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We investigate the spatial dynamics of two-disease epidemics reaching a three-species cyclic model. Regardless of their species, all individuals are susceptible to being infected with two different pathogens, which spread through person-to-person contact. We consider that the simultaneous presence of multiple infections leads to a synergistic amplification in the probability of host mortality due to complications arising from any of the co-occurring diseases. Employing stochastic simulations, we explore the ramifications of this synergistic coinfection on spatial configurations that emerge from stochastic initial conditions. Under conditions of pronounced synergistic coinfection, we identify the emergence of zones inhabited solely by hosts affected by a singular pathogen. At the boundaries of spatial domains dominated by a single disease, interfaces of coinfected hosts appear. The dynamics of these interfaces are shaped by curvature-driven processes and display a scaling behavior reflective of the topological attributes of the underlying two-dimensional space. As the lethality linked to coinfection diminishes, the evolution of the interface network's spatial dynamics is influenced by fluctuations stemming from waves of coinfection that infiltrate territories predominantly occupied by a single disease. Our analysis extends to quantifying the implications of synergistic coinfection at both the individual and population levels Our outcomes show that organisms' infection risk is maximized if the coinfection increases the death due to disease by 30 % and minimized as the network dynamics reach the scaling regime, with species populations being maximum. Our conclusions may help ecologists understand the dynamics of epidemics and their impact on the stability of ecosystems. [ABSTRACT FROM AUTHOR]

Antipredator behavior is a self-preservation strategy present in many biological systems, where individuals join the effort in a collective reaction to avoid being caught by an approaching predator. We study a nonhierarchical tritrophic system, whose predator–prey interactions are described by the rock–paper–scissors game rules. We perform a set of spatial stochastic simulations where organisms of one out of the species can resist predation in a collective strategy. The drop in predation capacity is local, which means that each predator faces a particular opposition depending on the prey group size surrounding it. Considering that the interference in a predator action depends on the prey's physical and cognitive ability, we explore the role of a conditioning factor that indicates the fraction of the species apt to perform the antipredator strategy. Because of the local unbalancing of the cyclic predator–prey interactions, departed spatial domains mainly occupied by a single species emerge. Unlike the rock–paper–scissors model with a weak species because of a nonlocal reason, our findings show that if the predation probability of one species is reduced because individuals face local antipredator response, the species does not predominate. Instead, the local unbalancing of the rock–paper–scissors model results in the prevalence of the weak species' prey. Finally, the outcomes show that local unevenness may jeopardize biodiversity, with the coexistence being more threatened for high mobility. [ABSTRACT FROM AUTHOR]

Recognizing surrounding situations, such as enemy attacks, which can be realized by predator–prey relationships, is one of the common behaviors of the population in ecosystems. In this paper, we explore the relationship between such species' behavior and biodiversity in the spatial rock–paper–scissors game by employing the ecological concept "vigilance." In order to describe the vigilance process, we adopt a multiplex structure where two distinct layers describe virtual and physical interactions. By investigating the process of evolution in species, we also found that species with different vigilance go together. In addition, by utilizing the dynamic time warping method, we found that species with the same vigilance have consistent behavior, but species with different vigilance have diverse behavior. Our findings may lead to broader interpretations of mechanisms promoting biodiversity via vigilance in species ecosystems. [ABSTRACT FROM AUTHOR]

The role of mutation, which is an error process in gene evolution, in systems of cyclically competing species has been studied from various perspectives, and it is regarded as one of the key factors for promoting coexistence of all species. In addition to naturally occurring mutations, many experiments in genetic engineering have involved targeted mutation techniques such as recombination between DNA and somatic cell sequences and have studied genetic modifications through loss or augmentation of cell functions. In this paper, we investigate nonlinear dynamics with targeted mutation in cyclically competing species. In different ways to classic approaches of mutation in cyclic games, we assume that mutation may occur in targeted individuals who have been removed from intraspecific competition. By investigating each scenario depending on the number of objects for targeted mutation analytically and numerically, we found that targeted mutation can lead to persistent coexistence of all species. In addition, under the specific condition of targeted mutation, we found that targeted mutation can lead to emergences of bistable states for species survival. Through the linear stability analysis of rate equations, we found that those phenomena are accompanied by Hopf bifurcation which is supercritical. Our findings may provide more global perspectives on understanding underlying mechanisms to control biodiversity in ecological/biological sciences, and evidences with mathematical foundations to resolve social dilemmas such as a turnover of group members by resigning with intragroup conflicts in social sciences. [ABSTRACT FROM AUTHOR]

Scientific cooperation on an international level has been well studied in the literature. However, much less is known about this cooperation on an intercontinental level. In this paper, we address this issue by creating a collection of approximately 13.8 × 10 6 publications around the papers by one of the highly cited authors working in complex networks and their applications. The obtained rank-frequency distribution of the probability of sequences describing continents and the number of countries—with which authors of papers are affiliated—follows the power law with an exponent − 1.9108 (15). Such dependence is known in the literature as Zipf's law, and it has been originally observed in linguistics; later, it turned out that it is very commonly observed in various fields. The number of distinct "continent (number of countries)" sequences in a function of the number of analyzed papers grows according to power law with exponent 0.527 (14) ; i.e., it follows Heap's law. [ABSTRACT FROM AUTHOR]

