Chaos

In nonautonomous dynamical systems, rate-induced tipping (R-tipping) is a critical transition triggered by the rate of change of a time-varying parameter, rather than its absolute value. In recent years, there is a growing interest in R-tipping due to its relevance to significant problems of current interest, such as potential, catastrophic collapse of various ecosystems induced by climate change. This brief review provides an overview of the basic concept, theory, and real-world implications of R-tipping from a global phase-space point of view. The key quantity underlying the global approach is the probability of R-tipping defined with respect to initial conditions in the phase space. A recently discovered scaling law governing this probability and the rate of parameter change is introduced, which has so far been restricted to a class of high-dimensional, complex, and empirical ecological networks: pollinator–plant mutualistic networks. Issues such as prediction of tipping and protection of ecosystems from R-tipping are discussed.

This work finds new exact soliton solutions to the fractional space–time higher-order nonlinear Schrödinger equation, describing how tiny pulses move through a nonlinear system. First, we transform this nonlinear fractional differential equation into an ordinary differential framework using the beta derivative and a traveling wave transformation. Then, we find analytical solutions using the unified solver method. Along with this, a thorough stability analysis is done using the Hamiltonian technique. Afterward, we study the chaotic analysis of the stated model using planner dynamics and show two- and three-dimensional phase illustrations, Lyapunov exponents, Poincaré maps, bifurcation figures, fractal dimensions, strange attractors, recurrence plots, and return maps as graphical representations regarding this chaotic analysis. Finally, we ensure that these precise solitons provide the internal complex image of wave travel.

We present a novel method of reconstructing the phase–amplitude dynamics directly from signals of a network of oscillators to estimate the coupling between its nodes. For this purpose, we use the recent advances in the field of phase–amplitude reduction of oscillatory systems, which allow the representation of an uncoupled oscillatory system as a phase–amplitude oscillator in a unique form using transformations (parameterizations) related to the eigenfunctions of the Koopman operator. By combining the parameterization method and the Fourier–Laplace averaging method for finding the eigenfunctions of the Koopman operator, we developed a method of assessing the transformation functions from the signals of the interacting oscillatory systems. The resulting reconstructed dynamical system is a network of phase–amplitude oscillators with the interactions between them represented as coupling functions in phase and amplitude coordinates.

Shearless curves are characteristic of nontwist systems and are not expected to exist in twist systems. However, the appearance of secondary shearless curves in the central area of islands has been reported in a few studies where the twist condition is still satisfied. In addition to these studies, we present a scenario in which secondary shearless curves emerge when two independent resonances interact on the same resonant surface. By varying the magnitude of the perturbation parameters, we observe the emergence of multiple secondary shearless curves, which can appear in pairs or individually. Our results are obtained for two discrete systems—the two-harmonic standard map and the Ullmann map—as well as for the Walker–Ford Hamiltonian flow.

The dynamics in a primary spectral submanifold (SSM) constructed over the slowest modes of a dynamical system provide an ideal reduced-order model for nearby trajectories. Modeling the dynamics of trajectories further away from the primary SSM, however, is difficult if the linear part of the system exhibits strong non-normal behavior. Such non-normality implies that simply projecting trajectories onto SSMs along directions normal to the slow linear modes will not pair those trajectories correctly with their reduced counterparts on the SSMs. In principle, a well-defined nonlinear projection along a stable invariant foliation exists and would exactly match the full dynamics to the SSM-reduced dynamics. This foliation, however, cannot realistically be constructed from practically feasible amounts and distributions of experimental data. Here, we develop an oblique projection technique that is able to approximate this foliation efficiently, even from a single experimental trajectory of a significantly non-normal and nonlinear beam.

Public goods game serves as a valuable paradigm for studying the challenges of collective cooperation in human and natural societies. Peer punishment is often considered an effective incentive for promoting cooperation in such contexts. However, previous related studies have mostly ignored the positive feedback effect of collective contributions on individual payoffs. In this work, we explore global and local state-feedback, where the multiplication factor is positively correlated with the frequency of contributors in the entire population or within the game group, respectively. By using replicator dynamics in an infinite well-mixed population, we reveal that state-based feedback plays a crucial role in alleviating the cooperative dilemma by enhancing and sustaining cooperation compared to the feedback-free case. Moreover, when the feedback strength is sufficiently strong or the baseline multiplication factor is sufficiently high, the system with local state-feedback provides full cooperation, hence supporting the “think globally, act locally” principle. Besides, we show that the second-order free-rider problem can be partially mitigated under certain conditions when the state-feedback is employed. Importantly, these results remain robust with respect to variations in punishment cost and fine.

