Chaotic Nature Research Articles

Bifurcations, chaotic behavior, and optical solutions for the complex Ginzburg–Landau equation

This research focuses on investigating an extended version of the Ginzburg–Landau (GL) equation that describes the motion of particles in a plasma. The other applications of this model can be found in optics, and other related fields. As part of our approach, we seek to discover some new wave solutions to the model, and then use the Galilean transformation in order to determine the model’s dynamical properties. After that, we use planar dynamical system theory to perform an in-depth bifurcation analysis, which gives us valuable insight into the system. By conducting thorough mathematical and bifurcation analyses, we identify the fundamental dynamics and critical points that regulate the system’s behavior. Moreover we explore the chaotic nature of the system, uncovering potential chaotic tendencies and their consequences. Further, we obtain several categories of optical solutions of the complex GL equation by utilizing new logarithmic transformations, offering valuable insight into its behavior and potential applications in optics. Our analytical technique yields closed-form solutions expressing elementary functions. In order to ensure the reliability of our findings, we rigorously validate the obtained solutions by substituting them back into the original model. Our research helps us better understand this equation’s characteristics and its relevance across a wide range of disciplines.

Analyzing the impact of COVID-19 on global stock markets: An international comparative analysis

In 2020, during the first wave of COVID-19, a massive collapse was observed in the stock markets of all the economically wealthy and prominent countries all over the world. Consequently, the second and third waves of COVID-19 strike stock markets in 2021. The objective of the present study is to search for the effect of the pandemic on eight major stock exchanges across different continents, namely, SENSEX, Dow-Jones, NYSE Composite, FTSE 100, SSE Composite, Nikkei 225, MOEX and MASI, in 2020-2021. Various techniques assess nonlinear analysis, volatility, and the chaotic nature of these markets. The present study concludes that though all the indices are volatile and non-chaotic, some structural changes in pattern are identified in this time frame. The FTSE 100 stands out as a stable investment option, while the NYSE Composite is recommended primarily for long-term investments.

Orchestrating Harmony in the Chaos of Maltese Language Learning through the Lens of Chaos/Complexity Theory

Despite considerable studies in the field of second language acquisition (SLA), the precise cognitive processes occurring in an adult’s mind during second language (L2) learning remain elusive. This study investigates the intricate process of adult’s SLA, specifically focusing on Maltese. Grounded in chaos/complexity theory (C/CT), the research explores the non-linear, dynamic, and unpredictable nature of language learning, likening it to the emergence of order in seemingly chaotic systems. The study employs a qualitative approach, by employing various data collection methods such as structured tests, reflective journals, and interviews to track the learning progression of 35 adults learning Maltese over 15 months. The emphasis is on the acquisition of Maltese verbs and the emergence of non-linear learning curves, akin to the butterfly effect in C/CT. The findings underscore the chaotic yet harmonious nature of Maltese language learning, providing valuable insights for educators and learners alike.

Underwater implosion behavior of 3D-printed polymer structures

This study experimentally investigates the failure behavior of 3D-printed polymer tubes during underwater implosion. Implosion is a prevalent failure mechanism in the underwater domain, and the adaptation of new technology, such as 3D printing, allows for the rapid manufacturing of pressure vessels with complex geometries. This study analyzes the failure performance of 3D-printed polymer structures to aid the future development of 3D-printed pressure vessels. The 3D-printed tube specimens analyzed were fabricated using digital light synthesis (DLS) technology and included four different case geometries. The geometries consist of three cylindrical shells of varying diameter and thickness and one double hull structure with a cylindrical gyroid core filler. These specimens were submerged in a pressure vessel and subjected to increasing hydrostatic pressure until implosion failure occurred. High-speed photography and Digital Image Correlation (DIC) were employed to capture the collapse event to obtain full-field displacements. Local dynamic pressure histories during failure were recorded using piezoelectric transducers. The findings highlight that the 3D-printed polymers underwent significant deformation and failed at localized points due to material failure. The fracture of the specimens during failure introduced inconsistencies in pressure and impulse data due to the chaotic nature of the failure. Notably, the energy flow analysis revealed that the proportion of energy released via the pressure pulse was lower than in traditional aluminum structures. These findings contribute to our understanding of the behavior of 3D-printed polymers under hydrostatic pressure conditions.

