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

We propose an effective theory to describe the different regimes of decay of the survival probability of chaotic systems with dissipation. Our general results are illustrated through numerical simulations in the paradigmatic Henon-Heiles model. This is a joint work with Lachlan Burton and Holger Dullin.

The concept of rate induced delayed tipping to alternate states has been introduced in different nonautonomous systems in the last decade. Very recently, the effect of external impulses on system parameters has also been reported in model systems. In this backdrop, we select an insect population model to explore tipping near its two saddle-node bifurcation points against the time-varying carrying capacity of the system. The frozen system exhibits bistability with a coexisting refuge state of low population density and an outbreak state of large population. If the carrying capacity is varied linearly, the system does not show sharp transitions as expected immediately at the bifurcation points but tips to the alternate states after an elapse of time. The delay in tipping decreases with faster rate of change of the carrying capacity following a power law. We have also emphasized on the impact of external shocks modelled by a shock like impulse that varies the carrying capacity at a constant rate, however, withdrawn at an identical or different constant rate. Delayed tipping also occurs in such a case of single impulse but shows a dependence on the falling and rising rates of the impulse that pushes the carrying capacity to cross the bifurcation points. The assumption of impulse to represent an impulsive shock allows a flexibility of varying the rising and falling rate of the carrying capacity. The range of the rate of rising and falling of the impulse is thus identified in a phase diagram that clearly delineates the rate parameter zones of tipping and no tipping. The active time window of the external impulse on the carrying capacity plays a decisive role on the occurrence of tipping. We analytically derive the critical value of the exceedance time that depends upon the rate parameters (rising and falling rates) and the depth of the impulsive. Furthermore, we apply a second impulse, in case the first impulse is large yet so weak in strength that fails to induce any tipping to the desirable low population state from a large population outbreak state. The second impulse is assumed to be even weaker in strength compared to the first one. In such a situation of multiple impulses applied on the carrying capacity, tipping occurs as a consequence of past effect of the first impulse. The role of the rate parameters of both the impulses, strength of the impulses and most importantly, the time interval of the pulses are considered in detail to delineate the tipping zones in parameter space. We demonstrate the scenarios with numerical experiments, and the process of tipping using the dynamical change in the potential of the system.

Network structure or connectivity patterns are critical in determining collective dynamics among interacting species in ecosystems. Conventional research on species persistence in spatial populations has focused on static network structure, though most real network structures change in time, forming time-varying networks. This raises the question, in metacommunities, how does the pattern of synchrony vary with temporal evolution in the network structure. The synchronous dynamics among species are known to reduce metacommunity persistence. Here we consider a time-varying metacommunity small-world network consisting of a chaotic three-species food chain oscillator in each patch or node. The rate of change in the network connectivity is determined by the natural frequency or its subharmonics of the constituent oscillator to allow sufficient time for the evolution of species in between successive rewirings. We find that over a range of coupling strengths and rewiring periods, even higher rewiring probabilities drive a network from asynchrony towards synchrony. Moreover, in networks with a small rewiring period, an increase in average degree (more connected networks) pushes the asynchronous dynamics to synchrony. On the other hand, in networks with a low average degree, a higher rewiring period drives the synchronous dynamics to asynchrony resulting in increased species persistence. Our results also follow the calculation of synchronization time and are robust across other ecosystem models. Overall, our study opens the possibility of developing temporal connectivity strategies to increase species persistence in ecological networks.

This article investigates the emergence of phase synchronization in a network of randomly connected neurons by chemical synapses. The study uses the classic Hodgkin-Huxley model to simulate the neuronal dynamics under the action of a train of Poissonian spikes. In such a scenario, we observed the emergence of irregular spikes for a specific range of conductances, and also that the phase synchronization of the neurons is reached when the external current is strong enough to induce spiking activity but without overcoming the coupling current. Conversely, if the external current assumes very high values, then an opposite effect is observed, i.e. the prevention of the network synchronization. We explain such behaviors considering different mechanisms involved in the system, such as incoherence, minimization of currents, and stochastic effects from the Poissonian spikes. Furthermore, we present some numerical simulations where the stimulation of only a fraction of neurons, for instance, can induce phase synchronization in the non-stimulated fraction of the network, besides cases in which for larger coupling values it is possible to propagate the spiking activity in the network when considering stimulation over only one neuron.

