6 Synchronisation and Non-autonomicity
87
orbits. More recently, Hagos and coworkers investigated the effect of time-variability
of coupling functions when the total coupling strength is kept constant [18]. At the
network level, the solution of the Kuramoto model with slow frequency and coupling modulations was obtained analytically in [45], and the Ott-Antonsen theory
was then extended for time-varying parameters in [46]. Recent work on synchronisation in temporal networks includes [12, 33].
Bifurcation in non-autonomous systems has been considered from a mathematical
viewpoint [2], including in finite-time dynamical systems [50]. In [5, 31], nonautonomicity in the form of a parameter-shift that can pass through critical bifurcation
values is considered, and it is shown that the rate and the actual shape of the parametershift can have a dramatic impact on the resulting dynamics.
Building on the aforementioned studies, especially [22, 56], we have shown for a
low-dimensional system that the non-autonomicity can enlarge the region of stability
in parameter space [35, 40]. The results were then extended to driven networks of
identical oscillators [34]. In the present chapter, we provide a synthesis of this recent
work, and also build on this with a new result on time-varying coupling in networks.
The chapter is organised as follows. In Sect. 6.2, we define non-autonomous
systems and discuss their relation to thermodynamically open systems. In Sect. 6.3,
we review some methods of analysing the behaviour of dynamical systems on both
short and long timescales. In Sect. 6.4, we present an analysis of five theoretical
systems of increasing complexity. Finally, in Sect. 6.5, we summarise the results and
discuss potential future directions.
6.2 Non-autonomicity
In this section, we discuss the relation between the thermodynamic classification of
a system and the types of dynamical systems to model it.
6.2.1 Thermodynamically Open Versus Isolated
From the perspective of thermodynamics, the universe is the sum of two things: the
system under consideration, and its environment, i.e. everything else [9]. The system
can be of one of three classes, based on the nature of the system-environment interactions: open, closed, and isolated. A system is called isolated if it does not exchange
anything with its environment—neither energy, nor matter. An open system, on the
contrary, exchanges both energy and matter with its environment. Finally, a closed
system exchanges only energy and not matter with its environment. Figure 6.1a shows
a sketch of an isolated system whereas Fig. 6.1b, c depict an open one.
Isolated systems have proven useful as an idealised concept, but truly isolated
systems are virtually non-existent in nature. Living systems are an example that, by
nature, cannot be seen as isolated, and need to be modelled as open, as argued above.
87
orbits. More recently, Hagos and coworkers investigated the effect of time-variability
of coupling functions when the total coupling strength is kept constant [18]. At the
network level, the solution of the Kuramoto model with slow frequency and coupling modulations was obtained analytically in [45], and the Ott-Antonsen theory
was then extended for time-varying parameters in [46]. Recent work on synchronisation in temporal networks includes [12, 33].
Bifurcation in non-autonomous systems has been considered from a mathematical
viewpoint [2], including in finite-time dynamical systems [50]. In [5, 31], nonautonomicity in the form of a parameter-shift that can pass through critical bifurcation
values is considered, and it is shown that the rate and the actual shape of the parametershift can have a dramatic impact on the resulting dynamics.
Building on the aforementioned studies, especially [22, 56], we have shown for a
low-dimensional system that the non-autonomicity can enlarge the region of stability
in parameter space [35, 40]. The results were then extended to driven networks of
identical oscillators [34]. In the present chapter, we provide a synthesis of this recent
work, and also build on this with a new result on time-varying coupling in networks.
The chapter is organised as follows. In Sect. 6.2, we define non-autonomous
systems and discuss their relation to thermodynamically open systems. In Sect. 6.3,
we review some methods of analysing the behaviour of dynamical systems on both
short and long timescales. In Sect. 6.4, we present an analysis of five theoretical
systems of increasing complexity. Finally, in Sect. 6.5, we summarise the results and
discuss potential future directions.
6.2 Non-autonomicity
In this section, we discuss the relation between the thermodynamic classification of
a system and the types of dynamical systems to model it.
6.2.1 Thermodynamically Open Versus Isolated
From the perspective of thermodynamics, the universe is the sum of two things: the
system under consideration, and its environment, i.e. everything else [9]. The system
can be of one of three classes, based on the nature of the system-environment interactions: open, closed, and isolated. A system is called isolated if it does not exchange
anything with its environment—neither energy, nor matter. An open system, on the
contrary, exchanges both energy and matter with its environment. Finally, a closed
system exchanges only energy and not matter with its environment. Figure 6.1a shows
a sketch of an isolated system whereas Fig. 6.1b, c depict an open one.
Isolated systems have proven useful as an idealised concept, but truly isolated
systems are virtually non-existent in nature. Living systems are an example that, by
nature, cannot be seen as isolated, and need to be modelled as open, as argued above.