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Since 1970, the R\"ossler system has remained as a considerably simpler and minimal dimensional chaos serving system. Unveiling the dynamics of a system of two coupled chaotic oscillators that leads to the emergence of extreme events in the system is an engrossing and crucial scientific research area. Our present study focuses on the emergence of extreme events in a system of diffusively and bidirectionally two coupled R\"ossler oscillators and unraveling the mechanism behind the genesis of extreme events. We find the appearance of extreme events in three different observables: average velocity, synchronization error, and one transverse directional variable to the synchronization manifold. The emergence of extreme events in average velocity variables happens due to the occasional in-phase synchronization. The on-off intermittency plays for the crucial role in the genesis of extreme events in the synchronization error dynamics and in the transverse directional variable to the synchronization manifold. The bubble transition of the chaotic attractor due to the on-off intermittency is illustrated for the transverse directional variable. We use generalized extreme value theory to study the statistics of extremes. The extreme events data sets concerning the average velocity variable follow generalized extreme value distribution. The inter-event intervals of the extreme events in the average velocity variable spread well exponentially. The upshot of the interplay between the coupling strength and the frequency mismatch between the system oscillators in the genesis of extreme events in the coupled system is depicted numerically., Comment: 12 pages, 16 figures

Adaptive dynamical networks are ubiquitous in real-world systems. This paper aims to explore the synchronization dynamics in networks of adaptive oscillators based on a paradigmatic system of adaptively coupled phase oscillators. Our numerical observations reveal the emergence of synchronization cluster bursting, characterized by periodic transitions between cluster synchronization and global synchronization. By investigating a reduced model, the mechanisms underlying synchronization cluster bursting are clarified. We show that a minimal model exhibiting this phenomenon can be reduced to a phase oscillator with complex-valued adaptation. Furthermore, the adaptivity of the system leads to the appearance of additional symmetries and thus to the coexistence of stable bursting solutions with very different Kuramoto order parameters.

In this work, effects of constant and time-dependent vaccination rates on the Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) seasonal model are studied. Computing the Lyapunov exponent, we show that typical complex structures, such as shrimps, emerge for given combinations of constant vaccination rate and another model parameter. In some specific cases, the constant vaccination does not act as a chaotic suppressor and chaotic bands can exist for high levels of vaccination (e.g., $> 0.95$). Moreover, we obtain linear and non-linear relationships between one control parameter and constant vaccination to establish a disease-free solution. We also verify that the total infected number does not change whether the dynamics is chaotic or periodic. The introduction of a time-dependent vaccine is made by the inclusion of a periodic function with a defined amplitude and frequency. For this case, we investigate the effects of different amplitudes and frequencies on chaotic attractors, yielding low, medium, and high seasonality degrees of contacts. Depending on the parameters of the time-dependent vaccination function, chaotic structures can be controlled and become periodic structures. For a given set of parameters, these structures are accessed mostly via crisis and in some cases via period-doubling. After that, we investigate how the time-dependent vaccine acts in bi-stable dynamics when chaotic and periodic attractors coexist. We identify that this kind of vaccination acts as a control by destroying almost all the periodic basins. We explain this by the fact that chaotic attractors exhibit more desirable characteristics for epidemics than periodic ones in a bi-stable state.

Complex networks have played an important role in describing real complex systems since the end of the last century. Recently, research on real-world data sets reports intermittent interaction among social individuals. In this paper, we pay attention to this typical phenomenon of intermittent interaction by considering the state transition of network vertices between online and hidden based on the birth and death process. By continuous-time Markov theory, we show that both the number of each vertex's online neighbors and the online network size are stable and follow the homogeneous probability distribution in a similar form, inducing similar statistics as well. In addition, all propositions are verified via simulations. Moreover, we also present the degree distributions based on small-world and scale-free networks and find some regular patterns by simulations. The application in fitting real networks is discussed., Comment: 20 pages, 9 figures

Early warning signals (EWSs) forewarn a sudden transition (or tipping) from a desirable state to an undesirable state. However, we observe that EWSs detect an impending tipping past bifurcation points when control parameters are varied fast; this questions the applicability of EWSs in real-world systems. When a control parameter is changed at a finite rate, the tipping is also delayed, providing a borrowed stability (in the parameter space) before the system tips. In this study, we use the Hurst exponent as EWS in a thermoacoustic system - a horizontal Rijke tube. We find that upon receiving an EWS alert, a quick reversal of the control parameter within the region of borrowed stability cannot always prevent tipping in real-world systems. We show this failure is due to the (i) delay in receiving the EWS alert and (ii) dispersion observed in the warning points received. For fast variation of parameters, where preventive measures fall short, we demonstrate EWS-based control actions to rescue the system after tipping. Our results in a real-world system for a fast variation of parameter highlight the limits of applicability of EWSs in preventing tipping., Comment: 13 pages, 7 figures

Canard cascading (CC) is observed in dynamical networks with global adaptive coupling. It is a fast-slow phenomenon characterized by a recurrent sequence of fast transitions between distinct and slowly evolving quasi-stationary states. In this letter, we uncover the dynamical mechanisms behind CC, using an illustrative example of globally and adaptively coupled semiconductor lasers, where CC represents sequential switching on and off the lasers. Firstly, we show that CC is a robust and truly adaptive network effect that is scalable with network size and does not occur without adaptation. Secondly, we uncover multiple saddle slow manifolds (unstable quasi-stationary states) linked by heteroclinic orbits (fast transitions) in the phase space of the system. This allows us to identify CC with a novel heteroclinic canard orbit that organises different unstable quasi-stationary states into an intricate fast-slow limit cycle. Although individual quasi-stationary states are unstable (saddles), the CC cycle as a whole is attractive and robust to parameter changes.

