
the application of the method. They demonstrate that an accurately-derived phase
model can properly capture the collective dynamics and they discuss the effect of
biologically plausible connectivity structures on the network behaviour.
In Chap. 4, Kloeden and Yang outline relevant ideas from the mathematical
theory of non-autonomous attractors, explaining that the nature of time in a nonautonomous dynamical system is very different from that in autonomous systems.
They point out that this difference has profound consequences in terms of the interpretation of dynamical behaviour, and that many of the familiar concepts developed
for autonomous dynamical systems are either too restrictive, or invalid, in the nonautonomous context. Chapter 5 by MacKay presents a view of a non-autonomous
oscillator as a mapping from input functions of time to a circle of possible solutions
(state functions of time). The author indicates how this view encompasses chrono-