Non-autonomous Dynamics in Complex Systems: Theory and Applications to Critical Transitions

In this lecture, we will discuss ways to detect critical transitions in complex systems based on analyses of observations of their non-autonomous dynamics. Topics include segmentation techniques for investigating non-stationary dynamics, univariate and multivariate time-series-analysis techniques as well as surrogate-based concepts to evaluate reliability of detections. Special attention is paid to sources of errors and potential remedies.

Most complex and networked systems are out of equilibrium and externally driven such that their collective dynamics may change drastically. They may exhibit tipping, here taken to be a qualitative change induced by sufficiently strong driving, that often disrupts the systems' intended or desired functionality. Yet, standard linear response theory generically characterizes systems responses to such fluctuations only for small driving amplitudes and cannot capture response properties that are either due to strong driving signals or intrinsically nonlinear. Here we present an alternative perturbative technique that exploits a self-consistency criterion, captures the full nonlinear response dynamics and enables the approximate prediction of tipping points and the response dynamics before tipping.

In the age of climate warming, comprehension of ecosystems’ future is one of the pressing challenges to humanity. Whilst 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. 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.

Various physical phenomena can be modelled through the use of fast-slow systems in the form of PDEs perturbed by noise. A common technique for a proper study of their properties is the focus on the fast variable, hence fixing the slow one and considering it as a parameter whose value can affect the qualitative behaviour of the solution of the equations. Of particular importance are the bifurcation thresholds of the deterministic system and the early-warning signs that can be observed as the parameter approaches them. I will compare different types of early warning-signs for SPDEs (stochastic partial differential equations) and discuss how they are affected by the choice of the drift component. The nature of the noise is a generalized Q-Wiener process in order to replicate the heterogeneity in space often displayed in applications.

Kernel methods are a family of machine learning algorithms that have been applied successfully to a wide range of fields. Kernel methods have primarily been applied to Euclidean data. However in many applications - such as climate science, geometric mechanics and computer vision - the data naturally lies on a non-Euclidean manifold. We investigate the generalisation of kernel based algorithms to non-Euclidean spaces. Kernel methods often rely on embedding the data space into a reproducing kernel Hilbert space (RKHS), which in turn relies on a positive definite (PD) kernel. We find that the geodesic Gaussian kernel, that generalises the Euclidean Gaussian kernel to Riemannian manifolds, is usually not (PD), preventing its use for learning with RKHS. We provide alternative ways of finding PD kernels on Riemannian symmetric spaces. We also show that, on many Riemannian symmetric spaces, the geodesic Gaussian kernel satisfies the weaker condition of admitting a positive decomposition. This gives an embedding of the data space into a reproducing kernel Krein space (RKKS) instead, providing an alternative route for learning with this kernel.

In systems of stochastic dynamical systems, noise-induced transitions occur when the presence of noise prompts the state of the system to transition from one (meta)stable state to another, potentially as a rare event. Understanding a system’s behavior through this transition is a significant area of current interest. In this talk, we will describe recent analytical and computational tools to identify Most Probable Escape Paths (MPEPs) in Stochastic Differential Equations (SDEs) over periodic boundaries. The study of noise-induced tipping has been dominated by Freidlin-Wentzell (FW) theory. One limitation of the FW theory is that it necessitates vanishingly small noise. However, for several applications of interest, particularly in environmental, biological, or social contexts, intermediate noise is of more relevance. In focusing on the intermediate noise regime, the transient behavior of the underlying deterministic system will play a key role. We will thus use a dynamical system approach to identify MPEPs in SDEs. The Maslov index will help us distinguish which critical points of the FW functional are minimizers and help explain the effect of the interaction of noise and transient dynamics. The Onsager-Machlup (OM) functional, which is treated as a perturbation of the FW functional, will provide a selection mechanism to pick out a specific MPEP. Subsequently, our computations will be compared with Monte Carlo simulations to verify theoretical predictions in the Inverted Van der Pol system.

