Predicting transient dynamics in a model of reed musical instrument with slowly time-varying control parameter

When playing a self-sustained reed instrument (such as the clarinet), initial acoustical transients (at the beginning of a note) are known to be of crucial importance. Nevertheless, they have been mostly overlooked in the literature on musical instruments. We investigate here the dynamic behavior of a simple model of reed instrument with a time-varying blowing pressure accounting for attack transients performed by the musician. In practice, this means studying a one-dimensional non-autonomous dynamical system obtained by slowly varying in time the bifurcation parameter (the blowing pressure) of the corresponding autonomous systems, i.e., whose bifurcation parameter is constant. In this context, the study focuses on the case for which the time-varying blowing pressure crosses the bistability domain (with the coexistence of a periodic solution and an equilibrium) of the corresponding autonomous model. Considering the time-varying blowing pressure as a new (slow) state variable, the considered non-autonomous one-dimensional system becomes an autonomous two-dimensional fast-slow system. In the bistability domain, the latter has attracting manifolds associated with two stable branches of the bifurcation diagram of the system with constant parameter. In the framework of the geometric singular perturbation theory, we show that a single solution of the two-dimensional fast-slow system can be used to describe the global system behavior. Indeed, this allows us to determine, depending on the initial conditions and rate of change of the blowing pressure, which manifold is approached when the bistability domain is crossed and to predict whether a sound is produced during transient as a function of the musician's control.

Recurrent chaotic clustering and slow chaos in adaptive networks

Adaptive dynamical networks are network systems in which the structure co-evolves and interacts with the dynamical state of the nodes. We study an adaptive dynamical network in which the structure changes on a slower time scale relative to the fast dynamics of the nodes. We identify a phenomenon we refer to as recurrent adaptive chaotic clustering (RACC), in which chaos is observed on a slow time scale, while the fast time scale exhibits regular dynamics. Such slow chaos is further characterized by long (relative to the fast time scale) regimes of frequency clusters or frequency-synchronized dynamics, interrupted by fast jumps between these regimes. We also determine parameter values where the time intervals between jumps are chaotic and show that such a state is robust to changes in parameters and initial conditions.

Multistability and intermediate tipping of the Atlantic Ocean circulation

Tipping points (TP) in climate subsystems are usually thought to occur at a well-defined, critical forcing parameter threshold, via destabilization of the system state by a single, dominant positive feedback. However, coupling to other subsystems, additional feedbacks, and spatial heterogeneity may promote further small-amplitude, abrupt reorganizations of geophysical flows at forcing levels lower than the critical threshold. Using a primitive-equation ocean model, we simulate a collapse of the Atlantic Meridional Overturning Circulation (AMOC) due to increasing glacial melt. Considerably before the collapse, various abrupt, qualitative changes in AMOC variability occur. These intermediate tipping points (ITP) are transitions between multiple stable circulation states. Using 2.75 million years of model simulations, we uncover a very rugged stability landscape featuring parameter regions of up to nine coexisting stable states. The path to an AMOC collapse via a sequence of ITPs depends on the rate of change of the meltwater input. This challenges our ability to predict and define safe limits for TPs.

Rate and memory effects in bifurcation-induced tipping

A variation in the environment of a system, such as the temperature, the concentration of a chemical solution, or the appearance of a magnetic field, may lead to a drift in one of the parameters. If the parameter crosses a bifurcation point, the system can tip from one attractor to another (bifurcation-induced tipping). Typically, this stability exchange occurs at a parameter value beyond the bifurcation value. This is what we call, here, the shifted stability exchange. We perform a systematic study on how the shift is affected by the initial parameter value and its change rate. To that end, we present numerical simulations and partly analytical results for different types of bifurcations and different paradigmatic systems. We show that the nonautonomous dynamics can be split into two regimes. Depending on whether we exceed the numerical or experimental precision or not, the system may enter the nondeterministic or the deterministic regime. This is determined solely by the conditions of the drift. Finally, we deduce the scaling laws governing this phenomenon and we observe very similar behavior for different systems and different bifurcations in both regimes.

Systematic analysis and optimization of early warning signals for critical transitions using distribution data

Abrupt shifts between alternative regimes occur in complex systems, from cell regulation to brain functions to ecosystems. Several model-free early warning signals (EWS) have been proposed to detect impending transitions, but failure or poor performance in some systems have called for better investigation of their generic applicability. Notably, there are still ongoing debates whether such signals can be successfully extracted from data in particular from biological experiments. In this work, we systematically investigate properties and performance of dynamical EWS in different deteriorating conditions, and we propose an optimized combination to trigger warnings as early as possible, eventually verified on experimental data from microbiological populations. Our results explain discrepancies observed in the literature between warning signs extracted from simulated models and from real data, provide guidance for EWS selection based on desired systems and suggest an optimized composite indicator to alert for impending critical transitions using distribution data.

