The analysis of event time series is in general challenging. Most time series analysis tools are limited for the analysis of this kind of data. Recurrence analysis, a powerful concept from nonlinear time series analysis, provides several opportunities to work with event data and even for the most challenging task of comparing event time series with continuous time series. Here, the basic concept is introduced, the challenges are discussed, and the future perspectives are summarised.

Recent decades have seen a rise in the use of physics methods to study different societal phenomena. This development has been due to physicists venturing outside of their traditional domains of interest, but also due to scientists from other disciplines taking from physics the methods that have proven so successful throughout the 19th and the 20th century. Here we dub this field 'social physics' and pay our respect to intellectual mavericks who nurtured it to maturity. We do so by reviewing the current state of the art. Starting with a set of topics that are at the heart of modern human societies, we review research dedicated to urban development and traffic, the functioning of financial markets, cooperation as the basis for our evolutionary success, the structure of social networks, and the integration of intelligent machines into these networks. We then shift our attention to a set of topics that explore potential threats to society. These include criminal behaviour, large-scale migrations, epidemics, environmental challenges, and climate change. We end the coverage of each topic with promising directions for future research. Based on this, we conclude that the future for social physics is bright. Physicists studying societal phenomena are no longer a curiosity, but rather a force to be reckoned with. Notwithstanding, it remains of the utmost importance that we continue to foster constructive dialogue and mutual respect at the interfaces of different scientific disciplines.

Functional networks are powerful tools to study statistical interdependency structures in spatially extended or multivariable systems. They have been used to get insights into the dynamics of complex systems in various areas of science. In particular, percolation properties of correlation networks have been employed to identify early warning signals of critical transitions. In this work, we further investigate the corresponding potential of percolation measures for the anticipation of different types of sudden shifts in the state of coupled irregularly oscillating systems. As a paradigmatic model system, we study the dynamics of a ring of diffusively coupled noisy FitzHugh-Nagumo oscillators and show that, when the oscillators are nearly completely synchronized, the percolation-based precursors successfully provide very early warnings of the rapid switches between the two states of the system. We clarify the mechanisms behind the percolation transition by separating global trends given by the mean-field behavior from the synchronization of individual stochastic fluctuations. We then apply the same methodology to real-world data of sea surface temperature anomalies during different phases of the El Niño-Southern Oscillation. This leads to a better understanding of the factors that make percolation precursors effective as early warning indicators of incipient El Niño and La Niña events.

Climate mitigation decisions today affect future generations, raising questions of intergenerational equity. Integrated assessment models (IAMs) rely on discounting to evaluate long-term policy costs and benefits. Using the DICE model, we quantify how optimal pathways distribute abatement and damage costs across cohorts. Unconstrained optimization creates intergenerational inequality, with future generations bearing higher costs relative to GDP. Extending the model with stochastic discount rates, we show that discount-rate uncertainty significantly amplifies this inequality. We consider two independent extensions: the financing of abatement costs and the modeling of nonlinear financing costs under large damages. Both extensions can materially improve intergenerational equity by distributing mitigation efforts more evenly. As an illustration, we present a modified DICE model whose optimal pathway limits generational costs to 3 % of GDP, leading to more equitable effort sharing. Our proposed model extensions are model-agnostic, applicable across IAMs, and compatible with alternative intergenerational equity metrics.

Turbulent reacting flows confined to ducts are plagued by thermoacoustic instability, a state in which a positive feedback between flow, flame and acoustic perturbations leads to the emergence of catastrophically high-amplitude oscillatory dynamics in the sound and global heat release rate fluctuations. Modeling the interdependence between local interactions and the global emergence of order in such spatially extended complex systems is exacting. Here, we present a novel reduced-order model to capture the influence of the local interactions on the variables exhibiting global emergence of order in a turbulent reacting flow system. We represent each variable that exhibits global oscillatory instability as an oscillator with a cubic nonlinearity. The oscillator is driven by a forcing term that represents the holistic influence of the inter-subsystem interactions on the global behavior. The forcing term essentially couples the local interactions and the globally emergent dynamics in the model. Further, the influence of the inter-subsystem interactions on the behavior of each subsystem is different. Therefore, we use different forcing terms for each variable inspired by the physical interactions in the system. The nonlinear oscillators representing the acoustic and the heat release rate oscillations are hence forced using Wiener and Markov-modulated Poisson processes, respectively. Using this approach, we are able to reproduce (i) the multifractal characteristics of acoustic pressure fluctuations during chaotic dynamics, (ii) the loss of multifractality through the experimentally observed scaling law behavior during the transition from chaos to order and (iii) the emergence of periodicity and bifurcation in heat release rate dynamics.