The Atlantic Meridional Overturning Circulation is responsible for the comparatively temperate climate found in Western Europe, and its previous collapse thought to have triggered glacial periods seen in the paleo data. This is a system that has multiple stable states- referred to as ‘on’ when the circulation is strong as in the current climate, and ‘off’ when it is much weaker. The AMOC has tipping points between these states. Tipping points occur when a rapid shift in dynamics happens in response to a relatively small change in a parameter. Making future projections of AMOC response to the climate is essential for avoiding any anthropogenic caused tipping, but it is computationally expensive to calculate the full hysteresis for different scenarios. This work looks at a conceptual five box model of the AMOC [1] which is easy to understand and cheap to implement. Previous work has considered bifurcation and rate-dependent tipping [2]. This current work looks to estimate a realistic amount of noise from various GCM data sets and apply this to the model. We compare the covariance of the salinity data for a variety of CMIP6 models, and we compare the amount of noise covariance found in each data set, and how this can be input back into the box model. Ongoing work looks to find first passage times for the box models with the estimated noise and a small amount of freshwater forcing. (1) Wood, R et.al. (2019), Climate Dynamics, 53(11), 6815-6834 (2) Alkhayuon, H et.al. (2019), Proc. R. Soc. A, 475(2225)

Heteroclinic dynamics is a suitable framework for describing transient and reproducible dynamics such as cognitive processes in the brain. We demonstrate how heteroclinic units can act as pacemakers to entrain larger sets of units from a resting state to hierarchical heteroclinic motion such as fast oscillations modulated by slow oscillations. The entrainment range depends on the type of coupling, the spatial location of the pacemaker and the individual bifurcation parameters of the pacemaker and the driven units. Noise as well as a small back-coupling to the pacemaker considerably facilitate synchronization. Units can be synchronously entrained to different temporal patterns, depending on the selected path in the hierarchical heteroclinic network. Such patterns are believed to code information in brain dynamics. Depending on the number and the location of pacemakers on a two-dimensional grid, synchronization can be maintained in the presence of a large number of resting state units and mediated via target waves when the pacemakers are concentrated to a small area of such a grid. In view of brain dynamics, our results indicate a possibly ample repertoire for coding information in temporal patterns, produced by sets of synchronized units entrained by pacemakers, without finetuning of the parameters, and distributed in space.

Parametrized state space models in the form of recurrent neural networks are often used in machine learning to learn from data streams exhibiting temporal dependencies. I will talk about a framework for rigorous analysis of such state space models known as Echo State Networks (ESN). In particular, I will show that there is a deep connection between the state space and the notion of feature space in kernel machines (a well known class of machine learning methods). This viewpoint will lead to several rather surprising results linking the structure of the dynamical coupling in ESN with properties of the associated temporal feature space. I will also present statements regarding universality of ESNs as capable of approximating a large class of fading memory filters with a minimal structural complexity.

I will present a new concept "dynamics-based data science" in biology and medicine for studying dynamical processes and disease progressions, including dynamic network biomarkers (DNB) for early-warning signals of critical transitions, spatial-temporal information (STI) transformation for short-term time-series prediction, and partial cross-mapping (PCM) for causal inference among variables. These methods are all data-driven or model-free approaches but based on the theoretical frameworks of nonlinear dynamics. We show the principles and advantages of dynamics-based data-driven approaches as explicable, quantifiable, and generalizable. In particular, dynamics-based data science approaches exploit the essential features of dynamical systems in terms of data, e.g. critical collective fluctuation near a tipping point, low-dimensionality of a center manifold or an attractor, and phase-space reconstruction from a single variable by delay embedding theorem, and thus are able to provide different or additional information to the traditional approaches, i.e. statistics-based data science approaches. The dynamical data science approaches will further play an important role in the systematical research of various fields in biology and medicine as well as AI.

The analysis of the warming trend of surface temperature anomalies shows an abrupt acceleration of warming in the past 15 years in central Europe (ERA5 2mT). Employing change point analysis and quantifying the significance of trends in the presence of long range temporal correlations, we try to answer the question of whether this change of warming rate is related to overcoming tipping points in the release of greenhouses gases by correlating these results to greenhouse gas concentrations and emissions.

