Under mild assumptions, the SRB measure mu associated to an Axiom A attractor A has the following properties: (i) the empirical measure starting at a typical point near A converges weakly to mu ; (ii) the pushforward of any Lebesgue-absolutely continuous probability measure supported near A converges weakly to mu . In general, a measure with the first property is called a 'physical measure', and physical measures are recognised by many authors as being important in their own right. Comparatively little has been written about the second property, but we highlight that this is also important in its own right as it characterises the attraction of general absolutely continuous measures to a measure mu -in such a case, we say mu is an 'attracting measure'. Attracting measures represent a kind of decay of correlation of observables that is more accessible than classical mixing when the measure is singular with respect to the Lebesgue measure. We prove a result that serves as a topological abstraction of the original result establishing that mixing SRB measures on Axiom A attractors are attracting measures.

Nonlinear dynamical systems subjected to a combination of noise and time-varying forcing can exhibit sudden changes, critical transitions or tipping points where large or rapid dynamic effects arise from changes in a parameter that are small or slow. Noise-induced tipping can occur where extremes of the forcing causes the system to leave one attractor and transition to another. If this noise corresponds to unresolved chaotic forcing, there is a limit such that this can be approximated by a stochastic differential equation (SDE) and the statistics of large deviations determine the transitions. Away from this limit it makes sense to consider tipping in the presence of chaotic rather than stochastic forcing. In general we argue that close to a parameter value where there is a bifurcation of the unforced system, there will be a chaotic tipping window outside of which tipping cannot happen, in the limit of asymptotically slow change of that parameter. This window is trivial for a stochastically forced system. Entry into the chaotic tipping window can be seen as a boundary crisis/non-autonomous saddle-node bifurcation and corresponds to an exceptional case of the forcing, typically by an unstable periodic orbit. We discuss an illustrative example of a chaotically forced bistable map that highlight the richness of the geometry and bifurcation structure of the dynamics in this case. If a parameter is changing slowly we note there is a dynamic tipping window that can also be determined in terms of unstable periodic orbits.

Various definitions of an attractor for a nonlinear dynamical system have been proposed. These use various assumptions on the set of initial conditions that should converge (the basin), and various notions of convergence. A weak assumption on the basin is the measure attractor of Milnor, which requires that the basin has positive measure. A weak assumption of the notion of convergence is the statistical attractor due to Ilyashenko, which requires that limiting to the attractor occurs on a set of future times of full density. We point out that many examples of statistical attractors actually satisfy a stronger definition which we call a bounded-return-time attractor, and we investigate such attractors. We also give an improved definition for the notion of pullback measure attraction. This was originally developed to understand attractors in nonautonomous systems, but we note here that it is helpful for understanding convergence towards statistical attractors in the autonomous setting. We investigate implications between all these different notions of attractors. We also investigate which of these notions are fulfilled by a hyperbolic fixed point with a homoclinic loop.

Order parameters have proved a vital tool for simplifying and understanding complex dynamics in a range of physical systems. However, applying these approaches to the high-dimensional time-dependent variability inherent to thermodynamically open systems—such as those found in neural networks and climate dynamics—remains a challenge. We introduce a novel order parameter based on alignment of component frequencies, in contrast to the widely used Kuramoto order parameter’s alignment of process phases. We present numerical simulations comparing this new parameter to the Kuramoto order parameter in a range of models, including a prototypical phase-oscillator model relevant to many open systems, such as neuronal dynamics. These results show that the new parameter more accurately identifies synchronisation in conditions characteristic of open systems, revealing dynamics entirely missed by established methods.

Coherence measures the similarity of progression of phases between oscillations or waves. When applied to multi-scale, nonstationary dynamics with time-varying amplitudes and frequencies, high values of coherence provide a useful indication of interactions, which might otherwise go unnoticed. However, the choice of analyzing coherence based on phases and amplitudes (amplitude-weighted phase coherence) vs only phases (phase coherence) has long been seen as arbitrary. Here, we review the concept of coherence and focus on time-localized methods of analysis, considering both phase coherence and amplitude-weighted phase coherence. We discuss the importance of using time-localized analysis and illustrate the methods and their practicalities on both numerically modeled and real time-series. The results show that phase coherence is more robust than amplitude-weighted phase coherence to both noise perturbations and movement artifacts. The results also have wider implications for the analysis of real data and the interpretation of physical systems.

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.