Cross-channel color image encryption through 2D hyperchaotic hybrid map of optimization test functions

The security of chaos-based Image Encryption (IE) algorithms inherently depends on the permutation and diffusion strategy of the algorithm and the dynamic performance of chaotic maps. However, the existing studies suffer from low disordering capability, poor dynamic performance, and lack of sensitivity of the chaotic maps. In order to address these limitations, a Cross-Channel Color IE (CCC-IE) algorithm is presented, which has high shuffling ability with diagonal permutation and bi-directional sequential diffusion through a novel 2D hyperchaotic hybrid map of optimization test functions. The proposed 2D hyperchaotic map, namely 2D-RG map, is designed by hybridizing the Rastrigin and Griewank functions. Their chaotic nature, such as high complexity and fluctuation, makes them suitable for design of chaotic map. The 2D-RG map-based CCC-IE algorithm diagonally scrambles the pixels of the image across RGB channels and sequentially manipulates the pixel values in bi-direction for improving resistance of the algorithm to the cyberthreats. The dynamic performance of the 2D-RG map and the reliability of the CCC-IE algorithm are separately investigated and validated through a detailed comparison with the state of the art. The results manifest that the CCC-IE not only visually but also numerically is the more reliable algorithm thanks to the superior chaotic characteristics of the 2D-RG map.

Tracking dynamics change parameters of chaotic infectious disease outbreak with bifurcated time-series long short-term memory model

In the dire need to develop a robust epidemiological surveillance model, this study aimed to determine if the dynamics change parameters of past infectious disease outbreaks would enable early accurate detection of future outbreaks. An optimized regular LSTM could not accurately learn the non-linear relationship of the epidemiological data in the peak stage of the pre-vaccination era measles outbreak, which hence influenced the decrease predictability at increasing time series. The result of this study shows that Pitchfork bifurcation of LSTM at 1.5 bifurcation point matches the dynamic change inherent in California's pre-vaccination era measles outbreak at the improved prediction performances of root mean squared error (RMSE) = 0.1684 for the 1962 outbreak, and root mean squared error (RMSE)= 0.2776 for 1964 outbreak. In conclusion, it was observed that the dynamics change parameters are dependent on the stochastic nature of the optimization algorithm engaged, and the chaotic nature of epidemiological data.

A Non‐Intrusive Machine Learning Framework for Debiasing Long‐Time Coarse Resolution Climate Simulations and Quantifying Rare Events Statistics

AbstractDue to the rapidly changing climate, the frequency and severity of extreme weather is expected to increase over the coming decades. As fully‐resolved climate simulations remain computationally intractable, policy makers must rely on coarse‐models to quantify risk for extremes. However, coarse models suffer from inherent bias due to the ignored “sub‐grid” scales. We propose a framework to non‐intrusively debias coarse‐resolution climate predictions using neural‐network (NN) correction operators. Previous efforts have attempted to train such operators using loss functions that match statistics. However, this approach falls short with events that have longer return period than that of the training data, since the reference statistics have not converged. Here, the scope is to formulate a learning method that allows for correction of dynamics and quantification of extreme events with longer return period than the training data. The key obstacle is the chaotic nature of the underlying dynamics. To overcome this challenge, we introduce a dynamical systems approach where the correction operator is trained using reference data and a coarse model simulation nudged toward that reference. The method is demonstrated on debiasing an under‐resolved quasi‐geostrophic model and the Energy Exascale Earth System Model (E3SM). For the former, our method enables the quantification of events that have return period two orders longer than the training data. For the latter, when trained on 8 years of ERA5 data, our approach is able to correct the coarse E3SM output to closely reflect the 36‐year ERA5 statistics for all prognostic variables and significantly reduce their spatial biases.