Thermal effects on neurons during stimulation of the brain

All electric and magnetic stimulation of the brain deposits thermal energy in the brain. This occurs through either Joule heating of the conductors carrying current through electrodes and magnetic coils, or through dissipation of energy in the conductive brain.Objective.Although electrical interaction with brain tissue is inseparable from thermal effects when electrodes are used, magnetic induction enables us to separate Joule heating from induction effects by contrasting AC and DC driving of magnetic coils using the same energy deposition within the conductors. Since mammalian cortical neurons have no known sensitivity to static magnetic fields, and if there is no evidence of effect on spike timing to oscillating magnetic fields, we can presume that the induced electrical currents within the brain are below the molecular shot noise where any interaction with tissue is purely thermal.Approach.In this study, we examined a range of frequencies produced from micromagnetic coils operating below the molecular shot noise threshold for electrical interaction with single neurons.Main results.We found that small temperature increases and decreases of 1∘C caused consistent transient suppression and excitation of neurons during temperature change. Numerical modeling of the biophysics demonstrated that the Na-K pump, and to a lesser extent the Nernst potential, could account for these transient effects. Such effects are dependent upon compartmental ion fluxes and the rate of temperature change.Significance.A new bifurcation is described in the model dynamics that accounts for the transient suppression and excitation; in addition, we note the remarkable similarity of this bifurcation's rate dependency with other thermal rate-dependent tipping points in planetary warming dynamics. These experimental and theoretical findings demonstrate that stimulation of the brain must take into account small thermal effects that are ubiquitously present in electrical and magnetic stimulation. More sophisticated models of electrical current interaction with neurons combined with thermal effects will lead to more accurate modulation of neuronal activity.

Critical rates of climate warming and abrupt collapse of ecosystems

In the age of climate warming, comprehension of ecosystems’ future is one of the pressing challenges to humanity. While most studies on climate warming focus on the ‘magnitude of change’ of the Earth’s temperature, the ‘rate’ at which it is increasing cannot be ruled out. Rapid warming has already caused sudden ecosystem transitions at numerous biodiversity hot spots; a mechanistic understanding of such transitions is crucial. Here, we study a slow–fast consumer–resource ecosystem interacting in rapid warming scenarios. Employing geometric singular perturbation theory, we find that while a gradual change in mean temperature may accord population persistence, a critical warming rate can drive the resource’s sudden collapse, termed a warming-induced abrupt transition. This further triggers the bottom-up effect, resulting in the extinction of the consumer. The difference between the optimum temperature of the resource’s growth rate and the habitat temperature is crucial in deciding the critical rate of warming. Consequently, species inhabiting extreme temperature regions are more susceptible to warming-induced collapse than those within intermediate temperature ranges. We find that stochastic fluctuations in the system can advance warming-induced transitions, and the efficacy of generic early warning signals to anticipate sudden transitions is challenged.

Rate-induced collapse in evolutionary systems

Recent work has highlighted the possibility of 'rate-induced tipping', in which a system undergoes an abrupt transition when a perturbation exceeds a critical rate of change. Here, we argue that this is widely applicable to evolutionary systems: collapse, or extinction, may occur when external changes occur too fast for evolutionary adaptation to keep up. To bridge existing theoretical frameworks, we develop a minimal evolutionary-ecological model showing that rate-induced extinction and the established notion of 'evolutionary rescue' are fundamentally two sides of the same coin: the failure of one implies the other, and vice versa. We compare the minimal model's behaviour with that of a more complex model in which the large-scale dynamics emerge from the interactions of many individual agents; in both cases, there is a well-defined threshold rate to induce extinction, and a consistent scaling law for that rate as a function of timescale. Due to the fundamental nature of the underlying mechanism, we suggest that a vast range of evolutionary systems should in principle be susceptible to rate-induced collapse. This would include ecosystems on all scales as well as human societies; further research is warranted.

The Balance of Nature: A Global Marine Perspective

The ancient idea of the balance of nature continues to influence modern perspectives on global environmental change. Assumptions of stable biogeochemical steady states and linear responses to perturbation are widely employed in the interpretation of geochemical records. Here, we review the dynamics of the marine carbon cycle and its interactions with climate and life over geologic time, focusing on what the record of past changes can teach us about stability and instability in the Earth system. Emerging themes include the role of amplifying feedbacks in producing past carbon cycle disruptions, the importance of critical rates of change in the context of mass extinctions and potential Earth system tipping points, and the application of these ideas to the modern unbalanced carbon cycle. A comprehensive dynamical understanding of the marine record of global environmental disruption will be of great value in understanding the long-term consequences of anthropogenic change.

