Search Results (41)

This paper introduces a rigorous class of two-dimensional Markov-switching autoregressive moving-average (2D MS-ARMA) models for spatial lattice data exhibiting regime-dependent dynamics. The switching mechanism is governed by a latent causal Markov random field that drives spatial transitions between regime-specific autoregressive and moving-average structures. We provide sufficient conditions for the existence of a strictly stationary solution through the top Lyapunov exponent associated with a sequence of random matrices obtained from a state-space representation constructed along the lexicographic order. For the first-order bidirectional specification, we derive explicit spectral conditions linking stationarity to the regime-dependent spectral radii. Sufficient conditions ensuring the existence of finite second-order moments are also provided. Parameter estimation is carried out using a variational expectation–maximization (VEM) algorithm based on a mean-field approximation of the posterior distribution of the hidden regimes. The E-step yields closed-form coordinate ascent updates, while the M-step relies on gradient-based numerical optimization with derivatives computed via recursive differentiation. Under increasing-domain asymptotics, we discuss the consistency and asymptotic behavior of the variational estimator. The proposed framework fills a methodological gap between classical one-dimensional Markov-switching ARMA models and spatial autoregressive structures by extending regime-switching theory to multi-indexed processes with rigorous probabilistic foundations. It provides a comprehensive basis for statistical inference, model diagnostics, and prediction in spatially heterogeneous environments. Full article

We study the planar repulsive two-center Coulomb system in the presence of a uniform magnetic field perpendicular to the plane, taking the inter-center separation a and the magnetic field strength B as independent control parameters. The free-field system $B=0$ is Liouville integrable and the motion is unbounded. The magnetic confinement introduces nonlinear coupling that breaks integrability and gives rise to chaotic bounded dynamics. Using Poincaré sections and maximal Lyapunov exponents, we characterize the transition from regular motion at $aB=0$ to mixed regular–chaotic dynamics for $aB\ne 0$. To probe the recoverability of the dynamics, we apply sparse regression techniques to numerical trajectories and assess their ability to capture the equations of motion across mixed dynamical regimes. Our results clarify how magnetic confinement competes with two-center repulsive interactions in governing the emergence of chaos and delineate fundamental limitations of data-driven model discovery in nonintegrable Hamiltonian systems. We further identify an organizing mechanism whereby the repulsive two-center system exhibits locally one-center-like dynamics in the absence of any static confining barrier. Full article

To address wind power fluctuations causing curtailment and high costs, this study proposes an integrated method combining wind power forecasting with substation optimization. An enhanced Bidirectional Gated Recurrent Unit (BiGRU) model is developed by incorporating chaotic features (maximum Lyapunov exponent) and sliding-window statistical features (mean, standard deviation), significantly improving short-term prediction accuracy. Based on these high-precision forecasts, a dynamic transformer switching optimization model is established to maximize the wind farm’s net profit. This model finely balances power generation revenue, wind curtailment penalties, and transformer losses (no-load and load) at a 15 min timescale. Experimental results from a wind farm in Xinjiang demonstrate that the proposed method effectively enhances the economic efficiency of wind farm operations. The study provides a valuable framework for optimizing energy storage configuration and improving profitability by leveraging accurate forecasting. Full article

This paper investigates the complex dynamics and fractal attractors that arise in a 60-dimensional ring lattice system of electrically coupled nonchaotic Rulkov neurons. While networks of chaotic Rulkov neurons have been widely studied, systems of nonchaotic Rulkov neurons have not been extensively explored due to the piecewise complexity of the nonchaotic Rulkov map. Here, we find that rich dynamics emerge from the electrical coupling of regular-spiking Rulkov neurons, including chaotic spiking, synchronized chaotic bursting, and synchronized hyperchaos. By systematically varying the electrical coupling strength between neurons, we also uncover general trends in the maximal Lyapunov exponent across the system’s dynamical regimes. By means of the Kaplan–Yorke conjecture, we examine the fractal geometry of the ring system’s high-dimensional chaotic attractors and find that these attractors can occupy as many as 45 of the 60 dimensions of state space. We further explore how variations in chaotic behavior—quantified by the full Lyapunov spectra—correspond to changes in the attractors’ fractal dimensions. This analysis advances our understanding of how complex collective behavior can emerge from the interaction of multiple simple neuron models and highlights the deep interplay between dynamics and geometry in high-dimensional systems. Full article

