Multistability and Tipping: From Mathematics and Physics to Climate and Brain

Ashwin et al [1] have suggested a definition of R-tipping appropriate for parameter shift non-autonomous systems by using pullback attractors which limit to equilibrium points backward in time. This study aims to generalize that definition in order to cover wide range of invariant sets. We considered the pullback attractors that can limit to any invariant set, backward in time, and we used the concept of upper limit of sets to come up with more general definition. The new definition will allow us to study the phenomena of R-tipping that may occur when an invariant set, such as periodic orbit, is moved as an effect of the parameter shift. [1] P. Ashwin, C. Perryman and S. Wieczorek. Parameter shift for nonautonomous systems in low dimension: bifurcation- and rate- induced tipping. arXiv:1506.07734.

The microscopic principles organizing dynamic units in complex networks—from proteins to power generators—can be understood in terms of network ‘motifs’: small interconnection patterns that appear much more frequently in real networks than expected in random networks. When considered as small subgraphs isolated from a large network, these motifs are more robust to parameter variations, easier to synchronize than other possible subgraphs, and can provide specific functionalities. But one can isolate these subgraphs only by assuming, for example, a significant separation of timescales, and the origin of network motifs and their functionalities when embedded in larger networks remain unclear. In this talk, I will show how most motifs emerge from interconnection patterns that best exploit the intrinsic stability characteristics at different scales of interconnection, from simple nodes to whole modules. This functionality suggests an efficient mechanism to stably build complex systems by recursively interconnecting nodes and modules as motifs. Joint work with Yang-Yu Liu (Harvard) and Jean-Jacques Slotine (MIT).

A problem of the analysis and prevention of catastrophic shifts in ecosystems with stochastic environment is studied. Mathematically, ecological shifts can be explained by the presence of coexisting attractors. Some of these attractors corresponding to extinction or population explosion are undesirable. A constructive method for the prediction of noise-induced transitions between basins of attraction is suggested and applied to some forced predator-prey models. To provide a stable coexistence of both species and prevent catastrophic noise-induced transitions, a control technique is suggested.

We explore and experimentally demonstrate the transition between different synchronization scenarios and amplitude death in coupled time-delay systems having {\em intrinsic time-delays}, that are coupled through linear, bidirectional coupling scheme. We identify novel synchronization transition scenarios comprising of generalized lag (GLS), generalized anticipatory (GAS), complete (CS) synchronization and amplitude death (AD) in the chaotic and hyperchaotic regime. The system is characterized under the variation of {\em intrinsic time-delays} and the {\em coupling strength}. Upon the variation of the intrinsic time-delay the system encountered the transitions among different generalized (complete) synchronization and AD. The effect of variation of the coupling strength produced the transitions among unsynchronized states (NS) to AD via generalized synchronized states for some range of values of the intrinsic delays. The transition mediated from the variation of intrinsic time-delays and coupling strength {\em has no analogue in non-delayed systems or oscillators coupled through delay coupling scheme}. We employ the Krasovskii-Lyapunov theorem to estimate the parametric zone for the onset of GAS, GLS and CS. A suitable approximate linear stability analysis is considered for the occurrence of AD. The GAS and GLS is supported in quantitative way by means of {\em modified similarity function} and {\em time averaged Euclidean distance}. All the proposed scenarios have been realized in electronic circuit experiment. The analytical, numerical and experimental results are in well agreement.

In this work we present an approach to generate multi-scroll chaotic attractors based on piecewise linear systems. This class of systems is constructed with a switching control law by changing the equilibrium point of an unstable dissipative system. The switching control law that governs the position of the equilibrium point changes according to the number of scrolls that is displayed in the attractor. Switched systems have been widely used in many different areas in science and engineering. There is some interest in generating chaotic or hyperchaotic attractors with multiple scroll with this kind of systems. We present a generalized theory that is capable of explaining different approaches as saturation, threshold and step functions in $R^3$. This class of systems is constructed with unstable dissipative systems (UDS) and a control law to display various multi-scroll strange attractors. Interesting phenomena have been observed with these multi-scroll strange attractors, for example, multistability via master-slave synchronization, generation of deterministic Brownian motion, chaotic attractors without equilibria.

