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

There are several classes of dynamical systems which are known to exhibit a multitude of coexisting stable states or attractors. Those attractors can be of different nature, ranging from equilibria via periodic and quasiperiodic motion to chaotic attractors depending on the specific values of the parameters or external forcing. The simplest class of multistable systems can be constructed by taking any conservative system and adding a small amount of damping. While a conservative system possesses infinitely many different marginally stable solutions, weakly dissipative systems exhibit a large but finite number of attractors, in fact their number scales as 1/damping.² Hence, the smaller the damping, the larger is the number of coexisting attractors. The second important class is coupled systems, where typically a diffusive coupling between the elements is used. As a prominent example, we mention coupled map lattices.³ Another widely studied phenomenon in coupled systems is related to different kinds of synchronization,⁴ where often the dynamics in the synchronization manifold coexists with dynamics outside that manifold. Here, the emergence of multistability is often related to the loss of synchrony^5,6 or the formation of clusters.^7,8 While in the aforementioned coupled systems the number of coexisting attractors is finite, one can also find an infinite number of attractors when two arbitrary, but identical systems are coupled in a specific way.⁹ This particular kind of multistability has been termed uncertain destination dynamics¹⁰ or extreme multistability,¹¹ and it is related to the emergence of a conserved quantity in the long-term limit. The third class of multistable systems is delayed feedback-systems, where additional attractors appear depending on the variation of the delay.^12,13 These systems are particularly important when a control in the form of a delayed feedback control is applied.¹⁴ Here, in this issue, two new classes of multistable systems are discussed. First, Dudkowski et al.¹⁵ introduce a class of systems possessing hidden attractors. These are attractors which do not appear as a result of bifurcations starting from a fixed point. Instead, they can be found by locating the so-called perpetual points in state space where the acceleration is zero. Together with other recently introduced regular points,¹⁶ the authors demonstrate that chaotic attractors can be localized and multistability can be identified in very different system classes such as chaotic flows, mechanical oscillators, or semiconductor super lattices. The second new class of multistability is described by Anzo-Hernández et al.,¹⁷ who show that systems composed of piecewise-linear and unstable dissipative systems exhibit the coexistence of multi-scroll attractors in a certain range of system parameters.

The coexistence of a multitude of different attractors implies that it depends crucially on the initial condition to which of the final stable states the system will converge or in other words which kind of dynamics corresponding to one of the attractors will be realized in the long-term limit. The set of initial conditions which all converge to the same attractor, called basin of attraction, can have a rather complicated fractal structure.¹⁸ An even more complicated structure of basin boundaries—riddled basins—has been found particularly in the context of bistability and multistability in coupled systems exhibiting certain symmetries which are characterized by a synchronization manifold.^19,20 In this issue, Saha and Feudel²¹ have shown that this kind of basin boundary exists even in an infinite-dimensional system, namely, in two FitzHugh-Nagumo systems coupled by two different time-delays. When such a basin structure is present, then any arbitrarily small perturbation will kick the system out of the current attractor. For the considered system, this property is particularly dangerous since the attractor with the riddled basin of attraction involves the emergence of extreme events, so that no control strategy could be found to prevent their occurrence.

As outlined above, the possibly complexly interwoven basins of attraction make multistable systems extremely sensitive to any single perturbation or to noise, responding to which the system can switch from one attractor to any other, adding a new feature to the dynamical behavior consisting of attractor hopping²² or chaotic itinerancy.^23,24 On the other hand, multistable systems offer great flexibility in different behaviors, taking into account that each attractor represents a different system performance. Therefore, one can also take advantage of multistability by applying control techniques that allow for a defined switching between different attractors, i.e., different system performances (for a review, cf. Ref. 25). Due to the sensitivity to perturbations, control is needed to stabilize the attractors against noise if a certain system performance is desired. If multistability contains an undesired state, which is true for many applications, it is even necessary to develop control strategies to avoid the convergence to those states or even to suppress multistability itself.

Besides the hopping between different attractors, which can be induced either by noise or any other perturbation as mentioned above, there are some other mechanisms, called critical transitions, which lead to switches from one stable state to another.²⁶ Such transitions have been studied in physics in the 80s, where they have been often called phase transitions. Nowadays, critical transitions have become a major focus of research in view of climate change, where the transition from one state to another might not be desired but could be related to a performance of the system that could be dangerous in the future. Since these scientific developments started largely independent of the physics literature, the terminology used is different: in climate science, those critical transitions are named “tipping points,” while in ecology, they are denoted as “regime shifts.”²⁷ Examples from ecology comprise changes in various ecosystems,^28,29 such as shallow lakes,³⁰ coral reefs,^31,32 or vegetation patterns in semiarid areas^33,34 or in the Sahara.³⁵ In climate dynamics, various regions in the world have been classified as tipping elements as they are considered to be under the threat of a critical transition,³⁶ such as the possible shutdown of the thermohaline ocean circulation (THC),^37,38 the transition from a wet to a dry Indian monsoon system,³⁹ or the current discussion about the loss of sea ice in the Arctic.^40,41

There are several possibilities of how critical transitions can occur. A classification from a mathematical point of view has been given by Ashwin et al.,⁴² who distinguish three kinds of critical transitions depending on the mechanism of how they occur.

