Synchronisation is a well-known phenomenon that plays a key role in physics, biology, and neuroscience, to cite a few. A typical example is when people in an audience clap in unison, because of their (audio and visual) interactions, or when heart cells synchronise to make the heart beat. A lot of theoretical work exists on synchronisation, but mostly considers the frequencies of each unit to be constant. In nature, and in living systems in particular, however, frequencies change all the time. Imagine how our heartbeat changes when we are calm, stressed, or scared.

This is what motivates our studies on synchronisation between oscillators with time-varying parameters. In particular, in different studies, we consider time-varying frequencies, coupling strenghts, and network structure.

More formally, an example system we considered is the following: a network driven by an oscillator with a time-varying frequency,

We introduce a new mathematical framework for the qualitative analysis of dynamical stability, designed particularly for finite-time processes subject to slow-timescale external influences. In particular, our approach is to treat finite-time dynamical systems in terms of a slow–fast formalism in which the slow time only exists in a bounded interval, and consider stability in the singular limit. Applying this to one-dimensional phase dynamics, we provide stability definitions somewhat analogous to the classical infinite-time definitions associated with Aleksandr Lyapunov. With this, we mathematically formalize and generalize a phase-stabilization phenomenon previously described in the physics literature for which the classical stability definitions are inapplicable and instead our new framework is required.

Nonautonomous driving induces stability in network of identical oscillators

Nonautonomous driving of an oscillator has been shown to enlarge the Arnold tongue in parameter space, but little is known about the analogous effect for a network of oscillators. To test the hypothesis that deterministic nonautonomous perturbation is a good candidate for stabilizing complex dynamics, we consider a network of identical phase oscillators driven by an oscillator with a slowly time-varying frequency. We investigate both the short- and long-term stability of the synchronous solutions of this nonautonomous system. For attractive couplings we show that the region of stability grows as the amplitude of the frequency modulation is increased, through the birth of an intermittent synchronization regime. For repulsive couplings, we propose a control strategy to stabilize the dynamics by altering very slightly the network topology. We also show how, without changing the topology, time-variability in the driving frequency can itself stabilize the dynamics. As a byproduct of the analysis, we observe chimeralike states. We conclude that time-variability-induced stability phenomena are also present in networks, reinforcing the idea that this is a quite realistic scenario for living systems to use in maintaining their functioning in the face of ongoing perturbations.

The synchronous dynamics of an array of excitable oscillators, coupled via a generic graph, is studied. Non-homogeneous perturbations can grow and destroy synchrony, via a self-consistent instability which is solely instigated by the intrinsic network dynamics. By acting on the characteristic time-scale of the network modulation, one can make the examined system to behave as its (partially) averaged analogue. This result is formally obtained by proving an extended version of the averaging theorem, which allows for partial averages to be carried out. As a byproduct of the analysis, oscillation death is reported to follow the onset of the network-driven instability.

Stabilization of dynamics of oscillatory systems by nonautonomous perturbation

Synchronization and stability under periodic oscillatory driving are well understood, but little is known about the effects of aperiodic driving, despite its abundance in nature. Here, we consider oscillators subject to driving with slowly varying frequency, and investigate both short-term and long-term stability properties. For a phase oscillator, we find that, counterintuitively, such variation is guaranteed to enlarge the Arnold tongue in parameter space. Using analytical and numerical methods that provide information on time-variable dynamical properties, we find that the growth of the Arnold tongue is specifically due to the growth of a region of intermittent synchronization where trajectories alternate between short-term stability and short-term neutral stability, giving rise to stability on average. We also present examples of higher-dimensional nonlinear oscillators where a similar stabilization phenomenon is numerically observed. Our findings help support the case that in general, deterministic nonautonomous perturbation is a very good candidate for stabilizing complex dynamics.

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 time-varying 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.

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.

PhD thesis

Synchronisation and stability in nonautonomous oscillatory systems

M. Lucas

Lancaster University and University of Florence, 2019

Many natural and artificial systems can be modelled by ensembles of coupled oscillators. These types of systems can exhibit various synchronisation phenomena, where the interaction between the oscillators leads them to some kind of coherent behaviour, despite heterogeneities in the system. Moreover, many such systems are subject to a timevariable environment which effectively drives them. Many examples can be found in living systems, e.g., the dynamics of a cell is strongly dependent on the ever-changing intra- and extra-cellular ionic concentrations.Motivated by these considerations, this thesis investigates the effect of time-varying parameters on synchronisation and stability in ensembles of coupled oscillators. Timevariability is a crucial ingredient of the dynamics of many real-life systems, and interest in it is only recently starting to grow. Such systems are in general described by nonautonomous equations, which are hard to treat in general. This present work aims at answering questions such as: Can time-variability be detrimental/beneficial to synchronisation? If so, under which conditions? Can time-variability seed new dynamical phenomena? How can one best treat nonautonomous systems?The systems studied can be divided into two categories. First, the effect of a driving oscillator with a time-varying frequency is investigated. It is shown that increasing the amplitude of the frequency modulation can increase the size of the stability region in parameter space, under general assumptions. Short-term dynamics and stability properties are also investigated, and their dynamics is shown to be of importance. Second, the effect of time-varying couplings between the oscillators is considered. This is shown to be able to make the synchronous state unstable and yield oscillation death.Overall, the thesis illustrates that time-variability can be either beneficial or detrimental to synchronous dynamics, and investigates in detail and gives insight about cases of both. It argues towards the general fact that short-term dynamics is often crucial to a physically relevant understanding of nonautonomous systems.