Tipping behavior can occur when an equilibrium of a dynamical system loses stability in response to a slowly varying parameter crossing a bifurcation threshold, or where noise drives a system from one attractor to another, or some combination of these effects. Similar behavior can be expected when a multistable system is forced by a chaotic deterministic system rather than by noise. In this context, the chaotic tipping window was recently introduced and investigated for discrete-time dynamics. In this paper, we find tipping windows for continuous-time nonlinear systems forced by chaos. We characterize the tipping window in terms of forcing by unstable periodic orbits of the chaos, and we show how the location and structure of this window depend on the relative timescales between the forcing and the responding system. We illustrate this by finding tipping windows for two examples of coupled bistable ODEs forced with chaos. Additionally, we describe the dynamic tipping window in the setting of a changing system parameter.

In a nonautonomous nonlinear dynamical system, generic critical transitions (tipping points) are not limited to slow passage through fold bifurcations. They can also correspond to slow passage through other generic bifurcations, such as subcritical Hopf or even (in the multiscale setting) singular Hopf bifurcation. In systems such as these, generic early warning signals associated with critical slowing down, such as observing an increase in variance and lag-1 autocorrelation, can be obscured by oscillatory behaviour. Here, we present a method using vector autoregression (VAR) that aims to identify multiple eigenvalues of the system from multiple time series. With this, we are not only able to deduce the system's stability but also identify the type of bifurcation we are approaching and, therefore, the underlying dynamics, which gives us insight into the system's future state post-tipping. In practice, this can be beneficial for estimation and mitigation of risk. We look at three bifurcations in particular: the fold, the subcritical Hopf bifurcation, and the singular Hopf bifurcation in the presence of an extra time-scale separation. We hope to show that VAR is potentially a powerful tool in an EWS toolkit and can be more enlightening than simply looking at AR(1) of individual time series, especially if there is more than one timescale present in the system.

Regions of fast-flowing ice in ice sheets, known as ice streams, have been theorized to be able to exhibit build-up/surge oscillatory variability due to thermomechanical coupling at the base of the ice. A simple model of three coupled ice streams is constructed to replicate the spatial configuration of a single ice stream being bisected into two termini. The model is constructed to mimic existing branching ice streams in northern Greenland. This model is shown to exhibit both steady-flow and build-up/surge oscillations. Further, the variability can be chaotic due to the nonlinear coupling of three incommensurate frequencies. This provides a mode of chaotic internal variability for ice sheets that contain these types of ice streams.

This Focus Issue forms one part (NDS-G) of the Double Focus Issue on nonautonomous dynamical systems in the sciences that was published together with a second Focus Issue on nonautonomous dynamical systems in climate (NDS-C). Nonlinear dynamics has achieved a pre-eminent status in the mathematical, physical, and life sciences over the last couple of decades. A very significant recent direction is dealing with the role of time dependence in the forcing and parameters of the models under study. Paramount examples concern the role of anthropogenic effects on climate and biodiversity as well as machine learning and biomedical problems. The mathematical theory of such effects is developing rapidly via the study of nonautonomous and random dynamical systems. Applications are blossoming and have in return been driving further developments of the theory, as we discuss in this Focus Issue.

The future behavioural fate of a forced nonlinear system can depend sensitively on the forcing profile as well as natural fluctuations within the system. This is especially the case for rate-induced tipping, where the forcing pushes the system to a basin boundary of a future behaviour and small changes in the forcing can lead to drastically different eventual behaviours. This sensitivity may be present only for a limited period of time, for example when the forcing is most rapidly changing. Moreover, critical slowing down based methods fail to be informative in such cases. We investigate a geometric early warning to evaluate when a system is in such a sensitive state. This involves computing the R-tipping indicator, namely the signed distance to an approximate R-tipping threshold. The latter is a dynamic state that embeds knowledge of the system and future behaviour of the forcing. We contrast this with early warnings of bifurcation-induced tipping, where tipping is associated with passing a threshold on slow variation of forcing. As an example, we consider methods of early prediction of the future state for a 3-box model of the Atlantic Meridional Overturning Circulation (AMOC) with specified rapid forcing. We show that the skill of the geometric early warning compares favourably with simple thresholds.

