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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]

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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

The state of a classical point-particle system may often be specified by giving the position and momentum for each constituent particle. For non-pointlike particles, the center-of-mass position may be augmented by an additional coordinate that specifies the internal state of each particle. The internal state space is typically topologically simple, in the sense that the particle's internal coordinate belongs to a suitable symmetry group. In this paper, we explore the idea of giving internal complexity to the particles, by attributing to each particle an internal state space that is represented by a point on a strange (or otherwise) attracting set. It is, of course, very well known that strange attractors arise in a variety of nonlinear dynamical systems. However, rather than considering strange attractors as emerging from complex dynamics, we may employ strange attractors to drive such dynamics. In particular, by using an attractor (strange or otherwise) to model each particle's internal state space, we present a class of matter coined "attractor-driven matter." We outline the general formalism for attractor-driven matter and explore several specific examples, some of which are reminiscent of active matter. Beyond the examples studied in this paper, our formalism for attractor-driven dynamics may be applicable more broadly, to model complex dynamical and emergent behaviors in a variety of contexts. [ABSTRACT FROM AUTHOR]

Research on cascading failures in power-transmission networks requires detailed data on the capacity of individual transmission lines. However, these data are often unavailable to researchers. Consequently, line limits are often modeled by assuming that they are proportional to some average load. However, there is scarce research to support this assumption as being realistic. In this paper, we analyze the proportional loading (PL) approach and compare it to two linear models that use voltage and initial power flow as variables and are trained on the line limits of a real power network that we have access to. We compare these artificial line-limit methods using four tests: the ability to model true line limits, the damage done during an attack, the order in which edges are lost, and accuracy ranking the relative performance of different attack strategies. We find that the linear models are the top-performing method or are close to the top in all the tests we perform. In comparison, the tolerance value that produces the best PL limits changes depending on the test. The PL approach was a particularly poor fit when the line tolerance was less than two, which is the most commonly used value range in cascading failure research. We also find indications that the accuracy of modeling line limits does not indicate how well a model will represent grid collapse. The findings of this paper provide an understanding of the weaknesses of the PL approach and offer an alternative method of line-limit modeling. [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]

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]

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]

In this paper, nonlinear piezoelectric energy harvesting induced by a Duffing oscillator is studied, and the bifurcation trees of period-1 motions to chaos for such a piezoelectric energy-harvesting system are obtained analytically. Distributed-parameter electromechanical modeling of a piezoelectric energy harvester is presented first, and the electromechanically coupled circuit equation excited by infinitely many vibration modes is developed. The governing electromechanical equations are reduced to ordinary differential equations in modal coordinates, and eventually an infinite set of algebraic equations is obtained for the complex modal vibration responses and the complex voltage responses of the energy harvester beam. One single mode case is considered in this paper, and periodic motions with bifurcation trees are obtained through an implicit discrete mapping method. The frequency–amplitude characteristics of periodic motions are obtained for the nonlinear piezoelectric energy-harvesting systems, which provide a better understanding of where and how to achieve the best energy harvesting. This study describes about how the nonlinear oscillator induces piezoelectric energy harvesting through a beam system. The nonlinear piezoelectric energy harvesting is presented through a nonlinear oscillator. Due to the nonlinear oscillator, chaotic piezoelectric energy-harvesting states can get more energy compared to the linear piezoelectric energy-harvesting system. [ABSTRACT FROM AUTHOR]

This paper reports the first finding of cupolets in a chaotic Hindmarsh–Rose neural model. Cupolets (chaotic, unstable, periodic, orbit-lets) are unstable periodic orbits that have been stabilized through a particular control scheme by applying a binary control sequence. We demonstrate different neural dynamics (periodic or chaotic) of the Hindmarsh–Rose model through a bifurcation diagram where the external input current, I , is the bifurcation parameter. We select a region in the chaotic parameter space and provide the results of numerical simulations. In this chosen parameter space, a control scheme is applied when the trajectory intersects with either of the two control planes. The type of the control is determined by a bit in a binary control sequence. The control is either a small microcontrol (0) or a large macrocontrol (1) that adjusts the future dynamics of the trajectory by a perturbation determined by the coding function r N (x). We report the discovery of many cupolets with corresponding control sequences and comment on the differences with previously reported cupolets in the double scroll system. We provide some examples of the generated cupolets and conclude by discussing potential implications for biological neurons. [ABSTRACT FROM AUTHOR]