Authors: Timo Bröhl [a,b], Thorsten Rings [a,b], Klaus Lehnertz [a,b,c] [a] Department of Epileptology, University of Bonn Medical Centre, Bonn, Germany [b] Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Bonn, Germany [c] Interdisciplinary Center for Complex Systems, University of Bonn, Bonn, Germany Abstract: Centrality is a fundamental network-theoretical concept that allows one to assess the role a constituent plays in the larger network. Previous research [1,2] has indicated that time-dependent changes of centrality of specific vertices (brain regions) as well as the weight of specific edges (strength of interactions between brain regions) in evolving large-scale human epileptic brain networks carry information predictive of an impending transition into an epileptic seizure. It remains unclear if constituents, carrying predictive information, are also the most central ones, and if connections between these constituents also carry predictive information. Our newly developed edge centrality concepts [3,4] together with well-known vertex centrality concepts allow us to answer these questions. To this end, we retrospectively investigate — in a time-resolved manner — evolving large-scale epileptic brain networks that we derived from multi-day, multi-electrode intracranial electroencephalographic recordings from subjects with pharmacoresistant epilepsies with different anatomical origins. Making use of the concept of seizure-time surrogates [5] and depending on the employed centrality concept, we find that temporal changes of centrality values of 4-10% of vertices (predictive vertices) and of 4-6% of edges (predictive edges) to carry predictive information (presumed duration of transitory phase: 4h). We assign vertices to functional brain modules (seizure onset zone (S), neighborhood (N), other brain regions (O)) and observe the majority of predictive vertices to be confined to module O and only rarely to modules S and N. Likewise, predictive edges largely connect vertices from module O and only rarely vertices from modules O and S. Interestingly, these findings hold even though the different centrality concepts identify different vertices and edges as central. Our findings [6] confirm earlier proposals for network-based mechanisms underlying the transitions to the pre-seizure phase, put into perspective the role of the seizure onset zone for this transition, and highlight the necessity to reassess current concepts for seizure generation and prevention. References: [1] Rings, T. et al. (2019). Precursors of seizures due to specific spatial-temporal modifications of evolving large-scale epileptic brain networks. Scientific reports, 9:10623. [2] Fruengel, R. et al. (2020). Reconfiguration of human evolving large-scale epileptic brain networks prior to seizures: an evaluation with node centralities. Scientific Reports, 10:21921 [3] Bröhl, T., & Lehnertz, K. (2019). Centrality-based identification of important edges in complex networks. Chaos, 29:033115. [4] Bröhl, T., & Lehnertz, K. (2022). A straightforward edge centrality concept derived from generalizing degree and strength. Scientific Reports, 12:4407. [5] Andrzejak, R. G. et al. (2003). Testing the null hypothesis of the nonexistence of a preseizure state. Physical Review E, 67:010901. [6] Bröhl, T. et al. submitted.

