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

Partial tipping in bistable ecological systems under periodic environmental variability

Periodic environmental variability is a common source affecting ecosystems and regulating their dynamics. This paper investigates the effects of periodic variation in species growth rate on the population dynamics of three bistable ecological systems. The first is a one-dimensional insect population model with coexisting outbreak and refuge equilibrium states, the second one describes two-species predator-prey interactions with extinction and coexistence states, and the third one is a three-species food chain model where chaotic and limit cycle states may coexist. We demonstrate with numerical simulations that a periodic variation in species growth rate may cause switching between two coexisting attractors without crossing any bifurcation point. Such a switchover occurs only for a specific initial population density close to the basin boundary, leading to partial tipping if the frozen system is non-chaotic. Partial tipping may also occur for some initial points far from the basin boundary if the frozen system is chaotic. Interestingly, the probability of tipping shows a frequency response with a maximum for a specific frequency of periodic forcing, as noticed for equilibrium and non-equilibrium limit cycle systems. The findings suggest that unexpected outbreaks or abrupt declines in population density may occur due to time-dependent variations in species growth parameters. Depending on the selective frequency of the periodic environmental variation, this may lead to species extinction or help the species to survive.

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.

Tipping points, multistability, and stochasticity in a two-dimensional traffic network dynamics

Mitigating traffic jams is a critical step for the betterment of the urban transportation system, which comprises a large number of interconnected routes to form an intricate network. To understand distinct features of vehicular traffic flow on a network, a macroscopic two-dimensional traffic network model is proposed incorporating intra-nodal and inter-nodal vehicular interaction. Utilizing the popular techniques of nonlinear dynamics, we investigate the impact of different parameters like occupancy, entry rates, and exit rates of vehicles. The existence of saddle-node, Hopf, homoclinic, Bogdanov-Takens, and cusp bifurcations have been shown using single or biparametric bifurcation diagrams. The occurrences of different multistability (bistability/tristability) phenomena, stochastic switching, and critical transitions are explored in detail. Further, we calculate the possibility of achieving each alternative state using the basin stability metric to characterize multistability. In addition, critical transitions from free flow to congestion are identified at different magnitudes of stochastic fluctuations. The applicability of critical slowing down based generic indicators, e.g., variance, lag-1 autocorrelation, skewness, kurtosis, and conditional heteroskedasticity are investigated to forewarn the critical transition from free flow to traffic congestion. It is demonstrated through the use of simulated data that not all of the measures exhibit sensitivity to rapid phase transitions in traffic flow. Our study reveals that traffic congestion emerges because of either bifurcation or stochasticity. The result provided in this study may serve as a paradigm to understand the qualitative behavior of traffic jams and to explore the tipping mechanisms occurring in transport phenomena.

Variation in environmental stochasticity dramatically affects viability and extinction time in a predator-prey system with high prey group cohesion

Understanding how tipping points arise is critical for population protection and ecosystem robustness. This work evaluates the impact of environmental stochasticity on the emergence of tipping points in a predator-prey system subject to the Allee effect and Holling type IV functional response, modeling an environment in which the prey has high group cohesion. We analyze the relationship between stochasticity and the probability and time that predator and prey populations in our model tip between different steady states. We evaluate the safety from extinction of different population values for each species, and accordingly assign extinction warning levels to these population values. Our analysis suggests that the effects of environmental stochasticity on tipping phenomena are scenario-dependent but follow a few interpretable trends. The probability of tipping towards a steady state in which one or both species go extinct generally monotonically increased with noise intensity, while the probability of tipping towards a more favorable steady state (in which more species were viable) usually peaked at intermediate noise intensity. For tipping between two equilibria where a given species was at risk of extinction in one equilibrium but not the other, noise affecting that species had greater impact on tipping probability than noise affecting the other species. Noise in the predator population facilitated quicker tipping to extinction equilibria, whereas prey noise instead often slowed down extinction. Changes in warning level for initial population values due to noise were most apparent near attraction basin boundaries, but noise of sufficient magnitude (especially in the predator population) could alter risk even far away from these boundaries. Our model provides critical theoretical insights for the conservation of population diversity: management criteria and early warning signals can be developed based on our results to keep populations away from destructive critical thresholds.

Time-scale synchronisation of oscillatory responses can lead to non-monotonous R-tipping

Rate-induced tipping (R-tipping) describes the fact that, for multistable dynamic systems, an abrupt transition can take place not only because of the forcing magnitude, but also because of the forcing rate. In the present work, we demonstrate through the case study of a piecewise-linear oscillator (PLO), that increasing the rate of forcing can make the system tip in some cases but might also prevent it from tipping in others. This counterintuitive effect is further called non-monotonous R-tipping (NMRT) and has already been observed in recent studies. We show that, in the present case, the reason for NMRT is the peak synchronisation of oscillatory responses operating on different time scales. We further illustrate that NMRT can be observed even in the presence of additive white noise of intermediate amplitude. Finally, NMRT is also observed on a van-der-Pol oscillator with an unstable limit cycle, suggesting that this effect is not limited to systems with a discontinuous right-hand side such as the PLO. This insight might be highly valuable, as the current research on tipping elements is shifting from an equilibrium to a dynamic perspective while using models of increasing complexity, in which NMRT might be observed but hard to understand.

