Chaos Analysis Research Articles

Investigating the Dynamics of a Unidirectional Wave Model: Soliton Solutions, Bifurcation, and Chaos Analysis

The current work investigates a recently introduced unidirectional wave model, applicable in science and engineering to understand complex systems and phenomena. This investigation has two primary aims. First, it employs a novel modified Sardar sub-equation method, not yet explored in the literature, to derive new solutions for the governing model. Second, it analyzes the complex dynamical structure of the governing model using bifurcation, chaos, and sensitivity analyses. To provide a more accurate depiction of the underlying dynamics, they use quantum mechanics to explain the intricate behavior of the system. To illustrate the physical behavior of the obtained solutions, 2D and 3D plots, along with a phase plane analysis, are presented using appropriate parameter values. These results validate the effectiveness of the employed method, providing thorough and consistent solutions with significant computational efficiency. The investigated soliton solutions will be valuable in understanding complex physical structures in various scientific fields, including ferromagnetic dynamics, nonlinear optics, soliton wave theory, and fiber optics. This approach proves highly effective in handling the complexities inherent in engineering and mathematical problems, especially those involving fractional-order systems.

Self-sustained chaotic movement of electrothermal responsive liquid crystal elastomer pendulum

Self-sustained movement has the ability to absorb energy from the external environment to maintain its own movement. In the present work, a self-sustained chaotic simple pendulum system featuring an electrothermal responsive liquid crystal elastomers fiber and a mass ball is proposed and examined. The power-on state is achieved in the system through the infusion of liquid metal into this fiber. Which enables the fiber to shrink, feeding energy into the system to compensate for the energy dissipation resulting from damping, thus allowing the system to maintain its self-sustained movement. On the basis of the existing dynamic liquid crystal elastomer model, the dynamic governing equations are established, and the movement behavior of the system under electrothermal stimulation is theoretically explored. Numerical findings suggest that the simple pendulum system presents three different movement patterns, namely stationary pattern, self-sustained oscillatory pattern, and self-sustained chaotic pattern. Moreover, five system parameters of elastic coefficient, viscosity coefficient, gravitational acceleration, shrinkage coefficient and electrothermal strength are examined. The present work can contribute to a better understanding of self-sustained chaotic systems driven by electrothermal responsive materials and offer guidance for further exploration of applications such as chaos machine, and mind-brain chaos analysis.

Dynamic of bifurcation, chaotic structure and multi soliton of fractional nonlinear Schrödinger equation arise in plasma physics

In this study, we examine the third-order fractional nonlinear Schrödinger equation (FNLSE) in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+1)$$\\end{document}-dimensional, by employing the analytical methodology of the new extended direct algebraic method (NEDAM) alongside optical soliton solutions. In order to better understand high-order nonlinear wave behaviors in such systems, the researched model captures the physical and mathematical properties of nonlinear dispersive waves, with applications in plasma physics and optics. With the aid of above mentioned approach, we rigorously assess the novel optical soliton solutions in the form of dark, bright–dark, dark–bright, periodic, singular, rational, mixed trigonometric and hyperbolic forms. Additionally, stability assessments using conserved quantities, such as Hamiltonian property, and consistency checks were used to validate the solutions. The dynamic structure of the governing model is further examined using chaos, bifurcation, and sensitivity analysis. With the appropriate parameter values, 2D, 3D, and contour plots can all be utilized to graphically show the data. This work advances our knowledge of nonlinear wave propagation in Bose–Einstein condensates, ultrafast fibre optics, and plasma physics, among other areas with higher-order chromatic effects.

Chaos and feasibility analysis of a healthy lifestyle’s sustainability using fractional-order polycystic ovarian syndrome’s effect on infertility

In this study, we developed a fractional-order model for polycystic ovarian syndrome issue transmission in women through four compartments with the Mittag-Leffler kernel. For biological feasibility, we discussed the positivity and boundedness of solutions with fixed point mappings in Banach spaces. Sensitivity analysis is performed using different variables as response functions each time, employing Latin Hyper-cube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) with 200 iterations. Further, the reproductive number is determined for the model’s biological feasibility with an analysis of parameter impact through Matlab software. The solutions are examined for global stability using the Lyapunov function with a first derivative. Chaos Control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium. Numerical solutions are derived by using a Newton polynomial in the generalized form of the Mittag-Leffler kernel, which illustrates the effect of the fractional operator and parameters involved in the system. Results are also analyzed with different assumptions of parameter values and initial conditions to observe the different impacts of treatment to overcome the risk of infertility in women. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies to overcome the risk of problems and best management care pre-detection of the issue and for treatment.