Effective frequency control in power grids has become increasingly important with the increasing demand for renewable energy sources. Here, we propose a novel strategy for resolving this challenge using graph convolutional proximal policy optimization (GC-PPO). The GC-PPO method can optimally determine how much power individual buses dispatch to reduce frequency fluctuations across a power grid. We demonstrate its efficacy in controlling disturbances by applying the GC-PPO to the power grid of the UK. The performance of GC-PPO is outstanding compared to the classical methods. This result highlights the promising role of GC-PPO in enhancing the stability and reliability of power systems by switching lines or decentralizing grid topology., Comment: 11 pages, 6 figures

Ever since the Barab\'{a}si-Albert (BA) scale-free network has been proposed, network modeling has been studied intensively in light of the network growth and the preferential attachment (PA). However, numerous real systems are featured with a dynamic evolution including network reduction in addition to network growth. In this paper, we propose a novel mechanism for evolving networks from the perspective of vertex degree. We construct a queueing system to describe the increase and decrease of vertex degree, which drives the network evolution. In our mechanism, the degree increase rate is regarded as a function positively correlated to the degree of a vertex, ensuring the preferential attachment in a new way. Degree distributions are investigated under two expressions of the degree increase rate, one of which manifests a ``long tail'', and another one varies with different values of parameters. In simulations, we compare our theoretical distributions with simulation results and also apply them to real networks, which presents the validity and applicability of our model.

With the prevalence of COVID-19, the modeling of epidemic propagation and its analyses have played a significant role in controlling epidemics. However, individual behaviors, in particular the self-protection and migration, which have a strong influence on epidemic propagation, were always neglected in previous studies. In this paper, we mainly propose two models from the individual and population perspectives. In the first individual model, we introduce the individual protection degree that effectively suppresses the epidemic level as a stochastic variable to the SIRS model. In the alternative population model, an open Markov queueing network is constructed to investigate the individual number of each epidemic state, and we present an evolving population network via the migration of people. Besides, stochastic methods are applied to analyze both models. In various simulations, the infected probability, the number of individuals in each state and its limited distribution are demonstrated.

Most complex systems are nonlinear, relying on emergent behavior from interacting subsystems, often characterized by oscillatory dynamics. Collective oscillatory behavior is essential for the proper functioning of many real world systems. Complex networks have proven efficient in elucidating the topological structures of both natural and artificial systems and describing diverse processes occurring within them. Recent advancements have significantly enhanced our understanding of emergent dynamics in complex networks. Among various processes, a substantial body of work explores the dynamical robustness of complex networks, their ability to withstand degradation in network constituents while maintaining collective oscillatory dynamics. Many physical and biological systems experience a decline in dynamic activities due to natural or environmental factors. The impact of such damages on network performance can be significant, and the system's robustness indicates its capability to maintain functionality despite dynamic changes, often termed aging. This review provides a comprehensive overview of notable research examining how networks sustain global oscillation despite increasing inactive dynamical units. We present contemporary research dedicated to the theoretical understanding and enhancement mechanisms of dynamical robustness in complex networks. Our focus includes various network structures and coupling functions, elucidating the persistence of networked systems. We cover system characteristics from heterogeneity in network connectivity to heterogeneity in dynamical units. Finally, we discuss challenges in this field and open areas for future studies., Comment: 52 pages, 33 figures

El Ni\~no-Southern Oscillation (ENSO) exhibits significant impacts on the frequency of extreme weather events and its socio-economic implications prevail on a global scale. However, a fundamental gap still exists in understanding the relationship between the ENSO and weather-related power outages in the continental United States. Through 24-year (2000-2023) composite and statistical analysis, our study reveals that higher power outage numbers (PONs) are observed from the developing winter to the decaying summer of La Ni\~na phases. In particular, during the decaying spring, high La Ni\~na intensity favors the occurrences of power outage over the west coast and east of the United States, by modulating the frequency of extreme precipitations and heatwaves. Furthermore, projected increasing heatwaves from the Coupled Model Intercomparison Project Phase 6 (CMIP6) indicate that spring-time PONs over the eastern United States occur about 11 times higher for the mid-term future (2041-2060) and almost 26 times higher for the long-term future (2081-2100), compared with 2000-2023. Our study provides a strong recommendation for building a more climate-resilient power system.

There are complex interactions between antibody levels and epidemic propagation, the antibody level of an individual influences the probability of infection, and the spread of the virus influences the antibody level of each individual. There exist some viruses that, in their natural state, cause antibody levels in an infected individual to gradually decay. When these antibody levels decay to a certain point, the individual can be reinfected, such as with COVID 19. To describe their interaction, we introduce a novel mathematical model that incorporates the presence of an antibody retention rate to investigate the infection patterns of individuals who survive multiple infections. The model is composed of a system of stochastic differential equations to derive the equilibrium point and threshold of the model and presents rich experimental results of numerical simulations to further elucidate the propagation properties of the model. We find that the antibody decay rate strongly affects the propagation process, and also that different network structures have different sensitivities to the antibody decay rate, and that changes in the antibody decay rate cause stronger changes in the propagation process in Barabasi Albert networks. Furthermore, we investigate the stationary distribution of the number of infection states and the final antibody levels, and find that they both satisfy the normal distribution, but the standard deviation is small in the Barabasi Albert network. Finally, we explore the effect of individual antibody differences and decay rates on the final population antibody levels, and uncover that individual antibody differences do not affect the final mean antibody levels. The study offers valuable insights for epidemic prevention and control in practical applications., Comment: 14 pages, 9 figures