The Atlantic Meridional Overturning Circulation (AMOC) was identified as a potential tipping element in the earth system. In a hierarchy of models of increasing complexity, ranging from simple box models to intermediate-complexity models to state-of-the-art climate models, the AMOC exhibits bistability with an 'on-state' characterised by a functioning overturning circulation and an 'off-state' in which the overturning circulation shuts down. The proximity of the present AMOC to a critical transition threshold is under debate. The impact of an AMOC collapse on the climate in many parts of the world would be enormous. In this study, a data-driven methodology for identifying, anticipating and predicting critical transitions in high-dimensional model or observational data sets is introduced, based on explicit non-stationary low-order modelling of the tipping dynamics. Unlike the more traditional early-warning signs such as rising autocorrelation (critical slowing down) or rising variance, this allows for dynamical understanding of the underlying tipping mechanism and genuine prediction of the future system state by extrapolation. A set of spatial modes carrying the tipping dynamics are identified and a stochastic model of appropriate complexity is estimated in the subspace spanned by these modes. Analysis of the reconstructed dynamics allows to determine the proximity to a bifurcation point and the type of the impending bifurcation. Different competing tipping mechanisms can be compared and assessed using likelihood inference and information criteria. The technique is inherently probabilistic with the randomness stemming from two sources: the statistical uncertainty in the model estimation and the inevitably stochastic nature of the low-order tipping model. As such, the method allows to quantify the likelihood or risk of a critical transition at some point in the future having observed a certain amount of data up to present. The methodology is here applied to a data set from a simulation of AMOC collapse with a complex climate model, actually a freshwater hosing experiment with the FAMOUS GCM. The AMOC on-state is found to lose stability via a subcritical Hopf bifurcation; however, the transition to the off-state occurs far ahead of the bifurcation point. The early collapse can be explained by a combination of rate-induced and noise-induced tipping. The finding is relevant in terms of safe operating zones as the risk of AMOC collapse in such a scenario would be higher than anticipated with conventional early-warning signs.

In the era of the data science, model-free techniques are developed overwhelmingly. When the experimentally-collected data are generated by dynamical systems, the missions of reconstruction, prediction, and modulation only based on these data are highly anticipated to be achieved for these systems. Here, we introduce several directions of progresses made by our research group in developing the model-free techniques using machine learning techniques and dynamical systems theory. We use representative systems of physical or/and biological significance to demonstrate the developed techniques. We hope that all the methods can shed a light on deciphering and controlling the hidden dynamics that dominate the evolutions of any systems in real-world.

The human brain is an open, dissipative, and adaptive nonstationary dynamical system composed of a large number of interacting subsystems. Its complicated spatial–temporal dynamics is still poorly understood. Epilepsy is a malfunction of the brain that affects about 50 million people worldwide. Epileptic seizures are the cardinal symptom of the disease and often appear as a transformation of otherwise normal brain dynamics. The exact mechanisms underlying seizure generation are still as uncertain as are mechanisms underlying seizure spreading and termination. Identifying early warning signals of seizures from brain dynamics could drastically improve therapeutic possibilities and thus, the quality of life of people with epilepsy. Methods from nonlinear dynamics, statistical physics, synchronization and network theory are capable of identifying seizure precursors from EEG recordings in a large number of people with epilepsy and with high sensitivity and specificity. In this talk, I will provide a brief overview of the current status of the field, will highlight shortcomings of recent approaches as well as unsovled issues, and will discuss possible research directions that may help to find better solutions.

Motion of stars is usually considered in time independent galactic potential. However, for various circumstances (e.g. encounters between the individual stars; stars of galaxies and globular clusters lose substantial quantities of mass) one have to take into account potential variations that are slow (or adiabatic) compared to a typical orbital frequency. In this case the action variables of stars are constant during such adiabatic changes of the potential. For this reason actions are often called adiabatic invariants. The nature of the problem posed by a time-varying potential depends on the speed with which the potential evolves. In slowly evolving potentials, angle-action variables enable us to predict how a stellar system will respond to changes in the gravitational field that confines it. We seek in our research a novel approach to describe the dynamics beyond the adiabatic limit, i.e. when the change of potential is comparable to the characteristic time scale of the problem. To our knowledge this is the first attempt to use contemporary results of nonautonomous Hamiltonian dynamics in celestial mechanics. We hope that this idea will enable us to explore the peculiar nature of stellar dynamics subjected to external forces.

