Search Results (54)

We present a novel and completely deterministic method to model chaotic orbits in nonlinear discrete dynamics, taking the quadratic map as an example. This method is based on the resurgent analysis developed by Écalle to perform the resummation of divergent power series given by asymptotic expansions in linear differential equations with variable coefficients. To determine the long-term behavior of the dynamics, we calculate the zeros of a function representing the unstable manifold of the system using Newton’s method. The asymptotic expansion of the function is expressed as a kind of negative power series, which enables the computation with high accuracy. By use of the obtained zeros, we visualize the set of homoclinic points. This set corresponds to the Julia set in one-dimensional complex dynamical systems. The presented method is easily extendable to two-dimensional nonlinear dynamical systems such as Hénon maps. Full article

In this paper, a pattern for visualizing fractals, namely Julia and Mandelbrot sets for complex functions of the form $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }\mathrm{for}\mathrm{all}u\in \mathbb{C}$ and $a\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \mathbb{C}$, $r,\rho \in \mathbb{C}\setminus \left\{0\right\}$ are created using novel fast convergent iterative techniques. The new iteration scheme discussed in this study uses s-convexity and improves earlier approaches, including the Mann and Picard–Mann schemes. Further, the proposed approach is amplified by unique escape conditions that regulate the convergence behavior and generate Julia and Mandelbrot sets. This new technique allows greater versatility in fractal design, influencing the shape, size, and aesthetic structure of the designs created. By modifying various parameters in the suggested scheme, a significant number of visually interesting fractals can be generated and evaluated. Furthermore, we provide numerical examples and graphic demonstrations to demonstrate the efficiency of this novel technique. Full article

Due to the increasing complexity of industrial systems, fault detection hinders the continuity of productivity. Also, many methods in industrial systems whose complexity increases over time have a mechanism based on human intervention. Therefore, the development of intelligent systems in fault detection is critical.. Avoiding false alarms in detecting real faults is one of the goals of these systems. Modern technology has the potential to improve strategies for detecting faults related to machine components. In this study, a hybrid approach was applied on two different datasets for fault detection. First, in this hybrid approach, data is given as input to the artificial neural network. Then, predictions are obtained as a result of training using the ANN mechanism with the feed forward process. In the next step, the error value calculated between the actual values and the estimated values is transmitted to the feedback layers. IWO (Invasive Weed Optimization) optimization algorithm is used to calculate the weight values in this hybrid structure. However the IWO optimization algorithm is designed to be initialized with fractal-based weighting. By this process sequence, it is planned to increase the global search power without getting stuck in local minima. Additionally, fractal-based initialization is an important part of the optimization process as it keeps the overall success and stability within a certain framework. Finally, a testing process is carried out on two separate datasets supplied by the Kaggle platform to prove the model’s success in failure detection. Test results exceed 98%. This success indicates that it is a successful model with high generalization ability. Full article

This paper introduces novel, non-classical Julia and Mandelbrot sets using the Jungck–Noor iterative method with s-convexity, and derives an escape criterion for higher-order complex polynomials of the form $z^n+\Im z^3-\Re z+\omega$, where $n\ge 4$ and $\Im ,\Re ,\omega \in \mathbb{C}$. The proposed method advances existing algorithms, enabling the visualization of intricate fractal patterns as Julia and Mandelbrot sets with enhanced complexity. Through graphical representations, we illustrate how parameter variations influence the color, size, and shape of the resulting images, producing visually striking and aesthetically appealing fractals. Furthermore, we explore the dynamic behavior of these sets under fixed input parameters while varying the degree n. The presented results, both methodologically and visually, offer new insights into fractal geometry and inspire further research. Full article

In this paper, we introduce a generalised formulation of the logistic map extended to the complex plane and correspondingly redefine the classical Mandelbrot and Julia sets within this broader framework. Central to our approach is the development of an escape criterion based on the Picard orbit, which underpins the escape-time algorithms employed for graphical approximations of these sets. We analyse the structural and dynamical properties of the resulting Mandelbrot and Julia sets, emphasising their inherent symmetries through detailed visualisations. Furthermore, we examine how variations in a key parameter of the generalised map affect two critical numerical metrics: the average escape time and the non-escaping area index. Our computational study reveals that, particularly for Julia sets, these dependencies are characterised by intricate, highly non-linear behaviour—highlighting the profound complexity and sensitivity of the system under this generalised mapping. Full article