Dynamic processes in complex networks are crucial for better understanding collective behavior in human societies, biological systems, and the internet. In this paper, we first focus on the continuous Markov-based modeling of evolving networks with the birth-death of individuals. A new individual arrives at the group by the Poisson process, while new links are established in the network through either uniform connection or preferential attachment. Moreover, an existing individual has a limited lifespan before leaving the network. We determine stationary topological properties of these networks, including their size and mean degree. To address the effect of the birth-death evolution, we further study the information dynamics in the proposed network model from the random drift and natural selection perspective, based on assumptions of total-stochastic and fitness-driven evolution, respectively. In simulations, we analyze the fixation probability of individual information and find that means of new connections affect the random drift process but do not affect the natural selection process., Comment: 14 pages, 9 figures

Dengue is a viral vector-borne infectious disease that affects many countries worldwide, infecting around 390 million people per year. The main outbreaks occur in subtropical and tropical countries. We study here the influence of climate on dengue in Natal (2016-2019), Brazil, Iquitos (2001-2012), Peru, and Barranquilla (2011-2016), Colombia. For the analysis and simulations, we apply Machine Learning (ML) techniques, especially the Random Forest (RF) algorithm. In addition, regarding a feature in the ML technique, we analyze three possibilities: only dengue cases (D); climate and dengue cases (CD); humidity and dengue cases (HD). Depending on the city, our results show that the climate data can improve or not the forecast. For instance, for Natal, D induces a better forecast. For Iquitos, it is better to use CD. Nonetheless, for Barranquilla, the forecast is better, when we include cases and humidity data. For Natal, when we use more than 64\% and less than 80\% of the time series for training, we obtain results with correlation coefficients ($r$) among 0.917 and 0.949 and mean absolute errors (MAE) among 57.783 and 71.768 for the D case in forecasting. The optimal range for Iquitos is obtained when 79\% up to 88\% of the time series is considered for training. For this case, the best case is CD, having a minimum $r$ equal to 0.850 and maximum 0.887, while values of MAE oscillate among 2.780 and 4.156. For Barranquilla, the optimal range occurs between 72\% until 82\% of length training. In this case, the better approach is HD, where the measures exhibit a minimum $r$ equal to 0.942 and a maximum 0.953, while the minimum and maximum MAE vary between 6.085 and 6.669. We show that the forecast of dengue cases is a challenging problem and climate variables do not always help. However, when we include the mentioned climate variables, the most important one is humidity.

Spiral waves are spatial-temporal patterns that can emerge in different systems as heart tissues, chemical oscillators, ecological networks and the brain. These waves have been identified in the neocortex of turtles, rats, and humans, particularly during sleep-like states. Although their functions in cognitive activities remain until now poorly understood, these patterns are related to cortical activity modulation and contribute to cortical processing. In this work, we construct a neuronal network layer based on the spatial distribution of pyramidal neurons. Our main goal is to investigate how local connectivity and coupling strength are associated with the emergence of spiral waves. Therefore, we propose a trustworthy method capable of detecting different wave patterns, based on local and global phase order parameters. As a result, we find that the range of connection radius (R) plays a crucial role in the appearance of spiral waves. For R < 20 {\mu}m, only asynchronous activity is observed due to small number of connections. The coupling strength (gsyn ) greatly influences the pattern transitions for higher R, where spikes and bursts firing patterns can be observed in spiral and non-spiral waves. Finally, we show that for some values of R and gsyn bistable states of wave patterns are obtained.

Adaptive dynamical networks are network systems in which the structure co-evolves and interacts with the dynamical state of the nodes. We study an adaptive dynamical network in which the structure changes on a slower time scale relative to the fast dynamics of the nodes. We identify a phenomenon we refer to as recurrent adaptive chaotic clustering (RACC), in which chaos is observed on a slow time scale, while the fast time scale exhibits regular dynamics. Such slow chaos is further characterized by long (relative to the fast time scale) regimes of frequency clusters or frequency-synchronized dynamics, interrupted by fast jumps between these regimes. We also determine parameter values where the time intervals between jumps are chaotic and show that such a state is robust to changes in parameters and initial conditions.

A central question of network science is how functional properties of systems arise from their structure. For networked dynamical systems, structure is typically quantified with network measures. A functional property that is of theoretical and practical interest for oscillatory systems is the stability of synchrony to localized perturbations. Recently, Graph Neural Networks (GNNs) have been shown to predict this stability successfully; at the same time, network measures have struggled to paint a clear picture. Here we collect 46 relevant network measures and find that no small subset can reliably predict stability. The performance of GNNs can only be matched by combining all network measures and nodewise machine learning. However, unlike GNNs, this approach fails to extrapolate from network ensembles to several real power grid topologies. This suggests that correlations of network measures and function may be misleading, and that GNNs capture the causal relationship between structure and stability substantially better., Comment: 30 pages (16 pages main section), 15 figures, 4 tables

In this paper, we consider the partial quantum consensus problem of a qubit network in a distributed view. The local quantum operation is designed based on the Hamiltonian by using the local information of each quantum system in a network of qubits. We construct the unitary transformation for each quantum system to achieve the partial quantum consensus, i.e., the directions of the quantum states in the Bloch ball will reach an agreement. A simple case of two-qubit quantum systems is considered first, and a minimum completing time of reaching partial consensus is obtained based on the geometric configuration of each qubit. Furthermore, we extend the approaches to deal with the more general N-qubit networks. Two partial quantum consensus protocols, based on the Lyapunov method for chain graphs and the geometry method for connected graphs, are proposed. The geometry method can be utilized to deal with more general connected graphs, while for the Lyapunov method, the global consensus can be obtained. The numerical simulation over a qubit network is demonstrated to verify the validity and the effectiveness of the theoretical results., Comment: 11 pages, 8 figures