Hypergraph is the model of relations lying in clusters of objects. Identifying vital nodes is a fundamental problem in the analysis of the hypergraph. To reflect the multilayer feature of the hypergraph, in this paper, we deconstruct the hypergraph into a simplicial complex and analyze the homological dual relations of boundary and coboundary between simplices. For clarity, these two relations are summarized into a bidirectional graph, called the simplicial diagram, which provides a global framework for the exploration of the hypergraph. To determine the node importance in the hypergraph, we propose a parameter-free eigenvector centrality for weighted hypergraphs in terms of a simplicial complex, named Simplicial DualRank centrality. For each simplex, we define two indices of importance, the inner centrality and the outer centrality. Inner centrality transmits according to the relation of coboundary, which converts to outer centrality at the hyperlinks; in duality, outer centrality transmits according to the relation of boundary, which converts to inner centrality at the nodes. Therefore, a circuit of centrality is constructed on the simplicial diagram, the steady state of which defines the Simplicial DualRank centrality of all the simplices in the hypergraph. Moreover, we apply the Simplicial DualRank centrality to weighted complex networks, which results in a variant of the classical eigenvector centrality. Finally, experimental results in a science collaboration dataset show that the Simplicial DualRank can identify Nobel laureates from the prize-winning papers in Physics, top scientists should select collaborators more carefully to maintain their research quality, and scholars tend to find relatively effective collaborations in their future research. [ABSTRACT FROM AUTHOR]

Machine learning (ML), a subset of artificial intelligence, refers to methods that have the ability to "learn" from experience, enabling them to carry out designated tasks. The main focus of this Focus Issue is on the new algorithms, strategies, and techniques with machine learning applied to complex systems and on applying complex system techniques to leverage the performance of machine learning techniques with high-efficiency. Prediction and manipulations of complex systems Machine Learning (ML) algorithms have provided great convenience in terms of analyzing and predicting the dynamics of chaotic and complex systems. [Extracted from the article]

This paper presents experimental switching between two attractors in the swinging bell. In the considered yoke–bell–clapper system, two coexisting solutions appear. In the first one, we observe a single impact between the bell and the clapper per one period of motion, and in the second solution, no impacts occur—no sound is produced. Based on the time-dependent stability margin method, we numerically detect parts of the trajectories where the system is most prone to perturbations. Using this knowledge, we experimentally investigate switching between attractors by applying the perturbation to the clapper. We show that we can easily enforce the change of attractor by properly timing the perturbation. The results prove that, based on the results from the time-dependent stability margin numerical method, we are able to effectively alter the wrong operation of the bell (lack of impact) to the correct operation (solution with impact). The analysis is conducted on the real-world mechanical system rather than paradigmatic examples. Therefore, it contributes to the subject of multistability and nonlinearity in engineering design. Novel, recently developed methods for analyzing multistable systems are successfully employed during the investigation. The paper shows that a complex phenomenon of multistability observed in the system, which is considered simple and undemanding from an engineering design point of view. [ABSTRACT FROM AUTHOR]

Generosity through donation plays a crucial role in reducing inequality and influencing human behavior. However, previous research on donation has overlooked individuals' acceptance of the extent of inequality, which acts as a trigger for donation. To address this gap, this paper systematically explores the impact of donation based on inequality tolerance on the evolution of cooperation in spatial public goods game. Specifically, donation occurs only when an individual's payoff advantage exceeds her inequality tolerance. The results show that donation patterns are crucial for the emergence and stability of cooperation. In the enduring period, the defector-to-cooperator donation pattern helps to form cooperative clusters. In the expanding period, cooperator-to-cooperator, defector-to-defector, and defector-to-cooperator donation patterns create a stable three-layer structure through self-organization, providing a payoff advantage to boundary cooperators. As donation ratio increases, the three-layer structure provides a greater payoff advantage to boundary cooperators, leading to an increase in cooperation. As inequality tolerance increases, changes in donation patterns weaken the three-layer structure, causing cooperation to decrease or disappear through discontinuous phase transitions. Subsequently, all critical points of discontinuous phase transitions are identified by specific spatial configurations. In addition, the influence of donation patterns on the evolution of cooperation is robust, even in heterogeneous small-world networks. This paper offers valuable insights into the dynamics of cooperation evolution and the role of donation in shaping behavior. [ABSTRACT FROM AUTHOR]

Given two unidirectionally coupled nonlinear systems, we speak of generalized synchronization when the responder "follows" the driver. Mathematically, this situation is implemented by a map from the driver state space to the responder state space termed the synchronization map. In nonlinear times series analysis, the framework of the present work, the existence of the synchronization map amounts to the invertibility of the so-called cross map, which is a continuous map that exists in the reconstructed state spaces for typical time-delay embeddings. The cross map plays a central role in some techniques to detect functional dependencies between time series. In this paper, we study the changes in the "noiseless scenario" just described when noise is present in the driver, a more realistic situation that we call the "noisy scenario." Noise will be modeled using a family of driving dynamics indexed by a finite number of parameters, which is sufficiently general for practical purposes. In this approach, it turns out that the cross and synchronization maps can be extended to the noisy scenario as families of maps that depend on the noise parameters, and only for "generic" driver states in the case of the cross map. To reveal generalized synchronization in both the noiseless and noisy scenarios, we check the existence of synchronization maps of higher periods (introduced in this paper) using recurrent neural networks and predictability. The results obtained with synthetic and real-world data demonstrate the capability of our method. [ABSTRACT FROM AUTHOR]

This paper introduces a rumor propagation model with saturation incidence, based on hypergraph theory. Hypergraphs can capture the higher-order interactions between nodes in a social network, where the node degree is substituted with hyperdegree. First, the threshold for rumor spreading model is obtained, the global asymptotically stable of the rumor-free equilibrium, and the global attractive and global asymptotically stable of the rumor-prevailing equilibrium are proved. Then, we incorporate impulsive vaccination into the model to represent the guided effect, simulating the periodic educational activities organized by official institutions to counteract rumor dissemination. By employing the comparison theorem, the paper demonstrates the global attractiveness of the rumor-free periodic solution and the persistence of the model. Finally, through numerical simulations, the paper compares the effects of higher-order and pairwise interactions on rumor spreading, validating the theoretical conclusions. The results indicate that higher-order interactions can promote rumor spreading, while impulsive vaccination can effectively decrease the scale of rumor spreading. [ABSTRACT FROM AUTHOR]