There are several mature topics in nonlinear science. The study of chaotic maps is one of these, where many considerable results have opened paths on chaos theory and impacted interdisciplinary areas. In this work we introduce a family of multimodal logistic maps with a single parameter. The maps domain is partitioned in subdomains according to the maximal number of modals to be generated and each subdomain contains one logistic map. Bifurcation diagrams and basins of attraction of fixed points are constructed for the family of chaotic logistic maps. The key ideas for the study of chaos can be understood by analyzing one-dimensional (1D) maps. As a consequence of maturity, there is an active new area regarding applications on technology, such as chaotic cryptography. we present a pseudo-random Bit Generator via unidimensional multi-modal discrete dynamical systems called k-modal maps. These multimodal maps are useful to generate pseudo-random sequences with longer period, i.e., in order to attend the problem of periodicity. In addition the pseudo-random sequences generated via multi-modal maps are evaluated with the statistical suite of test from NIST and satisfactory results are obtained when they are used as key stream. Furthermore, we show the impact of using these sequences in a stream cipher resulting in a better encryption quality correlated with the number of modals of the chaotic map.

Arindam Mishra1,2, Suman Saha1, Pinaki pal3, Hilda Cerdeira4, Syamal K. Dana1* 1 CSIR-Indian Institute of Chemical Biology, Jadavpur, Kolkata 700032, India 2Department of Physics, Jadavpur University, Kolkata 700032, India Department of Mathematics, National Institute of Technology, Durgapur, India 3Instituto de Física Teórica, UNESP, Universidade Estadual Paulista, 01140-070 São Paulo, Brazil *e-mail: syamaldana@gmail.com Extreme events such as tsunami, earthquake, flood, power blackout have devastating effect on life and society. Prediction of extreme events is thus a necessity to take a priori measures to mitigate disasters. It is basically defined as a large size event that deviates largely from the events of nominal size. Statistically more accurate criterion for detecting extreme events from a measured time series is available in the literature. For prediction of extreme events, its dynamic origin has to be understood. For this purpose, extreme events have been studied in simple dynamical systems including experiments in lasers and electronic circuits. Most of the extreme events are rare and reflected as long-tail non-Gaussian distribution of event sizes, small to very large and, thus become difficult to predict. On the other hand, a class of extreme events exists that are not so rare, more frequent, as examples, the bubbling of share market before a crash, population size of Paris [1]. The population size of Paris differs largely from the other cities in France. It showed a dragon-king behavior when the population size of all cities in France were plotted according to their rank size. It revealed a large departure of the population size of Paris from a power-law distribution followed by the small to intermediate size cities. Such an abnormal dragon-king like event, if occurs in any system, help distinguish small to intermediate size events from a large size event which has a significantly different dynamic origin. This enhances predictability of large size events or extreme events of such kind. Recently, the dragon-king behavior was observed in a simple model of two unidirectionally coupled oscillators [2, 3]. It showed bubbling of a trajectory as a large excursion from the synchronization manifold near the critical coupling when a small mismatch or noise was introduced. The large excursions of trajectories emerge as large size events that differ from the trajectories which remain confined to the synchronization manifold. The probability distribution of large size events depart largely from a power-law that is followed by small to intermediate size events mainly confined to the synchronization manifold. This work also indicates that a prediction and control of such extreme events is a possibility. Further investigations of such dragon-king-like events in simple dynamical models are necessary for deeper insights of such events that is still lacking. We report here a simple of model of Hindmarsh-Rose bursting neurons under a combination of excitatory and inhibitory synaptic coupling that shows presence of extreme events [4]. Two coupled bursting neurons establish antiphase (out-of-phase) synchronization when the inhibitory coupling specially plays a role in the occasional departure of the trajectory of the coupled system from the antiphase synchronization manifold. This large excursion of the trajectory from the antiphase synchronization manifold follows the dragon-king statistics with a large departure from a power law followed by the other trajectories which remained bounded to the antiphase synchronization manifold. Most importantly, we find a precursor to each large event that is encouraging for further investigation if we can derive a predictability measure. We extend this to a large ensemble to bursting neuron under the same coupling configuration. We successfully reproduce the dragon-king-like events in electronic analog of two Hindmarsh-Rose neuron systems. References 1. D. Sornette, Dragon-Kings, Black Swans and the Prediction of Crises, https://arxiv.org/ftp/arxiv/papers/0907/0907.4290.pdf. 2. H.L. D. de S. Cavalcante, M. O., D. Sornette, E. Ott, D. J. Gauthier, Predictability and Suppression of Extreme Events in a Chaotic System, Phys. Rev. Letts. 111, 198701 (2013). 3. G. F. de Oliveira, Jr., O. Di Lorenzo, T.P. de Silans, M. Chevrollier, M. Oriá, and H. L. D. de S. Cavalcante, Local instability driving extreme events in a pair of coupled chaotic electronic Circuits, Phys. Rev. E 93, 062209 (2016). 4. A.Mishra, S. Saha, H. Cerdeira, S. K. Dana, Extreme events in coupled bursting neurons, Opera Medica et Physiologica, Sppl S2, p.47, (2016)