Dynamical complexity and multifractal analysis of geomagnetic activities at high temporal scales over three solar cycles

Activities in geospace occur at different time scales. Understanding geomagnetic activity at high temporal scales will give insight into fast dynamics in geospace. This study aims to investigate dynamical complexities in geomagnetic activities at a high temporal scale across three solar cycles. Geomagnetic activities, as represented by 5-min SYM-H data, were considered in this study over three solar cycles (22–24) from 1986 to 2019. Chaos analysis using sample entropy, Lyapunov exponent, and correlation dimension indicates that the geomagnetic activities are driven by intrinsic complex and chaotic processes. Positive Lyapunov exponent values between 0.13 and 0.18, 0.15–0.18, and 0.16–0.19 were obtained for solar cycles 22, 23, and 24 respectively. Furthermore, geomagnetic activities were also found to have multifractal structures driven by high fractal exponents with fine structures. A positive relationship was obtained between the annual mean values of SYM-H and the degree of complexity. It is concluded that geomagnetic activities have a short prediction horizon.

Intelligent Adaptive Networks for Chaos Analysis in Stochastic SIS Differential Epidemic Model with the Impact of Unpredictable Environmental Factors

In this study, an intelligent computing framework is presented to explore the influence of unpredictable environmental factors on the spread of infections. A stochastic non-linear SIS epidemic model (SISEM) is examined that incorporates a non-linear incidence rate, and impact of random fluctuations on the disease transmission. The knacks of artificial neural networks Levenberg–Marquardt (ANNLMA) technique are exploited to illustrate the behavioral complexities of stochastic non-linear SISEM, which explores the dynamics of disease transmission in a stochastic environment and provides insights into the interactions between susceptible and infectious populations. The Milstein approach of order 1 generates a dataset for the proposed ANNLMA technique by varying integrated parameters, such as the rates of recovery, death, disease transmission, noise standard deviation and the entire population, for various SISEM scenarios. The referenced dataset is randomly divided into test, validation and train sets for the weight learning of the proposed ANNLMA. The viability of the ANNLMA is measured using intensive simulation-based investigations through analysis of mean squared errors, histograms, regression and autocorrelation studies. The investigations highlighted that the ANNLMA results closely match the reference results, illustrating convergence within the 10[Formula: see text] to 10[Formula: see text] range, indicating the validity of the proposed methodology.

Chaos in Bitcoin Cryptocurrency Metrics: Analysis and Forecasts

AbstractCryptocurrencies, particularly Bitcoin have attracted a lot of attention in the last decades of humanity. Analyzing cryptocurrencies algorithmic differences, chaotic behavior and self-similarity in cryptocurrency metrics might give significant insights for identifying risks and opportunities. Determining the degree of chaos in crypto metrics is critical for understanding complexity, improving prediction capabilities, and supporting decision-making. This study focuses on the analysis of chaos and self-similarity in Bitcoin dynamics for predictability perspective. Return, rate of return and volume quantities in different scales are analyzed with using rescaled range method to reveal the degree of self-similarity. Hurst parameter extracts a comprehensive summary providing information on how current values depend on previous ones to reveal any persistence in Bitcoin metrics. Daily rate of return and return give Hurst degree around 0.64 while they are in between 0.52–0.55 for minutely and hourly based prices. However, an increasing persistence is observed with the increasing time window. Although the largest Lyapunov exponents stay in the positive region for prices and returns of Bitcoin, they are approximately zero for inspected statistics. Periodic characteristics of Bitcoin are also investigated to reveal any dependencies on halving mechanism of Bitcoin. Detailed self-similarity analysis on specific periods shows that bull and bear market seasons don’t make any significant effect on the degree of Hurst parameter. Due to nonlinear and unpredictable characteristics of Bitcoin metrics, distribution fittings are applied to characterize BTC return and rate of return. While Wakeby distribution gives best fitting for daily return, Cauchy distribution gives best for hourly returns.

Data-driven system identification and model predictive control of pneumatic conveying using nonlinear dynamics analysis for optimised energy consumption

Pneumatic conveying systems provide secure transportation of particulate material and a dust-free environment. These systems face high energy consumption, material degradation, and pipeline blockages. This research presents an innovative solution by integrating nonlinear dynamics analysis of electrostatic sensor data, including chaos and recurrence quantification analysis, sparse identification of nonlinear dynamics with control (SINDYc) and model predictive control (MPC). The Lyapunov exponent, approximate entropy and recurrence rate of electrostatic sensor data reveal the chaotic nature of gas-solid flows. MPC framework was tailored for real-time optimisation of a pneumatic conveying system. SINDYc system models were developed using data collected from an open-loop control pneumatic conveying process to conduct MPC simulations and select an appropriate model for real-time control. This research illustrates the potential of integrating nonlinear dynamics analysis, SINDYc integrated and MPC for enhanced system performance, showcased a significant reduction in energy consumption without compromising the system's efficiency or reliability.