<p>Rate-induced phenomena can be mathematically modelled in terms of a dynamical system with a real-time (as opposed to quasistatic) parameter drift between two values; that is to say, the parameter converges to two different values as time tends to negative and positive infinity, giving rise to a nonautonomous dynamical system that is asymptotically autonomous. Representing stable climate states by attractors of the parameter-dependent autonomous system through which the parameter drift takes place, rate-induced tipping is modelled as the phenomenon that a trajectory of the nonautonomous system that starts in the past in the vicinity of one attractor lands in the vicinity of an attractor representing a different stable climate state in the future. However, if these attractors are chaotic, they exhibit sensitive dependence on initial conditions, which on the one hand makes investigation of any individually selected typical initial condition numerically impossible and physically irrelevant, but on the other hand makes a probabilistic description of long-term behaviour of trajectories an effective tool. This probabilistic description is provided by the "natural measure" on a chaotic attractor; in this poster, we consider the question of when this concept of "natural measures" can be extended from the classical setting of autonomous systems to the setting of asymptotically autonomous systems and hence used to provide a mathematically well-defined quantification of the "probability of tipping" between two stable climate states.</p>

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.

Although chaotic attractors for autonomous dynamical systems show sensitive dependence on initial conditions, they also typically support a physical or natural measure that characterizes the statistical behavior of almost all initial conditions near the attractor with respect to a background measure such as Lebesgue. In this paper, we identify conditions under which a nonautonomous system that limits as [Formula: see text] to an autonomous system with a physical measure is guaranteed to possess a “nonautonomous physical measure” that limits to the physical measure of the autonomous system.

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.

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.

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.

When modelling potential tipping elements of the earth system, one conventionally distinguishes "bifurcation-induced" and "noise-induced" tipping. The former occurs when an internal system parameter slowly crosses a critical threshold and external noise is negligible. The latter arises from forcing by noise well before a critical threshold for the internal dynamics is reached. The former comes with early warning signals, due to "critical slowing down" in the internal dynamics; but the latter occurs randomly without warning. However, these descriptions typically assume that the noise is Gaussian white noise, which arises as a limit of fast-timescale chaotic driving. We will instead consider, through a simple discrete-time prototype, finite-timescale bounded chaotic driving; this is a more suitable description of the subgrid forcing of turbulent geophysical fluid dynamics than uncorrelated noise. We will see that the phenomenon previously known as "noise-induced tipping" now corresponds to a deterministic bifurcation-induced tipping of the joint dynamics of the tipping element and the driving. Although "critical slowing down" does not occur in this bifurcation, early warning and near-exact prediction of the tipping event may still be possible. We also discuss the phenomenon of "noise-induced" prevention or delay of a tipping event, which cannot occur under conventional memoryless noise.

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.

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.

It has been shown by Le Jan that, given a memoryless-noise random dynamical system together with an ergodic distribution for the associated Markov transition probabilities, if the support of the ergodic distribution admits locally asymptotically stable trajectories, then there is a random attracting set consisting of finitely many points, whose basin of forward-time attraction includes a random full measure open set. In this paper, we present necessary and sufficient conditions for this attracting set to be a singleton; our result does not require the state space to be compact, but holds on general Lusin metric spaces.

For an autonomous dynamical system, an invariant measure is called physical or natural if it describes the statistics of a typically chosen trajectory that started an arbitrarily long time ago in the past, i.e. without transients. In order to apply such a concept to systems where there is time-varying forcing, we need to develop an analogous notion for such nonautonomous dynamical systems, where the measure is not fixed but evolves in time under the action of the nonautonomous system. The importance of such measures, and the pullback attractors on which they are supported, for interpreting climate statistics have been highlighted by Chekroun, Simmonet and Ghil (2011) Physica D 240:1685. We seek to gain a deeper understanding of these measures and implications for tipping points. We present some results for two classes of nonautonomous systems: autonomous random dynamical systems driven by stationary memoryless noise, and deterministic nonautonomous systems that are asymptotically autonomous in the negative-time limit. In both cases we show existence of a physical measure under suitable assumptions. We highlight further questions about defining rates of mixing in such a setting, as well as implications for prediction of tipping points. This work has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 820970 (TiPES).

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'.

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 asmallestnon-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.

Standard dynamical systems theory is centred around the coordinate-invariant asymptotic-time properties of autonomous systems. We identify three limitations of this approach. Firstly, we discuss how the traditional approach cannot take into account the time-varying nature of dynamics of open systems. Secondly, we show that models with explicit dependence on time exhibit stark dynamic phenomena, even when they cannot be defined for infinite time. We see a bifurcation occurring in nonautonomous finite-time systems that cannot be identified by classical methods for infinite-time autonomous systems. Thirdly, even when a time-varying model can be extended to infinite time, the classical infinite-time approach is likely to miss dynamical phenomena that are more readily understood within the framework of finite-time dynamics. We conclude the potentially crucial importance of a nonautonomous finite-time approach to real-world, open systems.