Climate response and sensitivity: time scales and late tipping points

Climate response metrics are used to quantify the Earth’s climate response to anthropogenic changes of atmospheric CO 2 . Equilibrium climate sensitivity (ECS) is one such metric that measures the equilibrium response to CO 2 doubling. However, both in their estimation and their usage, such metrics make assumptions on the linearity of climate response, although it is known that, especially for larger forcing levels, response can be nonlinear. Such nonlinear responses may become visible immediately in response to a larger perturbation, or may only become apparent after a long transient period. In this paper, we illustrate some potential problems and caveats when estimating ECS from transient simulations. We highlight ways that very slow time scales may lead to poor estimation of ECS even if there is seemingly good fit to linear response over moderate time scales. Moreover, such slow processes might lead to late abrupt responses (late tipping points) associated with a system’s nonlinearities. We illustrate these ideas using simulations on a global energy balance model with dynamic albedo. We also discuss the implications for estimating ECS for global climate models, highlighting that it is likely to remain difficult to make definitive statements about the simulation times needed to reach an equilibrium.

A probabilistic framework for windows of opportunity: the role of temporal variability in critical transitions

The establishment of young organisms in harsh environments often requires a window of opportunity (WoO). That is, a short time window in which environmental conditions drop long enough below the hostile average level, giving the organism time to develop tolerance and transition into stable existence. It has been suggested that this kind of establishment dynamics is a noise-induced transition between two alternate states. Understanding how temporal variability (i.e. noise) in environmental conditions affects establishment of organisms is therefore key, yet not well understood or included explicitly in the WoO framework. In this paper, we develop a coherent theoretical framework for understanding when the WoO open or close based on simple dichotomous environmental variation. We reveal that understanding of the intrinsic timescales of both the developing organism and the environment is fundamental to predict if organisms can or cannot establish. These insights have allowed us to develop statistical laws for predicting establishment probabilities based on the period and variance of the fluctuations in naturally variable environments. Based on this framework, we now get a clear understanding of how changes in the timing and magnitude of climate variability or management can mediate establishment chances.

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.

Rethinking the definition of rate-induced tipping

The current definition of rate-induced tipping is tied to the idea of a pullback attractor limiting in forward and backward time to a stable quasi-static equilibrium. Here, we propose a new definition that encompasses the standard definition in the literature for certain scalar systems and includes previously excluded N-dimensional systems that exhibit rate-dependent critical transitions.

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.

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.

The snowball Earth transition in a climate model with drifting parameters: Splitting of the snapshot attractor

Using an intermediate complexity climate model (Planet Simulator), we investigate the so-called snowball Earth transition. For certain values (including its current value) of the solar constant, the climate system allows two different stable states: one of them is the snowball Earth, covered by ice and snow, and the other one is today's climate. In our setup, we consider the case when the climate system starts from its warm attractor (the stable climate we experience today), and the solar constant is changed according to the following scenario: it is decreased continuously and abruptly, over one year, to a state, where only the Snowball Earth's attractor remains stable. This induces an inevitable transition or climate tipping from the warm climate. The reverse transition is also discussed. Increasing the solar constant back to its original value in a similar way, in individual simulations, depending on the rate of the solar constant reduction, we find that either the system stays stuck in the snowball state or returns to warm climate. However, using ensemble methods, i.e., using an ensemble of climate realizations differing only slightly in their initial conditions we show that the transition from the snowball Earth to the warm climate is also possible with a certain probability, which depends on the specific scenario used. From the point of view of dynamical systems theory, we can say that the system's snapshot attractor splits between the warm climate's and the snowball Earth's attractor.

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.

Basin bifurcations, oscillatory instability and rate-induced thresholds for Atlantic meridional overturning circulation in a global oceanic box model

The Atlantic meridional overturning circulation (AMOC) transports substantial amounts of heat into the North Atlantic sector, and hence is of very high importance in regional climate projections. The AMOC has been observed to show multi-stability across a range of models of different complexity. The simplest models find a bifurcation associated with the AMOC 'on' state losing stability that is a saddle node. Here, we study a physically derived global oceanic model of Wood et al. with five boxes, that is calibrated to runs of the FAMOUS coupled atmosphere-ocean general circulation model. We find the loss of stability of the 'on' state is due to a subcritical Hopf for parameters from both pre-industrial and doubled CO₂ atmospheres. This loss of stability via subcritical Hopf bifurcation has important consequences for the behaviour of the basin of attraction close to bifurcation. We consider various time-dependent profiles of freshwater forcing to the system, and find that rate-induced thresholds for tipping can appear, even for perturbations that do not cross the bifurcation. Understanding how such state transitions occur is important in determining allowable safe climate change mitigation pathways to avoid collapse of the AMOC.

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.