We study three different strategies of vaccination in a SEIRS (Susceptible--Exposed--Infected--Recovered--Susceptible) seasonal forced model, which are: ($i$) continuous vaccination; ($ii$) periodic short time localized vaccination and ($iii$) periodic pulsed width campaign. Considering the first strategy, we obtain an expression for the basic reproduction number and infer a minimum vaccination rate necessary to ensure the stability of the disease-free equilibrium (DFE) solution. In the second strategy, the short duration pulses are added to a constant baseline vaccination rate. The pulse is applied according to the seasonal forcing phases. The best outcome is obtained by locating the intensive immunization at inflection of the transmissivity curve. There, a vaccination rate of $44.4\%$ of susceptible individuals is enough to ensure DFE. For the third vaccination proposal, additionally to the amplitude, the pulses have a prolonged time width. We obtain a non-linear relationship between vaccination rates and the duration of the campaign. Our simulations show that the baseline rates, as well as the pulse duration, can substantially improve the vaccination campaign effectiveness. These findings are in agreement with our analytical expression. We show a relationship between the vaccination parameters and the accumulated number of infected individuals, over the years and show the relevance of the immunisation campaign annual reaching for controlling the infection spreading. Regarding the dynamical behaviour of the model, our simulations shows that chaotic and periodic solutions, as well as bi-stable regions, depend on the vaccination parameters range.

The generalized recurrence plot is a modern tool for quantification of complex spatial patterns. Its application spans the analysis of trabecular bone structures, Turing patterns, turbulent spatial plankton patterns, and fractals. Determinism is a central measure in this framework quantifying the level of regularity of spatial structures. We show by basic examples of fully regular patterns of different symmetries that this measure underestimates the orderliness of circular patterns resulting from rotational symmetries. We overcome this crucial problem by checking additional structural elements of the generalized recurrence plot which is demonstrated with the examples. Furthermore, we show the potential of the extended quantity of determinism applying it to more irregular circular patterns which are generated by the complex Ginzburg-Landau-equation and which can be often observed in real spatially extended dynamical systems. So, we are able to reconstruct the main separations of the system's parameter space analyzing single snapshots of the real part only, in contrast to the use of the original quantity. This ability of the proposed method promises also an improved description of other systems with complicated spatio-temporal dynamics typically occurring in fluid dynamics, climatology, biology, ecology, social sciences, etc., Comment: 10 pages, 7 figures

The increasing availability of highly resolved spatio-temporal data leads to new opportunities as well as challenges in many scientific disciplines such as climatology, ecology or epidemiology. This allows more detailed insights into the investigated spatially extended systems. However, this development needs advanced techniques of data analysis which go beyond standard linear tools since the more precise consideration often reveals nonlinear phenomena, for example threshold effects. One of these tools is the recurrence plot approach which has been successfully applied to the description of complex systems. Using this technique's power of visualization, we propose the analysis of the local minima of the underlying distance matrix in order to display driving forces of spatially extended systems. The potential of this novel idea is demonstrated by the analysis of the chlorophyll concentration and the sea surface temperature in the Southern California Bight. We are able not only to confirm the influence of El Ni\~no events on the phytoplankton growth in this region but also to confirm two discussed regime shifts in the California current system. This new finding underlines the power of the proposed approach and promises new insights into other complex systems., Comment: 14 pages, 10 figures

Irregular sampling is a common problem in palaeoclimate studies. We propose a method that provides regularly sampled time series and at the same time a difference filtering of the data. The differences between successive time instances are derived by a transformation costs procedure. A subsequent recurrence analysis is used to investigate regime transitions. This approach is applied on speleothem based palaeoclimate proxy data from the Indonesian-Australian monsoon region. We can clearly identify Heinrich events in the palaeoclimate as characteristic changes in the dynamics., Comment: 12 pages, 3 figures

Partially synchronized solitary states occur frequently when a synchronized system of networked oscillators is perturbed locally. Several asymptotic states of different frequencies can coexist at the same node. Here, we reveal the mechanism behind this multistability: additional solitary frequencies arise from the coupling between network modes and the solitary oscillator's frequency, leading to significant energy transfer. This can cause the solitary node's frequency to resonate with a Laplacian eigenvalue. We analyze which network structures enable this resonance and explain longstanding numerical observations. Another solitary state is characterized by the effective decoupling of the synchronized network and the solitary node at the natural frequency. Our framework unifies the description of solitary states near and far from resonance, allowing to predict the behavior of complex networks.

We show that delay-differential equations (DDE) exhibit universal bifurcation scenarios, which are observed in large classes of DDEs with a single delay. Each such universality class has the same sequence of stabilizing or destabilizing Hopf bifurcations. These bifurcation sequences and universality classes can be explicitly described by using the asymptotic continuous spectrum for DDEs with large delays. Here, we mainly study linear DDEs, provide a general transversality result for the delay-induced bifurcations, and consider three most common universality classes. For each of them, we explicitly describe the sequence of stabilizing and destabilizing bifurcations. We also illustrate the implications for a nonlinear Stuart-Landau oscillator with time-delayed feedback., Comment: 16 figures