Multiple timescales coexist ubiquitously throughout nature; consequently, when modelling such processes it is important to carefully consider their varying timescales, since a subtle change in their interplay can inhibit (alternatively, induce) the onset of critical transitions. In this study we present how vulnerability to tipping - within a low-dimensional stochastic fast-slow system - changes as both the intrinsic degree of timescale separation and correlation time of external perturbations vary. The chosen model is a modified Truscott-Brindley system, describing the interaction between phytoplankton and zooplankton in marine ecology. It can be bistable, possessing two stable states with low and high phytoplankton biomass, or in an excitable monostable regime, where phytoplankton blooms form following perturbations. When subjected to white Gaussian noise, we show that the system’s quasipotential height at the edge state - which quantifies resilience to a state transition or excitation - depends non-monotonically on the intrinsic timescale separation parameter $\xi$. We explore how residence times within competing basins of attraction scale as the correlation time $\tau$ of the perturbations increases, sliding the prescribed noise term between an uncorrelated Gaussian process and a highly correlated Ornstein-Uhlenbeck process. We find that increased correlations most drastically affect the expected residence time when the underlying deterministic system is close to the bifurcation separating bistability and excitability. We then study these timescales in conjunction by considering a non-autonomous system with $\xi$ and $\tau$ time-dependent, and discuss whether the effect (in resilience to tipping) of slowly changing one variable can be counterbalanced by simultaneously changing the other variable at a comparable rate. Finally, we present how most probable transition paths for the system change with different $\xi$ and $\tau$ values, and comment on how an augmented Lagrangian method can be applied to resolve the constrained optimisation problem behind computing some of these paths. This work can be applied to studying the effect of ocean heatwaves on marine life cycles.

TBA

Climate policy has become increasingly politicized in many countries including the US, with some political parties unwilling to pursue strong measures. Decision makers with strong climate ambitions need to find policies that are feasible and robust against political instability. Climate mitigation pathways are currently explored in so-called Integrated Assessment Models (IAMs). These models evaluate climate policies from an economic perspective, typically focusing on cost-effectiveness. However, the economy is not an isolated system. Rather, it is intertwined with the political system, in which policymakers impose economic policies, but are (in democracies) dependent on public opinion, which in turn can be influenced by economic performance. Changing governments can lead to disruptions in climate policy if some parties are much more ambitious in climate protection than others and climate policy will also influence voting behaviour. In this study, we assess the effect of party politics on the green transition by coupling existing agent-based models of relevant subsystems: economy, climate, and society to analyze the feasibility of a set of green policies in case some parties are largely unwilling to protect the climate. We show that this simple additional social layer of complexity strongly affects the outcome of the abatement measures. In particular, we conclude that a (high) pure carbon tax is vulnerable to abrupt interruptions and does not represent an optimal policy in this coupled scenario. Finally, by classifying the different policies according to their economic, social and environmental impact, we propose a strategy for a green party in succeeding in decarbonisation.

NN are abundant in applications across a range of fields and their importance in advanced technology is becoming ever more clear. However, there is a fundamental lack of transparency in the mechanisms underpinning their intelligent behaviour even in the simplest systems, namely, those responsible for memory formation and decision making. Many attempts to combat this issue have been so far unsuccessful and hence we propose a different approach to the NN by considering it as a non-autonomous dynamical system and analysing it’s velocity vector field which we believe may hold the key to unlocking its transparency. Here we study an 81-dimensional version of the Hopfield network, one of the simplest neural models available which, at the time of its inception was considered a major breakthrough in the field of artificial intelligence and has since informed many of the more advanced models in use today. We use a number of visualisation methods to view both the formation of the memories of a network but also their 81-dimensional basins of attraction which play a significant role in the network's decision-making.

Evolving phenomena are called dynamic systems, and in many cases, their complete description can be achieved through generic mappings. A traditional example is the simplified Fermi-Ulam model. From the Fermi-Ulam model, five attractors and their respective attraction basins have been identified. Furthermore, preliminary results indicate a possibility of fractal boundaries for these basins, with apparent coexistence among 3 or more basins.