This paper presents a survey and critical analysis of the mathematical literature on modeling of dynamic populations living in a fluid medium. The present review paper is divided into two main parts: The first part deals with the multiscale derivation of deterministic and stochastic cross-diffusion systems governed by the incompressible Navier–Stokes equations. The derivation is obtained from the underlying description at the microscopic scale in kinetic theory models according to the micro–macro decomposition method. In the second part of this review, we are delighted to present a new variety of mathematical models describing different applications, namely, the pursuit–evasion dynamics, cancer invasion, and virus dynamics. Finally, critical analysis and future research perspectives are discussed. [ABSTRACT FROM AUTHOR]

Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system. [ABSTRACT FROM AUTHOR]

The Edward N. Lorenz Early Career Award was given to three young researchers for their outstanding papers published in Chaos in 2022. The winners are Tiemo Pedergnana from ETH Zurich, Thomas Lilienkamp from Nuremberg Institute of Technology, and Yuzuru Kato from the University of Tokyo. Each winner will receive a $2000 honorarium and is invited to contribute a perspective article to the journal. The winning papers cover topics such as seeking hidden exact potentials in dynamical systems, developing low-energy defibrillation strategies for cardiac arrhythmias, and defining an asymptotic phase function for quantum nonlinear oscillators. The authors were eligible for the award if they received their Ph.D. within the past five years, their master's degree within the past eight years, or if they are current students. [Extracted from the article]

In this paper, we address the issue of testing two-dimensional Gaussian processes with a defined cross-dependency structure. Multivariate Gaussian processes are widely used in various applications; therefore, it is essential to identify the theoretical model that accurately describes the data. While it is relatively straightforward to do so in a one-dimensional case, analyzing multi-dimensional vectors requires considering the dependency between the components, which can significantly affect the efficiency of statistical methods. The testing methodology presented in this paper is based on the sample cross-covariance function and can be considered a natural generalization of the approach recently proposed for testing one-dimensional Gaussian processes based on the sample autocovariance function. We verify the efficiency of this procedure on three classes of two-dimensional Gaussian processes: Brownian motion, fractional Brownian motion, and two-dimensional autoregressive discrete-time process. The simulation results clearly demonstrate the effectiveness of the testing methodology, even for small sample sizes. The theoretical and simulation results are supported by analyzing two-dimensional real-time series that describe the main risk factors of a mining company, namely, copper price and exchange rates (USDPLN). We believe that the introduced methodology is intuitive and relatively simple to implement, and thus, it can be applied in many real-world scenarios where multi-dimensional data are examined. [ABSTRACT FROM AUTHOR]

This is an introductory paper of the Focus Issue Recent advances in modeling complex systems: Theory and applications, where papers presenting new advances and insights into chaotic dynamics, fractional dynamics, complex oscillations, complex traffic dynamics, and complex networks, as well as their applications, are collected. All these different problems share common ideas and methods and provide new perspectives for further progress in the modeling of complex systems. [ABSTRACT FROM AUTHOR]

Inappropriate human activities contribute to the degradation of ecosystems in arid or semi-arid regions. Therefore, emphasizing the importance of strategies for restoring vegetation in these areas cannot be overstated. However, there has been insufficient research on how to develop effective restoration strategies at minimal cost. This paper addresses this gap by studying how optimizing the spatiotemporal distribution of human activities through local and boundary controls can reduce the level of desertification in vegetation pattern structures, thereby facilitating the recovery of arid land vegetation. The results indicate that vegetation restoration depends on the proportion and number of human activity areas, with a trade-off between them. Furthermore, consistent conclusions were obtained on circular regions, demonstrating the robustness of the approach to boundary shapes. This paper aims to offer new insights into the restoration of arid land vegetation and the prevention of catastrophic ecosystem changes from the perspective of optimal control. [ABSTRACT FROM AUTHOR]

Analysis of correlation between time series is an essential step for complex system studies and dynamical characteristics extractions. Martingale difference correlation (MDC) theory is mainly concerned with the correlation of conditional mean values between response variables and predictor variables. It is the generalization and deepening of the Pearson correlation coefficient, Spearman correlation coefficient, Kendall correlation coefficient, and other statistics. In this paper, on the basis of phase space reconstruction, the generalized dependence index (GDI) is proposed by using MDC and martingale difference divergence matrix theories, which can measure the degree of dependence between time series more effectively. Moreover, motivated by the theoretical framework of the refined distance correlation method, the corresponding dependence measure (DE) is employed in this paper to construct the DE-GDI plane, so as to comprehensively and intuitively distinguish different types of data and deeply explore the operating mechanism behind the relevant time series and complex systems. According to the performances tested by the different simulated and real-world data, our proposed method performs relatively reasonably and reliably in dependence measuring and data distinguishing. The proposal of this complex data clustering method can not only recognize the features of complex systems but also distinguish them effectively so as to acquire more relevant detailed information. [ABSTRACT FROM AUTHOR]

Parameter estimation is important in data-driven fractional chaotic systems. Less work has been reported due to challenges in discretization of fractional calculus operators. In this paper, several numerical schemes are newly derived for delay fractional difference equations of Caputo and Riemann–Liouville types. Then, loss functions are constructed and unknown parameters of the discrete fractional chaotic system are estimated by a neural network method. Parameter estimation results demonstrate high accuracy compared with real values. Robust analysis is provided under different noise levels. It can be concluded that this paper provides an efficient deep learning method based on fractional discrete-time systems. [ABSTRACT FROM AUTHOR]

The channel delay in the game process has an important influence on its evolutionary dynamics. This paper aims to optimize the strategy game with general information delays, including the state delay in the previous work, and the control delay that is introduced for the first time to depict the time consumed by strategy propagation in reality. Specifically, the dynamics of networked evolutionary games is transformed into an algebraic form by use of the newly proposed semi-tensor product of matrices, which extends the ordinary matrix multiplication. Subsequently, according to the values of control and state delays, the strategy optimization problem can be divided into six different cases, and then via the constructed algebraic equation, we can obtain the sufficient and necessary conditions for the existence of the strategy optimization. Meanwhile, based on a reachable set method, the corresponding feedback controllers are further designed. Last, one illustrative example is taken to demonstrate the feasibility of our model. The results of this paper will be helpful to investigate the game-based control issues in the complex networked environment. [ABSTRACT FROM AUTHOR]