The flow of granular material through pipes is characterized by significant variations in solid fraction (density waves) along the pipe. Previously, it has been shown that this intermittent flow behavior can be mitigated by adding a texture to the pipe’s inner wall. This work shows that adding surface roughness can lead to particle size segregation as a side effect in bidisperse systems. Using particle simulations, we characterize the parameter ranges in which segregation occurs.

Two dynamical indicators, the local dimension and the extremal index, used to quantify persistence in phase space have been developed and applied to different data across various disciplines. These are computed using the asymptotic limit of exceedances over a threshold, which turns to be a generalized Pareto distribution in many cases. However, the derivation of the asymptotic distribution requires mathematical properties, which are not present even in highly idealized dynamical systems and unlikely to be present in the real data. Here, we examine in detail the issues that arise when estimating these quantities for some known dynamical systems. We focus on how the geometry of an invariant set can affect the regularly varying properties of the invariant measure. We demonstrate that singular measures supported on sets of the non-integer dimension are typically not regularly varying and that the absence of regular variation makes the estimates resolution dependent. We show as well that the most common extremal index estimation method is not well defined for continuous time processes sampled at fixed time steps, which is an underlying assumption in its application to data.

This work explores the relationship between state space methods and Koopman operator-based methods for predicting the time evolution of nonlinear dynamical systems. We demonstrate that extended dynamic mode decomposition with dictionary learning (EDMD-DL), when combined with a state space projection, is equivalent to a neural network representation of the nonlinear discrete-time flow map on the state space. We highlight how this projection step introduces nonlinearity into the evolution equations, enabling significantly improved EDMD-DL predictions. With this projection, EDMD-DL leads to a nonlinear dynamical system on the state space, which can be represented in either discrete or continuous time. This system has a natural structure for neural networks, where the state is first expanded into a high-dimensional feature space followed by linear mapping that represents the discrete-time map or the vector field as a linear combination of these features. Inspired by these observations, we implement several variations of neural ordinary differential equations (ODEs) and EDMD-DL, developed by combining different aspects of their respective model structures and training procedures. We evaluate these methods using numerical experiments on chaotic dynamics in the Lorenz system and a nine-mode model of turbulent shear flow, showing comparable performance across methods in terms of short-time trajectory prediction, reconstruction of long-time statistics, and prediction of rare events. These results highlight the equivalence of the EDMD-DL implementation with a state space projection to a neural ODE representation of the dynamics. We also show that these methods provide comparable performance to a non-Markovian approach in terms of the prediction of extreme events.

In this paper, we illustrate the innovative techniques to explore the dynamical properties of the hybrid complex systems by various parameters and demonstrate the clusters under the chaotic-condensation peculiarities to accentuate the different technological challenges in the modern world. The complex dynamic characteristics of the contemplated systems are explored with interferometry techniques, which play key roles in examining the consequences of intricate features to interpreting the dynamics of an assemblage that possesses the ability to traverse along dependable tracks of particles, and thus, the solutions within the framework of quantum perturbation for partially chaotic structures are explored with substantial peculiar outcomes in the expanding active matter systems. Correlation graphs with chaotic parameters illustrate the significance of coherent stochastic generation for quasi-granular systems at finite momenta that possess sufficient fractions of instability fluxes. The distribution of temperature profiles is demonstrated by employing specific techniques to account for the different asymmetries and distinct formulas that characterize the structure of the dynamical system using realistic interference methods. The analytical solution contained distinctive information about the response of chaotic and probabilistic droplets within the multiplicities throughout hot and cold particulates, which are triggered by an influenced time crossover phase that occurs continually under the emissions of various sources that proliferate. Our results indicate that the newly developed phase encompasses the partially coherent collection of active matter with the temperature, which probes the rapidity of its transformation, and such phases exhibit significant mutual relationships. The findings accentuate the significance of contemplating correlations and bestowing extraordinary farsightedness about the meticulous manifestation of complex systems. The current methods are unique in obtaining new forms of solutions that appear beneficial for researchers to further understand nonlinear dynamical problems. The acquired techniques are also applicable to examine the solutions of other types of chaotic systems with mathematical analysis through machine learning.