Soliton wave profiles and dynamical analysis of fractional Ivancevic option pricing model

This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions.

Chaos analysis and traveling wave solutions for fractional (3+1)-dimensional Wazwaz Kaur Boussinesq equation with beta derivative

Wazwaz Kaur Boussinesq (WKB) equation can effectively simulate the behavior of water waves in shallow water, including the nonlinear effect and dispersion phenomenon of waves, which is of great significance for understanding the dynamic process of ocean, river and other water bodies. To enrich the wave equation theory, the (3+1)-dimensional integer order derivative of WKB equation is changed to the fractional one with beta derivative. The current work deals with the fractional (3+1)-dimensional WKB equation for discussing its chaotic behavior and establishing some new analytic solutions. The chaotic properties of the equation are verified by the trend of evolution along with time, Lyapunov exponents and initial sensitivity analysis. And then complete discrimination system for polynomial method is applied to derive some trigonometric, hyperbolic, Jacobi elliptic and other solutions. The graphical demonstrations are provided for part of these solutions. From these visualized graphs, the solitary, periodic and quasi-periodic wave are shown and the effect of fractional derivatives on the equation can be seen intuitively.

Dynamical Study of Nonlinear Fractional-order Schrödinger Equations with Bifurcation, Chaos and Modulation Instability Analysis

Dynamical Study of Nonlinear Fractional-order Schrödinger Equations with Bifurcation, Chaos and Modulation Instability Analysis

Optical solitons, dynamics of bifurcation, and chaos in the generalized integrable (2+1)-dimensional nonlinear conformable Schrödinger equations using a new Kudryashov technique

In the present paper, the new Kudryashov approach is utilized to construct several novel optical soliton solutions for the generalized integrable (2 + 1)-dimensional nonlinear Schrödinger system with conformable derivative. Additionally, the dynamics of bifurcation behavior and chaos analysis in this system are investigated. We applied bifurcation and chaos theories to enhance our understanding of the planar dynamical system derived from the current model while we obtained and illustrated the chaotic solutions for the perturbed dynamical system using graphs. The study yields a class of new optical soliton solutions, including bell-shaped, wave, dark, dark-bright, dark, multi-dark, and singular soliton solutions. Three-dimensional, two-dimensional, and contour plots are presented to visually demonstrate the physical implications and dynamic characteristics of the current conformable equation system. Further, an analysis is discussed on how the conformable derivative parameter and the parameter of time impact the present optical solutions, demonstrating the system’s importance. It is believed that the solutions analyzed in this study are entirely new and have not been previously reported. These discoveries have the potential to significantly enhance our understanding of nonlinear physical phenomena, especially in nonlinear optics and traffic signaling effects with optical dromion transmission.

The fractional solitary wave profiles and dynamical insights with chaos analysis and sensitivity demonstration

This work investigates the dynamics of solitary waves in a thin-film ferroelectric material model, an essential component of optical systems. The study recognizes the limitations of the inverse scattering transform in solving the Cauchy problem for this conformable thin-film ferroelectric material model and applies a new Kudryashov approach and Bernoulli’s equation technique to ordinary differential equations. We apply these methods to the present model and obtain closed form analytical solutions. A variety of traveling wave and soliton solutions are constructed, including kink, bell, bright, and dark soliton solutions. It is acknowledged that the nonlinear evolution equation can be transformed into an ordinary differential equation by applying traveling wave transformations. Suitable physical parameters are chosen to create 2D, 3D, and contour plots that graphically depict the graphical dispersion of obtained fractional soliton solutions. The impact of the conformable fractional parameter is further demonstrated by the graphical representation of solitons propagation. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the system’s total energy in terms of momentum and position. While a chaos analysis describes chaotic, quasi-periodic, and periodic behaviors, a sensitivity analysis demonstrates how responsive the model is to various initial conditions. Due to their dual-purpose nature as nonlinear ferroelectric and dielectric materials, thin ferroelectric films are widely used in modern electrical devices.

Simulink Modeling and Analysis of a Three-Dimensional Discrete Memristor Map

The memristor, a novel device, has been widely utilized due to its small size, low power consumption, and memory characteristics. In this paper, we propose a new three-dimensional discrete memristor map based on coupling a one-dimensional chaotic map amplifier with a memristor. Firstly, we analyzed the memristor model to understand its characteristics. Then, a Simulink model for this three-dimensional discrete memristor map was developed. Lastly, the complex dynamical characteristics of the system were analyzed via equilibrium points, bifurcation diagrams, Lyapunov exponent spectra, complexity, and multistability. This study revealed the phenomena of coexisting attractors and hyperchaotic attractors. Simulink modeling confirmed that the discrete memristors effectively enhanced the chaos complexity in the three-dimensional discrete memristor map. This approach addresses the shortcomings of randomness, the lack of ergodicity, and the small key space in a one-dimensional chaotic map, thereby enriching the theoretical analysis and circuit implementation of chaos.