This paper reports the novel results on fractional order-induced bifurcation of a tri-neuron fractional-order neural network (FONN) with delays and instantaneous self-connections by the intersection of implicit function curves to solve the bifurcation critical point. Firstly, it considers the distribution of the root of the characteristic equation in depth. Subsequently, it views fractional order as the bifurcation parameter and establishes the transversal condition and stability interval. The main novelties of this paper are to systematically analyze the order as a bifurcation parameter and concretely establish the order critical value through an implicit function array, which is a novel idea to solve the critical value. The derived results exhibit that once the value of the fractional order is greater than the bifurcation critical value, the stability of the system will be smashed and Hopf bifurcation will emerge. Ultimately, the validity of the developed key fruits is elucidated via two numerical experiments. [ABSTRACT FROM AUTHOR]

The paper studies the finite-time stability (FTS) of equilibrium points (EPs) in fractional-order nonlinear systems (FONSs). Classifying the EPs into initial EP and finite-time EP innovatively, equations of the EPs in FONSs are investigated comprehensively, and a unified definition depicting the EPs in the FONSs is proposed. Furthermore, sufficient conditions of the FTS of EPs in FONSs are given. The proposed results are verified with an illustrative example. Thus, different from existing works that declare non-existence of finite-time stable equilibria in FONSs, the existence of finite-time stable EPs in FONSs is confirmed in this paper. [ABSTRACT FROM AUTHOR]

In this paper, the properties of equilibriums in planar polynomial dynamical systems are studied. The homoclinic networks of sources, sinks, and saddles in self-univariate polynomial systems are discussed, and the numbers of sources, sinks, and saddles are determined through a theorem, and the first integral manifolds are determined. The corresponding proof of the theorem is completed, and a few illustrations of networks for source, sinks, and saddles are presented for a better understanding of the homoclinic networks. Such homoclinic networks are without any centers even if the networks are separated by the homoclinic orbits. The homoclinic networks of positive and negative saddles with clockwise and counterclockwise limit cycles in crossing-univariate polynomial systems are studied secondly, and the numbers of saddles and centers are determined through a theorem, and the first integral manifolds are determined through polynomial functions. The corresponding proof of the theorem is given, and a few illustrations of networks of saddles and centers are given to show the corresponding geometric structures. Such homoclinic networks of saddles and centers are without any sources and sinks. Since the maximum equilibriums for such two types of planar polynomial systems with the same degrees are discussed, the maximum centers and saddles should be obtained, and maximum sinks, sources, and saddles should be achieved. This paper may provide a different way to determine limit cycles in the Hilbert 16th problem. [ABSTRACT FROM AUTHOR]

This paper proposes an approach for identifying unknown switching topology in a complex dynamical network. The setup is divided into two components: a primary drive network and a specialized response network equipped with switched topology observers. Each class of observers is dedicated to tracking a specific topology structure. The updating law for these observers is dynamically adjusted based on the operational status of the corresponding topology in the drive network—active if engaged and dormant if not. The sufficient conditions for successful identification are obtained by employing adaptive synchronization control and the Lyapunov function method. In particular, this paper abandons the generally used assumption of linear independence and adopts an easily verifiable condition for accurate identification. The result shows that the proposed identification method is applicable for any finite switching periods. By employing the chaotic Lü system and the Lorenz system as the local dynamics of the networks, numerical examples demonstrate the effectiveness of the proposed topology identification method. [ABSTRACT FROM AUTHOR]

It is known that when symbolizing a time series into ordinal patterns using the Bandt-Pompe (BP) methodology, there will be ordinal patterns called forbidden patterns that do not occur in a deterministic series. The existence of forbidden patterns can be used to identify deterministic dynamics. In this paper, the ability to use forbidden patterns to detect determinism in irregularly sampled time series is tested on data generated from a continuous model system. The study is done in three parts. First, the effects of sampling time on the number of forbidden patterns are studied on regularly sampled time series. The next two parts focus on two types of irregular-sampling, missing data and timing jitter. It is shown that forbidden patterns can be used to detect determinism in irregularly sampled time series for low degrees of sampling irregularity (as defined in the paper). In addition, comments are made about the appropriateness of using the BP methodology to symbolize irregularly sampled time series. [ABSTRACT FROM AUTHOR]

In this paper, a class of scalar quartic polynomial delay systems is investigated. We found rich dynamics in this system through numerical simulation, including chaotic attractors, chaotic saddles, and intermittent chaos. Moreover, this chaotic quartic system may serve as an approximation, through Taylor expansion, for a wide class of scalar delay differential equations. Thus, these nonlinear systems may exhibit chaotic behaviors, and the studies in our paper may provide an insight into the emergence of chaos in other time-delay nonlinear systems. We also conduct a detailed theoretical analysis of the system, including the stability of equilibria and Hopf bifurcation analysis based on the theory of normal form and center manifold. Additionally, a numerical analysis is provided to give numerical evidence for the existence of chaos. [ABSTRACT FROM AUTHOR]