We apply techniques from nonlinear dynamics to the study of propagating quantum matter waves. To this aim, we extend and adapt the recently developed concept of basin entropy to chaotic scattering situations and show how it enables to extract important information from the experimental data. In particular, we specify how to implement experimentally the equivalent of a Monte Carlo calculation of such quantities. We show how these techniques can be used to characterize the chaotic dynamics of the system and to demonstrate the presence of fractal structures in phase space. We also discuss how these methods allow to predict the efficiency of the switch and splitter regimes in a cross beam configuration. The escape time distribution can also be obtained and give access to the dynamical evolution of the system. These proposals can be implemented with current experimental techniques.

The understanding of natural climate variability as a simple linear stochastic process, a red noise, as proposed by Hasselmann (1976) works remarkably well for present days ocean temperature records. However, when it comes to the paleclimatic records obtained from the ice cores non-linearities and abrupt changes are apparent. Furthermore, careful analysis of the high resolution dust record from the GRIP ice core shows that if this is to be generated by a stochastic process the noise needs to be $\alpha$-stable, that is so extreme that the central limit theorem no longer holds.

We carried out an experimental study of a route to synchronization of a bistable chaotic piecewise linear R\"ossler electronic circuit with a chaotic driving signal from another R\"ossler oscillator. Different stages in dynamical behavior of the system have been identified and characterized with respect to the coupling strength, including intermittent switches between coexisting attractors and intermittent phase synchronization inside time windows. In addition to traditional synchronization measures, such as the phase difference between the drive and response, we have calculated probability distribution of the phase difference. This distribution has a bimodal character, where the two modes correspond to phase differences inside and outside of the windows of phase synchronization. The mean phase differences and their maximum probabilities are present as functions on the coupling strength.

Robust frequency synchronization of all power sources and sinks is necessary to maintain stable operation of an electrical distribution grid. In order to quantify the stability of a power system, an optimization technique to analyze the optimal perturbation is applied. The optimal perturbation corresponds to the weakest disturbance which is able to trigger the transition from the static synchronized state to the non-synchronized state. This measure is then used to identify topological features which might enhance or deteriorate resilience properties of a grid.

Multistability is the defining characteristic of various dynamical systems with the capacity to archive multiple steady states in response to perturbations. Functional capabilities often rest on local and long-range communication between functional units, but it is still questionable how to estimate a direct activation induced by perturbation of one functional unit to others. Here to cope with this question, we quantify the probability of the direct activation influence between pairwise units in terms of a new measure, termed the second order stability, illustrating conditional likelihood of perturbation propagation through complex networks, and we implement it on smart grids and the human brain, resulting respectively abnormal plants and assemblies of neurons. We anticipate that the measure will deepen the understanding the spreading of perturbations in complex networks, of highest actual importance from various aspects (economy, future society, sustainability, etc.