Phase space reconstruction (PSR) methods allow for the analysis of low-dimensional data with methods from dynamical system theory, but their application to prediction models, like those from machine learning (ML), is limited. Therefore, we present here a model adaptive phase space reconstruction (MAPSR) method that unifies the process of PSR with the modeling of the dynamical system. MAPSR is a differentiable PSR method that enables the use of ML methods and is based on the idea of time delay embedding. For achieving differentiable, continuous, real-valued delays, which can be optimized using gradient descent, the discrete time signal is converted to a continuous time signal. The delay vector, which stores all potential embedding delays and the trainable parameters of the model, are simultaneously updated to achieve an optimal time delay embedding for the observed system. MAPSR does not rely on any threshold or statistical criterion for determining the dimension and the set of delay values for the embedding process. The quality of the reconstruction is evaluated by the prediction loss. We apply the proposed approach to uni- and multivariate time series stemming from regular and chaotic dynamical systems and a turbulent combustor to test the generalizability of the method and compare our results with established PSR methods. We find that for the Lorenz system, the model trained with the MAPSR method is able to predict chaotic time series for nearly 7 to 8 Lyapunov time scales, which is found to be much better compared to other PSR methods (AMI-FNN and PECUZAL methods). For the univariate time series from the turbulent combustor, the long-term prediction error of the model trained using the MAPSR method stays in between that of AMI-FNN and PECUZAL methods for the regime of chaos, and for the regime of intermittency, the MAPSR method outperforms the AMI-FNN and PECUZAL methods.

Earth system tipping elements have been predominantly investigated for their potential to trigger \textit{negative} ecological, climatic, and societal shifts. Yet, an overlooked but seminal avenue exists in the form of \textit{positive} tipping phenomena, whose underlying mechanisms and benefits remain largely underexplored. To bridge this gap, our research introduces a fundamental percolation-based framework to assess the criticality and resilience of planetary terrestrial vegetation systems. Leveraging high-resolution satellite data, we focus on greening-induced positive tipping dynamics driven by global warming. We feature the Qinghai-Tibetan Plateau (QTP) and the Sahel region as contrasting yet analogous case studies. Our analysis uncovers an intriguing phenomenon where vegetation fragmentation aligns with a percolation threshold, exhibiting a scale-invariant pattern characterized by nearly perfect power laws with three critical exponents. Remarkably, contrary to conventional destructive tipping elements, these regions act as favorable tipping elements, transitioning from fragmented to cohesive vegetation patterns due to anthropogenic climate change and afforestation efforts. Furthermore, we propose an \textit{optimal resilience enhancement model} to reinforce vegetation robustness while minimizing socio-economic costs. This study provides valuable insights into the favorable aspects of tipping elements under climate change and offers effective strategies for enhancing ecological resilience against environmental threats.

Spatio-temporal forecasting in various domains, like traffic prediction and weather forecasting, is a challenging endeavor, primarily due to the difficulties in modeling propagation dynamics and capturing high-dimensional interactions among nodes. Despite the significant strides made by graph-based networks in spatio-temporal forecasting, there remain two pivotal factors closely related to forecasting performance that need further consideration: time delays in propagation dynamics and multi-scale high-dimensional interactions. In this work, we present a Series-Aligned Multi-Scale Graph Learning (SAMSGL) framework, aiming to enhance forecasting performance. In order to handle time delays in spatial interactions, we propose a series-aligned graph convolution layer to facilitate the aggregation of non-delayed graph signals, thereby mitigating the influence of time delays for the improvement in accuracy. To understand global and local spatio-temporal interactions, we develop a spatio-temporal architecture via multi-scale graph learning, which encompasses two essential components: multi-scale graph structure learning and graph-fully connected (Graph-FC) blocks. The multi-scale graph structure learning includes a global graph structure to learn both delayed and non-delayed node embeddings, as well as a local one to learn node variations influenced by neighboring factors. The Graph-FC blocks synergistically fuse spatial and temporal information to boost prediction accuracy. To evaluate the performance of SAMSGL, we conduct experiments on meteorological and traffic forecasting datasets, which demonstrate its effectiveness and superiority., Comment: Accepted by Chaos

Self-sustained order can emerge in complex systems due to internal feedback between coupled subsystems. Here, we present our discovery of a non-monotonic emergence of order amidst chaos in a turbulent thermo-acoustic fluid system. Fluctuations play a vital role in determining the dynamical state and transitions in a system. In this work, we use complex networks to encode jumps in amplitude scales owing to fluctuations as links between nodes representing amplitude bins. The number of possible amplitude transitions at a fixed timescale reflects the complexity of dynamics at that timescale. The network entropy quantifies the number of and uncertainty associated with such transitions. Using network entropy, we show that the uncertainty in fluctuations first increases and then decreases as the system transitions from chaos via intermittency to order. The competition between turbulence and nonlinear interactions leads to such non-monotonic emergence of order amidst chaos in turbulent thermo-acoustic fluid systems.

Oscillatory instability (OI) emerges amidst turbulent states in experiments in various turbulent fluid and thermo-fluid systems such as aero-acoustic, thermoacoustic and aeroelastic systems. For the time series of the relevant dynamic variable at the onset of the OI, universal scaling behavior have been discovered in experiments via the Hurst exponent and certain spectral measures. By means of a center manifold reduction, the spatiotemporal dynamics of these real systems can be mapped to a complex Ginzburg-Landau equation with a linear global coupling (GCGLE). In this work, we show that the GCGLE is able to capture the universal behavior of the route to OI, elucidating it as a transition from defect to phase turbulence mediated by the global coupling.

Swarmalators are oscillators that can swarm as well as sync via a dynamic balance between their spatial proximity and phase similarity. We present a generalized D-dimensional swarmalator model, which is more realistic and versatile, that captures the self-organizing behaviors of a plethora of real-world collectives. This allows for modeling complicated processes such as flocking, schooling of fish, cell sorting during embryonic development, residential segregation, and opinion dynamics in social groups. We demonstrate its versatility by capturing the manoeuvers of the school of fish and traveling waves of gene expression, both qualitatively and quantitatively, embryonic cell sorting, microrobot collectives, and various life stages of slime mold by a suitable extension of the original model to incorporate appropriate features besides a gallery of its intrinsic self-organizations for various interactions. We expect this high-dimensional model to be potentially useful in describing swarming systems in a wide range of disciplines including physics of active matter, developmental biology, sociology, and engineering.