This paper develops new mathematical techniques to identify temporal shifts among a collection of US equities partitioned into a new and more detailed set of market sectors. Although conceptually related, our three analyses reveal distinct insights about financial markets, with meaningful implications for investment managers. First, we explore a variety of methods to identify nonlinear shifts in a market sector structure and describe the mathematical connection between the measure used and the captured phenomena. Second, we study a network structure with respect to our new market sectors and identify meaningfully connected sector-to-sector mappings. Finally, we conduct a series of sampling experiments over different sample spaces and contrast the distribution of Sharpe ratios produced by long-only, long-short, and short-only investment portfolios. In addition, we examine the sector composition of the top-performing portfolios for each of these portfolio styles. In practice, the methods proposed in this paper could be used to identify regime shifts, optimally structured portfolios, and better communities of equities. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the intricate local dynamics at equilibria within a two-dimensional model of hepatitis C virus (HCV) alongside hepatocyte homeostasis. The study investigates the existence of bifurcation sets and conducts a comprehensive bifurcation analysis to elucidate the system's behavior under varying conditions. A significant focus lies on understanding how changes in parameters can lead to bifurcations, which are pivotal points where the qualitative behavior of the system undergoes fundamental transformations. Moreover, the paper introduces and employs hybrid control feedback and Ott–Grebogi–Yorke strategies as tools to manage and mitigate chaos inherent within the HCV model. This chaos arises due to the presence of flip and Neimark–Sacker bifurcations, which can induce erratic behavior in the system. Through the implementation of these control strategies, the study aims to stabilize the system and restore it to a more manageable and predictable state. Furthermore, to validate the theoretical findings and the efficacy of the proposed control strategies, extensive numerical simulations are conducted. These simulations serve as a means of confirming the theoretical predictions and provide insight into the practical implications of the proposed control methodologies. By combining theoretical analysis with computational simulations, the paper offers a comprehensive understanding of the dynamics of the HCV model and provides valuable insights into potential strategies for controlling and managing chaos in such complex biological systems. [ABSTRACT FROM AUTHOR]

In order to better study the interaction between epidemic propagation and information diffusion, a new coupling model on multiplex networks with time delay is put forward in this paper. One layer represents the information diffusion about epidemics. There is not only information about the positive prevention of infectious diseases but also negative preventive information. Meanwhile, the dissemination of information at this layer will be influenced by the mass media, which can convey positive and reliable preventive measures to help the public avoid exposure to contagion. The other layer represents the transmission of infectious diseases, and the public in this layer no longer only exchange information related to infectious diseases in the virtual social network like the information layer but spread infectious diseases through contact among people. The classical SIR model is used to model for epidemic propagation. Since each infected individual needs to spend enough time to recover, the infected one at one time does not necessarily change to the recovered one at the next time, so time delay is an essential factor to be considered in the model. Based on the microscopic Markov chain approach, this paper obtains an explicit expression for epidemic threshold in the two-layered multiplex networks with time delay, which reveals some main factors affecting epidemic threshold. In particular, the time delay has a noticeable effect on the epidemic threshold to some extent. Finally, the influence of these main factors on the epidemic threshold and their interaction are proved through numerical simulations. [ABSTRACT FROM AUTHOR]

This paper introduces a complex network of interaction between human behavior and virus transmission, in which group synchronous behavior influences cure rates. The study examines the influence of individual group behavior on virus transmission, the reciprocal influence of virus transmission on individual group behavior, and the effects of evolving network structures on cluster synchronization. It also analyzes the conditions necessary for virus extinction or the occurrence of a pandemic, as well as the conditions for achieving individual group synchronization. The paper provides discriminant conditions to distinguish between aggregation behavior and virus extinction. The proposed model effectively captures the phenomenon of resurgence observed in many viruses. The conclusions drawn are rigorously validated through simulations conducted under various conditions, confirming the validity and reliability of the findings. [ABSTRACT FROM AUTHOR]

The Fermi–Pasta–Ulam–Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work, we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear α -FPUT system in more detail. We aim to reconstruct the quasiperiodic behavior observed in the original paper from the canonical transformation used to remove the three-wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work, we find that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three-wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the α -FPUT system and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the α -FPUT problem. We argue that certain aspects of the α -FPUT system such as thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies. [ABSTRACT FROM AUTHOR]

Accordingly, point (ii) in the conclusion of Ref. [1] should be: noise in the second layer is rescaled in order to have a variance in the second layer that is HT ht times smaller than in the first one. The conclusions of the paper remain valid as the shorter first escape time of the second layer compared to the first one happens for noise amplitudes that are smaller than the noise in the first layer. This erratum corrects two mistakes in my previously published paper.[1] The first mistake concerns Figs. [Extracted from the article]

This paper analytically and numerically investigates the dynamical characteristics of a fractional Duffing–van der Pol oscillator with two periodic excitations and the distributed time delay. First, we consider the pitchfork bifurcation of the system driven by both a high-frequency parametric excitation and a low-frequency external excitation. Utilizing the method of direct partition of motion, the original system is transformed into an effective integer-order slow system, and the supercritical and subcritical pitchfork bifurcations are observed in this case. Then, we study the chaotic behavior of the system when the two excitation frequencies are equal. The necessary condition for the existence of the horseshoe chaos from the homoclinic bifurcation is obtained based on the Melnikov method. Besides, the parameters effects on the routes to chaos of the system are detected by bifurcation diagrams, largest Lyapunov exponents, phase portraits, and Poincaré maps. It has been confirmed that the theoretical predictions achieve a high coincidence with the numerical results. The techniques in this paper can be applied to explore the underlying bifurcation and chaotic dynamics of fractional-order models. [ABSTRACT FROM AUTHOR]

Link prediction has been widely studied as an important research direction. Higher-order link prediction has gained, in particular, significant attention since higher-order networks provide a more accurate description of real-world complex systems. However, higher-order networks contain more complex information than traditional pairwise networks, making the prediction of higher-order links a formidable challenging task. Recently, researchers have discovered that local features have advantages over long-range features in higher-order link prediction. Therefore, it is necessary to develop more efficient and concise higher-order link prediction algorithms based on local features. In this paper, we proposed two similarity metrics via local information, simplicial decomposition weight and closed ratio weight, to predict possible future higher-order interactions (simplices) in simplicial networks. These two algorithms capture local higher-order information at two aspects: simplex decomposition and cliques' state (closed or open). We tested their performance in eight empirical simplicial networks, and the results show that our proposed metrics outperform other benchmarks in predicting third-order and fourth-order interactions (simplices) in most cases. In addition, we explore the robustness of the proposed algorithms, and the results suggest that the performance of these novel algorithms is advanced under different sizes of training sets. [ABSTRACT FROM AUTHOR]

