Transient chimera states emerging from dynamical trapping in chaotic saddles.

Nonlinear systems possessing nonattracting chaotic sets, such as chaotic saddles, embedded in their state space may oscillate chaotically for a transient time before eventually transitioning into some stable attractor. We show that these systems, when networked with nonlocal coupling in a ring, are capable of forming chimera states, in which one subset of the units oscillates periodically in a synchronized state forming the coherent domain, while the complementary subset oscillates chaotically in the neighborhood of the chaotic saddle constituting the incoherent domain. We find two distinct transient chimera states distinguished by their abrupt or gradual termination. We analyze the lifetime of both chimera states, unraveling their dependence on coupling range and size. We find an optimal value for the coupling range yielding the longest lifetime for the chimera states. Moreover, we implement transversal stability analysis to demonstrate that the synchronized state is asymptotically stable for network configurations studied here.

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.

Bifurcations in rotating spherical shell convection under the influence of differential rotation.

The bifurcations of thermal convection in a rotating spherical shell heated from the inner sphere and driven by the buoyancy of a central gravity field are studied numerically. This model of spherical Rayleigh-Bénard convection describes large-scale convection in planets and in the outer zones of celestial bodies. In this work, the influence of an additionally imposed differential rotation of the inner sphere with respect to the outer one on the heat transfer and, more generally, on the whole bifurcation structure is investigated. In addition to numerical simulations, path-following techniques are applied in order to compute both stable and unstable solution branches. The dynamics and the heat transfer are essentially determined by a global bifurcation, which we have identified as a homoclinic bifurcation that consists of a collision of a stable modulated rotating with an unstable rotating wave.

Improved Mass Accuracy and Isotope Confirmation through Alignment of Ultrahigh-Resolution Mass Spectra of Complex Natural Mixtures.

Fourier-transform ion cyclotron resonance mass spectrometry (FT-ICR-MS) is one of the state-of-the-art methods to analyze complex natural organic mixtures. The precision of detected masses is crucial for molecular formula attribution. Random errors can be reduced by averaging multiple measurements of the same mass, but because of limited availability of ultrahigh-resolution mass spectrometers, most studies cannot afford analyzing each sample multiple times. Here we show that random errors can be eliminated also by averaging mass spectral data from independent environmental samples. By averaging the spectra of 30 samples analyzed on our 15 T instrument we reach a mass precision comparable to a single spectrum of a 21 T instrument. We also show that it is possible to accurately and reproducibly determine isotope ratios with FT-ICR-MS. Intensity ratios of isotopologues were improved to a degree that measured deviations were within the range of natural isotope fractionation effects. In analogy to δ13C in environmental studies, we propose Δ13C as an analytical measure for isotope ratio deviances instead of widely employed C deviances. In conclusion, here we present a simple tool, extensible to Orbitrap-based mass spectrometers, for postdetection data processing that significantly improves mass accuracy and the precision of intensity ratios of isotopologues at no extra cost.

ICBM-OCEAN: Processing Ultrahigh-Resolution Mass Spectrometry Data of Complex Molecular Mixtures.

Untargeted molecular analyses of complex mixtures are relevant for many fields of research, including geochemistry, pharmacology, and medicine. Ultrahigh-resolution mass spectrometry is one of the most powerful tools in this context. The availability of open scripts and online tools for specific data processing steps such as noise removal or molecular formula assignment is growing, but an integrative tool where all crucial steps are reproducibly evaluated and documented is lacking. We developed a novel, server-based tool (ICBM-OCEAN, Institute for Chemistry and Biology of the Marine Environment, Oldenburg-complex molecular mixtures, evaluation & analysis) that integrates published and novel approaches for standardized processing of ultrahigh-resolution mass spectrometry data of complex molecular mixtures. Different from published approaches, we offer diagnostic and validation tools for all relevant steps. Among other features, we included objective and reproducible reduction of noise and systematic errors, spectra recalibration and alignment, and identification of likeliest molecular formulas. With 15 chemical elements, the tool offers high flexibility in formula attribution. Alignment of mass spectra among different samples prior to molecular formula assignment improves mass error and facilitates molecular formula confirmation with the help of isotopologues. The online tool and the detailed instruction manual are freely accessible at www.icbm.de/icbm-ocean.

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

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

When very slow is too fast - collapse of a predator-prey system.

