Synchronization transitions caused by time-varying coupling functions Hagos, Zeray; Stankovski, Tomislav; Newman, Julian ...
Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences,
12/2019, Volume:
377, Issue:
2160
Journal Article
Peer reviewed
Open access
Interacting dynamical systems are widespread in nature. The influence that one such system exerts on another is described by a coupling function; and the coupling functions extracted from the ...time-series of interacting dynamical systems are often found to be time-varying. Although much effort has been devoted to the analysis of coupling functions, the influence of time-variability on the associated dynamics remains largely unexplored. Motivated especially by coupling functions in biology, including the cardiorespiratory and neural delta-alpha coupling functions, this paper offers a contribution to the understanding of effects due to time-varying interactions. Through both numerics and mathematically rigorous theoretical consideration, we show that for time-variable coupling functions with time-independent net coupling strength, transitions into and out of phase- synchronization can occur, even though the frozen coupling functions determine phase-synchronization solely by virtue of their net coupling strength. Thus the information about interactions provided by the shape of coupling functions plays a greater role in determining behaviour when these coupling functions are time-variable. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.
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For a composition of independent and identically distributed random maps or a memoryless stochastic flow on a compact space
$X$
, we find conditions under which the presence of locally asymptotically ...stable trajectories (e.g. as given by negative Lyapunov exponents) implies almost-sure mutual convergence of any given pair of trajectories (‘synchronization’). Namely, we find that synchronization occurs and is ‘stable’ if and only if the system exhibits the following properties: (i) there is a smallest non-empty invariant set
$K\subset X$
; (ii) any two points in
$K$
are capable of being moved closer together; and (iii)
$K$
admits asymptotically stable trajectories.
Physical measures are invariant measures that characterise “typical” behaviour of trajectories started in the basin of chaotic attractors for autonomous dynamical systems. In this paper, we make some ...steps towards extending this notion to more general nonautonomous (time-dependent) dynamical systems. There are barriers to doing this in general in a physically meaningful way, but for systems that have autonomous limits, one can define a physical measure in relation to the physical measure in the past limit. We use this to understand cases where rate-dependent tipping between chaotic attractors can be quantified in terms of “tipping probabilities”. We demonstrate this for two examples of perturbed systems with multiple attractors undergoing a parameter shift. The first is a double-scroll system of Chua et al., and the second is a Stommel model forced by Lorenz chaos.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Defining the wavelet bispectrum Newman, Julian; Pidde, Aleksandra; Stefanovska, Aneta
Applied and computational harmonic analysis,
March 2021, 2021-03-00, Volume:
51
Journal Article
Peer reviewed
Open access
Bispectral analysis is an effective signal processing tool for investigating interactions between oscillations, and has been adapted to the continuous wavelet transform for time-evolving analysis of ...open systems. However, one unaddressed question for the wavelet bispectrum is quantification of the bispectral content of an area of scale-scale space. Without this, interpretation of wavelet bispectrum computations is merely qualitative. We now overcome this limitation by providing suitable normalisations of the wavelet bispectrum formula that enable it to be treated as a density to be integrated. These are roughly analogous to the normalisation for second-order wavelet spectral densities. We prove that our definition of the wavelet bispectrum matches the traditional bispectrum of sums of sinusoids, in the limit as the frequency resolution tends to infinity. We illustrate the improved quantitative power of our definition with numerical and experimental data. We also discuss notions of bicoherence and its practical implementation.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Oscillatory dynamics pervades the universe, appearing in systems of all scales. Whilst autonomous oscillatory dynamics has been extensively studied and is well understood, the very important problem ...of non-autonomous oscillatory dynamics is less well understood. Here, we provide a framework for non-autonomous oscillatory dynamics, within which we can define intermittent phenomena such as intermittent phase synchronisation. Moreover, we demonstrate this framework with a coupled pair of non-autonomous phase oscillators as well as a higher-dimensional system comprising of two interacting phase-oscillator networks.
•A new framework for phase-dynamics phenomena in non-autonomous oscillatory systems.•Interactions between pairs of non-autonomous oscillators and networks are considered.•Intermittent synchronisation is formulated mathematically.•Relevant for many real-world problems.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Time-dependent dynamics is ubiquitous in the natural world and beyond. Effectively analysing its presence in data is essential to our ability to understand the systems from which it is recorded. ...However, the traditional framework for dynamics analysis is in terms of time-independent dynamical systems and long-term statistics, as opposed to the explicit tracking over time of time-localised dynamical behaviour. We review commonly used analysis techniques based on this traditional statistical framework—such as the autocorrelation function, power-spectral density, and multiscale sample entropy—and contrast to an alternative framework in terms of finite-time dynamics of networks of time-dependent cyclic processes. In time-independent systems, the net effect of a large number of individually intractable contributions may be considered as noise; we show that time-dependent oscillator systems with only a small number of contributions may appear noise-like when analysed according to the traditional framework using power-spectral density estimation. However, methods characteristic of the time-dependent finite-time-dynamics framework, such as the wavelet transform and wavelet bispectrum, are able to identify the determinism and provide crucial information about the analysed system. Finally, we compare these two frameworks for three sets of experimental data. We demonstrate that while techniques based on the traditional framework are unable to reliably detect and understand underlying time-dependent dynamics, the alternative framework identifies deterministic oscillations and interactions.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
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.
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Synchronization transitions caused by time-varying coupling functions Hagos, Zeray; Stankovski, Tomislav; Newman, Julian ...
Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences,
12/2019, Volume:
377, Issue:
2160
Journal Article
Peer reviewed
Interacting dynamical systems are widespread in nature. The influence that one such system exerts on another is described by a coupling function; and the coupling functions extracted from the ...time-series of interacting dynamical systems are often found to be time-varying. Although much effort has been devoted to the analysis of coupling functions, the influence of time-variability on the associated dynamics remains largely unexplored. Motivated especially by coupling functions in biology, including the cardiorespiratory and neural delta-alpha coupling functions, this paper offers a contribution to the understanding of effects due to time-varying interactions. Through both numerics and mathematically rigorous theoretical consideration, we show that for time-variable coupling functions with time-independent net coupling strength, transitions into and out of phase-synchronization can occur, even though the frozen coupling functions determine phase-synchronization solely by virtue of their net coupling strength. Thus the information about interactions provided by the shape of coupling functions plays a greater role in determining behaviour when these coupling functions are time-variable.
This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.
Full text
Available for:
BFBNIB, NMLJ, NUK, PNG, SAZU, UL, UM, UPUK
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