In this paper, based on the feedback mechanism from the perspective of network groups, the evolutionary characteristics and laws of group networks under information dissemination are studied. First, the network is divided into groups of different sizes, and each group is given a dynamically changing group activity and a positive degree of response to different categories of information. Second, a feedback-based model of information dissemination in group networks is developed, which takes into account the differences between same-group and cross-group dissemination of information. Next, the model is applied to a scale-free network and a small-world network for simulation experiments. The experimental results show that, under the feedback mechanism, the main factor affecting the final evolutionary results of each group size in the small-world network is group positivity, which has little relationship with the initial size; similarly, the main factor affecting the average degree of each group in the scale-free network is also group positivity, which has nothing to do with the initial average degree. Finally, the method is applied to a real network to verify the rationality and effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

The paper is devoted to the parameter identification problem for two-neuron FitzHugh–Nagumo models under condition when only one variable, namely, the membrane potential, is measured. Another practical assumption is that both variable derivatives cannot be measured. Finally, it is assumed that the sensor measuring the membrane potential is imprecise, and all measurements have some unknown scaling factor. These circumstances bring additional difficulties to the parameters' estimation problem, and therefore, such case was not studied before. To solve the problem first, the model is transformed to a more simple form without unmeasurable variables. Variables obtained from applying a second-order real filter-differentiator are used instead of unmeasurable derivatives. Then, an adaptive system, parameters of which are estimates of original system parameters, is designed. The estimation (identification) goal is to properly adjust parameter estimates. To this end, the speed-gradient method is employed. The correctness of the obtained solution is proved theoretically and illustrated by computer simulation in the Simulink environment. The sufficient conditions of asymptotically correct identification for the speed-gradient method with integral objective function are formulated and proved. The novelty of the paper is that in contrast to existing solutions to the FitzHugh–Nagumo identification problem, we take into account a systematic error of the membrane potential measurement. Furthermore, the parameters are estimated for a system composed of two FitzHugh–Nagumo models, which open perspectives for using the proposed results for modeling and estimation of parameters for neuron population. [ABSTRACT FROM AUTHOR]

Interdependent networks are susceptible to catastrophic consequences due to the interdependence between the interacting subnetworks, making an effective recovery measure particularly crucial. Empirical evidence indicates that repairing the failed network component requires resources typically supplied by all subnetworks, which imposes the multivariate dependence on the recovery measures. In this paper, we develop a multivariate recovery coupling model for interdependent networks based on percolation theory. Considering the coupling structure and the failure–recovery relationship, we propose three recovery strategies for different scenarios based on the local stability of nodes. We find that the supporting network plays a more important role in improving network resilience than the network where the repaired component is located. This is because the recovery strategy based on the local stability of the supporting nodes is more likely to obtain direct benefits. In addition, the results show that the average degree and the degree exponent of the networks have little effect on the superior performance of the proposed recovery strategies. We also find a percolation phase transition from first to second order, which is strongly related to the dependence coefficient. This indicates that the more the recovery capacity of a system depends on the system itself, the more likely it is to undergo an abrupt transition under the multivariate recovery coupling. This paper provides a general theoretical frame to address the multivariate recovery coupling, which will enable us to design more resilient networks against cascading failures. [ABSTRACT FROM AUTHOR]

Synchronization is one of the key issues in three-phase AC power systems. Its characteristics have been dramatically changed with the large-scale integration of power-electronic-based renewable energy, mainly including a permanent magnetic synchronous generator (PMSG) and a double-fed induction generator (DFIG) for wind energy and a photovoltaic (PV) generator for solar energy. In this paper, we review recent progresses on the synchronization stability and multi-timescale properties of the renewable-dominated power system (RDPS), from nodes and network perspectives. All PMSG, DFIG, and PV are studied. In the traditional synchronous generator (SG) dominated power system, its dynamics can be described by the differential–algebraic equations (DAEs), where the dynamic apparatuses are modeled by differential equations and the stationary networks are described by algebraic equations. Unlike the single electromechanical timescale and DAE description for the SG-dominated power system, the RDPS dynamics should be described by the multiscale dynamics of both nodes and networks. For three different timescales, including the AC current control, DC voltage control, and rotor electromechanical timescales, their corresponding models are well established. In addition, for the multiscale network dynamics, the dynamical network within the AC current control timescale, which should be described by differential equations, can also be simplified as algebraic equations. Thus, the RDPS dynamics can be put into a similar DAE diagram for each timescale to the traditional power system dynamics, with which most of power electrical engineers are familiar. It is also found that the phase-locked loop for synchronization plays a crucial role in the whole system dynamics. The differences in the synchronization and multiscale characteristics between the traditional power system and the RDPS are well uncovered and summarized. Therefore, the merit of this paper is to establish a basic physical picture for the stability mechanism in the RDPS, which still lacks systematic studies and is controversial in the field of electrical power engineering. [ABSTRACT FROM AUTHOR]

Continuous-time memristors have been used in numerous chaotic circuit systems. Similarly, the discrete memristor model applied to a discrete map is also worthy of further study. To this end, this paper first proposes a discrete memristor model and analyzes the voltage–current characteristics of the memristor. Also, the discrete memristor is coupled with a one-dimensional (1D) sine chaotic map through different coupling frameworks, and two different two-dimensional (2D) chaotic map models are generated. Due to the presence of linear fixed points, the stability of the 2D memristor-coupled chaotic map depends on the choice of control parameters and initial states. The dynamic behavior of the chaotic map under different coupled map frameworks is investigated by using various analytical methods, and the results show that different coupling frameworks can produce different complex dynamical behaviors for memristor chaotic maps. The dynamic behavior based on parameter control is also investigated. The numerical experimental results show that the change of parameters can not only enrich the dynamic behavior of a chaotic map, but also increase the complexity of the memristor-coupled sine map. In addition, a simple encryption algorithm is designed based on the memristor chaotic map under the new coupling framework, and the performance analysis shows that the algorithm has a strong ability of image encryption. Finally, the numerical results are verified by hardware experiments. [ABSTRACT FROM AUTHOR]