Dansgaard-Oeschger (DO) events are millennial-scale climate oscillations that occurred throughout the last glacial period. A DO event is characterised by a rapid increase in temperature (10-16°C in high Northern latitudes) over decadal timescales, followed by cooling over a longer period of centuries to millennia. To date, there is little consensus on the precise mechanism that causes these mysterious oscillations. Our focus is on how ice shelves and sea ice, each with their intrinsic timescales, interact with ocean circulation to explain DO events. We extend the conceptual model presented by Boers et al. [2018] by replicating sea ice trajectories using transitions through bifurcations. The goal is to detect early warning signals for DO events in model simulations after introducing noise into the system. Speculation suggests that a rapidly changing climate subsystem undergoing a bifurcation may be responsible for the early warning signals observed in high-frequency bands prior to DO events. Through a combination of fold bifurcations, we are able to produce oscillations in North Atlantic sea ice coverage that emulates the two-stage cooling process observed in temperature proxy data.

Under current emission trajectories, at least temporarily overshooting the Paris global warming limit of 1.5 °C above pre-industrial levels is a distinct possibility. Permanently exceeding this limit would substantially increase the risks of triggering several climate tipping elements with associated high-end impacts on human societies and the Earth system. It is essential to assess this risk under emission pathways that temporarily overshoot 1.5 °C. Here, we investigate the tipping risks associated with a number of policy-relevant future emission scenarios, using a stylised Earth system model that comprises four interconnected core tipping elements. We find that tipping risks until 2300 increase by 5 % for every tenth of a degree above 1.5 °C if global warming levels do not return to 1.5 °C or lower by the end of this century. If overshoots above 1.5 °C are limited in both peak temperature and duration (return below 1.5 °C in 2100), tipping risks until 2300 can be kept below 3 %. We show that achieving and maintaining at least net-zero greenhouse gas emissions is paramount to minimise tipping risks on multi-century timescales. Our results underscore that stringent emission reductions in the current decade in line with the Paris Agreement 1.5 °C limit are critical for planetary stability.

We present a structure-preserving learning method for gradient systems, including Hamiltonian systems and port-Hamiltonian systems, with Gaussian processes. Firstly, we establish the reproducing properties of differentiable kernels on unbounded sets, which enables us to embed the RKHS to $C^1_b$ space. Then we are able to formulate the learning problem as a statistical inverse learning problem and provide operator-theoretic framework to show that the Gaussian posterior coincides to minimizer of an empirical optimization problem. Finally, the convergence analysis and error bounds with respect to the RKHS norm for the Gaussian posterior are obtained.

When studying chaotic non-autonomous systems, the traditional "one-trajectory" method becomes unreliable, since the dynamics changes in every instant. This has been well understood in climate dynamics, where numerous studies in recent decades have been focusing on analysing an ensemble of trajectories (parallel climate realizations), leading to the construction of the so-called snapshot attractor of the climate. In the theory of low-dimensional non-autonomous chaotic dissipative systems, this refers to a time-dependent object in phase space which all trajectories converge to; a concept that has been well worked out and understood over the decades. However, the same cannot be said about Hamiltonian systems of the same kind, about which vitrually no literature existed before, despite such systems having a number of useful applications. In my talk, I present some of the concepts we developed over the past years to describe chaos in these systems. One of the most important of these concepts is that of the snapshot torus. This is the generalisation of KAM tori of autonomous Hamiltonian systems, its shape being time-dependent, and its dynamics having a rate-induced transition from regular to chaotic through a break-up process. In the same manner, we can generalise chaotic seas, obtaining snapshot chaotic seas of time-dependent shape. When investigating the dynamical instability, we introduce an instantaneous Lyapunov exponent, which can be obtained through a quantity called ensemble-averaged pairwise distance (EAPD). Besides these, we generalise a number of other important concepts, such as elliptic and hyperbolic fixed points, stable and unstable manifolds, as well as non-hyperbolic zones (so-called sticky regions). Our results already found their application is low-dimensional systems related to fusion plasma physics and celestial mechanics, as well as climate models of intermediate complexity.