We use the minimum description length (MDL) principle, which is an information-theoretic criterion for model selection, to determine echo-state network readout subsets. We find that this method of MDL subset selection improves accuracy when forecasting the Lorenz, Rössler, and Thomas attractors. It also improves the performance benefit that occurs when higher-order terms are included in the readout layer. We provide an explanation for these improvements in terms of decreased linear dependence and improved consistency.

Reinforcement learning technology has been empirically demonstrated to facilitate cooperation in game models. However, traditional research has primarily focused on two-strategy frameworks (cooperation and defection), which inadequately captures the complexity of real-world scenarios. To address this limitation, we integrated Q-learning into the prisoner's dilemma game, incorporating three strategies: cooperation, defection, and going it alone. We defined each agent's state based on the number of neighboring agents opting for cooperation and included social payoff in the Q-table update process. Numerical simulations indicate that this framework significantly enhances cooperation and average payoff as the degree of social-attention increases. This phenomenon occurs because social payoff enables individuals to move beyond narrow self-interest and consider broader social benefits. Additionally, we conducted a thorough analysis of the mechanisms underlying this enhancement of cooperation.

Collective decision-making is a process by which a group of individuals determines a shared outcome that shapes societal dynamics; from innovation diffusion to organizational choices. A common approach to model these processes is using binary dynamics, where the choices are reduced to two alternatives. One of the most popular models in this context is the q-voter model, which assumes that opinion changes are driven by peer pressure from a unanimous group. However, real-world decisions are also shaped by prior personal choices and external influences, such as mass media, which introduce biases that can favor certain options over others. To address this, we propose a generalized q-voter model that incorporates these biases. In our model, when the influence group is not unanimous, the probability that an individual changes its opinion depends on its current state, breaking the symmetry between opinions. In limiting cases, our model recovers both the original q-voter model and several recently introduced modifications of the q-voter model, while extending the framework to capture a broader range of scenarios. We analyze the model on a complete graph using analytical methods and Monte Carlo simulations. Our results highlight two key findings: (1) for larger influence groups ( q > 3), a phase emerges where both adopted and partially adopted states coexist, (2) in small systems, greater initial support for an opinion does not necessarily increase its likelihood of widespread adoption, as reflected in the unique form of the exit probability. These results point to one of the key issues in social science, the importance of group size in collective action.

While stochastic resetting (or total resetting) is a less young and more established concept in stochastic processes, partial stochastic resetting (PSR) is a relatively new field. PSR means that, at random moments in time, a stochastic process gets multiplied by a factor between 0 and 1, thus approaching but not reaching the resetting position. In this paper, we present new results on PSR highlighting the main similarities and discrepancies with total resetting. Specifically, we consider both symmetric α-stable Lévy processes (Lévy flights) and Brownian motion with PSR in arbitrary d dimensions. We derive explicit expressions for the propagator and its stationary measure and discuss in detail their asymptotic behavior. Interestingly, while approaching to stationarity, a dynamical phase transition occurs for the Brownian motion, but not for Lévy flights. We also analyze the behavior of the process around the resetting position and find significant differences between PSR and total resetting.

This paper proposes a novel fast online methodology for outlier detection called the exception maximization outlier detection (EMOD) algorithm, which employs probabilistic models and statistical algorithms to detect abnormal patterns from the outputs of complex systems. The EMOD is based on a two-state Gaussian mixture model and demonstrates strong performance in probability anomaly detection working on real-time raw data rather than using special prior distribution information. We confirm this using the synthetic data from two numerical cases. For the real-world data, we have detected the short-circuit pattern of the circuit system using EMOD by the current and voltage output of a three-phase inverter. The EMOD also found an abnormal period due to COVID-19 in the insured unemployment data of 53 regions in the United States from 2000 to 2024. The application of EMOD to these two real-life datasets demonstrated the effectiveness and accuracy of our algorithm.

