Chapter 5
Normal Hyperbolicity for
Non-autonomous Oscillators
and Oscillator Networks
Robert S. MacKay
Abstract In this chapter, non-autonomous oscillators are considered as mappings
from input functions of time to a circle’s worth of functions giving the state as
a function of time. This view is justified using the theory of normally hyperbolic
submanifolds of dynamical systems. It illuminates the phenomena of phase-locking,
synchronisation and chimera; it allows an extension of the concept of coupling;
and it allows a hierarchical aggregation treatment of synchronisation in networks of
oscillators. The view extends to excitable and chaotic oscillators.
5.1 Introduction
Aneta Stefanovska expressed a vision “to build a self-consistent theory of nonautonomous oscillators” (June 2014). In this direction she introduced the class of
“chronotaxic” systems [27], defined as “oscillatory systems with time-varying, but
stable, amplitudes and frequencies”.
This chapter 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). It
indicates how this view encompasses chronotaxic systems and enables one, at least
conceptually, to understand the extent of synchronisation in networks of oscillators, whether autonomous or not. For the latter a hierarchical aggregation scheme is
introduced.
The approach is based on the theory of normal hyperbolicity [10, 15]. This theory is the mathematical expression of Haken’s slaving principle [13], the idea that
R. S. MacKay (B)
Mathematics Institute and Centre for Complexity Science, University of Warwick,
Coventry CV4 7AL, UK
e-mail: R.S.MacKay@warwick.ac.uk
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_5
71
Normal Hyperbolicity for
Non-autonomous Oscillators
and Oscillator Networks
Robert S. MacKay
Abstract In this chapter, non-autonomous oscillators are considered as mappings
from input functions of time to a circle’s worth of functions giving the state as
a function of time. This view is justified using the theory of normally hyperbolic
submanifolds of dynamical systems. It illuminates the phenomena of phase-locking,
synchronisation and chimera; it allows an extension of the concept of coupling;
and it allows a hierarchical aggregation treatment of synchronisation in networks of
oscillators. The view extends to excitable and chaotic oscillators.
5.1 Introduction
Aneta Stefanovska expressed a vision “to build a self-consistent theory of nonautonomous oscillators” (June 2014). In this direction she introduced the class of
“chronotaxic” systems [27], defined as “oscillatory systems with time-varying, but
stable, amplitudes and frequencies”.
This chapter 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). It
indicates how this view encompasses chronotaxic systems and enables one, at least
conceptually, to understand the extent of synchronisation in networks of oscillators, whether autonomous or not. For the latter a hierarchical aggregation scheme is
introduced.
The approach is based on the theory of normal hyperbolicity [10, 15]. This theory is the mathematical expression of Haken’s slaving principle [13], the idea that
R. S. MacKay (B)
Mathematics Institute and Centre for Complexity Science, University of Warwick,
Coventry CV4 7AL, UK
e-mail: R.S.MacKay@warwick.ac.uk
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_5
71