Standard macroeconomic models assume that households are rational in the sense that they are perfect utility maximizers, and explain economic dynamics in terms of shocks that drive the economy away from the stead-state. Here we build on a standard macroeconomic model in which a single rational representative household makes a savings decision of how much to consume or invest. In our model households are myopic boundedly rational heterogeneous agents embedded in a social network. From time to time each household updates its savings rate by copying the savings rate of its neighbor with the highest consumption. If the updating time is short, the economy is stuck in a poverty trap, but for longer updating times economic output approaches its optimal value, and we observe a critical transition to an economy with irregular endogenous oscillations in economic output, resembling a business cycle. In this regime households divide into two groups: Poor households with low savings rates and rich households with high savings rates. Thus inequality and economic dynamics both occur spontaneously as a consequence of imperfect household decision making. Our work here supports an alternative program of research that substitutes utility maximization for behaviorally grounded decision making.

We develop a framework within which to conceptualize World-Earth System resilience. Our notion of World-Earth System resilience emphasizes the need to move beyond the basin of attraction notion of resilience as we are not in a basin we can stay in. We are on a trajectory to a new basin and we have to avoid falling into undesirable basins. We thus focus on `pathway resilience', i.e. the relative number of paths that allow us to move from the transitional operating space we occupy now as we leave the Holocene basin to a safe and just operating space in the Anthropocene. We develop a mathematical model to formalize this conceptualization and demonstrate how interactions between earth system resilience (biophysical processes) and world system resilience (social processes) impact pathway resilience. Our findings show that building earth system resilience is probably our only chance to reach a safe and just operating space. We also illustrate the importance of world system dynamics by showing how the notion of fairness coupled with regional inequality affects pathway resilience.

A pervasive trend in economic development is the shift from agricultural to manufacturing and finally to service economies. Because the type of employment associated with each sector influences natural resource use this global trend may manifest itself in predictable land use transitions. We relate these to changes in urban, agricultural, and natural lands. We find that the economic transition is common in most counties and across counties. Agricultural land expands to natural areas but is eventually replaced by urban. Relating sectors to land cover, we see a strong relationship between increases in service employment and the growth of urban areas. However, we find that both large agricultural areas and large natural areas also occur with high service employment. Finally, we test if the 100th Meridian is associated with predictable relationships between land cover and economic sectors and find strong results that suggest different development patterns in the East and West.

Decarbonizing China's energy system necessitates both greening the power supply and end-use electrification. However, there are concerns that electrification may be premature while coal power dominates. Using a global climate mitigation model, we examine multiple high electrification scenarios with different coal phase-out timelines. On an aggregate level, the pace of Chinese power sector decarbonization is climate significant. A ten-year delay in coal phase-out could alone increase global warming by around 0.011°C. However, on energy service and sectoral level there is no evidence of large-scale premature electrification even under slower coal phase-out. This challenges the sequential interpretation of the "order of abatement" - electrification can begin only when the power sector is almost decarbonized. As long as power emission intensity reduces to below 150 kgCO2/MWh by 2040, even with the current power supply mix, early scale-up of electrification brings a huge gain in CO2 abatement in the medium- to long-term, equivalent to approximately 0.04°C avoided warming.

The last decade has witnessed a number of important and exciting developments that had been achieved for improving recurrence plot based data analysis and to widen its application potential. We will give a brief overview about important and innovative developments, such as computational improvements, alternative recurrence definitions (event-like, multiscale, heterogeneous, and spatio-temporal recurrences) and ideas for parameter selection, theoretical considerations of recurrence quantification measures, new recurrence quantifiers (e.g., for transition detection and causality detection), and correction schemes. New perspectives have recently been opened by combining recurrence plots with machine learning. We finally show open questions and perspectives for futures directions of methodical research.

We analyze solutions to the stochastic skeleton model, a minimal nonlinear oscillator model for the Madden-Julian Oscillation (MJO). This model has been recognized for its ability to reproduce several large-scale features of the MJO. In previous studies, the model's forcings were predominantly chosen to be mathematically simple and time-independent. Here, we present solutions to the model with time-dependent observation-based forcing functions. Our results show that the model, with these more realistic forcing functions, successfully replicates key characteristics of MJO events, such as their lifetime, extent, and amplitude, whose statistics agree well with observations. However, we find that the seasonality of MJO events and the spatial variations in the MJO properties are not well reproduced. Having implemented the model in the presence of time-dependent forcings, we can analyze the impact of temporal variability at different time scales. In particular, we study the model's ability to reflect changes in MJO characteristics under the different phases of ENSO. We find that it does not capture differences in studied characteristics of MJO events in response to differences in conditions during El Ni\~no, La Ni\~na, and neutral ENSO.