Jammed granular media and glasses exhibit spatial long-range correlations as a result of mechanical equilibrium. However, the existence of such correlations in the flowing matter, where the mechanical equilibrium is unattainable, has remained elusive. Here, we investigate this problem in the context of the percolation of interparticle forces in flowing granular media. We find that the flow rate introduces an effective long-range correlation, which plays the role of a relevant perturbation giving rise to a spectrum of varying exponents on a critical line as a function of the flow rate. Remarkably, our numerical simulations along with analytical arguments predict a crossover flow rate $\dot{\gamma}_c \simeq 10^{-5}$ below which the effect of induced disorder is weak and the universality of the force chain structure is shown to be given by the standard rigidity percolation. We also find a power-law behavior for the critical exponents with the flow rate $\dot{\gamma}>\dot{\gamma}_c$., Comment: 26 pages, 5 figures, to appear in Science Advances (2023)

We study two coupled active rotators with Kuramoto-type coupling and focus our attention to specific transitional regimes where the coupling is neither attractive nor repulsive. We show that certain such situations at the edge of synchronization can be characterized by the existence of a time-reversal symmetry of the system. We identify two different cases with such a time-reversal symmetry. The first case is characterized by a non-reciprocal attractive/repulsive coupling. The second case is a reciprocal coupling exactly at the edge between attraction and repulsion. We give a detailed description of possible different types of dynamics and bifurcations for both cases. In particular, we show how the time-reversible coupling can induce both oscillation death and oscillation birth to the active rotators. Moreover, we analyse the coexistence of conservative and dissipative regions in phase space, which is a typical feature of systems with a time-reversal symmetry. We show also, how perturbations breaking the time-reversal symmetry and destroying the conservative regions can lead to complicated types of dissipative dynamics such as the emergence of long-period cycles showing a bursting-like behavior., Comment: 29 pages, 14 figures

It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical networks.

A variation in the environment of a system, such as the temperature, the concentration of a chemical solution or the appearance of a magnetic field, may lead to a drift in one of the parameters. If the parameter crosses a bifurcation point, the system can tip from one attractor to another (bifurcation-induced tipping). Typically, this stability exchange occurs at a parameter value beyond the bifurcation value. This is what we call here the stability exchange shift. We study systematically how the shift is affected by the initial parameter value and its change rate. To that end, we present numerical and analytical results for different types of bifurcations and different paradigmatic systems. Finally, we deduce the scaling laws governing this phenomenon. We show that increasing the change rate and starting the drift further from the bifurcation can delay the tipping process. Furthermore, if the change rate is sufficiently small, the shift becomes independent of the initial condition (no memory) and the shift tends to zero as the square root of the change rate. Thus, the bifurcation diagram for the system with fixed parameters is recovered.

Reservoir computing (RC), a particular form of recurrent neural network, is under explosive development due to its exceptional efficacy and high performance in reconstruction or/and prediction of complex physical systems. However, the mechanism triggering such effective applications of RC is still unclear, awaiting deep and systematic exploration. Here, combining the delayed embedding theory with the generalized embedding theory, we rigorously prove that RC is essentially a high dimensional embedding of the original input nonlinear dynamical system. Thus, using this embedding property, we unify into a universal framework the standard RC and the time-delayed RC where we novelly introduce time delays only into the network's output layer, and we further find a trade-off relation between the time delays and the number of neurons in RC. Based on this finding, we significantly reduce the network size of RC for reconstructing and predicting some representative physical systems, and, more surprisingly, only using a single neuron reservoir with time delays is sometimes sufficient for achieving those tasks.

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems like the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges, and give perspectives on future research directions, looking to inspire interdisciplinary approaches., Comment: 46 pages, 9 figures

We introduce a new model consisting of globally coupled high-dimensional generalized limit-cycle oscillators, which explicitly incorporates the role of amplitude dynamics of individual units in the collective dynamics. In the limit of weak coupling, our model reduces to the $D$-dimensional Kuramoto phase model, akin to a similar classic construction of the well-known Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators. For the practically important case of $D=3$, the incoherence of the model is rigorously proved to be stable for negative coupling $(K<0)$ but unstable for positive coupling $(K>0)$; the locked states are shown to exist if $K>0$; in particular, the onset of amplitude death is theoretically predicted. For $D\geq2$, the discrete and continuous spectra for both locked states and amplitude death are governed by two general formulas. Our proposed $D$-dimensional model is physically more reasonable, because it is no longer constrained by fixed amplitude dynamics, which puts the recent studies of the $D$-dimensional Kuramoto phase model on a stronger footing by providing a more general framework for $D$-dimensional limit-cycle oscillators., Comment: Accepted for Publication in Physical Review Letters

Extreme multistability (EM) is characterized by the emergence of infinitely many coexisting attractors or continuous families of stable states in dynamical systems. EM implies complex and hardly predictable asymptotic dynamical behavior. We analyse a model for pendulum clocks coupled by springs and suspended on an oscillating base, and show how EM can be induced in this system by a specifically designed coupling. First, we uncover that symmetric coupling can increase the dynamical complexity. In particular, the coexistence of multiple isolated attractors and continuous families of stable periodic states is generated in a symmetric cross-coupling scheme of four pendulums. These coexisting infinitely many states are characterized by different levels of phase synchronization between the pendulums, including anti-phase and in-phase states. Some of the states are characterized by splitting of the pendulums into groups with silent sub-threshold and oscillating behavior, respectively. The analysis of the basins of attraction further reveals the complex dependence of EM on initial conditions., Comment: 13 pages, 9 figures