The permutation largest slope entropy (PLSE) algorithm has been shown to be effective to distinguish between regular and non-regular dynamics from time series analysis. However, as it is the case for many non-linear time series analysis algorithms, such a characterization is locally made and does not allow one to capture some micro-phenomena, such as intermittency, that may occur in the system behavior. This paper presents a PIC micro-controller based implementation of the PLSE for a real-time monitoring of system dynamics. The PLSE algorithm is optimized to fit the program and data memory of low-end processors using the XC8 compiler and the MPLAB X IDE. The resulting algorithm is implemented on the PIC16F18446 and deployed on the Explorer 8 development board. The effectiveness of the developed tool is validated by considering an electrical circuit of the Duffing oscillator that can generate both periodic and chaotic dynamics. By comparing the PLSE values with the phase portraits and previous results on the Duffing oscillator circuit, the developed tool efficiently allows one to monitor the behavior of dynamical systems. [ABSTRACT FROM AUTHOR]

Entropy, as a nonlinear feature in information science, has drawn much attention for time series analysis. Entropy features have been used to measure the complexity behavior of time series. However, traditional entropy methods mainly focus on one-dimensional time series originating from single-channel transducers and are incapable of handling the multidimensional time series from multi-channel transducers. Previously, the multivariate multiscale sample entropy (MMSE) algorithm was introduced for multi-channel data analysis. Although MMSE generalizes multiscale sample entropy and provides a new method for multidimensional data analysis, it lacks necessary theoretical support and has shortcomings, such as missing cross-channel correlation information and having biased estimation results. This paper proposes an improved multivariate multiscale sample entropy (IMMSE) algorithm to overcome these shortcomings. This paper highlights the existing shortcomings in MMSE under the generalized algorithm. The rationality of IMMSE is theoretically proven using probability theory. Simulations and real-world data analysis have shown that IMMSE is capable of effectively extracting cross-channel correlation information and demonstrating robustness in practical applications. Moreover, it provides theoretical support for generalizing single-channel entropy methods to multi-channel situations. [ABSTRACT FROM AUTHOR]

In social science, we consider systems that consist of an usually large number of individuals whose actions and behaviors are mediated by the interactions among them. • B An agent-based model of multi-dimensional opinion dynamics and opinion alignment b , Schweighofer I et al. i [13] This is also an Individual Based Model paper on opinion dynamics addressing the question of the emergence of opinion alignment and polarization which is of much relevance in the political context. Social interaction and consumption choices b , Caravaggio and Sodini.[17] The study of consumption behavior is a very interesting problem in the dynamics of social systems, as it combines historical, social, and psychological dimensions. DATA DRIVEN RESEARCH • B Scaling laws and dynamics of hashtags on Twitter b , Chen I et al. i [28] Massive Twitter data is freely available and, as a consequence, is widely used by the academic community to study collective phenomena in social networks. [Extracted from the article]

This paper investigates dynamical behaviors and controllability of some nonautonomous localized waves based on the Gross–Pitaevskii equation with attractive interatomic interactions. Our approach is a relation constructed between the Gross–Pitaevskii equation and the standard nonlinear Schrödinger equation through a new self-similarity transformation which is to convert the exact solutions of the latter to the former's. Subsequently, one can obtain the nonautonomous breather solutions and higher-order rogue wave solutions of the Gross–Pitaevskii equation. It has been shown that the nonautonomous localized waves can be controlled by the parameters within the self-similarity transformation, rather than relying solely on the nonlinear intensity, spectral parameters, and external potential. The control mechanism can induce an unusual number of loosely bound higher-order rogue waves. The asymptotic analysis of unusual loosely bound rogue waves shows that their essence is energy transfer among rogue waves. Numerical simulations test the dynamical stability of obtained localized wave solutions, which indicate that modifying the parameters in the self-similarity transformation can improve the stability of unstable localized waves and prolong their lifespan. We numerically confirm that the rogue wave controlled by the self-similarity transformation can be reproduced from a chaotic initial background field, hence anticipating the feasibility of its experimental observation, and propose an experimental method for observing these phenomena in Bose–Einstein condensates. The method presented in this paper can help to induce and observe new stable localized waves in some physical systems. [ABSTRACT FROM AUTHOR]

A time-delayed virus dynamic model is proposed with general monotonic incidence, different nonlinear CTL (cytotoxic T lymphocyte) responses [CTL elimination function p y g 1 (z) and CTL stimulation function c y g 2 (z) ], and immune impairment. Indeed, the different CTL responses pose challenges in obtaining the dissipativeness of the model. By constructing appropriate Lyapunov functionals with some detailed analysis techniques, the global stability results of all equilibria of the model are obtained. By the way, we point out that the partial derivative f v (x , 0) is increasing (but not necessarily strictly) in x > 0 for the general monotonic incidence f (x , v). However, some papers defaulted that the partial derivative was strictly increasing. Our main results show that if the basic reproduction number R 0 ≤ 1 , the infection-free equilibrium E 0 is globally asymptotically stable (GAS); if CTL stimulation function c y g 2 (z) = 0 for z = 0 and the CTL threshold parameter R 1 ≤ 1 < R 0 , then the immunity-inactivated infection equilibrium E 1 is GAS; if the immunity-activated infection equilibrium E + exists, then it is GAS. Two specific examples are provided to illustrate the applicability of the main results. The main results acquired in this paper improve or extend some of the existing results. [ABSTRACT FROM AUTHOR]

