1 Introduction
3
taxic systems and provides a picture of synchronisation in networks of oscillators,
whether autonomous or not.
In Chap. 6, Lucas et al. start from the viewpoint that the concept of thermodynamic
openness is key to the functioning of living systems. The authors model openness
in coupled oscillators through an external driving force with time-varying parameters. They consider a single, driven oscillator with a periodic, noisy frequency and
a time-varying driving frequency, followed by driven networks with time-varying
frequency and coupling. They characterise system stability by short- and long-time
Lyapunov exponents, both analytically and numerically, and they also describe the
different dynamical regimes in time-frequency representations. They show that timevariation of parameters can enlarge the parameter space within which synchronous
behaviour is stable, as well as yielding additional phenomena such as intermittent
synchronisation. The authors also demonstrate that the stabilising effect of deterministic non-autonomous driving is similar to that of bounded noise over longer times,
although the short-time dynamics is very different.
Chapter 7 by Newman et al. addresses the question of non-asymptotic-time
dynamics where the usual assumption of long-time-asymptotic properties like traditionally defined notions of stability and neutral stability, as well as asymptotic Lyapunov exponents, are inapplicable. By consideration of the non-autonomous Adler
equation with slowly-varying forcing, they illustrate three limitations of the traditional approach. They then propose an alternative, “finite-time slow-fast” approach,
that is more suitable for slowly time-dependent one-dimensional phase dynamics,
and likely to be suitable for describing the dynamics of open systems involving two
or more timescales.
In the final chapter of the Theory Part, Yuvan and Bier discuss synchronisation phenomena from yet another point of view. They come to the topic through a
consideration of phase transitions in large systems of interacting units, leading to
a mathematical description of an order parameter’s power-law behaviour near the
critical temperature of the system. The authors also discuss the phenomenon from
an entropy point of view and indicate implications for real-life experiments on the
oscillatory behaviour of yeast cells (cf. Chap. 13).
1.2 Model-Driven and Data-Driven Approaches
The relationship between (often idealised and abstract) mathematical theory and phenomena measured in the real world almost invariably involves modelling, the usual
aim being to build the simplest possible model capable of encompassing the observations. Sometimes the model is created mainly on the basis of physical intuition,
and several may be considered before arriving at the seemingly optimal one. In other
cases, the model can emerge directly from the observations i.e. from the data that
are measured. Part II comprises four chapters in which modelling plays a key role.
In Chap. 9 Kova˘ ci˘ c et al. introduce a mechanics perspective by considering the
oscillations on a chain of masses connected by linear springs and focus, in partic-
3
taxic systems and provides a picture of synchronisation in networks of oscillators,
whether autonomous or not.
In Chap. 6, Lucas et al. start from the viewpoint that the concept of thermodynamic
openness is key to the functioning of living systems. The authors model openness
in coupled oscillators through an external driving force with time-varying parameters. They consider a single, driven oscillator with a periodic, noisy frequency and
a time-varying driving frequency, followed by driven networks with time-varying
frequency and coupling. They characterise system stability by short- and long-time
Lyapunov exponents, both analytically and numerically, and they also describe the
different dynamical regimes in time-frequency representations. They show that timevariation of parameters can enlarge the parameter space within which synchronous
behaviour is stable, as well as yielding additional phenomena such as intermittent
synchronisation. The authors also demonstrate that the stabilising effect of deterministic non-autonomous driving is similar to that of bounded noise over longer times,
although the short-time dynamics is very different.
Chapter 7 by Newman et al. addresses the question of non-asymptotic-time
dynamics where the usual assumption of long-time-asymptotic properties like traditionally defined notions of stability and neutral stability, as well as asymptotic Lyapunov exponents, are inapplicable. By consideration of the non-autonomous Adler
equation with slowly-varying forcing, they illustrate three limitations of the traditional approach. They then propose an alternative, “finite-time slow-fast” approach,
that is more suitable for slowly time-dependent one-dimensional phase dynamics,
and likely to be suitable for describing the dynamics of open systems involving two
or more timescales.
In the final chapter of the Theory Part, Yuvan and Bier discuss synchronisation phenomena from yet another point of view. They come to the topic through a
consideration of phase transitions in large systems of interacting units, leading to
a mathematical description of an order parameter’s power-law behaviour near the
critical temperature of the system. The authors also discuss the phenomenon from
an entropy point of view and indicate implications for real-life experiments on the
oscillatory behaviour of yeast cells (cf. Chap. 13).
1.2 Model-Driven and Data-Driven Approaches
The relationship between (often idealised and abstract) mathematical theory and phenomena measured in the real world almost invariably involves modelling, the usual
aim being to build the simplest possible model capable of encompassing the observations. Sometimes the model is created mainly on the basis of physical intuition,
and several may be considered before arriving at the seemingly optimal one. In other
cases, the model can emerge directly from the observations i.e. from the data that
are measured. Part II comprises four chapters in which modelling plays a key role.
In Chap. 9 Kova˘ ci˘ c et al. introduce a mechanics perspective by considering the
oscillations on a chain of masses connected by linear springs and focus, in partic-