This paper introduces a novel three-dimensional financial risk system that exhibits complex dynamical behaviors, including chaos, multistability, and a butterfly attractor. The proposed system is an extension of the Zhang financial risk model (ZFRM), with modifications that enhance its applicability to real-world economic stability assessments. Through numerical simulations, we confirm the system’s chaotic nature using Lyapunov exponents (LE), with values calculated as $L_1=3.5547,$ $L_2=0,$ $L_3=-22.5642$, indicating a positive Maximal Lyapunov Exponent (MLE) that confirms chaos. The Kaplan–Yorke Dimension (KYD) is determined as D_k = 2.1575, reflecting the system’s fractal characteristics. Bifurcation analysis (BA) reveals parameter ranges where transitions between periodic, chaotic, and multistable states occur. Additionally, the system demonstrates coexisting attractors, where different initial conditions lead to distinct long-term behaviors, emphasizing its sensitivity to market fluctuations. Offset Boosting Control (OBC) is implemented to manipulate the chaotic attractor, shifting its amplitude without altering the underlying system dynamics. These findings provide deeper insights into financial risk modeling and economic stability, with potential applications in financial forecasting, risk assessment, and secure economic data transmission. Full article

A fractional-order Lorenz system is optimized to maximize its maximum Lyapunov exponent and Kaplan-York dimension using the Non-dominated Sorting Genetic Algorithm II (NSGA-II) algorithm. The fractional-order Lorenz system is integrated with a recent process called the “modified two-stage Runge-Kutta” (M2sFRK) method, which is very fast and efficient. A Pseudo-Random Number Generator (PRNG) was built using one of the optimized systems that was obtained. The M2sFRK method allows for obtaining a very fast optimization time and also designing a very efficient PRNG with linear complexity, $O\left(n\right)$. The designed PRNG generates 24 random bits at each iteration step, and the random sequences pass all the National Institute of Standards and Technology (NIST) and TestU01 statistical tests, making the PRNG suitable for cryptographic applications. The presented methodology could be extended to any other chaotic system. Full article

This paper introduces a novel chaotic finance system derived by incorporating a modeling uncertainty with an absolute function nonlinearity into existing financial systems. The new system, based on the works of Gao and Ma, and Vaidyanathan et al., demonstrates enhanced chaotic behavior with a maximal Lyapunov exponent (MLE) of 0.1355 and a fractal Lyapunov dimension of 2.3197. These values surpass those of the Gao-Ma system (MLE = 0.0904, Lyapunov dimension = 2.2296) and the Vaidyanathan system (MLE = 0.1266, Lyapunov dimension = 2.2997), signifying greater complexity and unpredictability. Through parameter analysis, the system transitions between periodic and chaotic regimes, as confirmed by bifurcation diagrams and Lyapunov exponent spectra. Furthermore, multistability is demonstrated with coexisting chaotic attractors for p = 0.442 and periodic attractors for p = 0.48. The effects of offset boosting control are explored, with attractor positions adjustable by varying a control parameter k, enabling transitions between bipolar and unipolar chaotic signals. These findings underline the system’s potential for advanced applications in secure communications and engineering, providing a deeper understanding of chaotic finance models. Full article

Currently, despite advances in the analysis of dynamical systems, there are still doubts about the transition between both stable and chaotic behaviors. In this research, we will explain the transition of a system that develops between two dynamic systems that have already been studied: the classical logistic model and a new chaotic system. This research addresses the study of the transition of both the system and its behaviors using computational techniques, where cobweb diagrams, time series, bifurcation diagrams, and even a graphical visualization for the maximum Lyapunov exponent will be visualized. Using a graphical and numerical methodology, bifurcation points were identified that revealed the transition of behaviors at different points. This resulted in a deep understanding of the dynamics of the system, thus highlighting the importance of incorporating computational analysis in dynamic systems, which greatly contributes to the efficient modeling of natural phenomena. Full article