Sets of saddle equilibria connected by trajectories are known as heteroclinic networks. Trajectories near a heteroclinic network typically spend a long period of time near one of the saddles before rapidly transitioning to the neighborhood of a different saddle. The sequence of saddles visited by a trajectory can be considered a stochastic sequence of states. In the presence of small-amplitude noise, this sequence may be either Markovian or non-Markovian, depending on the appearance of a phenomenon called lift-off at one or more saddles of the network. In this paper, we investigate how lift-off occurring at one saddle affects the dynamics near the next saddle visited, how we might determine the order of the associated Markov chain of states, and how we might calculate the transition probabilities of that Markov chain. We first review methods developed by Bakhtin to determine the map describing the dynamics near a linear saddle in the presence of noise and extend the results to include three different initial probability distributions. Using Bakhtin's map, we determine conditions under which the effect of lift-off persists as the trajectory moves past a subsequent saddle. We then propose a method for finding a lower bound for the order of this Markov chain. Many of the theoretical results in this paper are only valid in the limit of small noise, and we numerically investigate how close simulated results get to the theoretical predictions over a range of noise amplitudes and parameter values. [ABSTRACT FROM AUTHOR]

Vaccination is the most effective way to control the epidemic spreading. However, the probability of people getting vaccinated changes with the epidemic situation due to personal psychology. Facing various risks, some people are reluctant to vaccinate and even prefer herd immunity. To encourage people to get vaccinated, many countries set up reward mechanisms. In this paper, we propose a disease transmission model combining vaccination behaviors based on the SIR (Susceptible-Infected-Recovered) model and introduce three vaccination mechanisms. We analyze the impact of the infection rate and the recovery rate on the total cost and the epidemic prevalence. Numerical simulations fit with our intuitive feelings. Then, we study the impact of vaccination rewards on the total social cost. We find that when vaccination rewards offset vaccination costs, both the total cost and the epidemic prevalence reach the lowest levels. Finally, this paper suggests that encouraging people to get vaccinated at the beginning of an epidemic has the best effect. [ABSTRACT FROM AUTHOR]

This paper comments on a recent paper by R. Rhouma and B. Safya [Chaos 17, 033117 (2007)]. They claimed to find some security weakness of the spatiotemporal chaotic cryptosystem suggested by G. Tang et al. [Phys. Lett. A 318, 388 (2003)] and proposed a chosen-plaintext attack to analyze this system. We find that in their analysis, called a “chosen-plaintext attack,” they actually act as a legal receiver (with a machine in their hands during the entire decryption process) rather than an attacker and, therefore, the whole reasoning is not valid. [ABSTRACT FROM AUTHOR]

This paper introduces the Graphics Processing Unit (GPU)-based tool Geo-Temporal eXplorer (GTX), integrating a set of highly interactive techniques for visual analytics of large geo-referenced complex networks from the climate research domain. The visual exploration of these networks faces a multitude of challenges related to the geo-reference and the size of these networks with up to several million edges and the manifold types of such networks. In this paper, solutions for the interactive visual analysis for several distinct types of large complex networks will be discussed, in particular, time-dependent, multi-scale, and multi-layered ensemble networks. Custom-tailored for climate researchers, the GTX tool supports heterogeneous tasks based on interactive, GPU-based solutions for on-the-fly large network data processing, analysis, and visualization. These solutions are illustrated for two use cases: multi-scale climatic process and climate infection risk networks. This tool helps one to reduce the complexity of the highly interrelated climate information and unveils hidden and temporal links in the climate system, not available using standard and linear tools (such as empirical orthogonal function analysis). [ABSTRACT FROM AUTHOR]

Synchronization stability is one of central problems in power systems, and it is becoming much more complicated with the high penetration of renewable energy and power electronics devices. In this paper, we review recent work by several nonlinear models for renewable-dominated power systems in terms of multiple timescales, in particular, grid-tied converters within the DC voltage timescale. For the simplest model, a second-order differential equations called the generalized swing equation by considering only the phase-locked loop (PLL) is obtained, which shows a similar form with the well-known swing equation for a synchronous generator in the traditional power systems. With more outer controllers included, fourth-order and fifth-order models can be obtained. The fourth-order model is called the extended generalized swing equation, exhibiting the combined function of grid synchronization and active power balance on the DC capacitor. In addition, a nonlinear model for a two coupled converter system is given. Based on these studies, we find that the PLL plays a key role in synchronization stability. In summary, the value of this paper is to clarify the key concept of the synchronization stability in renewable-dominated power systems based on different nonlinear models, which still lacks systematic studies and is controversial in the field of electrical power engineering. Meanwhile, it clearly uncovers that the synchronization stability of converters has its root in the phase synchronization concept in nonlinear sciences. [ABSTRACT FROM AUTHOR]

Transient dynamics leading to the synchrony of a type of pulse-coupled oscillators, so-called scrambler oscillators, has previously been studied as an aggregation process of synchronous clusters, and a rate equation for the cluster size distribution has been proposed. However, the evolution of the cluster size distribution for general cluster sizes has not been fully understood yet. In this paper, we study the evolution of the cluster size distribution from the perspective of a percolation model by regarding the number of aggregations as the number of attached bonds. Specifically, we derive the scaling form of the cluster size distribution with specific values of the critical exponents using the property that the characteristic cluster size diverges as the percolation threshold is approached from below. Through simulation, it is confirmed that the scaling form well explains the evolution of the cluster size distribution. Based on the distribution behavior, we find that a giant cluster of all oscillators is formed discontinuously at the threshold and also that further aggregation does not occur like in a one-dimensional bond percolation model. Finally, we discuss the origin of the discontinuous formation of the giant cluster from the perspective of global suppression in explosive percolation models. For this, we approximate the aggregation process as a cluster–cluster aggregation with a given collision kernel. We believe that the theoretical approach presented in this paper can be used to understand the transient dynamics of a broad range of synchronizations. [ABSTRACT FROM AUTHOR]