Community detection in time-evolving networks is a longstanding problem in network theory. In particular, the graph supra-Laplacian of the corresponding multiplex network has been used to characterise clusters or communities. In this work we port the inflated dynamic Laplacian constructed by Froyland and Koltai to graphs, which was used recently in the continuous domain to discover the birth and death of coherent sets. We look at the eigenproblem of the graph inflated dynamic Laplacian and present results on community detection and discovering balanced graph-cuts, motivated by classical isoperimetric problems. We also demonstrate our method on space-time graphs with different types of community transitions.

The coexistence of multiple limit cycles occurs in many nonlinear dynamical systems. Oscillators that exhibit the coexistence of two stable limit cycles separated by an unstable limit cycle are commonly known as birhythmic oscillators, examples include the modified van der Pol oscillator and the Decroly-Goldbeter glycolysis model. A sudden transition or tipping from one limit cycle to another limit cycle has rarely been studied. In this work, we shall present rate-dependent tipping in birhythmic oscillators, i.e. rate-dependent tipping from one stable limit cycle to another stable limit cycle. We employ the concept of basin instability to determine the parameter path pertaining to the observed rate-dependent tipping.

A discrete Hamiltonian mapping described by the variables action $I$ and angle $\theta$ is considered and the diffusion of the particles in the chaotic sea is studied using the diffusion equation, where we can observe a scaling invariance. The scaling invariance is a signature that the system is passing through a phase transition. We investigate a set of four questions that characterize a phase transition: (i) identify the broken symmetry; (ii) define the order parameter; (iii) identify what are the elementary excitations and; (iv) detect the topological defects which impact the transport of the particles.

When considering future warming scenarios of the Earth’s climate and the potential critical transitions that might occur because of them, it is important to investigate not just the amplitude of the warming but also its rate. In certain systems, a critical transition may occur for lower thresholds when the rate of parameter increase is faster - a phenomenon known as rate-induced tipping. We investigate the rate-induced tipping of the Greenland ice sheet under warming scenarios using the ice sheet model Yelmo. Unlike the west Antarctic ice sheet, which is expected to display rate-induced tipping due to interplay of critical feedbacks such as the marine ice sheet instability and glacial isostatic adjustment, in Greenland these feedbacks are much less prevalent and we do not expect to see a strong rate-induced effect. However, for a range of warming rates and amplitudes, we see highly non-monotonic tipping behaviour. Not only do certain lower rates tip where faster ones do not, but the time it takes to tip also does not depend strictly on warming amplitude.

The phase-isostable description, or augmented phase reduction, is a powerful tool for studying limit-cycle dynamics, as it considers amplitude dynamics. Each limit-cycle system has an unambiguous phase-isostable representation valid in the basin of the limit cycle, but finding the appropriate transformation is almost always a numerically demanding task. However, the reverse approach of choosing a transformation from phase-isostable coordinates to some observation coordinates allows for the exact design of a dynamical system with preferred properties of the limit cycle. These include frequency, stability, signal form, and response curves. We demonstrate this method on a two-dimensional system designing a generalized Stuart-Landau model. Furthermore, we present a new approach to using augmented phase reduction to derive higher terms in the phase reduction of coupled two-dimensional limit-cycle oscillators. We demonstrate the method for the well-known Stuart-Landau model and use them to determine the borders of Arnold's tongue in the second order.

Nonlinear dynamical systems often exhibit many stable coexistent solutions with different qualitative features. Trivially, the convergence to one of the available attractors depends solely on the system's initial configuration. However, when subject to parameter drifts, this scenario is altered, giving a central role to unstable invariant sets with a typical escape time rather than stable ones. Here, we show how parameter drifts can mitigate multistability, demonstrating the required timescale differences between the drifts and the escape time of the unstable invariant sets for it to occur.

Consider a synchronized network where each unit presents only a periodic attractor with a chaotic transient. Depending on the instant that a perturbation is applied, we observe two possible network long-term states: (i) The network neutralizes the perturbation effects and returns to its synchronized configuration. (ii) The perturbation leads the network to an alternative desynchronized state. We show that this time-dependent vulnerability of synchronized state is due to the existence of a fractal set of initial conditions conducing the dynamic to a chaotic set in which trajectories persist for times indefinitely long. We argue that this phenomenon is general and illustrate with a complex network composed of electronic circuits.