Critical transitions or regime shifts are sudden and unexpected changes in the state of an ecosystem, that are usually associated with dangerous levels of environmental change. However, recent studies show that critical transitions can also be triggered by dangerous rates of environmental change. In contrast to classical regime shifts, such rate-induced critical transitions do not involve any obvious loss of stability, or a bifurcation, and thus cannot be explained by the linear stability analysis. In this work, we demonstrate that the well-known Rosenzweig-MacArthur predator-prey model can undergo a rate-induced critical transition in response to a continuous decline in the habitat quality, resulting in a collapse of the predator and prey populations. Rather surprisingly, the collapse occurs even if the environmental change is slower than the slowest process in the model. To explain this counterintuitive phenomenon, we combine methods from geometric singular perturbation theory with the concept of a moving equilibrium, and study critical rates of environmental change with dependence on the initial state and the system parameters. Moreover, for a fixed rate of environmental change, we determine the set of initial states that undergo a rate-induced population collapse. Our results suggest that ecosystems may be more sensitive to how fast environmental conditions change than previously assumed. In particular, unexpected critical transitions with dramatic ecological consequences can be triggered by environmental changes that (i) do not exceed any dangerous levels, and (ii) are slower than the natural timescales of the ecosystem. This poses an interesting research question whether regime shifts observed in the natural world are predominantly rate-induced or bifurcation-induced.

Riddled basins of attraction in systems exhibiting extreme events.

Using a system of two FitzHugh-Nagumo units, we demonstrate the occurrence of riddled basins of attraction in delay-coupled systems as the coupling between the units is increased. We characterize riddled basins using the uncertainty exponent which is a measure of the dimensions of the basin boundary. Additionally, we show that the phase space can be partitioned into pure and mixed regions, where initial conditions in the pure regions certainly avoid the generation of extreme events, while initial conditions in the mixed region may or may not exhibit such events. This implies that any tiny perturbation of initial conditions in the mixed region could yield the emergence of extreme events because the latter state possesses a riddled basin of attraction.

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

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

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

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

Complexity and irreducibility of dynamics on networks of networks.

We study numerically the dynamics of a network of all-to-all-coupled, identical sub-networks consisting of diffusively coupled, non-identical FitzHugh-Nagumo oscillators. For a large range of within- and between-network couplings, the network exhibits a variety of dynamical behaviors, previously described for single, uncoupled networks. We identify a region in parameter space in which the interplay of within- and between-network couplings allows for a richer dynamical behavior than can be observed for a single sub-network. Adjoining this atypical region, our network of networks exhibits transitions to multistability. We elucidate bifurcations governing the transitions between the various dynamics when crossing this region and discuss how varying the couplings affects the effective structure of our network of networks. Our findings indicate that reducing a network of networks to a single (but bigger) network might not be accurate enough to properly understand the complexity of its dynamics.

The impact of a predator on the outcome of competition in the three-trophic food web.

We study the effects of predation on the competition of prey populations for two resources in a chemostat. We investigate a variety of small food web compositions: the bi-trophic food web (two resources-two competing prey) and the three-trophic food web (two resources-two prey-generalist predator) comparing different model formulations: substitutable resources and essential resources, namely Liebig's minimum law model (perfect essential resources) and complementary resources formulations. The prediction of the outcome of competition is solely based on bifurcation analysis in which the inflow of resources into the chemostat is used as the bifurcation parameter. We show that the results for different bi-trophic food webs are very similar, as only equilibria are involved in the long-term dynamics. In the three-trophic food web, the outcome of competition is manifested largely by non-equilibrium dynamics, i.e., in oscillatory behavior. The emergence of predator-prey cycles leads to strong deviations between the predictions of the outcome of competition based on Liebig's minimum law and the complementary resources. We show that the complementary resources formulation yields a stabilization of the three-trophic food web by decreasing the existence interval of oscillations. Furthermore, we find an exchange of a region of oscillatory co-existence of all three species in Liebig's formulation by a region of bistability of two limit cycles containing only one prey and the predator in the complementary formulation.

Desertification by front propagation?

Understanding how desertification takes place in different ecosystems is an important step in attempting to forecast and prevent such transitions. Dryland ecosystems often exhibit patchy vegetation, which has been shown to be an important factor on the possible regime shifts that occur in arid regions in several model studies. In particular, both gradual shifts that occur by front propagation, and abrupt shifts where patches of vegetation vanish at once, are a possibility in dryland ecosystems due to their emergent spatial heterogeneity. However, recent theoretical work has suggested that the final step of desertification - the transition from spotted vegetation to bare soil - occurs only as an abrupt shift, but the generality of this result, and its underlying origin, remain unclear. We investigate two models that detail the dynamics of dryland vegetation using a markedly different functional structure, and find that in both models the final step of desertification can only be abrupt. Using a careful numerical analysis, we show that this behavior is associated with the disappearance of confined spot-pattern domains as stationary states, and identify the mathematical origin of this behavior. Our findings show that a gradual desertification to bare soil due to a front propagation process can not occur in these and similar models, and opens the question of whether these dynamics can take place in nature.

Edge anisotropy and the geometric perspective on flow networks.

Spatial networks have recently attracted great interest in various fields of research. While the traditional network-theoretic viewpoint is commonly restricted to their topological characteristics (often disregarding the existing spatial constraints), this work takes a geometric perspective, which considers vertices and edges as objects in a metric space and quantifies the corresponding spatial distribution and alignment. For this purpose, we introduce the concept of edge anisotropy and define a class of measures characterizing the spatial directedness of connections. Specifically, we demonstrate that the local anisotropy of edges incident to a given vertex provides useful information about the local geometry of geophysical flows based on networks constructed from spatio-temporal data, which is complementary to topological characteristics of the same flow networks. Taken both structural and geometric viewpoints together can thus assist the identification of underlying flow structures from observations of scalar variables.