The first one is a bifurcation-induced transition (B-tipping), which happens when an external forcing or an internal parameter crosses a certain threshold value, leading to a transition from one stable state to another. While plotting a usual bifurcation diagram, it is assumed that the parameter is changed infinitesimally slowly; however, the real change in nature would happen at a certain time scale, which would actually change the dynamics. This problem has been addressed in the past as slow passage through a bifurcation, leading to a delayed bifurcation in cases of a pitchfork⁴³ and a period doubling.⁴⁴ Such bifurcation-induced transitions have been extensively discussed in the literature during the last decade, particularly in applications dealing with systems possessing a hysteresis related to bistability sandwiched between two saddle-node bifurcations. To name a few cases dealing with two alternative stable states in ecology and climate science, we mention here, as examples, studies on dominance changes of species in the ocean sediment of the North Sea⁴⁵ or the loss of the summer sea ice cover in the Arctic.⁴⁰ It has been further shown that for hysteresis which occurs in between two saddle-node bifurcations of limit cycles, the critical transitions are much more difficult to detect because of the ghost limit cycle which still influences the dynamics even beyond the bifurcation.⁴⁶ 

The second critical transition corresponds to a noise-induced transition (N-tipping).⁴⁷ As already mentioned, multistable systems are very sensitive to noise. This behavior is even more pronounced if the basins of attraction were fractal. The stability of a stable state is then measured in terms of the mean escape time, i.e., the mean time after which the system will leave the neighborhood of this stable state and either switch to another one or, in the case of fractal basin boundaries, dwell around the chaotic saddle embedded in the basin boundary and then return to the same or switch to another stable state after some time.⁴⁸ A crucial concept used to quantify the time that it takes on average to leave a particular stable state is the mean first passage time or Kramer's law.^49–51 Noise-induced critical transitions appear, e.g., in laser systems,⁵² in neuron populations,⁵³ or in the climate system.⁵⁴ Different aspects, which are related to such noise-induced dynamics in multistable systems, are outlined in Refs. 55–57 and discussed in more detail below.

The third class of critical transitions is related to a new kind of tipping, namely, rate-induced tipping (R-tipping).⁴² This class is particularly of importance for all processes in which environmental parameters or external forcings are varied on a certain time scale which is different from that of the internal dynamics of the system. Whenever a problem addresses the impact of global change on ecosystems or on processes in the ocean and atmosphere, this consideration of external forcings exhibiting a trend has to be taken into account.⁵⁸ It has been shown that the speed with which the parameter is varied plays an important role for the question whether the current stable state can be successfully tracked or not^42,59 and if not, what unusual behavior is to be expected instead. Ashwin et al.⁴² derived a critical speed below which tracking is possible. With a speed above this critical speed, the system will tip, i.e., it undergoes a qualitative change in the dynamics. In contrast to the previous literature dealing mainly with equilibria, Alkhayoun and Ashwin⁶⁰ have extended this study successfully to periodic attractors, as discussed in this issue. Employing the concept of pullback attractors,^61,62 they discover a novel kind of partial tipping: While some trajectories track the quasistatic attractor, others tip. Under certain conditions, this partial tipping is accompanied by the formation of global connecting orbits in an extended system.

Rate-dependent tipping can be considered as a special case of analyzing dynamical systems with an arbitrary time-dependence. This idea of a continuous drift of a parameter in time and hence considering a non-autonomous system is also exploited by Kaszás et al.,⁶³ who study the paradigmatic example of a pendulum with a periodically forced suspension point, where this forcing is switched off exponentially in time until no forcing is applied anymore and the pendulum approaches the resting state. They show that the time-dependent topology of the dynamics in phase space can be effectively described by snapshot saddles and their stable and unstable foliations. Furthermore, they introduce the concept of leaking in history space to explore details of the dynamics of time-dependent systems.