This paper focus on the analysis and control of multistability behaviour (i.e. coexistence of different attractors) in the well known improved Colpitts oscillator. The state equations of the oscillator are described by a continuous time four-dimensional autonomous system with smooth nonlinearity. Using the two-parameter diagram, the regions of periodic and chaotic behaviour in the system are identified. The parameters’ space regions where the system exhibits the coexistence of multiple attractors are depicted by performing forward and backward bifurcation analysis of the model. Various phase portraits are plotted to illustrate the coexistence of different attractors. Basins of attraction of these coexisting attractors are also computed showing complex basin boundaries. Futhermore, the control of multistability dynamics which consist to control the system dynamics from one attractor to another (desired state) is addressed using a parametric method. Numerical simulations are performed to show the effectiveness of the control method.

In this paper, the dynamics of a system of two coupled van der Pol oscillators is investigated. The coupling between the two oscillators consists of adding to each one’s amplitude a perturbation proportional to the other one. The coupling between two laser oscillators and the coupling between two vacuum tube oscillators are examples of physical/experimental systems related to the model considered in this paper. The stability of fixed points and the symmetries of the model equations are discussed. The bifurcations structures of the system are analyzed with particular attention on the effects of frequency detuning between the two oscillators. It is found that the system exhibits a variety of bifurcations including symmetry breaking, period-doubling, and crises when monitoring the frequency detuning parameter in tiny steps. The multistability property of the system for special sets of its parameters is also analyzed. An experimental study of the coupled system is carried out in this work. An appropriate electronic simulator is proposed for the investigations of the dynamic behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results yields a very good agreement.

Extreme events in forced multistable Liénard system S. Leo Kingston1, K. Thamilamaran1, Pinaki Pal2, Ulrike Feudel3, and Syamal K. Dana4 1Centre for Nonlinear Dynamics, School of Physics, Bharatidasan University, Tiruchirappalli, Tamilnadu 620024, India, 2Department of Mathematics, National Institute of Technology, Durgapur 713209, India, 3Institute for Chemistry and Biology of the Marine Environment, University of Oldenburg, Oldenburg, Germany and 4CSIR-Indian Institute of Chemical Biology, Kolkata 700032, India. Abstract We observe extremely large amplitude intermittent spiking in a dynamical variable of a periodically forced Liénard type oscillator and characterize them as extreme events with a long-tail distribution of the large excursions. We locate the extreme events in a broad parameter space of the system by plotting a bifurcation diagram which shows sudden expansion of either a periodic or a chaotic attractor by an interplay of interior crisis and intrinsic multistability of the system at two extreme values of the forcing frequency. A kurtosis measure confirms the presence of rare to large events in the parameter space. We observe large phase slips between the forced dynamics and the forcing signal whenever the large amplitude excursions cross a well-defined threshold and the event qualifies as extreme. We evidence the extreme events in an electronic experiment of the Liénard oscillator that shows perfect agreement with the numerical results. References 1. A. N. Pisarchik, R. Jaimes-Reategui, R. Sevilla-Escoboza, G. Huerta-Cuellar, and M. Taki, Phys. Rev. Lett. 107, 274101 (2011). 2. J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. R. Leite, C. Masoller, and J. R. Tredicce, Phys. Rev. A 87, 035802 (2013). 3. R. Karnatak, G. Ansmann, U. Feudel, and K. Lehnertz, Phys. Rev. E 90, 022917 (2014). 4. F. Selmi, S. Coulibaly, Z. Loghmari, I. Sagnes, G. Beaudoin, M. G. Clerc, and S. Barbay, Phys. Rev. Lett. 116, 198701 (2016).

An approach to predicting critical transitions from time series is introduced. A non-stationary low-order stochastic dynamical model of appropriate complexity to capture the transition mechanism under consideration is estimated from data. In the simplest case, the model is a one-dimensional effective Langevin equation, but also higher-dimensional dynamical reconstructions based on time-delay embedding and local modelling are considered. Integrations with the non-stationary models are performed beyond the learning data window to predict the nature and timing of critical transitions. The technique is generic, not requiring detailed a priori knowledge about the underlying dynamics of the system. The method is demonstrated to successfully predict a fold and a Hopf bifurcation well beyond the learning data window.