Extreme multistability: Attractor manipulation and robustness.

The coexistence of infinitely many attractors is called extreme multistability in dynamical systems. In coupled systems, this phenomenon is closely related to partial synchrony and characterized by the emergence of a conserved quantity. We propose a general design of coupling that leads to partial synchronization, which may be a partial complete synchronization or partial antisynchronization and even a mixed state of complete synchronization and antisynchronization in two coupled systems and, thereby reveal the emergence of extreme multistability. The proposed design of coupling has wider options and allows amplification or attenuation of the amplitude of the attractors whenever it is necessary. We demonstrate that this phenomenon is robust to parameter mismatch of the coupled oscillators.

Local control for the collective dynamics of self-propelled particles.

Utilizing a paradigmatic model for the motion of interacting self-propelled particles, we demonstrate that local accelerations at the level of individual particles can drive transitions between different collective dynamics, leading to a control process. We find that the ability to trigger such transitions is hierarchically distributed among the particles and can form distinctive spatial patterns within the collective. Chaotic dynamics occur during the transitions, which can be attributed to fractal basin boundaries mediating the control process. The particle hierarchies described in this paper offer decentralized capabilities for controlling artificial swarms.

Temperature-dependent stochastic dynamics of the Huber-Braun neuron model.

The response of a four-dimensional mammalian cold receptor model to different implementations of noise is studied across a wide temperature range. It is observed that for noisy activation kinetics, the parameter range decomposes into two regions in which the system reacts qualitatively completely different to small perturbations through noise, and these regions are separated by a homoclinic bifurcation. Noise implemented as an additional current yields a substantially different system response at low temperature values, while the response at high temperatures is comparable to activation-kinetic noise. We elucidate how this phenomenon can be understood in terms of state space dynamics and gives quantitative results on the statistics of interspike interval distributions across the relevant parameter range.

Noise-induced precursors of tonic-to-bursting transitions in hypothalamic neurons and in a conductance-based model.

The dynamics of neurons is characterized by a variety of different spiking patterns in response to external stimuli. One of the most important transitions in neuronal response patterns is the transition from tonic firing to burst discharges, i.e., when the neuronal activity changes from single spikes to the grouping of spikes. An increased number of interspike-interval sequences of specific temporal correlations was detected in anticipation of temperature induced tonic-to-bursting transitions in both, experimental impulse recordings from hypothalamic brain slices and numerical simulations of a stochastic model. Analysis of the modelling data elucidates that the appearance of such patterns can be related to particular system dynamics in the vicinity of the period-doubling bifurcation. It leads to a nonlinear response on de- and hyperpolarizing perturbations introduced by noise. This explains why such particular patterns can be found as reliable precursors of the neurons' transition to burst discharges.

Asymmetry-induced order in multilayer networks.

Symmetries naturally occur in real-world networks and can significantly influence the observed dynamics. For instance, many synchronization patterns result from the underlying network symmetries, and high symmetries are known to increase the stability of synchronization. Yet here we find that general macroscopic features of network solutions such as regularity can be induced by breaking their symmetry of interactions. We demonstrate this effect in an ecological multilayer network where the topological asymmetries occur naturally. These asymmetries rescue the system from chaotic oscillations by establishing stable periodic orbits and equilibria. We call this phenomenon asymmetry-induced order and uncover its mechanism by analyzing both analytically and numerically the absence of dynamics on the system's synchronization manifold. Moreover, the bifurcation scenario describing the route from chaos to order is also disclosed. We demonstrate that this result also holds for generic node dynamics by analyzing coupled paradigmatic Rössler and Lorenz systems.

Food quality in producer-grazer models: a generalized analysis.

Stoichiometric constraints play a role in the dynamics of natural populations but are not explicitly considered in most mathematical models. Recent theoretical works suggest that these constraints can have a significant impact and should not be neglected. However, it is not yet resolved how stoichiometry should be integrated in population dynamical models, as different modeling approaches are found to yield qualitatively different results. Here we investigate a unifying framework that reveals the differences and commonalities between previously proposed models for producer-grazer systems. Our analysis reveals that stoichiometric constraints affect the dynamics mainly by increasing the intraspecific competition between producers and by introducing a variable biomass conversion efficiency. The intraspecific competition has a strongly stabilizing effect on the system, whereas the variable conversion efficiency resulting from a variable food quality is the main determinant for the nature of the instability once destabilization occurs. Only if the food quality is high can an oscillatory instability, as in the classical paradox of enrichment, occur. While the generalized model reveals that the generic insights remain valid in a large class of models, we show that other details such as the specific sequence of bifurcations encountered in enrichment scenarios can depend sensitively on assumptions made in modeling stoichiometric constraints.