Chaos Analysis and Prediction of Monthly Runoff Using a Two-Stage Variational Mode Decomposition Framework

Chaos Analysis and Prediction of Monthly Runoff Using a Two-Stage Variational Mode Decomposition Framework

The analysis of chaos and fractals

Chaos and fractals are closely interconnected ideas in the field of mathematics. The behavior of complex systems is characterized by chaos, which is the result of excessive sensitivity to initial conditions, leading to unpredictable and random consequences. Fractals are geometric patterns that display self-similarity at many scales. This paper examines the complex connection between chaos and fractals using a method of theoretical analysis and a comprehensive examination of relevant literature. Studying the connection between chaos science and fractals can provide valuable insights into chaotic events in nonlinear dynamical systems. Furthermore, this investigation enables readers to understand the underlying fractal nature of such systems. Gaining insight into this correlation not only enhances comprehension of the fundamental principles that control intricate systems, but also provides opportunities for utilization in other domains, spanning from physics to biology and beyond.

A Two-Dimensional Discrete Memristor Map: Analysis and Implementation

In this paper, we present a novel two-dimensional discrete memristor map that is based on a discrete memristor model and a sine–arcsine one-dimensional map. First, an analysis is conducted on the memristor model to understand its characteristics. Then, the model is coupled with the sine–arcsine one-dimensional map to achieve the two-dimensional discrete memristor map. Our investigation reveals the presence of coexisting attractors and hyperchaotic attractors as the bifurcation parameters vary. Numerical simulations show that the discrete memristors effectively enhance the complexity of chaos in the sine–arcsine map. Furthermore, a digital circuit is designed to experimentally verify the new chaotic system. The research results can enrich the theoretical analysis and circuit implementation of chaos.

Bifurcation and Chaos Analysis of a Supersonic Slipper–Track System

Bifurcation and Chaos Analysis of a Supersonic Slipper–Track System

A Comprehensive Review on Fractional Order Control

Fractional order control is one of the most important enablers of advanced closed loop industrial automation systems. This paper presents a detailed bibliometric study of fractional order control related research published during the past two decades (2000–2022) in the Web of Science (WoS) and Scopus indexed sources. The WoS analysis included yearly publications, publication types and top contributing publishing sources, organizations, researchers, countries and research areas. VoS viewer based full counting methodology was used to analyze citation, co-citation and co-authorship based collaborations among authors, organizations, countries, sources and documents. Co-occurrence, timeline and burst detection analyses of keywords and published articles were also carried out to unravel significant fractional control research trends in the last two decades. The Scopus analysis firstly included annual publication counts, subject areas, document types and top contributing authors, sources, nations, institutions and funding agencies. Secondly, dedicated Scopus searches were carried out for hardware implementation, non integer control and patents. Results indicate that the fractional order controller research has been primarily focused in engineering domain, followed by mathematics, physics, automation control systems and computer science. Materials, mechanics and telecommunications are some of the lesser explored areas. There is a lot of scope of fractional order control research related to patents, hardware implementation, non integer control, anomalous diffusion, chaos and task analysis. Future investigations will also include neural networks based machine learning in fractional order control solutions. Key Words: Fractional order controller, FOPID, Fractional calculus

A Lie symmetry approach to traveling wave solutions, bifurcation, chaos and sensitivity analysis of the geophysical Korteweg–de Vries equation

Strong waves known as tsunamis are caused by earthquakes, landslides, or volcanic eruptions that traverse oceans. This article examines the geophysical Korteweg–de Vries (GPKdV) equation, which controls the propagation of tsunami waves in seas. The study involves exploring symmetry diminution exerted Lie group analysis, examining the properties of the dynamical structure with the help of bifurcation phase pictures, and researching the dynamic demeanor of the perturbed dynamical system utilizing chaos theory. Techniques such as 3D and 2D phase portraits, time series analysis, poincaré maps, looking into the existence of multistability in the autonomous structure under diverse beginning circumstances, lyapunov exponent, and bifurcation diagram are applied to identify chaotic demeanor. Furthermore, the study establishes general forms of solitary wave results, containing periodic, trigonometric, and singular soliton results, by using the unified Riccati equation expansion approach to address the examined problem analytically. These results are visually represented as 2D and 3D graphs with cautiously chosen parameters, along with their accompanying constraint conditions. Moreover, the sensitivity evaluation of the investigated equation is discussed and demonstrated pictorially. The discoveries revealed are intriguing, novel, and potentially helpful in understanding a wide range of physical events in engineering and science.