Bootstrap percolation is a widely studied model to investigate the robustness of a network for cascading failures. Extensive real-world data analysis has revealed the existence of higher-order interactions among elements, i.e., the interactions beyond pairwise, which are usually described by hypergraphs. In this paper, we propose a generalized bootstrap percolation model on hypergraph, which assumes that the activation of an inactive node depends on the number of active neighbors through its hyperedges. Through numerical simulation and theoretical analysis, we found that the bootstrap percolation threshold and the phase transition type are closely related to the infection threshold and the proportion of higher-order edges. When the infection threshold is significant, for any initial activation probability, the size of the giant active component (GAC) shows continuous growth with increasing occupation probability. When the infection threshold is small, with the increase of the initial activation probability, the size of the GAC changes from continuous growth to discontinuous growth. In addition, we found that in the case of a fixed network average degree, increasing the proportion of higher-order edges will reduce the percolation threshold, which is conducive to enhancing the robustness of the network. At the same time, higher-order edges create more opportunities for inactive nodes to be activated, and increasing the proportion of higher-order edges under the same conditions will change the size of the GAC from continuous growth to discontinuous growth.

Network provides a low-dimensional representation of the heart through a sparse adjacency matrix, which ushers in a new opportunity to conduct cardiac simulation. We discovered that a self-organizing network encodes and resembles complex heart geometry. This, in turn, helps characterize the structure–function relationship of the heart through network theory. However, very little has been done to investigate the simulation of electrical activity on a self-organizing network. Thus, this paper presents a new self-organizing network approach for simulating cardiac electrical dynamics. We formulate and solve reaction–diffusion equations on the self-organizing network to simulate the propagation and turbulent behavior of electrical waves. The proposed methodology is evaluated and validated on both 2D cardiac tissues, consisting of healthy and infarcted cells, and the whole heart. Experimental results show that the proposed approach not only yields a compact network representation that resembles the heart geometry but also provides an effective simulation of spatiotemporal dynamics when benchmarking with traditional finite element method simulations.

We describe completely the dynamics of the three-dimensional Muthuswamy–Chua systems x ˙ = y , y ˙ = − x 3 + y 2 − y z 2 2 , z ˙ = y − α z − y z, in the infinity of the Poincaré ball for all values of α ∈ R. Additionally, we provide a complete description of the dynamics of this differential system when α = 0.

The discrete memristive chaotic system is characterized by discontinuous phase trajectories. To address the limitations of the ideal integer-order discrete memristor model, which fails to accurately reflect the characteristics of practical devices, this study introduces a Grunwald–Letnikov type quadratic trivariate fractional discrete memristor model to enhance the nonlinearity and memory properties of memristors. Simultaneously, it is demonstrated that our model satisfies the essential characteristics of the generalized memristor. Based on this newly proposed fractional discrete memristor, a new four-dimensional fractional discrete memristive hyperchaotic system is constructed by coupling non-uniform, incommensurate-order memristors. This system advances the structure of existing discrete chaotic systems and provides a more flexible strategy for optimizing memory effects. The dynamical behaviors are analyzed using attractor phase diagrams, bifurcation diagrams, Lyapunov exponent spectra, and permutation entropy complexity. Numerical simulation results show that the system can exhibit a larger hyperchaotic region, higher complexity, and rich multistable behaviors, such as the coexistence of infinitely symmetric attractors and enhanced offset. Additionally, the impact of the incommensurate-order parameter on the system’s chaotic behavior is revealed, with order serving as a tunable control variable that dynamically reconfigures bifurcation paths as needed, thereby enabling transitions between hyperchaotic, chaotic, and non-chaotic states. Furthermore, a simulation circuit was designed to validate the numerical simulation results.

Discerning chaos in quantum systems is an important problem as the usual route of Lyapunov exponents in classical systems is not straightforward in quantum systems. A standard route is the comparison of statistics derived from model physical systems to those from random matrix theory (RMT) ensembles, of which the most popular is the nearest-neighbor-spacing distribution, which almost always shows good agreement with chaotic quantum systems. However, even in these cases, the long-range statistics (like number variance and spectral rigidity), which are also more difficult to calculate, often show disagreements with RMT. As such, a more stringent test for chaos in quantum systems, via an analysis of intermediate-range statistics, is needed, which will additionally assess the extent of agreement with RMT universality. In this paper, we deduce the effective level-repulsion parameters and the corresponding Wigner–surmise-like results of the next-nearest-neighbor-spacing distribution (nNNSD) for integrable systems (semi-Poissonian statistics) as well as the three classical quantum chaotic Wigner–Dyson regimes, by stringent comparisons to numerical RMT models and benchmarking against our exact analytical results for 3 × 3 Gaussian matrix models, along with a semi-analytical form for the nNNSD in the orthogonal-to-unitary symmetry crossover. To illustrate the robustness of these RMT based results, we test these predictions against the nNNSD obtained from quantum chaotic models as well as disordered lattice spin models. This reinforces the Bohigas–Giannoni–Schmit and the Berry–Tabor conjectures, extending the associated universality to longer-range statistics. In passing, we also highlight the equivalence of nNNSD in the apparently distinct orthogonal-to-unitary and diluted-symplectic-to-unitary crossovers.