This article investigates and analyzes the diverse patterns of Julia sets generated by new classes of generalized exponential and sine rational functions. Using a generalized viscosity approximation-type iterative method, we derive escape criteria to visualize the Julia sets of these functions. This approach enhances existing algorithms, enabling the visualization of intricate fractal patterns as Julia sets. We graphically illustrate the variations in size and shape of the images as the iteration parameters change. The new fractals obtained are visually appealing and attractive. Moreover, we observe fascinating behavior in Julia sets when certain input parameters are fixed, while the values of n and m vary. We believe the conclusions of this study will inspire and motivate researchers and enthusiasts with a strong interest in fractal geometry. Full article

This paper presents a novel Graphics Processing Unit (GPU) accelerated implementation of a modified Particle Swarm Optimization (PSO) algorithm specifically designed to solve large-scale Systems of Nonlinear Equations (SNEs). The proposed GPU-based parallel version of the PSO algorithm uses the inherent parallelism of modern hardware architectures. Its performance is compared against both sequential and multithreaded Central Processing Unit (CPU) implementations. The primary objective is to evaluate the efficiency and scalability of PSO across different hardware platforms with a focus on solving large-scale SNEs involving thousands of equations and variables. The GPU-parallelized and multithreaded versions of the algorithm were implemented in the Julia programming language. Performance analyses were conducted on an NVIDIA A100 GPU and an AMD EPYC 7643 CPU. The tests utilized a set of challenging, scalable SNEs with dimensions ranging from 1000 to 5000. Results demonstrate that the GPU accelerated modified PSO substantially outperforms its CPU counterparts, achieving substantial speedups and consistently surpassing the highly optimized multithreaded CPU implementation in terms of computation time and scalability as the problem size increases. Therefore, this work evaluates the trade-offs between different hardware platforms and underscores the potential of GPU-based parallelism for accelerating SNE solvers. Full article

Active inference under the Free Energy Principle has been proposed as an across-scales compatible framework for understanding and modelling behaviour and self-maintenance. Crucially, a collective of active inference agents can, if they maintain a group-level Markov blanket, constitute a larger group-level active inference agent with a generative model of its own. This potential for computational scale-free structures speaks to the application of active inference to self-organizing systems across spatiotemporal scales, from cells to human collectives. Due to the difficulty of reconstructing the generative model that explains the behaviour of emergent group-level agents, there has been little research on this kind of multi-scale active inference. Here, we propose a data-driven methodology for characterising the relation between the generative model of a group-level agent and the dynamics of its constituent individual agents. We apply methods from computational cognitive modelling and computational psychiatry, applicable for active inference as well as other types of modelling approaches. Using a simple Multi-Armed Bandit task as an example, we employ the new ActiveInference.jl library for Julia to simulate a collective of agents who are equipped with a Markov blanket. We use sampling-based parameter estimation to make inferences about the generative model of the group-level agent, and we show that there is a non-trivial relationship between the generative models of individual agents and the group-level agent they constitute, even in this simple setting. Finally, we point to a number of ways in which this methodology might be applied to better understand the relations between nested active inference agents across scales. Full article

This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply s-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem that determines the condition when a sequence diverges to infinity, known as the escape criterion, for the transcendental sine function $sin\left(u^m\right)-\alpha u+\beta$, where $u,\alpha ,$$\beta \in \mathbb{C}$ and $m\ge 2$. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Visual representations are included to demonstrate how various parameters influence the coloration and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set’s structure. Full article