Evolutionary rescue can prevent rate-induced tipping

The transformation of ecosystems proceeds at unprecedented rates. Recent studies suggest that high rates of environmental change can cause rate-induced tipping. In ecological models, the associated rate-induced critical transition manifests during transient dynamics in which populations drop to dangerously low densities. In this work, we study how indirect evolutionary rescue—due to the rapid evolution of a predator’s trait—can save a prey population from the rate-induced collapse. Therefore, we explicitly include the time-dependent dynamics of environmental change and evolutionary adaptation in an eco-evolutionary system. We then examine how fast the evolutionary adaptation needs to be to counteract the response to environmental degradation and express this relationship by means of a critical rate. Based on this critical rate, we conclude that indirect evolutionary rescue is more probable if the predator population possesses a high genetic variation and, simultaneously, the environmental change is slow. Hence, our results strongly emphasize that the maintenance of biodiversity requires a deceleration of the anthropogenic degradation of natural habitats.

Phase tipping: how cyclic ecosystems respond to contemporary climate

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.

Canard solutions in neural mass models: consequences on critical regimes

Mathematical models at multiple temporal and spatial scales can unveil the fundamental mechanisms of critical transitions in brain activities. Neural mass models (NMMs) consider the average temporal dynamics of interconnected neuronal subpopulations without explicitly representing the underlying cellular activity. The mesoscopic level offered by the neural mass formulation has been used to model electroencephalographic (EEG) recordings and to investigate various cerebral mechanisms, such as the generation of physiological and pathological brain activities. In this work, we consider a NMM widely accepted in the context of epilepsy, which includes four interacting neuronal subpopulations with different synaptic kinetics. Due to the resulting three-time-scale structure, the model yields complex oscillations of relaxation and bursting types. By applying the principles of geometric singular perturbation theory, we unveil the existence of the canard solutions and detail how they organize the complex oscillations and excitability properties of the model. In particular, we show that boundaries between pathological epileptic discharges and physiological background activity are determined by the canard solutions. Finally we report the existence of canard-mediated small-amplitude frequency-specific oscillations in simulated local field potentials for decreased inhibition conditions. Interestingly, such oscillations are actually observed in intracerebral EEG signals recorded in epileptic patients during pre-ictal periods, close to seizure onsets.

Tipping points induced by parameter drift in an excitable ocean model

Numerous systems in the climate sciences and elsewhere are excitable, exhibiting coexistence of and transitions between a basic and an excited state. We examine the role of tipping between two such states in an excitable low-order ocean model. Ensemble simulations are used to obtain the model's pullback attractor (PBA) and its properties, as a function of a forcing parameter [Formula: see text] and of the steepness [Formula: see text] of a climatological drift in the forcing. The tipping time [Formula: see text] is defined as the time at which the transition to relaxation oscillations (ROs) arises: at constant forcing this occurs at [Formula: see text]. As the steepness [Formula: see text] decreases, [Formula: see text] is delayed and the corresponding forcing amplitude decreases, while remaining always above [Formula: see text]. With periodic perturbations, that amplitude depends solely on [Formula: see text] over a significant range of parameters: this provides an example of rate-induced tipping in an excitable system. Nonlinear resonance occurs for periods comparable to the RO time scale. Coexisting PBAs and total independence from initial states are found for subsets of parameter space. In the broader context of climate dynamics, the parameter drift herein stands for the role of anthropogenic forcing.

A resilience concept based on system functioning: A dynamical systems perspective

We introduce a new framework for resilience, which is traditionally understood as the ability of a system to absorb disturbances and maintain its state, by proposing a shift from a state-based to a system functioning-based approach to resilience, which takes into account that several different coexisting stable states could fulfill the same functioning. As a consequence, not every regime shift, i.e., transition from one stable state to another, is associated with a lack or loss of resilience. We emphasize the importance of flexibility-the ability of a system to shift between different stable states while still maintaining system functioning. Furthermore, we provide a classification of system responses based on the phenomenological properties of possible disturbances, including the role of their timescales. Therefore, we discern fluctuations, shocks, press disturbances, and trends as possible disturbances. We distinguish between two types of mechanisms of resilience: (i) tolerance and flexibility, which are properties of the system, and (ii) adaptation and transformation, which are processes that alter the system's tolerance and flexibility. Furthermore, we discuss quantitative methods to investigate resilience in model systems based on approaches developed in dynamical systems theory.

Triple point of synchronization, phase singularity, and excitability along the transition between unbounded and bounded phase oscillations in a forced nonlinear oscillator

We report the discovery of a codimension-two phenomenon in the phase diagram of a second-order self-sustained nonlinear oscillator subject to a constant external periodic forcing, around which three regimes associated with the synchronization phenomenon exist, namely phase-locking, frequency-locking without phase-locking, and frequency-unlocking states. The triple point of synchronization arises when a saddle-node homoclinic cycle collides with the zero-amplitude state of the forced oscillator. A line on the phase diagram where limit-cycle solutions contain a phase singularity departs from the triple point, giving rise to a codimension-one transition between the regimes of frequency unlocking and frequency locking without phase locking. At the parameter values where the critical transition occurs, the forced oscillator exhibits a separatrix with a π phase jump, i.e., a particular trajectory in phase space that separates two distinct behaviors of the phase dynamics. Close to the triple point, noise induces excitable pulses where the two variants of type-I excitability, i.e., pulses with and without 2π phase slips, appear stochastically. The impacts of weak noise and some other dynamical aspects associated with the transition induced by the singular phenomenon are also discussed.

