Chapter 7
Non-asymptotic-time Dynamics
Julian M. I. Newman, Maxime Lucas, and Aneta Stefanovska
Abstract Traditional analysis of dynamics concerns coordinate-invariant features of
the long-time-asymptotic behaviour of a system. Using the non-autonomous Adler
equation with slowly varying forcing, we illustrate three of the limitations of this
traditional approach. We discuss an alternative, “slow-fast finite-time dynamical
systems” approach, that is more suitable for slowly time-dependent one-dimensional
phase dynamics, and is likely to be suitable for more general dynamics of open
systems involving two or more timescales.
7.1 Introduction
In recent decades, the limitations of long-time-asymptotic approaches to dynamical
systems analysis have been increasingly recognised, giving rise to the birth of finitetime dynamical systems (FTDS) theory. In this chapter, building upon considerations
from the chapter [24] of this same volume, we illustrate these limitations through
the dynamics of a non-autonomous Adler equation with slowly varying forcing.
Furthermore, we briefly introduce the concept of dynamics analysis via a slow-fast
limit of finite-time dynamical systems, which is more suitable for describing this
system than traditional long-time-asymptotic analysis is.
J. M. I. Newman (B)
Exeter University, Exeter, UK
e-mail: J.M.I.Newman@exeter.ac.uk
M. Lucas
Aix-Marseille University, Marseille, France
e-mail: maxime.lucas.1@univ-amu.fr
A. Stefanovska
Lancaster University, Lancaster, UK
e-mail: aneta@lancaster.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_7
111
Non-asymptotic-time Dynamics
Julian M. I. Newman, Maxime Lucas, and Aneta Stefanovska
Abstract Traditional analysis of dynamics concerns coordinate-invariant features of
the long-time-asymptotic behaviour of a system. Using the non-autonomous Adler
equation with slowly varying forcing, we illustrate three of the limitations of this
traditional approach. We discuss an alternative, “slow-fast finite-time dynamical
systems” approach, that is more suitable for slowly time-dependent one-dimensional
phase dynamics, and is likely to be suitable for more general dynamics of open
systems involving two or more timescales.
7.1 Introduction
In recent decades, the limitations of long-time-asymptotic approaches to dynamical
systems analysis have been increasingly recognised, giving rise to the birth of finitetime dynamical systems (FTDS) theory. In this chapter, building upon considerations
from the chapter [24] of this same volume, we illustrate these limitations through
the dynamics of a non-autonomous Adler equation with slowly varying forcing.
Furthermore, we briefly introduce the concept of dynamics analysis via a slow-fast
limit of finite-time dynamical systems, which is more suitable for describing this
system than traditional long-time-asymptotic analysis is.
J. M. I. Newman (B)
Exeter University, Exeter, UK
e-mail: J.M.I.Newman@exeter.ac.uk
M. Lucas
Aix-Marseille University, Marseille, France
e-mail: maxime.lucas.1@univ-amu.fr
A. Stefanovska
Lancaster University, Lancaster, UK
e-mail: aneta@lancaster.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_7
111