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.

The saddle-node bifurcation is the simplest example of a generic bifurcation in smooth ordinary differential equations, and is associated with the creation or destruction of a pair of equilibria. In this paper we examine the unfolding of the dynamics that occur when two generically coupled systems have simultaneous saddle-node bifurcations. We note that four parameters are required to generically unfold the interactions, and the dynamics are surprisingly complicated relative to the simplicity of a single saddle-node bifurcation. In the unfolding, in addition to saddle-node, Hopf and codimension-two local bifurcations, we also find a variety of global bifurcations, including homoclinic, saddle-node on invariant circle (SNIC), SNICeroclinic and non-central SNIC bifurcations. The latter two are codimension-two bifurcations that occur at the termination of a curve of SNIC bifurcations. A further contribution of this work is the development of numerical continuation techniques for the tracking of these codimension-two bifurcations through parameter space.

In this paper, we investigate saddle-node to saddle separatrix-loops that we term SNICero clinic bifurcations. They are generic codimension-two bifurcations involving a hetero clinic loop between one nonhyperbolic and one hyperbolic saddle. A particular co dimension-three case is the noncentral SNICero clinic bifurcation. We unfold this bifurcation in the minimal dimension (planar) case where the nonhyperbolic point is assumed to undergo a saddle-node bifurcation. Applying the method of Poincare' return maps, we present a minimal set of perturbations that captures all qualitatively distinct behaviors near a noncentral SNICero clinic loop. Specifically, we study how variation of the three unfolding parameters leads to transitions from hetero clinic and homo clinic loops; saddle-node on an invariant circle (SNIC); and periodic orbits, as well as equilibria. We show that although the bifurcation has been largely unexplored in applications, it can act as an organizing center for transitions between various types of saddle-node and saddle separatrix loops. It is also a generic route to oscillations that are both born and destroyed via global bifurcations compared to the commonly observed scenarios involving local (Hopf) and, in some cases, global (homo clinic or SNIC) bifurcations.

This Double Focus Issue consists of two linked Focus Issues-(a) Nonautonomous dynamical systems: Theory, methods, and applications and (b) Nonautonomous dynamics in the climate sciences, which we refer to as NDS-G and NDS-C, for short, where "G" stands for general and "C" for climate. Nonlinear dynamics has achieved a leading status in the mathematical, physical, and life sciences over the last couple of decades. A very significant recent direction is dealing with the role of time dependence in the external forcing and the parameters of the models under study. Paramount examples concern the role of anthropogenic effects on climate and biodiversity or of ageing on biomedical problems. The mathematical theory of such effects is developing rapidly via the study of NDSs and random dynamical systems. Applications are blossoming and have in return been driving further developments of the theory. These theoretical advances and their application to an increasing number of areas of the physical, life, and socioeconomic sciences were on view in two recent meetings: a three-day minisymposium within the framework of Dynamics Days Europe 2023 in Naples, Italy, and an International Seminar and Workshop in Dresden, Germany, in October 2023. Discussions at these two events led to this Double Focus Issue, taking contributions from a wide community of researchers in the relevant areas.

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.

In networked systems, the interplay between the dynamics of individual subsystems and their network interactions has been found to generate multistability in various contexts. Despite its ubiquity, the specific mechanisms and ingredients that give rise to multistability from such interplay remain poorly understood. In a network of coupled excitable units, we demonstrate that this interplay generating multistability occurs through a competition between the units' transient dynamics and their coupling. Specifically, the diffusive coupling between the units reinjects them into the excitability region of their individual state space, effectively trapping them there. We show that this trapping mechanism leads to the coexistence of multiple types of oscillations: periodic, quasi-periodic, and even chaotic, although the units separately do not oscillate. Interestingly, we find that the attractors emerge through different types of bifurcations-in particular, the periodic attractors emerge through either saddle-node of limit cycles bifurcations or homoclinic bifurcations-but in all cases, the reinjection mechanism is present.