This paper applies new and recently introduced approaches to study trends in gun violence in the United States. We use techniques in both the time and frequency domain to provide a more complete understanding of gun violence dynamics. We analyze gun violence incidents on a state-by-state basis as recorded by the Gun Violence Archive. We have numerous specific phenomena of focus, including periodicity of incidents, locations in time where behavioral changes occur, and shifts in gun violence patterns since April 2020. First, we implement a recently introduced method of spectral density estimation for nonstationary time series to investigate periodicity on a state-by-state basis, including revealing where periodic behaviors change with time. We can also classify different patterns of behavioral changes among the states. We then aim to understand the most significant shifts in gun violence since numerous key events in 2020, including the COVID-19 pandemic, lockdowns, and periods of civil unrest. Our dual-domain analysis provides a more thorough understanding and challenges numerous widely held conceptions regarding the prevalence of gun violence incidents. [ABSTRACT FROM AUTHOR]

A three-layer prey–predator model with two time-delays and one delay-dependent parameter is established in this paper. To begin, the paper calculates the conditions for each population in the model to maintain the quantity stable and Hopf bifurcation when τ 1 = τ 2 = 0 , τ 1 = 0 , τ 2 ≠ 0 , τ 2 is in the stable interval, and τ 1 > 0. The crossing curves, their type, and the direction of the crossing curves are then obtained using the crossing curve method, which is composed of the threshold values of the dynamic behavior change on the two time-delays plane when τ 1 , τ 2 > 0. The real data from the forage grass–Ochotona curzoniae–Buteo hemilasius food chain is used to conduct an empirical study of the model. When τ 1 , τ 2 > 0 , the feasible region of the crossing curves is open-ended, and the model's crossing curves on the (τ 2 , τ 1) plane are truncated. This indicates that the model's threshold distribution of dynamic behavior change is a regular curve made of several curves. The simulation using the time-delay value on the crossing curves shows that the model produces different dynamic behaviors such as stability, bifurcation, and chaos depending on the time-delay value on both sides of the curves. The critical values of dynamic behavior change are time-delay values on the crossing curves. The empirical study shows that increasing Ochotona curzoniae's environmental capacity can easily cause Hopf bifurcation of the system. At this time, the number of each population in the Plateau ecosystem constantly fluctuates, and Ochotona curzoniae is vulnerable to extinction. [ABSTRACT FROM AUTHOR]

We study a dynamical system defined by a repeated game on a 1D lattice, in which the players keep track of their gross payoffs over time in a bank. Strategy updates are governed by a Boltzmann distribution, which depends on the neighborhood bank values associated with each strategy, relative to a temperature scale, which defines the random fluctuations. Players with higher bank values are, thus, less likely to change strategy than players with a lower bank value. For a parameterized rock–paper–scissors game, we derive a condition under which communities of a given strategy form with either fixed or drifting boundaries. We show the effect of a temperature increase on the underlying system and identify surprising properties of this model through numerical simulations. [ABSTRACT FROM AUTHOR]

The interaction between the swarm individuals affects the dynamic behavior of the swarm, but it is difficult to obtain directly from outside observation. Therefore, the problem we focus on is inferring the structure of the interactions in the swarm from the individual behavior trajectories. Similar inference problems that existed in network science are named network reconstruction or network inference. It is a fundamental problem pervading research on complex systems. In this paper, a new method, called Motion Trajectory Similarity, is developed for inferring direct interactions from the motion state of individuals in the swarm. It constructs correlations by combining the similarity of the motion trajectories of each cross section of the time series, in which individuals with highly similar motion states are more likely to interact with each other. Experiments on the flocking systems demonstrate that our method can produce a reliable interaction inference and outperform traditional network inference methods. It can withstand a high level of noise and time delay introduced into flocking models, as well as parameter variation in the flocking system, to achieve robust reconstruction. The proposed method provides a new perspective for inferring the interaction structure of a swarm, which helps us to explore the mechanisms of collective movement in swarms and paves the way for developing the flocking models that can be quantified and predicted. [ABSTRACT FROM AUTHOR]