The Atlantic Meridional Overturning Circulation (AMOC) is believed to possess two stable flow regimes, thus qualifying as a proposed tipping element in the Earth system. Abrupt climate changes in paleo records are often associated with shifts between different AMOC flow patterns. To better understand the mechanisms and probability of such transitions, it is important to understand their transition pathway in the state space of climate models. Specifically, in weakly perturbed dynamical systems, it is expected that sample transitions will cross from one to another basin of attraction via special regions on the basin boundary, termed Melancholia states. Here we implement an edge-tracking algorithm in PlaSim-LSG, a global climate model of intermediate complexity, to identify a Melancholia state between the vigorous and collapsed states of the AMOC. We discuss the properties of such a state, its role for critical transitions of the AMOC, and its dependence on time-dependent forcing. Knowing what lies in between the competing metastable states could help elucidate the risk and most likely scenario of potential future AMOC tipping.

Syamal Kumar Dana Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India Abstract: We report tipping in an insect outbreak model near two saddle-node bifurcation points of the model system. The model defines outbreak of budworm population from a desirable low population state against the carrying capacity. If the carrying capacity is varied at a linear rate, the system does not show sharp transitions immediately at the bifurcation points but tips to the alternate states after a lapse of time. This delay in tipping decreases with faster rate of change of the carrying capacity. We check the impact of a shock (external effect) modelled by a triangular shape impulse that slowly varies the carrying capacity at a constant rate, however, withdrawn at an identical or different constant rate. The tipping occurs in such cases but shows a dependence on the falling and rising rates of the impulse that influences the carrying capacity. We identify the rate of rising and falling in a phase diagram that provides information when a tipping may occur. An interplay of the dissipation rate of the system and the rate parameter decides the occurrence of tipping. In case of insufficient strength of a single impulse that may fail to induce any tipping to the desirable low population state from a large population or an outbreak state, a second impulse is applied, but weaker in strength compared to the first one and it is able to induce a tipping that prevents an insect outbreak. Therefore, in a situation of multiple shocks or impulses applied on the carrying capacity, tipping may occur as a consequence of past environmental changes. We explain the scenarios with numerical experiments and using the dynamical change in the potential function of the system.

Abrupt and irreversible winter Arctic sea ice loss may occur under anthropogenic warming due to the disappearance of a sea ice equilibrium at a threshold value of CO$_2$, commonly referred to as a tipping point. Previous work has been unable to conclusively identify whether a tipping point in winter Arctic sea ice exists because fully coupled climate models are too computationally expensive to run to equilibrium for many CO2 values. Here, we explore the non-autonomous behavior of sea ice under realistic rates of CO2 increase to demonstrate for the first time how a few time-dependent CO2 experiments can be used to predict the existence and timing of sea ice tipping points without running the model to steady state. This work highlights the inefficacy of using a single experiment with slow-changing CO2 to discover changes in the sea-ice steady state and provides a novel alternate method that can be developed for the identification of tipping points in realistic climate models.

An excitable low-order quasigeostrophic model [1] captures key features of intrinsic low-frequency variability of the oceans’ wind-driven circulation. This double-gyre model is used as a prototype of an excitable nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The present studies rely on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), namely the time-dependent invariant sets that attract all trajectories initialized in the remote past. Ensemble simulations help us explore these PBAs. In an excitable dynamical system, a relaxation oscillation connects a basic state to an unstable excited state, from which spontaneous returns to the original state occur. Such back-and-forth oscillations are self-sustained in a certain parameter range of the autonomous system, and they can be excited by a suitable external time-dependent forcing outside this range. In this presentation, by using model [1] in its excitable state, we discuss: (i) the effect of a periodic forcing on synchronization scenarios and on the onset of chaos in an otherwise non-chaotic system [2,3]; (ii) the effect of an aperiodic forcing mimicking a time dependence dominated by interdecadal and high-frequency variability [4,5]; (iii) the tipping points due to parameter drift in the presence of small periodic perturbations [6]. The latter situation can be thought of as the seasonal-to-interannual variability in the wind stress, while the monotonically increasing component stands for the effect of reduction in the midlatitude winds due to anthropogenic warming. References: [1] Pierini S., 2011. J. Phys. Oceanogr. 41, 1585-1604. [2] Pierini S., 2014. J. Phys. Oceanogr. 441, 3245-3254. [3] Pierini S., Chekroun M.D., Ghil M., 2018. Nonlin. Processes Geophys. 25, 671-692. [4] Pierini S., Ghil M., Chekroun M.D., 2016. J. Climate 29, 4185-4202. [5] Pierini S., 2020. J. Stat. Phys. 179, 1475-1495. [6] Pierini S., Ghil M., 2021. Sci. Rep. 11, 11126