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.

The organisation and cellular architecture of islets of Langerhans are critical to the physiological regulation of hormone secretion but it is debated whether human islets adhere to the characteristic mantle–core (M-C) structure seen in rodents. It is also unclear whether inherent architectural changes contribute to islet dysfunction in type 1 diabetes, aside from the loss of beta cells. Therefore, we have exploited advances in immunostaining, spatial biology and machine learning to undertake a detailed, systematic analysis of adult human islet architecture in health and type 1 diabetes, by a quantitative analysis of a dataset of >250,000 endocrine cells in >3500 islets from ten individuals. Formalin-fixed paraffin-embedded pancreatic sections (4 μm) from organ donors without diabetes and living donors with recent-onset type 1 diabetes were stained for all five islet hormones and imaged prior to analysis, which employed a novel automated pipeline using QuPath software, capable of running on a standard laptop. Whole-slide image analysis involved segmentation classifiers, cell detection and phenotyping algorithms to identify islets, specific cell types and their locations as (x,y)-coordinates in regions of interest. Each endocrine cell was categorised into binary variables for cell type (i.e. beta or non-beta) and position (mantle or core). A χ2 test for independence of these properties was performed and the OR was considered to estimate the effect size of the potential association between position and cell type. A quantification of the M-C structure at islet level was performed by computing the probability, r, that the observed number of non-beta cells in the mantle is due to a random arrangement. The distribution of the r values for the islets in the study was contrasted against the r values of a digital population of equivalent randomly arranged islets, termed digital siblings. Both distributions of r values were compared using the earth mover’s distance (EMD), a mathematical tool employed to describe differences in distribution patterns. The EMD was also used to contrast the distribution of islet size and beta cell fraction between type 1 diabetes and control islets. The χ2 test supports the existence of a significant (p<0.001) relationship between cell position and type. The effect size was measured via the OR <0.8, showing that non-beta cells are more likely to be found at the mantle (and vice versa). At the islet level, the EMD between the distributions of r values of the observed islets and the digital siblings was emd-1d=0.10951 (0

We establish conditions for the existence of a family of piecewise linear invariant curves in a two-parameter family of piecewise isometries on the upper half-plane known as Translated Cone Exchange Transformations. We show that these curves are embeddings of interval exchange transformations and give rise to layers of invariant regions. We also show the existence of a trapezoidal piecewise isometry for which the dynamics on the top and bottom edges are distinct 2-interval exchange transformations.

In the spirit of Hasselmann’s program, Climate Tipping Points are often studied in systems described by stochastic differential equations where a combination of noise and a parameter approaching or crossing a bifurcation threshold leads to tipping. However, in some cases, a multistable system forced by another potentially chaotic system is a more appropriate description and might give rise to unexpected effects. We show how tipping in such chaotically forced systems is affected by varying the relative timescale between the chaotic forcing and the forced multistable system. Further, we explain how periodic orbits of the forcing system can help to understand this effect and how they can be used to characterise the chaotic tipping window.

A non-autonomous system can undergo a rapid change of state in response to a small or slow change in forcing, due to the presence of nonlinear processes that give rise to critical transitions or tipping points. Such transitions are thought possible in various subsystems (tipping elements) of the Earth's climate system. The Atlantic Meridional Overturning Circulation (AMOC) is considered a particular tipping element where models of varying complexity have shown the potential for bi-stability and tipping. We consider both transient and stochastic forcing of a simple but data-adapted model of the AMOC. We propose and test a geometric early warning signal to predict whether tipping will occur for large transient forcing, based on the dynamics near an edge state. For stochastic forcing, we quantify mean times between noise-induced tipping in the presence of stochastic forcing using an Ordered Line Integral Method of Cameron (2017) to estimate the quasipotential. We calculate minimum action paths between stable states for various scenarios. Finally, we discuss the problem of finding early warnings in the presence of both transient and stochastic forcing.