An important task besides the detection of critical transitions is their prediction not only from simulations but also from real observation data in nature or experiments, which is even more difficult because of the observational noise. One method that has been developed in the physics and chemistry literature is the critical slowing down (CSD) of the restoring forces when a transition is approached.^64–67 As a result of the smaller restoring forces bringing the system back to its stable state after a perturbation, the response to inevitable noise is amplified, leading to an increase in the standard deviation^68,69 and the lag-1 autocorrelation when approaching the bifurcation.^70,71 These methods—critical slowing down and noise amplification—have become extremely popular over the last decade particularly in the ecological literature as possible early-warning signals, but they have also been critically discussed from various perspectives.^72,73 Nevertheless, they have been successfully applied to many different observations in nature or simulated data such as pollinator collapse,⁷⁴ the Atlantic Meridional Overturning Circulation collapse,⁷⁵ or the decline of salt marsh resilience,⁷⁶ to name only a few. Nowadays, there is a large number of methods available on how to detect thresholds in nonlinear dynamical systems.^77,78 In this Focus Issue, a novel method to predict abrupt changes in the system dynamics with the help of information theory is proposed by Rubido et al.⁷⁹ This algorithm provides an entropy-based encoding of detected signals, transforming them into simple symbolic sequences which contain relevant information about the system behavior. Approximate Generating Markov Partitions can be used for the estimation of statistical dynamical invariants, essential for understanding the system properties near a tipping point. This allows the prediction of the system tendency to drift towards the critical transition. The new complexity measure proposed by the authors can be efficiently used for characterizing such complex behaviors as brain activity and climate variability.

Although the importance of detecting critical transitions in spatially extended systems has been stressed,⁷⁸ only a few methods have been developed until now.^80–82 In this focus issue, Kwasniok⁸³ discusses different approaches for data-based detecting, anticipating, and predicting critical transitions or tipping points in spatially extended systems. As outlined above, in natural systems, the proximity to such transitions can be detected in two ways: an increasingly slow system recovery from noisy perturbations due to critical slowing down before the bifurcation and/or an increasing variance of random fluctuations of the system variable due to noise amplification near the bifurcation. Exploring the stochastic Swift-Hohenberg equation as an example, the author performs the nonlinear principal oscillation pattern (POP) analysis to predict critical transitions by extrapolation of the model beyond the learning data window.

Multistability was first mentioned in 1971 by Atteneave⁸⁴ in the context of perception, and since then, it has been considered as one important mechanism for cognitive processes in the brain,^85,86 such as activity patterns inside and outside the visual cortex related to multistable perception manifested in binocular rivalry (alternation of two different images, one presented to each eye) or in the perception of ambiguous figures (figures allowing for two or more interpretations),⁸⁷ or auditory stream segregation (distinction of different sound sources).⁸⁸ Ambiguous figures provide an excellent object to examine the resulting brain activity as a response to their presentation.^89,90 In this issue, Hramov et al.⁹¹ investigate the bistable perception of the Necker cube—a specific ambiguous figure. They consider the brain as a multistable dynamical system and analyze neurophysiological data of brain activity using an artificial neural network (ANN). The bistable brain states are successfully recognized in experimental magnetoencephalographic (MEG) data recorded during decision-making on interpretation of ambiguous images. The authors demonstrate that ANN can be efficiently used for the detection of human uncertainty in decision-making. The proposed approach is promising for the development of a new generation of brain-computer interfaces.

Not only cognitive processes are related to multistability in the brain. Also the resting state of the brain without any external stimuli is considered to be not random but highly structured, involving different spatial neuronal activity patterns.^92,93 Such switching between different spatial activity patterns, which can be intermittent, has recently been modelled by networks of FitzHugh-Nagumo units⁹⁴ or Morris-Lecar neurons.⁹⁵ In this issue, Pisarchik et al.⁹⁶ suggest a possible mechanism for the intermittent character of neurophysiological data. The origin of such a behavior can lie in multistability induced by an asymmetry in the electrical coupling between neurons. By analyzing the dynamics of coupled neuronal oscillators, they find the coexistence of three different oscillatory regimes in a certain range of the coupling strengths. The authors also provide experimental evidence of asymmetry-induced multistability in equivalent electronic oscillators built on the basis of the Hindmarsh-Rose equations.

The first experimental evidence for multistability in physics has been discovered by Arecchi et al.⁹⁷ in a Q-switched gas laser followed by many theoretical and experimental findings in various kinds of lasers such as optically induced semiconductor lasers,⁹⁸ semiconductor ring lasers,⁹⁹ lasers with optical feedback,¹⁰⁰ or erbium-doped fiber lasers.¹⁰¹ Furthermore, multistable laser systems have been shown to exhibit also extreme events, i.e., extremely rare large amplitude events,¹⁰² or chimeras, i.e., stable states which comprise coherent and incoherent behaviors.¹⁰³ A review about a new field of multistability in physics is outlined in this issue. Wang et al.¹⁰⁴ present a novel class of multistable dynamical systems composed of relativistic quantum and classical subsystems, namely, a topological insulator and an insulating ferromagnet. Besides multistability, this hybrid system exhibits a very rich dynamical behavior, including different types of bifurcations and chaos. The authors review main concepts of condensed matter physics and material science and analyze the dynamical properties of topological quantum materials. They stress that multistability in ferromagnetic materials is promising for further development of memory devices, which can be used for binary operations to process and store information.