The Complexities in the R&D Competition Model with Spillover Effects in the Supply Chain

This study aims to investigate the research and development (R&D) competition within the supply chain, focusing on two aspects: R&D competition at the manufacturing level and competition in pricing strategies. This paper establishes a dynamic game model of R&D competition, comprising two manufacturers and two retailers, with both manufacturers exhibiting bounded rationality. The key findings are as follows: (1) an increase in the adjustment speed positively affects the chaotic nature of the R&D competition system, leading to a state of disorder. This chaotic state has adverse implications for manufacturing profitability. (2) The spillover effect exhibits a positive relationship with the level of chaos in the R&D competition system. A greater spillover effect contributes to a more turbulent environment, which subsequently impacts the profitability of manufacturers. (3) R&D cost parameters exert a positive influence on the stability of the R&D competition system. When the system reaches a state of equilibrium, an escalation in the R&D cost parameters poses a threat to manufacturer profitability. (4) Retailer costs play a detrimental role in the stability of the R&D competition system. As retailer costs increase, there is a decline in R&D levels, thereby diminishing manufacturer profitability. (5) To mitigate the chaotic state, we propose the implementation of the time-delayed feedback control (TDFC) method, which reflects a more stable state in the R&D competition system.

A 5-D memristive hyperchaotic system with extreme multistability and its application in image encryption

By coupling memristors to nonlinear circuits, more complex dynamical behaviors can be induced. However, to date, there has been insufficient attention given to high-dimensional chaotic systems based on memristors. In this paper, a magnetic-controlled memristor is combined with a three-dimensional chaotic system, resulting in a five-dimensional memristive chaotic system. Through dynamic analysis and numerical simulations, the chaotic nature of the system is elucidated based on fundamental system behaviors, including Lyapunov dimension, dissipativity, stability of equilibrium points, 0–1 test, and Poincaré mapping. During the complex dynamical analysis of this system, unique dynamical behaviors are discovered, including intermittent chaos, transient chaos, extreme multistability, and offset-boosting. Moreover, the consistency between numerical calculations and the physical implementation of the actual system is verified through equivalent circuit design. Finally, this system is applied to image encryption, leading to the design of an efficient and secure hyper-chaotic image encryption algorithm, whose effectiveness is confirmed through several security tests.

A novel crude oil price forecasting model using decomposition and deep learning networks

Crude oil prices are susceptible to a variety of elements, including military, political, financial, and diplomatic forces, resulting in price fluctuations that are random, abrupt, and chaotic. Based on the chaotic nature of crude oil price series, this paper proposes a model for predicting crude oil prices that incorporates variational modal decomposition algorithm (VMD), phase space reconstruction technique (PSR), convolutional neural network (CNN), and bidirectional long and short-term memory network (BiLSTM). Specifically, the noises in the original data are removed using VMD. Next, the crude oil price is reconstructed using PSR. Then the reconstructed and denoised phase space is fed into the CNN-BiLSTM model for multi-step predictions. The empirical results show that the proposed model achieves the lowest MAPEs as 0.662%, 2.283%, 3.829%, and 4.44% and the lowest MSEs as 0.335, 3.942, 10.047 and 13.681. The Diebold Mariano (DM) test also demonstrates the proposed model’s superiority in the domain of crude oil price forecasting.

The chaotic properties and circuit design of a generalized high-dimensional integer-domain system

The purpose of this study is to formulate an innovative approach for creating a higher-dimensional digital chaotic system (HDDCS) within the realm of finite precision. The suggested methodology revolves around generating random sequences to regulate the emergence of chaotic behavior. Rigorous mathematical proofs of the chaotic characteristics of the HDDCS are given, and the nonlinear characteristics of HDIDS are demonstrated by examples. Furthermore, an analog–digital hybrid circuit is developed to design the generalized HDIDS. This study offers substantial theoretical support for the chaotic nature of the HDIDS, and validates the feasibility of chaotic circuit design by simulation.