Electromagnetic radiation (EMR) affects the dynamical behavior of the nervous system, and appropriate EMR helps to study the dynamic mechanism of the nervous system. This paper uses a sophisticated four-dimensional Hopfield neural network (HNN) model augmented with one or more memristors to simulate the effects of EMR. We focus on the chaotic dynamics of HNN under the influence of EMR. Complex dynamical behaviors are found and transient chaotic phenomena have the same initial value sensitivity, showing how transient chaos is affected by EMR. Multiperiodic phenomena induced by quasi-periodic alternations are found in the dual EMR, as well as the suppression properties of the dual EMR for system chaos. This implies that the dynamical behavior of the HNN system can be controlled by varying the amount of EMR or the number of affected neurons in the HNN. Finally, a strong validation of our proposed model is provided by Multisim and Field Programmable Gate Array(FPGA) hardware. [ABSTRACT FROM AUTHOR]

When players are dissatisfied with their actual payoffs, they will change the actuality by learning strategy of neighbors. The more effort players put in, the more likely they are to succeed in learning. Inspired by this, this paper proposes a two-stage strategy update rule based on learning cost. The players first decide whether to learn strategy according to the updating willingness. If the players imitate the strategy of neighbors, they need to pay the learning cost. Results show that for the well-mixed population, if the updating willingness is homogeneous and remains unchanged, reducing the updating willingness or increasing the learning cost can extend the life cycle of cooperators. If the updating willingness is heterogeneous and dynamically adjusted based on the difference between the actual payoff and the expected payoff, increasing aspiration value and learning cost promotes cooperation. For the structured population, if the updating willingness is homogeneous and remains unchanged, the moderate learning cost is beneficial for cooperators to resist the temptation of defection, and reducing updating willingness makes the system maintain cooperation within a larger parameter range. If the updating willingness is heterogeneous and dynamically adjusted, the larger learning cost and the appropriate aspiration value promote cooperation. This study highlights the complex dynamics of cooperation in paid strategy learning, contributing to the theory of cooperation in the evolutionary game. [ABSTRACT FROM AUTHOR]

Cooperation between individuals is emergent in all parts of society; yet, mechanistic reasons for this emergence are ill understood in the literature. A specific example of this is insurance. Recent work has, though, shown that assuming the risk individuals face is proportional to their wealth and optimizing the time average growth rate rather than the ensemble average results in a non-zero-sum game, where both parties benefit from cooperation through insurance contracts. In a recent paper, Peters and Skjold present a simple agent-based model and show how, over time, agents that enter into such cooperatives outperform agents that do not. Here, we extend this work by restricting the possible connections between agents via a lattice network. Under these restrictions, we still find that all agents profit from cooperating through insurance. We, though, further find that clusters of poor and rich agents emerge endogenously on the two-dimensional map and that wealth inequalities persist for a long duration, consistent with the phenomenon known as the poverty trap. By tuning the parameters that control the risk levels, we simulate both highly advantageous and extremely risky gambles and show that despite the qualitative shift in the type of risk, the findings are consistent. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the data-driven rogue waves solutions of the focusing and the variable coefficient nonlinear Schrödinger (NLS) equations by the deep learning method from initial and boundary conditions. Specifically, first- and second-order rogue wave solutions for the focusing NLS equation and three deformed rogue wave solutions for the variable coefficient NLS equation are solved using physics-informed memory networks (PIMNs). The effects of optimization algorithm, network structure, and mesh size on the solution accuracy are discussed. Numerical experiments clearly demonstrate that the PIMNs can capture the nonlinear features of rogue waves solutions very well. This is of great significance for revealing the dynamical behavior of the rogue waves solutions and advancing the application of deep learning in the field of solving partial differential equations. [ABSTRACT FROM AUTHOR]

This paper focuses on achieving pathwise synchronization in stochastic differential equations with linear multiplicative rough noises, which are fractional Brownian rough paths with Hurst parameter H ∈ (1 3 , 1 2). Using rough paths theory, a useful transformation is introduced to convert the equations into random differential equations. Stability and dynamical behavior of the solutions to the equations are discussed, and pathwise synchronization of the solutions to the coupled system is proven. Also we have verified the synchronization results in Hölder space. And at the end, two alternative forms of noises are considered, and synchronization results are presented. Moreover, numerical simulations are provided to illustrate the results. [ABSTRACT FROM AUTHOR]