Over the last two decades, tipping points have become a hot topic due to the devastating consequences that they may have on natural and human systems. Tipping points are typically associated with a system bifurcation when external forcing crosses a critical level, causing an abrupt transition to an alternative, and often less desirable, state. However, the rate of change in forcing is arguably of even greater relevance in the human-dominated anthropocene, but is rarely examined as a potential sole mechanism for tipping points. Thus, I will introduce the related phenomenon of rate-induced tipping: an instability that occurs when external forcing varies across some critical rate, usually without crossing any bifurcations. First, I will explain when to expect rate-induced tipping. Then, using illustrating examples of differing complexity I will highlight universal and generic properties of rate-induced tipping in a range of natural and human systems.

The effect of climate warming on species' physiological parameters, including growth rate, mortality rate, and handling time, is well established from empirical data. However, with an alarming rise in global temperature more than ever, predicting the interactive influence of these changes on mutualistic communities remains uncertain. Using 139 real plant–pollinator networks sampled across the globe and a modeling approach, we study the impact of species’ individual thermal responses on mutualistic communities. We show that at low mutualistic strength plant–pollinator networks are at potential risk of rapid transitions at higher temperatures. Evidently, generalist species play a critical role in guiding tipping points in mutualistic networks. Further, we derive stability criteria for the networks in a range of temperatures using a two-dimensional reduced model. We identify network structures that can ascertain the delay of a community collapse. Until the end of this century, on account of increasing climate warming many real mutualistic networks are likely to be under the threat of sudden collapse, and we frame strategies to mitigate this. Together, our results indicate that knowing individual species' thermal responses and network structure can improve predictions for communities facing rapid transitions.

We investigate the phenomenon of extreme transient dynamics in four pendula coupled via the movable support. Our findings uncover a traveling phase state, within which the oscillators synchronize with phases evolving. The scenarios of possible traveling are discussed and the properties of patterns are determined. We report on the problem of transient dynamics, which can be observed on the road to final network synchronization, exhibiting that the accompanied transient windows can become extremely long. The behaviours are carefully studied in relation with initial conditions and parameters, showing how the system characteristics influence the transients. The results of our research underline the problem of slow dynamics evolution and its relation with mechanical setups.

We discuss the suppression of a collective rhythm produced by a population of interactive oscillatory units by pulsatile stimulation. The problem is motivated by the medical technique known as deep brain stimulation. To imitate beta-band brain activity observed in Parkinsonian patients, we introduce time-dependent connectivity and consider a feedback-based approach for controlling this non-autonomous system. We pay attention to related technical problems like a causal real-time estimation of the phase and amplitude of a signal mixed with strong observational noise. We demonstrate that an adaptive control scheme can successfully suppress an undesired rhythm.

Nonlinear dynamics teaches us that small stressors can lead to much larger catastrophes and even the collapse of entire complex systems. Modern global human civilisation is precisely such a system, and the stressors --- such as global climate change --- are not small. Catastrophic scenarios such as local or global societal collapse are increasingly being studied, yet essential ideas from nonlinear dynamics have not yet been incorporated. One of these ideas is rate-induced tipping. Systems break down when external changes overcome the capacity for adaptation; in other words, when things change too quickly. Using evidence from nature, dynamical systems, and agent-based models, I argue that rate-induced collapse is ubiquitous across complex systems. If true, rate-induced tipping is of fundamental relevance to some of the most consequential problems facing humanity this century. There is much exciting work to be done.