Energy distribution and chaotic pressure pulsation analysis of vortex ropes in Francis-99

Francis turbines, essential for stability in diverse operating conditions and variable-speed scenarios, encounter efficiency-compromising vortex rope formations in the draft tube, leading to substantial pressure fluctuations. This research delves into the analysis of energy loss and pressure fluctuations associated with these vortex ropes. Employing the local entropy generation rate (LEGR) method and chaos theory, we scrutinize the behaviour of vortex ropes and their resultant pressure fluctuations. Notably, vortex ropes exhibit maximum LEGR near the runner cone, with secondary vortices escalating instability downstream. In the elbow section, the collision of vortex ropes with the outer elbow amplifies LEGR, primarily driven by fluctuating velocities (approximately 90%). Leveraging the GWO-VMD algorithm, non-stationary signals are decomposed, unveiling a significant 1.6 Hz vortex rope frequency under partial load (PL) conditions and isolating external noise frequencies, such as the prominent 300 Hz. Following decomposition, chaos theory tools, including phase space reconstruction and phase trajectory graphs, unveil the chaotic nature of PL conditions attributed to spiral vortex ropes, resulting in profound pressure fluctuations. This study enhances our understanding of such systems and provides methodologies for improved noise reduction and optimization of turbine performance.

Low-altitude small target detection in sea clutter background based on improved CEEMDAN-IZOA-ELM

To effectively detect low-altitude small targets under complex sea surface environment, an innovative method has been developed. This method harnesses the chaotic characteristics of sea clutter and employs a combination of Adaptive Noise Complete Ensemble Empirical Modal Decomposition (CEEMDAN), Adaptive Wavelet Thresholding (AWT), and Polynomial Fitting Filtering (SG) for denoising sea clutter data. Subsequently, the Improved Zebra Optimization Algorithm-Extreme Learning Machine (IZOA-ELM) detector is utilized to identify low-altitude small targets amidst the sea clutter background. To begin, the CEEMDAN method is applied to disentangle the measured sea clutter data into a set of Intrinsic Mode Functions (IMFs). Afterwords, the Refined Composite Multiscale Dispersion Entropy (RCMDE) is computed for each individual IMF. This process categorizes the IMFs into three distinct components: noise-dominant, signal-noise mixture, and signal-dominant segments. The noise-dominate of IMF component is subjected to denoising through AWT, the signal-noise mixture of IMF components are processed using SG filtering, while the signal-dominant of IMF remains unaltered. The denoised sea clutter signal is reconstructed by concatenating the denoised and unprocessed IMFs. Based on the chaotic nature of sea clutter signals, first-order sea clutter data is transformed into high-dimensional data through phase space reconstruction. The initial weights and thresholds of the ELM are optimized through the IZOA to establish an optimal prediction model. This model is then used to detect small, low-altitude targets by analyzing the prediction error. The algorithm's effectiveness in noise removal is validated using IPIX and SPRR measured sea clutter data, demonstrating a significant improvement in the root mean square of prediction error (RMSE) by one order of magnitude after denoising compared to the pre-denoising state. Furthermore, we observed that the IZOA-ELM method can be effectively applied to detect small targets at low altitudes across various sea conditions. However, when the sea state is complex and greatly affected by the surrounding noise, an effective approach is to first employ CEEMDAN-AWT-SG to denoise the original signal, and then utilize IZOA-ELM for target detection.

Relationship between magnetic rigidity cutoff and chaotic behavior in cosmic ray time series using visibility graph and network analysis techniques.

Cosmic rays are highly energetic particles originating from astrophysical events outside the Solar System. In this study, we analyze the time series of cosmic ray flux measured by neutron detectors at 16 monitoring stations distributed worldwide. By applying visibility graph analysis, we explore the relationship between the magnetic rigidity cutoff (Rc) and the fractality exhibited from topology of the cosmic ray time series. Our results reveal a significant association between the magnetic rigidity cutoff and the fractality of the cosmic ray time series. Specifically, the analysis of visibility graphs and network properties demonstrates that the magnetic rigidity is inversely related to the magnetic rigidity cutoff. The identified relationship between magnetic rigidity and fractality provides insights into the chaotic nature of cosmic ray variations and their potential uses for predictability.