Risk of tipping the overturning circulation due to increasing rates of ice melt

Central elements of the climate system are at risk for crossing critical thresholds (so-called tipping points) due to future greenhouse gas emissions, leading to an abrupt transition to a qualitatively different climate with potentially catastrophic consequences. Tipping points are often associated with bifurcations, where a previously stable system state loses stability when a system parameter is increased above a well-defined critical value. However, in some cases such transitions can occur even before a parameter threshold is crossed, given that the parameter change is fast enough. It is not known whether this is the case in high-dimensional, complex systems like a state-of-the-art climate model or the real climate system. Using a global ocean model subject to freshwater forcing, we show that a collapse of the Atlantic Meridional Overturning Circulation can indeed be induced even by small-amplitude changes in the forcing, if the rate of change is fast enough. Identifying the location of critical thresholds in climate subsystems by slowly changing system parameters has been a core focus in assessing risks of abrupt climate change. This study suggests that such thresholds might not be relevant in practice, if parameter changes are not slow. Furthermore, we show that due to the chaotic dynamics of complex systems there is no well-defined critical rate of parameter change, which severely limits the predictability of the qualitative long-term behavior. The results show that the safe operating space of elements of the Earth system with respect to future emissions might be smaller than previously thought.

Weak tracking in nonautonomous chaotic systems

Previous studies have shown that rate-induced transitions can occur in pullback attractors of systems subject to "parameter shifts" between two asymptotically steady values of a system parameter. For cases where the attractors limit to equilibrium or periodic orbit in past and future limits of such an nonautonomous systems, these can occur as the parameter change passes through a critical rate. Such rate-induced transitions for attractors that limit to chaotic attractors in past or future limits has been less examined. In this paper, we identify a new phenomenon is associated with more complex attractors in the future limit: weak tracking, where a pullback attractor of the system limits to a proper subset of an attractor of the future limit system. We demonstrate weak tracking in a nonautonomous Rössler system, and argue there are infinitely many critical rates at each of which the pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor. We also state some necessary conditions that are needed for weak tracking.

Routes to global glaciation

Theory and observation suggest that Earth and Earth-like planets can undergo runaway low-latitude glaciation when changes in solar heating or in the carbon cycle exceed a critical threshold. Here, we use a simple dynamical-system representation of the ice-albedo feedback and the carbonate-silicate cycle to show that glaciation is also triggered when solar heating changes faster than a critical rate. Such 'rate-induced glaciations' remain accessible far from the outer edge of the habitable zone, because the warm climate state retains long-term stability. In contrast, glaciations induced by changes in the carbon cycle require the warm climate state to become unstable, constraining the kinds of perturbations that could have caused global glaciation in Earth's past. We show that glaciations can occur when Earth's climate transitions between two warm stable states; this property of the Earth system could help explain why major events in the development of life have been accompanied by glaciations.

Emergence of cascading dynamics in interacting tipping elements of ecology and climate

In ecology, climate and other fields, (sub)systems have been identified that can transition into a qualitatively different state when a critical threshold or tipping point in a driving process is crossed. An understanding of those tipping elements is of great interest given the increasing influence of humans on the biophysical Earth system. Complex interactions exist between tipping elements, e.g. physical mechanisms connect subsystems of the climate system. Based on earlier work on such coupled nonlinear systems, we systematically assessed the qualitative long-term behaviour of interacting tipping elements. We developed an understanding of the consequences of interactions on the tipping behaviour allowing for tipping cascades to emerge under certain conditions. The (narrative) application of these qualitative results to real-world examples of interacting tipping elements indicates that tipping cascades with profound consequences may occur: the interacting Greenland ice sheet and thermohaline ocean circulation might tip before the tipping points of the isolated subsystems are crossed. The eutrophication of the first lake in a lake chain might propagate through the following lakes without a crossing of their individual critical nutrient input levels. The possibility of emerging cascading tipping dynamics calls for the development of a unified theory of interacting tipping elements and the quantitative analysis of interacting real-world tipping elements.

Routes to extreme events in dynamical systems: Dynamical and statistical characteristics

Intermittent large amplitude events are seen in the temporal evolution of a state variable of many dynamical systems. Such intermittent large events suddenly start appearing in dynamical systems at a critical value of a system parameter and continues for a range of parameter values. Three important processes of instabilities, namely, interior crisis, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic motion, are most common as observed in many systems that lead to such occasional and rare transitions to large amplitude spiking events. We characterize these occasional large events as extreme events if they are larger than a statistically defined significant height. We present two exemplary systems, a single system and a coupled system, to illustrate how the instabilities work to originate extreme events and they manifest as non-trivial dynamical events. We illustrate the dynamical and statistical properties of such events.