We investigate the dynamics of bidirectionally coupled systems. The multistability, nonlinear amplification and synchronization are found in the dynamics of the coupled systems. The bifurcation diagram shows abrupt switchs between attractors. Mathematical demonstrations and numerical simulations are given to support our results.

Evidence from various ecosystems, ranging from lakes to semi-arid ecosystems, suggests that gradually changing drivers can cause abrupt shifts from one stable state to an alternative stable state. Such shifts can often be irreversible and may result in significant ecological and economic losses. Recently, studies have devised early warning signals of abrupt transitions. However, these signals cannot forecast the threshold value of the ecosystem state or the driver at which the transition will actually occur. In this work, we propose a method to analyze discrete spatial data such as data from satellite imagery to estimate this threshold. To do so, we employ a spatially explicit model of vegetation that includes local interactions such as facilitation between nearby vegetation and competition for resources. These models exhibit transitions in ecological state from a vegetated to a bare state. We propose a method to calculate variance of the spatial data at different levels of coarse-graining (smoothing) of data. Analytical arguments suggest that the spatial variance of the coarse-grained data should peak at the value of the state or the driver variable corresponding to a critical point. In contrast, the raw data (which is typically 0 or 1 at each location) exhibits peak of spatial variance at a mean density of half, irrespective of the nature of local interactions. This method could be applied to satellite or other imagery based data of vegetation cover along a gradient of rainfall or related drivers. An important implication of our work is using such relatively easily available data, we may be able to estimate the threshold value of the driver or the state variable at which the threshold is expected. Our future work will involve testing the method with real data and to understand its strengths as well as limitations.

The abundance of multistable dynamical systems calls for an appropriate quantification of the respective stability of the (stable) states of such systems. Motivated by the concept of ecological resilience, we propose a novel and pragmatic measure called ‘integral stability’ which integrates different aspects commonly addressed separately by existing local and global stability concepts. We demonstrate the potential of integral stability by using exemplary multistable dynamical systems such as the damped driven pendulum, a model of Amazonian rainforest as a known climate tipping element and the Daisyworld model. A crucial feature of integral stability lies in its potential of arresting a gradual loss of the stability of a system when approaching a tipping point, thus providing a potential early-warning signal sufficiently prior to a qualitative change of the system’s dynamics. Reference: Mitra, C. et al. An integrative quantifier of multistability in complex systems based on ecological resilience. Sci. Rep. 5 (2015).

We investigate the emergence of algal bloom in marine ecosystem in presence of temporal variation of nutrient input. The investigation is based on a phytoplankton-zooplankton model due to Truscott-Brindley with slow-fast setting. The nutrient input parameter in the model is allowed to vary temporally to incorporate the natural variation. Linear as well as logistic temporal variation or ramping of the nutrient are considered. For static setting of the nutrient input parameter, the model shows steady equilibrium. On the other hand, as the carrying capacity of the system varies temporally beyond a critical rate, plankton concentration starting from an equilibrium shows a sudden bloom for a significant period of time before following the equilibrium plankton concen- trations of the system with fixed carrying capacities. The mechanism of plankton bloom is then analyzed numerically.