Does an ecological community allow stable coexistence? In particular, what is the interplay between stability of coexistence and the network of competitive, mutualistic, or predator-prey interactions between the species of the community? These are fundamental questions of theoretical ecology, yet meaningful analytical progress is in most cases impossible beyond two-species communities. In this talk, I will therefore show how we addressed this problem statistically: For all non-trivial networks of interaction types of N ⩽ 5 species, we sampled Lotka–Volterra model parameters randomly and thus computed the probability of steady-state coexistence being stable and feasible with Lotka–Volterra dynamics. Surprisingly, our analysis reveals "impossible ecologies", very rare non-trivial networks of interaction types that do not allow stable and feasible steady-state coexistence. I will classify these impossible ecologies, and then prove, somewhat conversely, that any non-trivial ecology that has a possible subecology is itself possible. This theorem highlights the "irreducible ecologies" that allow stable and feasible steady-state coexistence, but do not contain a possible subecology. I will conclude by showing the classification of all irreducible ecologies of N ⩽ 5 species. Strikingly, this indicates that the proportion of non-trivial ecologies that are irreducible decreases exponentially with the number N of species. Our results thus suggest that interaction networks and stability of coexistence are linked crucially by the very small subset of ecologies that are irreducible.

Competition between macroalgae and corals for occupying the available space in sea bed is an important ecological process underlying coral-reef dynamic. We investigate coral-macroalgal phase shift in presence of macroalgal allelopathy and microbial infection on corals by means of an eco-epidemiological model under the assumption that the transmission of infection occurs through both contagious and non-contagious pathways. We found that the system is capable of exhibiting the existence of two stable configurations by saddle-node bifurcations.

Amazon forest dieback is seen as a potential tipping point under climate change. These concerns are partly based-on an early coupled climate-carbon cycle simulation that produced unusually strong drying and warming in Amazonia. In contrast, the 5th generation Earth System Models (CMIP5) produced few examples of Amazon dieback under climate change. This presentation examines the results from seven 6th generation models (CMIP6) which include interactive vegetation carbon, and in some cases interactive forest fires. Although these models typically project increases in area-mean forest carbon across Amazonia under CO2-induced climate change, five of the seven models also produce abrupt reductions in vegetation carbon which indicate localised dieback events. The Northern South America region (NSA), which contains most of the rainforest, is especially vulnerable in the models. These dieback events, some of which are mediated by fire, are preceded by an increase in the amplitude of the seasonal cycle in near surface temperature, which is consistent with more extreme dry seasons. Based on the ensemble mean of the detected dieback events we estimate that 7+/-5% of the NSA region will experience abrupt downward shifts in vegetation carbon for every degree of global warming past 1.5°C.

Many real world time series exhibit both significant short and long range temporal correlations. Such correlations enhance the errors of linear trend analysis. We provide a general framework for trend analysis under the consideration of such correlations. We derive analytical closed form results for the error bars of the least squares estimate of the trend for such time series highlighting the different effects of short and of long range correlations. We apply this framework to the study of warming trends on gridded temperature data in central Europe. We make use of the redundancy of the trend signal in adjacent grid points using methods of spatial averaging and the first principal component of Empirical Orthogonal Function analysis. We find a statistically significant decadal warming trend in Central Europe over the past 70 years, which shows a particularly dramatic increase over the past 20 years.