Modal analysis of vortex rope using dynamic mode decomposition

The decelerating swirling flow in the draft tube of hydraulic turbines at part load conditions often results in a self-induced instability known as vortex rope. This phenomenon is associated with detrimental pressure pulsations in the hydropower system that need to be mitigated. A deep understanding of such instability is essential for developing effective mitigation and control strategies. The current article exploits the dynamic mode decomposition (DMD) algorithm to perform an in-depth modal analysis of the physical aspects of the vortex rope. DMD can efficiently identify distinct coherent structures with isolated frequencies. The sparsity-promoting variant of DMD is exploited to extract the most influential modes. The computational fluid dynamics (CFD) data is generated via a resolved improved delayed detached eddy simulation using OpenFOAM. Frequency analysis of the CFD data uncovered peaks at the normalized frequencies of f/fn=0.56 and 0.63, whose origins seemed initially unclear. Nevertheless, the DMD modal analysis elucidates that these excitations are associated with the rotation of the reunited vortex and fluctuations of the separated boundary layer, respectively. The non-linear dynamics of the flow field are unveiled through a modal decomposition revealing distinctive coherent structures with isolated frequencies. These include rotational and plunging modes of the vortex rope, traveling wakes of the blades, boundary layer separation due to strong adverse pressure gradient, and a reunited vortex core. The flow field reconstruction through time dynamics of DMD modes highlights while it is possible to achieve a perfect flow field reconstruction considering all recovered modes, the model typically fails to predict future behavior with an acceptable level of accuracy. The chaotic nature of the resolved turbulent flow field presents a substantial challenge for predicting the future through a model built based on prior events. The current modal analysis not only provides a more comprehensive understanding of the physics underlying the vortex rope phenomenon but also lays the groundwork for potential future applications in controlling mechanisms.

Study on William Faulkner's Long Sentence Style Based on “The Sound and the Fury”

This paper provides an in-depth analysis of William Faulkner's narrative technique in "The Sound and the Fury," with a specific focus on his use of long sentences and multi-perspective storytelling. Faulkner's innovative approach to sentence structure is examined as a key element in shaping the novel's thematic depth and emotional resonance. Each section of the novel, presented through different characters' viewpoints, is characterized by unique sentence constructions that reflect individual mental states and emotional experiences. The paper explores how these long, complex sentences enhance the reader's understanding of themes such as time, memory, and identity. Additionally, the interaction between Faulkner's elongated sentences and the multi-perspective narrative is analyzed as a crucial device in conveying the fragmented and chaotic nature of the characters' lives and their broader social context. The paper concludes that Faulkner's work, through its stylistic complexity and depth, continues to resonate as a seminal exploration of the human condition.

A new chaos function development through the combination of Circle map and MS map

Digital data protection is very important to prevent manipulation of digital data by unauthorized parties. Reliable techniques for securing digital data are needed, safe and fast. One technique is to use cryptography. One of the cryptographic techniques that can be used to encode digital data is to use the chaos function. We propose in this paper a new chaos function which is a composition of Circle map and MS map functions. This function has chaotic nature and the result of which is named the MSI-Circle map. The sensitivity and randomness tests of the MSI-Circle map function are carried out using a bifurcation diagram, Lyapunov exponent, and NIST test suites. The analysis result of the bifurcation diagram shows that the MSI-Circle map has a good density at the value of r ∈ (−∞,−3] ∪ [3,−∞]. Lyapunov exponent has a non-negative value at x0 = 0.4, r = 3.8, Ω = 0.5, λ = 2.1, K = 4 which is the domain xn ∈ (0, 1) and parameter values r, Ω, λ and K are any real numbers. The results of the NIST randomness level test show that the MSI-Circle map function passed all the randomness test of 16 NIST tests.