A.L. Pankratov$^{1,2,3}$, E.V. Pankratova$^{3}$, V.A. Shamporov$^{1,2,3}$, S.V. Shitov$^{4,5}$ $^1$Institute for Physics of Microstructures of RAS, Nizhny Novgorod, Russia \\ $^2$Cryogenic Nanoelectronics Center, Nizhny Novgorod State Technical University n.a. R.E. Alexeyev, Nizhny Novgorod, Russia\\ $^{3}$N.I. Lobachevsky State University of Nizhni Novgorod, Nizhny Novgorod, Russia \\ $^4$Institute of Radio-engineering and Electronics of RAS, Moscow, Russia \\ $^5$Laboratory of Superconducting Metamaterials, National University of Science and Technology "MISIS", Moscow, Russia The joint action of the matching to a common RC-load and thermal noise on the spectral properties of parallel Josephson junction array is studied. It is demonstrated that proper matching may suppress the chaotic dynamics of the system. The efficiency of radiation was found to be highest within a limited frequency band, which corresponds to transformation of the shuttle soliton oscillating regime into the linear wave resonance synchronization mode. In this frequency band the spectral linewidth agrees well with a double of the linewidth for a shuttle fluxon oscillator, divided by a number of the oscillators in the chain. If the oscillations demonstrate strong amplitude modulation, it leads to increase of the linewidth roughly by a factor of five compared to this theoretical linewidth formula. This work is supported by RFBR (projects 14-02-31727 and 15-02-05869), by Ministry of Education and Science of the Russian Federation (projects 11.G34.31.0062, 14.740.12.0845 and 3.2054.2014/K) and in part by RSF grant 16-19-10478.

We present an influence of discontinuous coupling on the dynamics of multistable systems. Our model consists of two periodically forced oscillators that can interact via soft impacts. The controlling parameters are the distance between the oscillators and the difference in the phase of the harmonic excitation. When the distance is large two systems do not collide and a number of different possible solutions can be observed in both of them. When decreasing of the distance, one can observe some transient impacts and then the systems settle down on non-impacting attractor. It is shown that with the properly chosen distance and difference in the phase of the harmonic excitation, the number of possible solutions of the coupled systems can be reduced. The proposed method is robust and applicable in many different systems.

Populations of phase oscillators can display a variety of synchronization patterns. This depends on the coupling between oscillators and on intrinsic properties like the oscillators' natural frequency. If several populations with unimodal frequency distribution are coupled to one another, the resulting dynamics may resemble that of a single population with multimodally distributed frequencies; see, e.g., [3-5] for related conjectures. Using an Ott-Antonsen ansatz in the all-to-all coupled Kuramoto model [1], we have proven that in the case of two symmetric networks both the subpopulation approach and the bimodal approach are equivalent and lead to the same properties as regards sta- bility, dynamics, and bifurcations [2]. This equivalence strongly suggests a generalization to more complicated set-ups. On the one hand, our findings for two populations seem to be robust when refraining from perfect symmetry assumptions, see, e.g., [5]. On the other hand, the step to more than two populations appears to be not trivial at all. Even in the case of three populations, the network dynamics cease from being analyzed by means of the Ott-Antonsen ansatz. In our talk, we will address these three points step by step. First, we discuss the dynamics of two coupled symmetric populations of Kuramoto phase oscillators. It turns out that, compared to the bimodal network as has been studied by Martens and co-workers [3], an additional bifurcation parameter enters the system. However, we show that this parameter does not lead to new dynamical behavior, but both descriptions prove to be topologically equivalent. Second, we add asymmtry in the frequency distributions. A starting point here is the work by Pazo and Montbrio [5], which then will be compared against our results. Last, we give an outlook for investigating more than two populations, and discuss chances and problems when comparing the coupled subpopulation approach vis-a-vis the multimodal approach. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642563 (COSMOS). [1] E. Ott, T.M. Antonsen. Chaos 18 (3) pp. 037113, 2008. [2] B. Pietras, N. Deschle, A. Daertshofer. arXiv:1602.08368 [nlin.CD] [3] E. A. Martens, E. Barreto, S.H. Strogatz, E. Ott, P. So, T.M. Antonsen. PRE 79 (2) pp. 026204, 2009. [4] E. Barreto, B. Hunt, E. Ott, P. So. PRE 77 (3) pp.036107, 2008. [5] D. Pazo, E. Montbrio. PRE 80 (4) pp. 046215, 2009.