The investigation of synchronous patterns and their emergence in neuronal networks remains a subject of great interest in the field of neuroscience. In light of this, we have conducted a study using two neuronal networks composed of excitatory and inhibitory neurons. Our primary aim is to explore the association between specific interconnections among these networks and the manifestation of phase, anti-phase, and shift-phase synchronization. In our simulation, the identified connectivities from a sender-to-receiver network play a crucial role in the emergence of diverse synchronous patterns. Additionally, we investigated whether the synchronization level of the sender network impacts the synchronization in the receiver network. The results were compelling, revealing that a highly synchronous sender network has a greater influence in inducing various types of synchronization in the receiver network. Furthermore, we discovered that these connectivities affect the mean velocity of the resulting phase angle of the receiver network. Depending on the specific connectivities involved, the connectivities could either increase or decrease the average velocity. These findings contribute significantly to our understanding of how specific conductivities between neuronal networks can contribute to the emergence of different neuronal firings. In summary, our study sheds light on the intricate mechanisms underlying the emergence of distinct synchronous patterns in neuronal networks, emphasizing the crucial role played by specific connectivities. These results expand our comprehension of the complex dynamics observed in inter-network communication within the brain. P. R. Protachevicz, M. Hansen, K. C. Iarosz, I. L. Caldas, A. M. Batista, J. Kurths. Emergence of neuronal synchronization in coupled areas. Frontiers in Computational Neuroscience, 2021. https://doi.org/10.3389/fncom.2021.663408

We introduce a novel notion of physical measure on chaotic saddles in terms of Lebesgue measure. We explain how to approximate this measure by ”uniform sprinkling” according to the background measure, and we use the central limit theorem to find the convergence rate of the approximation. In an example, we compute the measure, and we use a formula from Sweet and Ott’s work from 2000 to compute its Information dimension from the finite-time Lyapunov exponents. We find it to be very close to the box-counting dimension of the approximated saddle.

Neural activity typically follows a series of transitions between well-defined states, in a regime generally called metastability. In this work, we review current observations and formulations of metastability to argue that they have been largely context-dependent, and a unified framework is still missing. To address this, we propose a context-independent framework that unifies the context-dependent formulations by defining metastability as an umbrella term encompassing regimes with transient but long-lived states. This definition can be applied directly to experimental data but also connects neatly to the theory of nonlinear dynamical systems, which allows us to extract a general dynamical principle for metastability: the coexistence of attracting and repelling directions in phase space. With this, we extend known mechanisms and propose new ones that can implement metastability through this general dynamical principle. We believe that our framework is an important advancement towards a better understanding of metastability in the brain, and can facilitate the development of tools to predict and control the brain’s behavior.

Studying the complex-network representation of time series has proven to be useful in many areas of dynamical systems theory but also in experimental time series analysis. Using symbolization, one can preserve dynamic information from the time series and encode it into a coarse- grained model of the system which is often easier to store and analyze than the original. Here we present a novel numerical toolkit we developed for analyzing so-called state-transition networks constructed from time series. Our Julia package, called StateTransitionNetworks.jl, implements already established methods combined with new approaches to create the symbolized version of the time series using grid-based discretization of the Poincaré surface of section. Transitions between states are counted and stored in weighted, directed networks as edge weights. Defining random walk processes on the network allows for computing simple complexity measures such as the mean and variance of total path lengths that, similarly to the largest Lyapunov exponent, can characterize the dynamics. Furthermore, the variance can signalize critical transitions analogously to the specific heat in the thermodynamic formalism. Tests on prototypical dynamical systems, the effect of user-defined parameters, and applications to experimental data are also outlined.

Over the last decade, many potential tipping elements have been identified in the climate system. However, some of these tipping elements are surrounded by large uncertainties. To address this, we do an updated analysis of abrupt changes in state-of-the-art climate models. We examine all CMIP6 models under the 1pctCO2 scenario using a Canny edge detection method — adapted for spatiotemporal dimensions — to detect abrupt shifts in climate data. We perform an automatic analysis of many two-dimensional variables of the ocean, atmosphere and land. We report statistics on number of abrupt changes detected, size (surface area) of abrupt changes, and critical global mean temperature at which these abrupt changes occur.