The rapidly shrinking Arctic sea ice is changing weather patterns and disrupting the balance of nature. Dynamics of Arctic weather variability (WV) plays a crucial role in weather forecasting and is closely related to extreme weather events. Yet, assessing and quantifying the WV for both local Arctic regions and its planetary impacts under anthropogenic climate change is still unknown. Here, we develop a complexity-based approach to systematically evaluate and analyze the dynamic behaviour of WV. We reveal that the WV within and around the Arctic is statistically correlated to the Arctic Oscillation at the intraseasonal time scale. We further find that the variability of the daily Arctic sea ice is increasing due to its dramatic decline under a warming climate. Unstable Arctic weather conditions can disturb regional weather patterns through atmospheric teleconnection pathways, resulting in higher risk to human activities and greater weather forecast uncertainty. A multivariate climate network analysis reveals the existence of such teleconnections and implies a positive feedback loop between the Arctic and global weather instabilities. This enhances the mechanistic understanding of the influence of Arctic amplification on mid-latitude severe weather. Our framework provides a fresh perspective on the linkage of complexity science, WV and the Arctic.

In this paper we show the possibility of creating and identifying the features of an artificial neural network (ANN) which consists of mathematical models of biological neurons. The FitzHugh--Nagumo (FHN) system is used as an example of model demonstrating simplified neuron activity. First, in order to reveal how biological neurons can be embedded within an ANN, we train the ANN with nonlinear neurons to solve a a basic image recognition problem with MNIST database; and next, we describe how FHN systems can be introduced into this trained ANN. After all, we show that an ANN with FHN systems inside can be successfully trained and its accuracy becomes larger. What has been done above opens up great opportunities in terms of the direction of analog neural networks, in which artificial neurons can be replaced by biological ones. \end{abstract}, Comment: 6 pages, 7 figures, 2 tables

The Texan electric network in the Gulf Coast of the United States is frequently hit by Tropical Cyclones (TC) causing widespread power outages, a risk that is expected to substantially increase under global warming. Here, we introduce a new approach of combining a probabilistic line fragility model with a network model of the Texas grid to simulate the temporal evolution of wind-induced failures of transmission lines and the resulting cascading power outages from seven major historical hurricanes. The approach allows reproducing observed supply failures. In addition, compared to a static approach, it provides a significant advantage in identifying critical lines whose failure can trigger large supply shortages. We show that protecting only 1% of total lines can reduce the likelihood of the most destructive type of outages by a factor of between 5 and 20. The proposed modelling approach could represent a tool so far missing to effectively strengthen the power grids against future hurricane risks even under limited knowledge.

Network (or graph) sparsification compresses a graph by removing inessential edges. By reducing the data volume, it accelerates or even facilitates many downstream analyses. Still, the accuracy of many sparsification methods, with filtering-based edge sampling being the most typical one, heavily relies on an appropriate definition of edge importance. Instead, we propose a different perspective with a generalized local-property-based sampling method, which preserves (scaled) local \emph{node} characteristics. Apart from degrees, these local node characteristics we use are the expected (scaled) number of wedges and triangles a node belongs to. Through such a preservation, main complex structural properties are preserved implicitly. We adapt a game-theoretic framework from uncertain graph sampling by including a threshold for faster convergence (at least $4$ times faster empirically) to approximate solutions. Extensive experimental studies on functional climate networks show the effectiveness of this method in preserving macroscopic to mesoscopic and microscopic network structural properties., Comment: 8 pages, 10 figures, to be published in IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, 2022 (ASONAM 2022)

We introduce a new formulation of local clustering coefficient for weighted correlation networks. This new formulation is based upon a definition introduced previously in the neuroscience context and aimed at compensating for spurious correlations caused by indirect interactions. We modify this definition further by replacing Pearson's pairwise correlation coefficients and three-way partial correlation coefficients by the respective Kendall's rank correlations. This reduces statistical sample size requirements to compute the correlations, which translates into the possibility of using shorter time windows and hence into a shorter response time of the real-time climate network analysis. We construct evolving climate networks of mean sea level pressure fluctuations and analyze anomalies of local clustering coefficient in these networks. We develop a broadly applicable statistical methodology to study association between spatially inhomogeneous georeferenced multivariate time series and binary-valued spatiotemporal data (or other data reducible to this representation) and use it to compare the newly proposed formulation of local clustering coefficient (for weighted correlation networks) to the conventional one (for unweighted graphs) in terms of the association of these measures in climate networks to tropical cyclones. Thus we substantiate the previously made observation that tropical cyclones are associated with anomalously high values of local clustering coefficient, and confirm that the new formulation shows a stronger association.