In the last decade, there has been a growing body of literature addressing the utilization of complex network methods for the characterization of dynamical systems based on time series. While both nonlinear time series analysis and complex network theory are widely considered to be established fields of complex systems sciences with strong links to nonlinear dynamics and statistical physics, the thorough combination of both approaches has become an active field of nonlinear time series analysis, which has allowed addressing fundamental questions regarding the structural organization of nonlinear dynamics as well as the successful treatment of a variety of applications from a broad range of disciplines. In this report, we provide an in-depth review of existing approaches of time series networks, covering their methodological foundations, interpretation and practical considerations with an emphasis on recent developments. After a brief outline of the state-of-the-art of nonlinear time series analysis and the theory of complex networks, we focus on three main network approaches, namely, phase space based recurrence networks, visibility graphs and Markov chain based transition networks, all of which have made their way from abstract concepts to widely used methodologies. These three concepts, as well as several variants thereof will be discussed in great detail regarding their specific properties, potentials and limitations. More importantly, we emphasize which fundamental new insights complex network approaches bring into the field of nonlinear time series analysis. In addition, we summarize examples from the wide range of recent applications of these methods, covering rather diverse fields like climatology, fluid dynamics, neurophysiology, engineering and economics, and demonstrating the great potentials of time series networks for tackling real-world contemporary scientific problems.

Intermittent renewable energies are increasingly dominating electricity grids and are forecasted to be the main force driving out fossil fuels from the grid in most major economies until 2040. However, grids based on intermittent renewables are challenged by diurnal and seasonal mismatch between supply of sun and wind and demand for electricity, including for heat pumps and electric two and four wheelers. Load management and demand response measures promise to adjust for this mismatch, utilizing information- and price-based approaches to steer demand towards times with high supply of intermittent renewables. Here, we systematically review the literature estimating CO2 savings from residential load management in developing and developed nations. We find that load management holds high potential, locally differentiated with energy mix (including the respective share of renewables and fossils), climate zone, and the regulatory environment and price mechanism. Most identified studies suggest a mitigation potential between 1 and 20%. Load management becomes more relevant with higher shares of intermittent renewables, and when electricity prices are high. Importantly, load management aligns consumers' financial incentives with climate change mitigation, thus rendering accompanying strategies politically feasible. We summarize key regulatory steps to facilitate load management in economies and to realize relevant consumer surplus and mitigation potential.

Declines in resilience have been observed in several climate tipping elements over the past decades, including the Atlantic Meridional Overturning Circulation (AMOC) and the Amazon rainforest (AR). Large-scale nonlinear and possibly irreversible changes in system state, such as AMOC weakening or rainforest-savanna transitions in the Amazon basin, would have severe impacts on ecosystems and human societies worldwide. In order to improve future tipping risk assessments, understanding interactions between tipping elements is crucial. The AMOC is known to influence the Intertropical Convergence Zone, potentially altering precipitation patterns over the AR and affecting its stability. However, AMOC-AR interactions are currently not well understood. Here, we identify a previously unknown stabilising interaction pathway from the AMOC onto the Southern AR, applying an established causal discovery and inference approach to tipping element interactions for the first time. Analysing observational and reanalysis data from 1982-2022, we show that AMOC weakening leads to increased precipitation in the Southern AR during the critical dry season, in line with findings from recent Earth system model experiments. Specifically, we report a 4.8% increase of mean dry season precipitation in the Southern AR for every 1 Sv of AMOC weakening. This finding is consistent across multiple data sources and AMOC strength indices. We show that this stabilising interaction has offset 17% of dry season precipitation decrease in the Southern AR since 1982. Our results demonstrate the potential of causal discovery methods for analysing tipping element interactions based on reanalysis and observational data. By improving the understanding of AMOC-AR interactions, we contribute toward better constraining the risk of potential climate tipping cascades under global warming.

Couplings in complex real-world systems are often nonlinear and scale-dependent. In many cases, it is crucial to consider a multitude of interlinked variables and the strengths of their correlations to adequately fathom the dynamics of a high-dimensional nonlinear system. We propose a recurrence based dependence measure that quantifies the relationship between multiple time series based on the predictability of their joint evolution. The statistical analysis of recurrence plots (RPs) is a powerful framework in nonlinear time series analysis that has proven to be effective in addressing many fundamental problems, e.g., regime shift detection and identification of couplings. The recurrence flow through an RP exploits artifacts in the formation of diagonal lines, a structure in RPs that reflects periods of predictable dynamics. By using time-delayed variables of a deterministic uni-/multivariate system, lagged dependencies with potentially many time scales can be captured by the recurrence flow measure. Given an RP, no parameters are required for its computation. We showcase the scope of the method for quantifying lagged nonlinear correlations and put a focus on the delay selection problem in time-delay embedding which is often used for attractor reconstruction. The recurrence flow measure of dependence helps to identify non-uniform delays and appears as a promising foundation for a recurrence based state space reconstruction algorithm.

