Chapter 6
Synchronisation and Non-autonomicity
Maxime Lucas, Julian M. I. Newman, and Aneta Stefanovska
Abstract Thermodynamic openness is key to the long-term stability of living systems and can yield rich dynamical behaviours. Here, we model openness in coupled
oscillator systems by introducing an external driving with time-varying parameters. Five systems of increasing complexity are considered: three cases of single
driven oscillators followed by two cases of driven networks. We show how the timevarying parameters can enlarge the range of other parameters for which synchronous
behaviour is stable. In addition, it can yield additional behaviours such as intermittent
synchronisation. The stability of these systems is analysed via short- and long-time
Lyapunov exponents, both analytically and numerically. The different dynamical
regimes are also described via time-frequency representation. Finally, we compare
the stabilising effect of deterministic non-autonomous driving to that of bounded
noise. All in all, we give an overview of some effects time-varying parameters can
have on synchronisation. These effects could be a key to understand how living
systems maintain stability in the face of their ever-changing environment.
6.1 Introduction
Many systems in nature are thermodynamically open, i.e. they exchange energy and
matter with their environment. In fact, their openness is often key to their maintenance
of life. How could an organism survive isolated from its environment? Without
external supplies of nutrients and the ability to get rid of waste products of metabolism
M. Lucas (B)
Aix-Marseille University, Marseille, France
e-mail: maxime.lucas.1@univ-amu.fr
J. M. I. . Newman
Exeter University, Exeter, UK
e-mail: J.M.I.Newman@exeter.ac.uk
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_6
85
Synchronisation and Non-autonomicity
Maxime Lucas, Julian M. I. Newman, and Aneta Stefanovska
Abstract Thermodynamic openness is key to the long-term stability of living systems and can yield rich dynamical behaviours. Here, we model openness in coupled
oscillator systems by introducing an external driving with time-varying parameters. Five systems of increasing complexity are considered: three cases of single
driven oscillators followed by two cases of driven networks. We show how the timevarying parameters can enlarge the range of other parameters for which synchronous
behaviour is stable. In addition, it can yield additional behaviours such as intermittent
synchronisation. The stability of these systems is analysed via short- and long-time
Lyapunov exponents, both analytically and numerically. The different dynamical
regimes are also described via time-frequency representation. Finally, we compare
the stabilising effect of deterministic non-autonomous driving to that of bounded
noise. All in all, we give an overview of some effects time-varying parameters can
have on synchronisation. These effects could be a key to understand how living
systems maintain stability in the face of their ever-changing environment.
6.1 Introduction
Many systems in nature are thermodynamically open, i.e. they exchange energy and
matter with their environment. In fact, their openness is often key to their maintenance
of life. How could an organism survive isolated from its environment? Without
external supplies of nutrients and the ability to get rid of waste products of metabolism
M. Lucas (B)
Aix-Marseille University, Marseille, France
e-mail: maxime.lucas.1@univ-amu.fr
J. M. I. . Newman
Exeter University, Exeter, UK
e-mail: J.M.I.Newman@exeter.ac.uk
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_6
85