Predicting critical ignition in slow-fast excitable models

Linearization around unstable traveling waves in excitable systems can be used to approximate strength-extent curves in the problem of initiation of excitation waves for a family of spatially confined perturbations to the rest state. This theory relies on the knowledge of the unstable traveling wave solution as well as the leading left and right eigenfunctions of its linearization. We investigate the asymptotics of these ingredients, and utility of the resulting approximations of the strength-extent curves, in the slow-fast limit in two-component excitable systems of FitzHugh-Nagumo type and test those on four illustrative models. Of these, two are with degenerate dependence of the fast kinetic on the slow variable, a feature which is motivated by a particular model found in the literature. In both cases, the unstable traveling wave solution converges to a stationary "critical nucleus" of the corresponding one-component fast subsystem. We observe that in the full system, the asymptotics of the left and right eigenspaces are distinct. In particular, the slow component of the left eigenfunction corresponding to the translational symmetry does not become negligible in the asymptotic limit. This has a significant detrimental effect on the critical curve predictions. The theory as formulated previously uses an heuristic to address a difficulty related to the translational invariance. We describe two alternatives to that heuristic, which do not use the misbehaving eigenfunction component. These new heuristics show much better predictive properties, including in the asymptotic limit, in all four examples.

Tipping positive change

Tipping points exist in social, ecological and climate systems and those systems are increasingly causally intertwined in the Anthropocene. Climate change and biosphere degradation have advanced to the point where we are already triggering damaging environmental tipping points, and to avoid worse ones ahead will require finding and triggering positive tipping points towards sustainability in coupled social, ecological and technological systems. To help with that I outline how tipping points can occur in continuous dynamical systems and in networks, the causal interactions that can occur between tipping events across different types and scales of system-including the conditions required to trigger tipping cascades, the potential for early warning signals of tipping points, and how they could inform deliberate tipping of positive change. In particular, the same methods that can provide early warning of damaging environmental tipping points can be used to detect when a socio-technical or socio-ecological system is most sensitive to being deliberately tipped in a desirable direction. I provide some example targets for such deliberate tipping of positive change. This article is part of the theme issue 'Climate change and ecosystems: threats, opportunities and solutions'.

Long-range dispersal promotes species persistence in climate extremes

Anthropogenic global warming in this century can act as a leading factor for large scale species extinctions in the near future. Species, in order to survive, need to develop dispersal strategies depending upon their environmental niche. Based on empirical evidence only a few previous studies have addressed how dispersal can evolve with changing temperature. However, for the analytical tractability, there is a need to develop an explicit model to ask how the temperature-dependent dispersal alters ecological dynamics. We investigate the persistence of species in a spatial ecological model, where dispersal is considered as a function of temperature. Spatial persistence is of major concern and dispersal is reasonably an important factor for extinction risk in the context of promoting synchrony. Our study yields how the temperature influences species decision of dispersal, resulting in either short-range or long-range dispersal. We examine synchronous or asynchronous behavior of species under their thermal dependence of dispersal. Moreover, we also analyze the transients to study the collective behavior of species away from their final or asymptotic dynamics. One of the key findings is at the most unfavorable environmental conditions long-range dispersal works out as the driving force for the persistence of species.

Characteristic disruptions of an excitable carbon cycle

The history of the carbon cycle is punctuated by enigmatic transient changes in the ocean's store of carbon. Mass extinction is always accompanied by such a disruption, but most disruptions are relatively benign. The less calamitous group exhibits a characteristic rate of change whereas greater surges accompany mass extinctions. To better understand these observations, I formulate and analyze a mathematical model that suggests that disruptions are initiated by perturbation of a permanently stable steady state beyond a threshold. The ensuing excitation exhibits the characteristic surge of real disruptions. In this view, the magnitude and timescale of the disruption are properties of the carbon cycle itself rather than its perturbation. Surges associated with mass extinction, however, require additional inputs from external sources such as massive volcanism. Surges are excited when [Formula: see text] enters the oceans at a flux that exceeds a threshold. The threshold depends on the duration of the injection. For injections lasting a time [Formula: see text] y in the modern carbon cycle, the threshold flux is constant; for smaller [Formula: see text], the threshold scales like [Formula: see text] Consequently the unusually strong but geologically brief duration of modern anthropogenic oceanic [Formula: see text] uptake is roughly equivalent, in terms of its potential to excite a major disruption, to relatively weak but longer-lived perturbations associated with massive volcanism in the geologic past.