Based on suggested interactions of potential tipping elements in the Earth's climate and in ecological systems, tipping cascades as possible dynamics are increasingly discussed and studied as their activation would impose a considerable risk for human societies and biosphere integrity. However, there are ambiguities in the description of tipping cascades within the literature so far. Here we illustrate how different patterns of multiple tipping dynamics emerge from a very simple coupling of two previously studied idealized tipping elements. In particular, we distinguish between a two phase cascade, a domino cascade and a joint cascade. While a mitigation of an unfolding two phase cascade may be possible and common early warning indicators are sensitive to upcoming critical transitions to a certain degree, the domino cascade may hardly be stopped once initiated and critical slowing down--based indicators fail to indicate tipping of the following element. These different potentials for intervention and anticipation across the distinct patterns of multiple tipping dynamics should be seen as a call to be more precise in future analyses on cascading dynamics arising from tipping element interactions in the Earth system.

The emergence of explosive synchronization has been reported as an abrupt transition in complex networks of first-order Kuramoto oscillators. In this Letter, we demonstrate that the nodes in a second-order Kuramoto model, perform a cascade of transitions toward a synchronous macroscopic state, which is a novel phenomenon that we call \textit{cluster explosive synchronization}. We provide a rigorous analytical treatment using a mean-field analysis in uncorrelated networks. Our findings are in good agreement with numerical simulations and fundamentally deepen the understanding of microscopic mechanisms toward synchronization.

High-resolution energy consumption and emissions datasets are essential for localized policy-making, resource optimization, and climate action planning. They enable municipalities to monitor mitigation strategies and foster engagement among governments, businesses, and communities. However, smaller municipalities often face data limitations that hinder tailored climate strategies. This study generates detailed final energy consumption and emissions data at the local administrative level for Germany and Spain. Using national datasets, we apply spatial disaggregation techniques with open data sources. A key innovation is the application of XGBoost for imputing missing data, combined with a stepwise spatial disaggregation process incorporating district- and province-level statistics. Prioritizing reproducibility, our open-data approach provides a scalable framework for municipalities to develop actionable climate plans. To ensure transparency, we assess the reliability of imputed values and assign confidence ratings to the disaggregated data.

Different collective behaviors emerging from the unknown have been examined in networks of mobile agents in recent years. Mobile systems, far from being limited to modeling and studying various natural and artificial systems in motion and interaction, offer versatile solutions across various domains, facilitating tasks ranging from navigation and communication to data collection and environmental monitoring. We examine the relative mobility between clusters, each composed of different elements in a multi-clusters network-a system composed of clusters interconnected to form a larger network of mobile oscillators. Each mobile oscillator exhibits both external (i.e., position in a 2D space) and internal dynamics (i.e., phase oscillations). Studying the mutual influence between external and internal dynamics, often leads the system towards a state of synchronization within and between clusters. We show that synchronization between clusters is affected by their spatial closeness. The stability of complete synchronization observed within the clusters is demonstrated through analytical and numerical methods.

Recurrence plots and their associated quantifiers provide a robust framework for detecting and characterising complex patterns in non-linear time-series. In this paper, we employ recurrence quantification analysis to investigate the dynamics of the cyclic, non-hierarchical May-Leonard model, also referred to as rock--paper--scissors systems, that describes competitive interactions among three species. A crucial control parameter in these systems is the species' mobility $m$, which governs the spatial displacement of individuals and profoundly influences the resulting dynamics. By systematically varying $m$ and constructing suitable recurrence plots from numerical simulations, we explore how recurrence quantifiers reflect distinct dynamical features associated with different ecological states. We then introduce an ensemble-based approach that leverages statistical distributions of recurrence quantifiers, computed from numerous independent realisations, allowing us to identify dynamical outliers as significant deviations from typical system behaviour. Through detailed numerical analyses, we demonstrate that these outliers correspond to divergent ecological regimes associated with specific mobility values, providing also a robust manner to infer the mobility parameter from observed numerical data. Our results highlight the potential of recurrence-based methods as diagnostic tools for analysing spatial ecological systems and extracting ecologically relevant information from their non-linear dynamical patterns.