Spatial self-organization of different types of vegetation is observed in mesic savannas. This pattern formation is key to understanding ecosystem response to changing climate. We developed a mesic savanna vegetation model to gain more understanding into these patterns. The model distinguishes itself from others as we take into account spatial effects from the head start. Indeed, when considering tipping phenomena, the role of spatial effects should not be overlooked. Model analysis reveals that recruitment delay of savanna trees induced by fire is one of the driving forces for the appearance of Turing patterns. We show mathematically that without this mechanism these types of patterns cannot occur. Furthermore, by means of simulation, we show that a Turing bifurcation in the full model can lead to stable Turing patterns, resembling the spatial self-organization of vegetation in mesic savannas. Moreover, we observed the occurrence of transient Turing patterns, where Turing patterns form the pathway of tipping from one steady state to another. This seems to strengthen a common rationale regarding vegetation patterns: the sudden occurrence of patterned vegetation is an early-warning sign for a drastic change of the system as a whole. The interpretation of patterned vegetation can also be shed in a different light. In fact, a state may undergo tipping in the corresponding non-spatial model, while our spatially model exhibits Turing patterns that remain unaffected under the parameter change that drives the collapse in the reduced model – a general phenomenon we call “Turing before tipping”. We show that this mechanism can occur in our model. Thus, also in savanna ecosystems, patterning of vegetation may strengthen the resilience of ecosystems evading critical transitions by spatial interactions.

Coherent sets are regions of the phase space in a dynamical system which resist mixing with the surrounding space. Typical applications come from complex physical systems such as atmospheric flows and ocean dynamics. They are determined by transport barriers which can be computed via level sets of eigenfunctions of the so-called dynamic Laplacian introduced by G. Froyland. Recently, Froyland and P. Koltai developed an extension of this theory and introduced the inflated dynamic Laplacian which detects emergence and decay of finite-time coherent sets via eigenfunctions of a Laplace-Beltrami operator in a time-expanded manifold. Towards efficient computation of these eigenfunctions, we consider a Trotter-product approximation of the operator semigroup associated with the inflated dynamic Laplacian.

Many complex systems are a combination of deterministic and stochastic dynamics. Some systems can be understood by applying random perturbation theory to the underlying deterministic dynamics. However, generally the randomness may not necessarily be small and could consist of any type of stochasticity. Here, we consider a simplest case where only two types (i.e., dichotomic) of dynamics are generated randomly in time, and explore in (1) a piecewise-linear random system where either an expanding or contracting map is selected at each time step, and (2) a system of coupled rotors kicked with temporally correlated random sequences. In both examples, we illustrate analytically and numerically their rich dynamical behaviours.

Transfer operators tracking the movement of probability densities in the phase space are at the basis of the statistical investigation of nonlinear chaotic systems. Correlation and response functions of chaotic, possibly noisy, systems, can be quite generally decomposed as a sum of exponentially decaying terms, with a rate given by the so called Ruelle-Pollicott (RP) resonances. The RP resonances are associated with the spectrum of the transfer operator acting on highly non trivial functional spaces that take into account the rate of expansion and contraction typical of the dynamics on chaotic attractors. We here provide some preliminary results on the investigation of mixing properties of chaotic systems by adopting an alternative perspective based on the theory of the Koopman operator. Such operator determines the time evolution of general observables of dynamical systems and is at the basis of sophisticated linear representations and dimension reduction methodologies for highly nonlinear systems. However, there is little cross-pollination between the Transfer Operator and Koopman operator communities and Koopman methods have not been extensively applied to the investigation of mixing properties of chaotic systems. We provide numerical evidence that the Extended Dynamic Mode Decomposition algorithm, a powerful method developed to approximate the action of the Koopman operator on selected classes of observables, is able to capture the RP resonances and their relative Koopman eigenfunctions from one single run of the chaotic dynamics. Moreover, by employing a decomposition of general observables in terms of the set of Koopman eigenfunctions, we show that we are able to reconstruct correlation functions between (virtually) any observable of interest.