In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t,$ and $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$ (which are analytic except at $z=0$) where $n\ge 2,$ $m,n\in \mathbb{N},$ $\alpha ,\beta ,\gamma \in \mathbb{C},c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1,$ by employing a viscosity approximation-type iterative method. We employ the proposed iterative method to establish an escape criterion for visualizing Julia sets. We provide graphical illustrations of Julia sets that emphasize their sensitivity to different iteration parameters. We present graphical illustrations of Julia sets; the color, size, and shape of the images change with variations in the iteration parameters. With fixed input parameters, we observe the intriguing behavior of Julia sets for different values of n and m. We hope that the conclusions of this study will inspire researchers with an interest in fractal geometry. Full article

In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step p-order method to order $p+3$. We obtained the convergence order of Cordero et al.’s method by assuming only up to the third-order derivative of the operator. Our analysis is in a more general commutative Banach algebra setting and provides a radius of the convergence ball. Finally, we validate our theoretical findings with several numerical examples. Also, the concept of basin of attraction is discussed with examples. Full article

This paper introduces hybrid models designed to analyze daily and weekly bitcoin return spanning the periods from 18 July 2010 to 28 December 2023 for daily data, and from 18 July 2010 to 24 December 2023 for weekly data. Firstly, the fractal and chaotic structure of the selected variables was explored. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z² −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. The R/S and Mandelbrot–Wallis tests confirmed long-term dependence and fractionality. The largest Lyapunov test, the Rosenstein, Collins and DeLuca, and Kantz methods of Lyapunov exponents, and the HCT and Shannon entropy tests tracked by the Kolmogorov–Sinai (KS) complexity test determined the evidence of chaos, entropy, and complexity. The BDS test of independence test approved nonlinearity, and the TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, the LR test for threshold nonlinearity, and White’s test and Engle test confirmed nonlinearity and heteroskedasticity, in addition to fractionality and chaos. In the second stage, the standard ARFIMA method was applied, and its results were compared to the LieNLS and LieOLS methods. The results showed that, under conditions of chaos, entropy, and complexity, the ARFIMA method did not yield successful results. Both baseline models, LieNLS and LieOLS, are enhanced by integrating them with deep learning methods. The models, LieLSTMOLS and LieLSTMNLS, leverage manifold-based approaches, opting for matrix representations over traditional differential operator representations of Lie algebras were employed. The parameters and coefficients obtained from LieNLS and LieOLS, and the LieLSTMOLS and LieLSTMNLS methods were compared. And the forecasting capabilities of these hybrid models, particularly LieLSTMOLS and LieLSTMNLS, were compared with those of the main models. The in-sample and out-of-sample analyses demonstrated that the LieLSTMOLS and LieLSTMNLS methods outperform the others in terms of MAE and RMSE, thereby offering a more reliable means of assessing the selected data. Our study underscores the importance of employing the LieLSTM method for analyzing the dynamics of bitcoin. Our findings have significant implications for investors, traders, and policymakers. Full article

Traditional population-based metaheuristic algorithms are effective in solving complex real-world problems but require careful strategy selection and parameter tuning. Metaphorless population-based optimization algorithms have gained importance due to their simplicity and efficiency. However, research on their applicability for solving large systems of nonlinear equations is still incipient. This paper presents a review and detailed description of the main metaphorless optimization algorithms, including the Jaya and enhanced Jaya (EJAYA) algorithms, the three Rao algorithms, the best-worst-play (BWP) algorithm, and the new max–min greedy interaction (MaGI) algorithm. This article presents improved GPU-based massively parallel versions of these algorithms using a more efficient parallelization strategy. In particular, a novel GPU-accelerated implementation of the MaGI algorithm is proposed. The GPU-accelerated versions of the metaphorless algorithms developed were implemented using the Julia programming language. Both high-end professional-grade GPUs and a powerful consumer-oriented GPU were used for testing, along with a set of hard, large-scale nonlinear equation system problems to gauge the speedup gains from the parallelizations. The computational experiments produced substantial speedup gains, ranging from 33.9× to 561.8×, depending on the test parameters and the GPU used for testing. This highlights the efficiency of the proposed GPU-accelerated versions of the metaphorless algorithms considered. Full article

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form $Q_{\mathbb{C}}\left(p\right)=a{p}^n+mp+c$, where $n\ge 2$. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals. Full article

This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of a Mandelbrot set and Julia sets of fractional order are determined. Full article