The Basener and Ross mathematical model is widely recognized for its ability to characterize the interaction between the population dynamics and resource utilization of Easter Island. In this study, we develop and investigate a discrete-time version of the Basener and Ross model. First, the existence and the stability of fixed points for the present model are investigated. Next, we investigated various bifurcation scenarios by establishing criteria for the occurrence of different types of codimension-one bifurcations, including flip and Neimark–Sacker bifurcations. These criteria are derived using the center manifold theorem and bifurcation theory. Furthermore, we demonstrated the existence of codimension-two bifurcations characterized by 1:2, 1:3, and 1:4 resonances, emphasizing the model’s complex dynamical structure. Numerical simulations are employed to validate and illustrate the theoretical predictions. Finally, through bifurcation diagrams, maximal Lyapunov exponents, and phase portraits, we uncover a diversity of dynamical characteristics, including limit cycles, periodic solutions, and chaotic attractors. Full article

The attractor complexity index (ACI) is a recently developed gait analysis tool based on nonlinear dynamics. This study assesses ACI’s sensitivity to attentional demands in gait control and its potential for characterizing age-related changes in gait patterns. Furthermore, we compare ACI with classical gait metrics to determine its efficacy relative to established methods. A 4 × 200 m indoor walking test with a triaxial accelerometer attached to the lower back was used to compare gait patterns of younger (N = 42) and older adults (N = 60) during normal and metronome walking. The other linear and non-linear gait metrics were movement intensity, gait regularity, local dynamic stability (maximal Lyapunov exponents), and scaling exponent (detrended fluctuation analysis). In contrast to other gait metrics, ACI demonstrated a specific sensitivity to metronome walking, with both young and old participants exhibiting altered stride interval correlations. Furthermore, there was a significant difference between the young and old groups (standardized effect size: −0.77). Additionally, older participants exhibited slower walking speeds, a reduced movement intensity, and a lower gait regularity. The ACI is likely a sensitive marker for attentional load and can effectively discriminate age-related changes in gait patterns. Its ease of measurement makes it a promising tool for gait analysis in unsupervised (free-living) conditions. Full article

This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger maximal Lyapunov exponents (MLE) compared to the original model, indicating increased system complexity. The existence, uniqueness, and Ulam–Hyers stability of the proposed system are verified. Additionally, we establish the global Mittag-Leffler attractive set (MLAS) and Mittag-Leffler positive invariant set (MLPIS) for the system. Numerical simulations and MATLAB phase portraits demonstrate the chaotic nature of the proposed system. Furthermore, a dynamical analysis achieves verification via the Lyapunov exponents, a bifurcation diagram, a 0–1 test, and a complexity analysis. A new numerical approximation method is proposed to solve non-linear fractional differential equations, utilizing fractional differentiation with a non-singular and non-local kernel. These numerical simulations illustrate the primary findings, showing that both external and internal factors can accelerate the process. Furthermore, a robust control scheme is designed to stabilize the system in finite time, effectively suppressing chaotic behaviors. The theoretical findings are supported by the numerical results, highlighting the effectiveness of the control strategy and its potential application in real-world supply chain management (SCM). Full article

Chaotic time series are widely present in practice, but due to their characteristics—such as internal randomness, nonlinearity, and long-term unpredictability—it is difficult to achieve high-precision intermediate or long-term predictions. Multi-layer perceptron (MLP) networks are an effective tool for chaotic time series modeling. Focusing on chaotic time series modeling, this paper presents a generalized degree of freedom approximation method of MLP. We then obtain its Akachi information criterion, which is designed as the loss function for training, hence developing an overall framework for chaotic time series analysis, including phase space reconstruction, model training, and model selection. To verify the effectiveness of the proposed method, it is applied to two artificial chaotic time series and two real-world chaotic time series. The numerical results show that the proposed optimized method is effective to obtain the best model from a group of candidates. Moreover, the optimized models perform very well in multi-step prediction tasks. Full article