Tipping phenomena in typical dynamical systems subjected to parameter drift

Tipping phenomena, i.e. dramatic changes in the possible long-term performance of deterministic systems subjected to parameter drift, are of current interest but have not yet been explored in cases with chaotic internal dynamics. Based on the example of a paradigmatic low-dimensional dissipative system subjected to different scenarios of parameter drifts of non-negligible rates, we show that a number of novel types of tippings can be observed due to the topological complexity underlying general systems. Tippings from and into several coexisting attractors are possible, and one can find fractality-induced tipping, the consequence of the fractality of the scenario-dependent basins of attractions, as well as tipping into a chaotic attractor. Tipping from or through an extended chaotic attractor might lead to random tipping into coexisting regular attractors, and rate-induced tippings appear not abruptly as phase transitions, rather they show up gradually when the rate of the parameter drift is increased. Since chaotic systems of arbitrary time-dependence call for ensemble methods, we argue for a probabilistic approach and propose the use of tipping probabilities as a measure of tipping. We numerically determine these quantities and their parameter dependence for all tipping forms discussed.

Multistability and tipping: From mathematics and physics to climate and brain-Minireview and preface to the focus issue

Multistability refers to the coexistence of different stable states in nonlinear dynamical systems. This phenomenon has been observed in laboratory experiments and in nature. In this introduction, we briefly introduce the classes of dynamical systems in which this phenomenon has been found and discuss the extension to new system classes. Furthermore, we introduce the concept of critical transitions and discuss approaches to distinguish them according to their characteristics. Finally, we present some specific applications in physics, neuroscience, biology, ecology, and climate science.

Leaking in history space: A way to analyze systems subjected to arbitrary driving

Our aim is to unfold phase space structures underlying systems with a drift in their parameters. Such systems are non-autonomous and belong to the class of non-periodically driven systems where the traditional theory of chaos (based e.g., on periodic orbits) does not hold. We demonstrate that even such systems possess an underlying topological horseshoe-like structure at least for a finite period of time. This result is based on a specifically developed method which allows to compute the corresponding time-dependent stable and unstable foliations. These structures can be made visible by prescribing a certain type of history for an ensemble of trajectories in phase space and by analyzing the trajectories fulfilling this constraint. The process can be considered as a leaking in history space-a generalization of traditional leaking, a method that has become widespread in traditional chaotic systems, to leaks depending on time.

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared with the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinize the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points, and it pinpoints the crucial need of considering this effect when investigating critical transitions.

Rate-induced tipping from periodic attractors: Partial tipping and connecting orbits

We consider how breakdown of the quasistatic approximation for attractors can lead to rate-induced tipping, where a qualitative change in tracking/tipping behaviour of trajectories can be characterised in terms of a critical rate. Associated with rate-induced tipping (where tracking of a branch of quasistatic attractors breaks down), we find a new phenomenon for attractors that are not simply equilibria: partial tipping of the pullback attractor where certain phases of the periodic attractor tip and others track the quasistatic attractor. For a specific model system with a parameter shift between two asymptotically autonomous systems with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system.

Experimental investigation on preconditioned rate induced tipping in a thermoacoustic system

Many systems found in nature are susceptible to tipping, where they can shift from one stable dynamical state to another. This shift in dynamics can be unfavorable in systems found in various fields ranging from ecology to finance. Hence, it is important to identify the factors that can lead to tipping in a physical system. Tipping can mainly be brought about by a change in parameter or due to the influence of external fluctuations. Further, the rate at which the parameter is varied also determines the final state that the system attains. Here, we show preconditioned rate induced tipping in experiments and in a theoretical model of a thermoacoustic system. We provide a specific initial condition (preconditioning) and vary the parameter at a rate higher than a critical rate to observe tipping. We find that the critical rate is a function of the initial condition. Our study is highly relevant because the parameters that dictate the asymptotic behavior of many physical systems are temporally dynamic.

Death and revival of chaos

We investigate the death and revival of chaos under the impact of a monotonous time-dependent forcing that changes its strength with a non-negligible rate. Starting on a chaotic attractor it is found that the complexity of the dynamics remains very pronounced even when the driving amplitude has decayed to rather small values. When after the death of chaos the strength of the forcing is increased again with the same rate of change, chaos is found to revive but with a different history. This leads to the appearance of a hysteresis in the complexity of the dynamics. To characterize these dynamics, the concept of snapshot attractors is used, and the corresponding ensemble approach proves to be superior to a single trajectory description, that turns out to be nonrepresentative. The death (revival) of chaos is manifested in a drop (jump) of the standard deviation of one of the phase-space coordinates of the ensemble; the details of this chaos-nonchaos transition depend on the ratio of the characteristic times of the amplitude change and of the internal dynamics. It is demonstrated that chaos cannot die out as long as underlying transient chaos is present in the parameter space. As a condition for a "quasistatically slow" switch-off, we derive an inequality which cannot be fulfilled in practice over extended parameter ranges where transient chaos is present. These observations need to be taken into account when discussing the implications of "climate change scenarios" in any nonlinear dynamical system.