We propose a novel Reservoir Computing (RC) based classification method that distinguishes between different chaotic time series. Our method is composed of two steps: (i) we use the reservoir as a feature extracting machine that captures the salient features of time series data; (ii) the readout layer of the reservoir is subsequently fed into a Convolutional Neural Network (CNN) to facilitate classification and recognition. One of the notable advantages is that the readout layer, as obtained by randomly generated empirical hyper-parameters within the RC module, provides sufficient information for the CNN to accomplish the classification tasks effectively. The quality of extracted features by RC is independently evaluated by the root mean square error, which measures how well the training signal may be reconstructed from the input time series. Furthermore, we propose two ways to implement the RC module, namely, a single shallow RC and parallel RC configurations, to further improve the classification accuracy. The important roles of RC in feature extraction are demonstrated by comparing the results when the CNN is provided with either ordinal pattern probability features or unprocessed raw time series directly, both of which perform worse than RC-based method. In addition to CNN, we show that the readout of RC is good for other classification tools as well. The successful classification of electroencephalogram recordings of different brain states suggests that our RC-based classification tools can be used for experimental studies.

Cover times quantify the speed of exhaustive search. In this work, we approximate the moments of cover times of a wide range of stochastic search processes in d-dimensional continuous space and on an arbitrary discrete network under frequent stochastic resetting. These approximations apply to a large class of resetting time distributions and search processes including diffusion, run-and-tumble particles, and Markov jump processes. We illustrate these results in several examples; in the case of diffusive search, we show that the errors of our approximations vanish exponentially fast. Finally, we derive a criterion for when endowing a discrete state search process with minimal stochastic resetting reduces the mean cover time.

With the increasing scale of power systems, their reliability analysis and calculation become more complex and difficult. Community structure, as an important topological characteristic of complex networks, plays a prominent role in power grid research and application. The current methods for community division of power networks are mainly based on the topological characteristics of the network, with less consideration of the power balance of the subnetwork, which requires larger-scale machine-cutting or load-cutting operations when the subnetwork operates independently after the grid is unbundled. To solve this problem, this paper proposes a community segmentation method for power networks based on graph neural networks that integrally considers the topology of the network and the power balance of the network. Node attributes such as node degree, betweenness, and power value are selected as node features to help the model capture more correlations between nodes. The traditional K-means algorithm is also optimized and improved, and the method of selecting generator nodes as the clustering centers is proposed to ensure that there are generator nodes supplying energy in each community. Experiments are conducted on the IEEE standard test systems, and the effectiveness of the method proposed in this paper is verified by comparing it with other community segmentation methods.

This paper explores the representational structure of linear Simple Cycle Reservoirs (SCRs) operating at the edge of stability. We view SCR as providing in their state-space feature representations of the input-driving time series. By endowing the state space with the canonical dot-product, we “reverse engineer” the corresponding kernel (inner product) operating in the original time series space. The action of this time series kernel is fully characterized by the eigenspace of the corresponding metric tensor. We demonstrate that when linear SCRs are constructed at the edge of stability, the eigenvectors of the time series kernel align with the Fourier basis. This theoretical insight is supported by numerical experiments.

This study explores the semiclassical limit of an integrable-chaotic bosonic many-body quantum system, providing nuanced insights into its behavior. We examine classical-quantum correspondences across different interaction regimes of bosons in a triple-well potential, ranging from the integrable to the self-trapping regime, and including the chaotic one. The close resemblance between the phase-space mean projections of classical trajectories and those of Husimi distributions evokes the principle of uniform semiclassical condensation of Wigner functions of eigenstates. Notably, the resulting figures also exhibit patterns reminiscent of Jason Gallas’s “shrimp” shapes.