The Ott-Antonsen (OA) ansatz [Chaos 18, 037113 (2008), Chaos 19, 023117 (2009)] has been widely used to describe large networks of coupled phase oscillators. If the coupling is sinusoidal and if the phase dynamics does not depend on the specific oscillator, then the macroscopic behavior of the network can be fully described by a low-dimensional system. Does the OA manifold remain attractive, when introducing an intrinsic dependence between an oscillator's phase and its dynamics by additional, oscillator specific parameters? To answer this we extended the OA ansatz and proved that parameter-dependent oscillatory networks continue to converge to the OA manifold under certain conditions. Our proof confirms recent numerical findings that hint at this convergence. It also provides a thorough mathematical underpinning for networks of theta neurons, where the OA ansatz has just been applied.

Using the synchrony factor and its periodicity, we are able to identify not only complete synchronization but also in-phase synchronization. The analysis of the periodicities of the synchrony factor allows to obtain a phase diagram that contains all the information of synchronous behavior in a wide range of parameters. This method constitutes a new and useful tool to study synchronization.

A general approach for the analysis of stochastically forced attractors (equilibrium, periodic, quasiperiodic) and noise-induced transitions between them is proposed. Mathematically, this approach is based on the stochastic sensitivity function technique and confidence domains method. An effectiveness of this approach is demonstrated in applications to the analysis of neuronal, climate and volcanic activity.

The concept of resilience and its applications are found in numerous disciplines, including engineering sciences, psychology, economics, social sciences and ecology. Its present significance, however, stems from a publication by the renowned ecologist C.S. Holling in 1973. He made clear that resilient systems are not only able to return to an equilibrium state after a temporary disturbance, what is his definition for stability. Resilience means that a system can maintain its relationships when state variables, driving variables and parameters change. Ever since, the concept of resilience became a paradigm and has been studied widely. In the process, many misconceptions and confusions have emerged, partly because of transdisciplinary applications. Many aspects remain unclear, for example, the connection between kinds of disturbances – such as fluctuations, perturbations or a change of those over time – and the corresponding system response of resilient systems. One objective of my work is to classify possible system responses to different kinds of disturbances.

The development of sustainable agricultural systems in drylands is currently an important issue in the context of mitigating the outcomes of population growth under the conditions of climatic changes. The need to meet the growing demand for food, fodder, and fuel under the threat of climate change, requires cross-disciplinary studies of ways to increase the livelihood while minimizing the impact on the environment. Practices of agroforestry systems, in which herbaceous species are intercropped between rows of woody species plantations, have shown to increase land productivity. As vegetation in drylands tends to self-organize in spatial patterns, it is important to explore the relationship between the patterns that agroforestry systems tend to form, and the productivity of these system in terms of biomass, their resilience to variability of water availability, and water use efficiency. A spatially-explicit vegetation model for two species that compete for water and light and may exploit soil layers of different depths will be introduced. Spatially-uniform and periodic solutions, and their stability properties, will be presented for different scenarios of species and environmental conditions. The implications for optimal intercropping in terms of biomass productivity, water use efficiency, and resilience to environmental changes, will be discussed.

We obtain chimeras, namely, the coexisting desynchronous and synchronized dynamical states, in an ensemble of identical chaotic oscillators by inducing multistability in the system. We observe that the ensemble of identical chaotic oscillators---under the influence of a common forcing or common noise and also with global coupling---may exhibit chimeric behavior at appropriate parameter values. This is due to the coexistence of multiple attractors of different nature on which both synchronized and incoherent motions are possible. We examine the mechanism of the emergence of these chimeric states and discuss the generality of the results.

The impacts of climate change are typically assumed to depend on the way average climate conditions change in time and space and on the occurrence of extreme events, but not on changes in the spectrum. Here, we show that temporal autocorrelation (redness) is an equally important feature when it comes to the likelihood of climate-induced transitions in nature and society. The analysis of three well-known ecological models with alternative stable states, reveals that as the temporal autocorrelation increases, the transition occurs at a lower value of the control parameter, independently of changes in variance. This indicates that the temporal autocorrelation itself could drive a system towards a (critical) threshold. Following these results, we should monitor systematically how redness is expected to change and how systems are sensitive to the spectrum of climate variability.