Towards a critical transition theory under different temporal scales and noise strengths

The mechanism of critical phenomena or critical transitions has been recently studied from various aspects, in particular considering slow parameter change and small noise. In this article, we systematically classify critical transitions into three types based on temporal scales and noise strengths of dynamical systems. Specifically, the classification is made by comparing three important time scales τ(λ), τ(tran), and τ(ergo), where τ(λ) is the time scale of parameter change (e.g., the change of environment), τ(tran) is the time scale when a particle or state transits from a metastable state into another, and τ(ergo) is the time scale when the system becomes ergodic. According to the time scales, we classify the critical transition behaviors as three types, i.e., state transition, basin transition, and distribution transition. Moreover, for each type of transition, there are two cases, i.e., single-trajectory transition and multitrajectory ensemble transition, which correspond to the transition of individual behavior and population behavior, respectively. We also define the critical point for each type of critical transition, derive several properties, and further propose the indicators for predicting critical transitions with numerical simulations. In addition, we show that the noise-to-signal ratio is effective to make the classification of critical transitions for real systems.

Semianalytical approach to criteria for ignition of excitation waves

We consider the problem of ignition of propagating waves in one-dimensional bistable or excitable systems by an instantaneous spatially extended stimulus. Earlier we proposed a method [I. Idris and V. N. Biktashev, Phys. Rev. Lett. 101, 244101 (2008)] for analytical description of the threshold conditions based on an approximation of the (center-)stable manifold of a certain critical solution. Here we generalize this method to address a wider class of excitable systems, such as multicomponent reaction-diffusion systems and systems with non-self-adjoint linearized operators, including systems with moving critical fronts and pulses. We also explore an extension of this method from a linear to a quadratic approximation of the (center-)stable manifold, resulting in some cases in a significant increase in accuracy. The applicability of the approach is demonstrated on five test problems ranging from archetypal examples such as the Zeldovich-Frank-Kamenetsky equation to near realistic examples such as the Beeler-Reuter model of cardiac excitation. While the method is analytical in nature, it is recognized that essential ingredients of the theory can be calculated explicitly only in exceptional cases, so we also describe methods suitable for calculating these ingredients numerically.

Predicting the onset of period-doubling bifurcations in noisy cardiac systems

Biological, physical, and social systems often display qualitative changes in dynamics. Developing early warning signals to predict the onset of these transitions is an important goal. The current work is motivated by transitions of cardiac rhythms, where the appearance of alternating features in the timing of cardiac events is often a precursor to the initiation of serious cardiac arrhythmias. We treat embryonic chick cardiac cells with a potassium channel blocker, which leads to the initiation of alternating rhythms. We associate this transition with a mathematical instability, called a period-doubling bifurcation, in a model of the cardiac cells. Period-doubling bifurcations have been linked to the onset of abnormal alternating cardiac rhythms, which have been implicated in cardiac arrhythmias such as T-wave alternans and various tachycardias. Theory predicts that in the neighborhood of the transition, the system's dynamics slow down, leading to noise amplification and the manifestation of oscillations in the autocorrelation function. Examining the aggregates' interbeat intervals, we observe the oscillations in the autocorrelation function and noise amplification preceding the bifurcation. We analyze plots--termed return maps--that relate the current interbeat interval with the following interbeat interval. Based on these plots, we develop a quantitative measure using the slope of the return map to assess how close the system is to the bifurcation. Furthermore, the slope of the return map and the lag-1 autocorrelation coefficient are equal. Our results suggest that the slope and the lag-1 autocorrelation coefficient represent quantitative measures to predict the onset of abnormal alternating cardiac rhythms.

Excitable neurons, firing threshold manifolds and canards

We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The model analyzed investigates the phenomenon of post-inhibitory rebound spiking due to propofol anesthesia and is adapted from McCarthy et al. (SIAM J. Appl. Dyn. Syst. 11(4):1674-1697, 2012). Propofol modulates the decay time-scale of an inhibitory GABAa synaptic current. Interestingly, this system gives rise to rebound spiking within a specific range of propofol doses. Using techniques from geometric singular perturbation theory, we identify geometric structures, known as canards of folded saddle-type, which form the firing threshold manifolds. We find that the position and orientation of the canard separatrix is propofol dependent. Thus, the speeds of relevant slow synaptic processes are encoded within this geometric structure. We show that this behavior cannot be understood using a static, inhibitory current step protocol, which can provide a single threshold for rebound spiking but cannot explain the observed cessation of spiking for higher propofol doses. We then compare the analyses of dynamic and static synaptic inhibition, showing how the firing threshold manifolds of each relate, and why a current step approach is unable to fully capture the behavior of this model.

Bifurcations of the climate system and greenhouse gas emissions

We propose a generalization of the classical Goody model by taking into account greenhouse gas emission effects. We develop an asymptotic approach that allows us to obtain an expression for the greenhouse gas flux via the temperature and fluid fields. We show that there is a possible tipping point in atmospheric dynamics resulting from greenhouse gas emissions, where the climate system becomes bistable under sufficiently intensive greenhouse gas emissions.