The brain is not static: neuronal networks shift between contrasting modes of activity, alternating between active and quiescent regimes known as up and down states. Together with rhythmic oscillations, such modes of activity are fundamental to perception, memory, and information processing. However, the dynamical principles underlying the diverse repertoire of activity patterns and their transitions remain poorly understood. Here, we identify a geometric structure that governs dynamic states emergence and organizes neuronal networks transitions. We derive the conditions for its existence and demonstrate that it emerges robustly across canonical models of neuronal population dynamics. Near this organizing center, switches between oscillations, bistability and up and down states are orchestrated by the excitation-inhibition balance in the neuronal network. Thus, we show that excitation and inhibition do not simply modulate network activity but define the dynamical landscape from which distinct brain states emerge. We also consider neuron-astrocyte interactions and reveal how astrocytes can tune excitatory-inhibitory balance, therefore modulating the transitions between neuronal activity regimes. Overall, our results identify a general dynamical blueprint underlying the emergence, organization, and control of brain states.

Travelling waves of neural firing activity are observed in brain tissue as a part of various sensory, motor and cognitive processes. They represent an object of major interest in the study of excitable networks, with analysis conducted in both neural field models and spiking neuronal networks. The latter class exposes the single-neuron dynamics directly, allowing us to study the details of their influence upon network-scale behaviour. Here we present a study of a laterally-inhibited network of leaky integrate-and-fire neurons modulated by a slow voltage-gated ion channel that acts as a linear adaptation variable. As the strength of the ion channel increases, we find that its interaction with the lateral inhibition increases wave speeds. The ion channel can enable subthreshold oscillations, with the intervals between the firing events of loosely-coupled travelling wave solutions structured around the neuron's natural period. These subthreshold oscillations also enable the occurrence of codimension-2 grazing bifurcations; along with the emergence of fold bifurcations along wave solution branches, the slow ion channel introduces a variety of intermediate structures in the solution space. These point towards further investigation of the role neighbouring solution branches play in the behaviour of waves forced across bifurcations, which we illustrate with the aid of simulations using a novel root-finding algorithm designed to handle uncertainty over the existence of firing solutions.

The Atlantic Meridional Overturning Circulation (AMOC) and polar ice sheets are considered susceptible to critical transitions under climate change. Identified as core tipping elements, their collapse would have global and drastic consequences. Furthermore, the AMOC and polar ice sheets form a complex interacting system, where the collapse of one component can heavily impact the stability of others. In the worst case, this could result in a large-scale domino effect, otherwise known as a cascading tipping event. In this presentation, our focus is on assessing the stability of the AMOC in the presence of tipping Greenland ice sheet (GIS) and West Antarctica ice sheet (WAIS). While most existing studies agree on the destabilizing impact of a GIS collapse on the AMOC, the consequences of a WAIS collapse remain uncertain. A previous conceptual study suggested that a WAIS tipping event might actually prevent an AMOC collapse against both climate warming and increased GIS meltwater fluxes. Using a better conceptual model of the AMOC, we demonstrate that both the melting rate and natural variability associated with surface meltwater fluxes are decisive factors for this phenomenon to occur. Finally, we present preliminary findings in which the relevance of this stabilizing effect is investigated in the model of intermediate complexity CLIMBER-X.

We investigate how nonlinear behaviour (both of forcing in time and of the system itself) can affect the skill of early warning signals to predict tipping in (directionally) coupled bistable systems when using measures based on critical slowing down due to the breakdown of extrapolation. We quantify the skill of early warnings with a time horizon using a receiver-operator methodology for ensembles where noise realisations and parameters are varied to explore the role of extrapolation and how it can break down. We highlight cases where this can occur in an accelerating cascade of tipping elements, where very slow forcing of a slowly evolving “upstream” system forces a more rapidly evolving “downstream” system. If the upstream system crosses a tipping point, this can shorten the timescale of valid extrapolation. In particular, “downstream-within-upstream” tipping will typically have warnings only on a timescale comparable to the duration of the upstream tipping process, rather than the timescale of the original forcing.