An Efficient Algorithm for Stock Market Prediction Using Attention Mechanism

Forecasting the stock market is a significant challenge in the financial industry due to its time series' complicated, noisy, chaotic, dynamic, volatile, and non-parametric nature. Nevertheless, due to computer advancements, an intelligent model can assist investors and expert analysts mitigate the risk associated with their investments. In recent years, substantial research has been conducted on deep learning models. Many studies have investigated using these techniques to anticipate stock values by analyzing historical data and technical indications. However, since the goal is to create predictions for the financial market, validating the model using profitability indicators and model performance is crucial. This article incorporates the attention mechanism model, incorporating attention from both feature and time perspectives. Utilize artificial neural networks. This approach addresses issues in time series prediction. The issue is the varying degrees of influence that many input features have on the target sequence. To tackle this, the method utilizes a feature attention mechanism to obtain the weights of distinct input features. An enhanced feature association relationship is achieved, whereas the data before and following the sequence exhibit a significant time correlation. An attention technique is employed to address this issue, allowing for the acquisition of weights at various time intervals to enhance robustness and temporal dependence. The system is applied to the three global SMs (TESLA, SP500, and NASDAQ) datasets, the best enhancement results are 99% in Acc, and the better results improvement to minimize error in MSE, MAPE, and RMSE are 0.004, 0.004 and 0.01 respectively.

Ринкова резильєнтність бізнесу: теоретичний конструкт

The aim of the article is to study the development and change of the conception of resilience of socioeconomic systems in the modern turbulent world, as well as to create a theoretical construct for the market resilience of business. The conception of BANI of the world, which demonstrates the chaotic and unpredictable nature of modern society and technological environment, is studied, so business organizations must not only be resistant to stress, but also ready for change and even successfully function in conditions of uncertainty and instability in order to adapt to the new reality. Thus, the central theme of world economic thought is the study of «resilience» in general, as well as its various aspects and potential ways to guarantee the sustainability of social, economic and ecological systems. Resilience is considered as an economic category, which is inherent in the subject of economic activity at the micro-, meso- and macro-levels. The main methodological approaches to resilience are proposed, which contain the following key components: crisis management of the socioeconomic system; change and innovation management; organizational culture and leadership; staff resilience. A new approach that changes the attitude to resilience as an economic category – the paradigm of Industry 5.0 – is considered. In the context of resilience, this paradigm aims to create viable, flexible and resilient systems that can adapt to different challenges. The types of resilience of different levels of the socioeconomic system are examined and systematized. The authors concluded that these kinds intersect, interact and complement at different levels, which is an important aspect of ensuring the stability and adaptability of this system. A definition of the concept of market resilience has been proposed, which is a characteristic of a business organization, which determines its ability to create new opportunities for sustainability, survival and development in conditions of total uncertainty, non-linear and unpredictable changes in the internal and external environment. The key elements of the formation of market resilience of a business entity are systematized and characterized.

The Intersection of Reality and Fiction in Tom Stoppard’s Rosencrantz and Guildenstern Are Dead: A Study of Absurdity and Metadrama

The article titled "The Intersection of Reality and Fiction in Tom Stoppard’s Rosencrantz and Guildenstern Are Dead: A Study of Absurdity and Metadrama" explores how Tom Stoppard's play transforms the minor characters from Shakespeare's Hamlet into central figures within an absurdist framework. This study examines the play’s themes of human identity, confusion, and helplessness, common in the Theatre of the Absurd, using postmodernist metadramatic techniques. By employing metadrama, Stoppard highlights the blurred lines between reality and fiction, as seen in the characters' struggles to understand their existence within the play. The paper delves into the philosophical implications of absurdity, drawing on the ideas of Albert Camus and other theorists to illustrate how Rosencrantz and Guildenstern Are Dead reflects the chaotic and purposeless nature of human life. Through various metadramatic devices like the play within a play, role-playing, and the breakdown of conventional narrative structures, Stoppard's work is analyzed as a profound commentary on the human condition and the search for meaning in an incomprehensible world.