Decision making in noisy bistable systems with time-dependent asymmetry

Our work draws special attention to the importance of the effects of time-dependent parameters on decision making in bistable systems. Here, we extend previous studies of the mechanism known as speed-dependent cellular decision making in genetic circuits by performing an analytical treatment of the canonical supercritical pitchfork bifurcation problem with an additional time-dependent asymmetry and control parameter. This model has an analogous behavior to the genetic switch. In the presence of transient asymmetries and fluctuations, slow passage through the critical region in both systems increases substantially the probability of specific decision outcomes. We also study the relevance for attractor selection of reaching maximum values for the external asymmetry before and after the critical region. Overall, maximum asymmetries should be reached at an instant where the position of the critical point allows for compensation of the detrimental effects of noise in retaining memory of the transient asymmetries.

Climate predictions: the influence of nonlinearity and randomness

The current threat of global warming and the public demand for confident projections of climate change pose the ultimate challenge to science: predicting the future behaviour of a system of such overwhelming complexity as the Earth's climate. This Theme Issue addresses two practical problems that make even prediction of the statistical properties of the climate, when treated as the attractor of a chaotic system (the weather), so challenging. The first is that even for the most detailed models, these statistical properties of the attractor show systematic biases. The second is that the attractor may undergo sudden large-scale changes on a time scale that is fast compared with the gradual change of the forcing (the so-called climate tipping).

Speed-dependent cellular decision making in nonequilibrium genetic circuits

Despite being governed by the principles of nonequilibrium transitions, gene expression dynamics underlying cell fate decision is poorly understood. In particular, the effect of signaling speed on cellular decision making is still unclear. Here we show that the decision between alternative cell fates, in a structurally symmetric circuit, can be biased depending on the speed at which the system is forced to go through the decision point. The circuit consists of two mutually inhibiting and self-activating genes, forced by two external signals with identical stationary values but different transient times. Under these conditions, slow passage through the decision point leads to a consistently biased decision due to the transient signaling asymmetry, whereas fast passage reduces and eventually eliminates the switch imbalance. The effect is robust to noise and shows that dynamic bifurcations, well known in nonequilibrium physics, are important for the control of genetic circuits.

Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system

Tipping points associated with bifurcations (B-tipping) or induced by noise (N-tipping) are recognized mechanisms that may potentially lead to sudden climate change. We focus here on a novel class of tipping points, where a sufficiently rapid change to an input or parameter of a system may cause the system to 'tip' or move away from a branch of attractors. Such rate-dependent tipping, or R-tipping, need not be associated with either bifurcations or noise. We present an example of all three types of tipping in a simple global energy balance model of the climate system, illustrating the possibility of dangerous rates of change even in the absence of noise and of bifurcations in the underlying quasi-static system.

Arctic climate tipping points

There is widespread concern that anthropogenic global warming will trigger Arctic climate tipping points. The Arctic has a long history of natural, abrupt climate changes, which together with current observations and model projections, can help us to identify which parts of the Arctic climate system might pass future tipping points. Here the climate tipping points are defined, noting that not all of them involve bifurcations leading to irreversible change. Past abrupt climate changes in the Arctic are briefly reviewed. Then, the current behaviour of a range of Arctic systems is summarised. Looking ahead, a range of potential tipping phenomena are described. This leads to a revised and expanded list of potential Arctic climate tipping elements, whose likelihood is assessed, in terms of how much warming will be required to tip them. Finally, the available responses are considered, especially the prospects for avoiding Arctic climate tipping points.

Stochastic tipping points in climate dynamics

A crucial question recently raised in climate dynamics concerns abrupt climate transitions: Are they due to a tipping point (TP) being exceeded, or is fast noisy dynamics the cause of their excitation? In this respect, a case study based on a low-order ocean model is developed to show that in an excitable dynamical system perturbed by noise (a possible climate condition) the TPs may have limited physical meaning, with the coherence resonance mechanism being predominant. The analysis is based on an operational definition of stochastic TP, which accounts for the effect of noise and reconciles formally the TP and coherence resonance views.

Solitary and coupled semiconductor ring lasers as optical spiking neurons

We theoretically investigate the possibility of generating pulses in an excitable (asymmetric) semiconductor ring laser (SRL) using optical trigger pulses. We show that the phase difference between the injected field and the electric field inside the SRL determines the direction of the perturbation in phase space. Due to the folded shape of the excitability threshold, this has an important influence on the ability to cross it. A mechanism for exciting multiple consecutive pulses using a single trigger pulse (i.e., multipulse excitability) is revealed. We furthermore investigate the possibility of using asymmetric SRLs in a coupled configuration, which is a first step toward an all-optical neural network using SRLs as building blocks.