The purpose of this study was to determine the test-retest repeatability of Blue Trident inertial measurement units (IMUs) and VICON Nexus kinematic modelling in analysing the Lyapunov Exponent (LyE) during a maximal effort 4000 m cycling bout in different body segments/joints. An additional aim was to determine if changes in the LyE existed across a trial. Twelve novice cyclists completed four sessions of cycling; one was a familiarisation session to determine a bike fit and become better accustomed to the time trial position and pacing of a 4000 m effort. IMUs were attached to the head, thorax, pelvis and left and right shanks to analyse segment accelerations, respectively, and reflective markers were attached to the participant to analyse neck, thorax, pelvis, hip, knee and ankle segment/joint angular kinematics, respectively. Both the IMU and VICON Nexus test-retest repeatability ranged from poor to excellent at the different sites. In each session, the head and thorax IMU acceleration LyE increased across the bout, whilst pelvic and shank acceleration remained consistent. Differences across sessions were evident in VICON Nexus segment/joint angular kinematics, but no consistent trend existed. The improved reliability and the ability to identify a consistent trend in performance, combined with their improved portability and reduced cost, advocate for the use of IMUs in analysing movement variability in cycling. However, additional research is required to determine the applicability of analysing movement variability during cycling. Full article

This paper considers a dynamic Stackelberg game model for a manufacturer-led supply chain with risk aversion. Cooperative advertising strategy is applied to the marketing decisions of supply chain participants. Based on Stackelberg game and system dynamic theory, the game and complex dynamical behaviors are studied through the use of several methods, such as the stability region of the system, bifurcation diagram, attractor diagram, and the largest Lyapunov exponent diagram. The expected utilities of participants are given and compared by numerical simulation. The results illustrate that a series of variations in adjustment speed of advertising expenditure, participation rate of local advertising expenditure by manufacturer, risk tolerance levels, and the effect coefficient of advertising expenditure may cause a loss of stability to the system and evolve into chaos. Meanwhile, the Nash equilibrium point and the expected utility of the manufacturer and retailer will change greatly. The parameter control method is further applied to control the chaos phenomenon of the system effectively. By means of analyzing the impact of relevant factors on the game model, the manufacturer and retailer can make optimal strategy decisions in the supply chain competition. The findings of this study mainly include the following three aspects. Firstly, for market stability and maximizing revenue, the manufacturer adjusts the participation rate appropriately, avoiding too high or too low values. Secondly, the manufacturer will try to reduce their own risk tolerance level for the economic revenue, and the retailer appropriately adjust the risk tolerance level to adapt to their own development according to their own enterprise strategy. Finally, both the manufacturer and retailer reduce their own effect coefficients of advertising expenditure. Meanwhile, they will attempt to increase their opponent’s effect coefficient to gain the most revenue. The research results of this study can provide important reference for the advertising expenditure decision and revenue maximization of participants in the context of risk aversion. Full article

Understanding neuron function may aid in determining the complex collective behavior of brain systems. To delineate the collective behavior of the neural network, we consider modified tabu learning neurons (MTLN) with magnetic flux. Primarily, we explore the rest points and stability of the isolated MTLN, as well as its dynamical characteristics using maximal Lyapunov exponents. Surprisingly, we discover that for a given set of parameter values with distinct initial conditions, the periodic and the chaotic attractors may coexist. In addition, experimental analysis is carried out using a microcontroller-based implementation technique to support the observed complex behavior of the MTLN. We demonstrate that the observed numerical results are in good agreement with the experimental verification. Eventually, the collective behaviors of the considered MTLN are investigated by extending them to the network of the lattice array. We discover that when the magnetic flux coupling coefficient is varied in the presence of an external stimulus, the transition from spiral waves to traveling plane waves occurs. Finally, we manifest the formation of spiral waves in the absence of an external stimulus in contrast to previous observations. Full article