The stickiness effect is a fundamental feature of quasi-integrable Hamiltonian systems. We propose the use of an entropy-based measure of the recurrence plots (RP), namely, the entropy of the distribution of the recurrence times (estimated from the RP), to characterize the dynamics of a typical quasi-integrable Hamiltonian system with coexisting regular and chaotic regions. We show that the recurrence time entropy (RTE) is positively correlated to the largest Lyapunov exponent, with a high correlation coefficient. We obtain a multi-modal distribution of the finite-time RTE and find that each mode corresponds to the motion around islands of different hierarchical levels., Comment: 16 pages, 7 figures

The study of chimera states or, more generally, coherence-incoherence patterns has led to the development of several tools for their identification and characterization. In this work, we extend the eigenvalue decomposition method to distinguish between single-well and double-well patterns. By applying our method, we are able to identify the following four types of dynamical patterns in a ring of nonlocally coupled Chua circuits and nonlocally coupled cubic maps: single-well cluster, single-well coherence-incoherence pattern, double-well cluster, and double-well coherence-incoherence. In a ring-star network of Chua circuits, we investigate the influence of adding a central node on the spatio-temporal patterns. Our results show that increasing the coupling with the central node favors the occurrence of single-well coherence-incoherence states. We observe that the boundaries of the attraction basins resemble fractal and riddled structures

Tipping elements of the Earth system may shift abruptly and irreversibly from one state to another at tipping points, resulting in a growing threat to our society. Yet, it is not fully clear how to assess and quantify the influence of a tipping element and how to explore the teleconnections between different tipping elements. To fill this knowledge gap, we propose a climate network approach to quantitatively analyze the global impacts of a prominent tipping element, the Amazon Rainforest Area (ARA). We find that regions, such as, the Tibetan Plateau (TP) and West Antarctic ice sheet, are characterized by higher network weighted links and exhibit strong correlations with the ARA. We then identify a teleconnection propagation path between the ARA and the TP. This path is robust under climate change as simulated by various climate models of CMIP5 and CMIP6. In addition, we detect early warning signals for critical transition in the snow cover extent on the Tibetan Plateau by applying critical slowing down indicators, lag-1 autocorrelation and detrended fluctuation analysis. We find that the snow cover of the TP has been losing stability since 2008, revealing that the TP is operating like a tipping element and approaching a potential tipping point. We further uncover that various climate extremes between the ARA and the TP are significantly synchronized under climate change. Our framework provides new insights into how tipping elements are linked to each other and into the potential predictability of cascading tipping dynamics.

A class of nucleation and growth models of a stable phase (S-phase) is investigated for various different growth velocities. It is shown that for growth velocities $v\sim s(t)/t$ and $v\sim x/\tau(x)$, where $s(t)$ and $\tau$ are the mean domain size of the metastable phase (M-phase) and the mean nucleation time respectively, the M-phase decays following a power law. Furthermore, snapshots at different time $t$ are taken to collect data for the distribution function $c(x,t)$ of the domain size $x$ of M-phase are found to obey dynamic scaling. Using the idea of data-collapse we show that each snapshot is a self-similar fractal. However, for $v={\rm const.}$ like in the classical Kolmogorov-Johnson-Mehl-Avrami (KJMA) model and for $v\sim 1/t$ the decay of the M-phase are exponential and they are not accompanied by dynamic scaling. We find a perfect agreement between numerical simulation and analytical results., Comment: 10 pages, 8 captioned figures

Cyclones are amongst the most hazardous extreme weather events on Earth. When two co-rotating cyclones come in close proximity, a possibility of complete merger (CM) arises due to their interactions. However, identifying the transitions in the interaction of binary cyclones and predicting the merger is challenging for weather forecasters. In the present study, we suggest an innovative approach to understand the evolving vortical interactions between the cyclones during two such CM events using time-evolving induced velocity based unweighted directed networks. We find that network-based indicators, namely, in-degree and out-degree, can quantify the changes during the interaction between two cyclones and are better candidates than the traditionally used separation distance to classify the interaction stages before a CM. The network indicators also help to identify the dominating cyclone during the period of interaction and quantify the variation of the strength of the dominating and merged cyclones. Finally, we show that the network measures also provide an early indication of the CM event well before its occurrence., Comment: 16 pages in double columns, 8 figures

We assume that the state of a number of nodes in a network could be investigated if necessary, and study what configuration of those nodes could facilitate a better solution for the diffusion-source-localization (DSL) problem. In particular, we formulate a candidate set which contains the diffusion source for sure, and propose the method, Percolation-based Evolutionary Framework (PrEF), to minimize such set. Hence one could further conduct more intensive investigation on only a few nodes to target the source. To achieve that, we first demonstrate that there are some similarities between the DSL problem and the network immunization problem. We find that the minimization of the candidate set is equivalent to the minimization of the order parameter if we view the observer set as the removal node set. Hence, PrEF is developed based on the network percolation and evolutionary algorithm. The effectiveness of the proposed method is validated on both model and empirical networks in regard to varied circumstances. Our results show that the developed approach could achieve a much smaller candidate set compared to the state of the art in almost all cases. Meanwhile, our approach is also more stable, i.e., it has similar performance irrespective of varied infection probabilities, diffusion models, and outbreak ranges. More importantly, our approach might provide a new framework to tackle the DSL problem in extreme large networks.

Percolation establishes the connectivity of complex networks and is one of the most fundamental critical phenomena for the study of complex systems. On simple networks, percolation displays a second-order phase transition; on multiplex networks, the percolation transition can become discontinuous. However, little is known about percolation in networks with higher-order interactions. Here, we show that percolation can be turned into a fully-fledged dynamical process when higher-order interactions are taken into account. By introducing signed triadic interactions, in which a node can regulate the interactions between two other nodes, we define triadic percolation. We uncover that in this paradigmatic model the connectivity of the network changes in time and that the order parameter undergoes a period-doubling and a route to chaos. We provide a general theory for triadic percolation which accurately predicts the full phase diagram on random graphs as confirmed by extensive numerical simulations. We find that triadic percolation on real network topologies reveals a similar phenomenology. These results radically change our understanding of percolation and may be used to study complex systems in which the functional connectivity is changing in time dynamically and in a non-trivial way, such as in neural and climate networks., Comment: 62 pages, 20 figures