Critical transitions between different stable states have been studied intensively in population dynamics since the seminal paper by May.¹⁰⁵ While most of those investigations deal with equilibria, this special issue offers a new type of dynamics—mixed-mode oscillations—rarely studied in population dynamics. Sadhu and Kuehn analyze oscillation patterns obtained in a predator-prey model, where they find mixed-mode oscillations in the intermediate dynamics between two cycles of population outbreaks. This interesting effect is interpreted as noise-induced multistate intermittency. In the paradigmatic example, they combine theoretical modeling and numerical simulations to gain insights into the interplay between noise and singularity in slow-fast systems.

Bashkirtseva⁵⁷ proposes a semi-analytical approach which allows the prediction of tipping points in a bistable Hassell-type population model with the Allee effect. The method is based on the stochastic sensitivity function and confidence domains. In the deterministic model, the tipping points are located at persistence zone borders (Allee threshold, crisis, and saddle-node bifurcation points), which are shifted in the stochastic model, so that the persistence zone decreases as the noise intensity increases. The author states that the proposed technique is general and can be used for detection of early-warning signals of noise-induced interior crisis in other systems.

A stochastic sensitivity analysis is also applied by Ryashko⁵⁶ to study noise-induced oscillatory multistability in glycolysis, where the author analyzes mixed-mode stochastic oscillations in different parameter ranges. Using the stochastic sensitivity formalism, he finds noise-induced transitions from order to chaos in the glycolytic system.

While most of the work on critical transitions in ecology is devoted to simple population dynamical systems, only a few examples of multiplicity of different spatial patterns have been investigated. The most prominent ecosystems in which different patterns coexist are dryland ecosystems,¹⁰⁶ distributions of bacteria and nutrients in ocean sediments,¹⁰⁷ and patterns of seagrass¹⁰⁸ and mussels.¹⁰⁹ For such spatial systems, it has been shown that critical transitions are not only abrupt changes in patterns but can also happen gradually due to the fact that critical transitions can apply only to small spatial parts of the system which conquer the whole spatial area slowly step by step.⁸¹ In this Focus Issue, Zelnik et al.¹¹⁰ investigate tristability in a dryland ecosystem model, where two homogeneous (full vegetation cover and bare soil) and one patterned states coexist. The authors discuss in particular the breakdown of a snaking region, i.e., a parameter region in which localized patterns exist. Furthermore, they identify a novel type of complex fronts between full vegetation and bare soil with different patterns in between. Additionally, they show that each of those front solutions has its own speed.

In view of global change, the coexistence of multiple stable states and possible critical transitions between them has become a new focus of research in climate science. Various aspects of climate thresholds are discussed¹¹¹ with respect to changes in the thermohaline ocean circulation (THC), deep ocean convection, ocean acidification, and changes in the Arctic ice sheets. Several areas in the world have been identified as tipping elements,³⁶ such as the Indian monsoon, ElNino-Southern Oscillation, and permafrost areas besides the THC. As mentioned above, the melting of Arctic ice, possibly leading to an almost ice-free summer in the Arctic, is debated.⁴⁰ Early warning signal analyses have been proven to be useful to study critical transitions in climate science.⁷¹ Rate-induced transitions have been studied, related to global change manifested as a warming trend.⁵⁸ A more general approach considers an Earth system possessing a warm climate or a global snowball state as two alternatives.⁵⁴ It has been shown that besides those two extremes, the system exhibits multistability, including a complex dynamical state on the basin boundary, which can be considered as a climate edge state possessing properties similar to a chaotic saddle.

In summary, multistability and critical transitions are very active fields of science, and this collection of papers can only reflect a small part of this activity.

U.F. acknowledges support from the German Science Foundation (DFG) for funding an international workshop at the Max Planck Institute for the Physics of Complex Systems (MPIPKS) in Dresden, Germany, with the same title (FE 359/16) as this Focus Issue. A.N.P. acknowledges support from the Ministry of Economy and Competitiveness (Spain) (Ref. SAF2016-80240-P). K.S. acknowledges financial support from the U.S. National Science Foundation (NSF), Grant No. CHE-1565665.