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Discusses chaotic behavior (movement) in a nonlinear system using LOGO programs. Presents several examples with program listings. (MVL)

The evolution of ideas about the concept of chaos is surveyed. Discussed are chaos in deterministic, dynamic systems; order in dissipative systems; and thermodynamics and irreversibility. Included are logistic and bifurcation maps to illustrate points made in the discussion. (CW)

Describes the underlying pattern that, once seen, will allow students to describe the shape of any hydrogen orbital. Involves the exponential form of the orbitals. Provides computer-generated plots of this concept. (CW)

Offers several LOGO programs to study the behavior of simple nonlinear systems. Suggests that LOGO is an excellent tool for studying chaotic systems. Offers suggestions for different forms of LOGO. Builds upon programs presented in a previous article. (MVL)

Described are the idea of chaos and the ability to control the chaotic behavior of a real-world physical system. Included is an explanation of the methodology and applications in biology and chemistry. (KR)

Surveys current research on learning and memory and associated brain functions. Suggests several models of olfactory recognition and memory. Explains several computer models being evaluated and lists possible flaws. Differentiates between noise and chaos in brain functions. Describes a challenge to the theory of chaos called adaptive resonance theory (ART). (CW)

This study investigated how data series length and gaps in human kinematic data impact the accuracy of Lyapunov exponents (LyE) calculations with and without cubic spline interpolation. Kinematic time series were manipulated to create various data series lengths (28% and 100% of original) and gap durations (0.05–0.20 s). Longer gaps generally resulted in significantly higher LyE% error values in each plane in noninterpolated data. During cubic spline interpolation, only the 0.20-second gap in frontal plane data resulted in a significantly higher LyE% error. Data series length did not significantly affect LyE% error in noninterpolated data. During cubic spline interpolation, sagittal plane LyE% errors were significantly higher at shorter versus longer data series lengths. These findings suggest that not interpolating gaps in data could lead to erroneously high LyE values and mischaracterization of movement variability. When applying cubic spline, a long gap length (0.20 s) in the frontal plane or a short sagittal plane data series length (1000 data points) could also lead to erroneously high LyE values and mischaracterization of movement variability. These insights emphasize the necessity of detailed reporting on gap durations, data series lengths, and interpolation techniques when characterizing human movement variability using LyE values. [ABSTRACT FROM AUTHOR]

The stochastic dynamical ρ4 equation is utilized as a robust framework for modeling the behavior of complex systems characterized by randomness and nonlinearity, with applications spanning various scientific fields. The aim of this paper is to employ an analytical method to identify stochastic traveling wave solutions of the dynamical ρ4 equation. Novel hyperbolic and rational functions are investigated through this method. A Galilean transformation is applied to reformulate the model into a planar dynamical system, which enables a comprehensive qualitative analysis. Additionally, the emergence of chaotic and quasi-periodic patterns following the introduction of a perturbation term is addressed. Simulation results indicate that significant changes in the systems’ dynamic behavior are caused by adjusting the amplitude and frequency parameters. Our findings indicate the impact of the method on system dynamics and its efficacy in analyzing solitons and phase behavior in nonlinear models. These discoveries provide fresh perspectives on how the suggested method can lead to notable shifts in the systems’ dynamic behavior. The effectiveness and practicality of the proposed methodology in scrutinizing soliton solutions and phase visualizations across diverse nonlinear models are underscored by these revelations. [ABSTRACT FROM AUTHOR]

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves. [ABSTRACT FROM AUTHOR]

Fog computing utilizes user premises resources to provide better services than traditional cloud computing. Due to the heterogeneity of fog devices, scheduling poses a challenge. This paper proposes a novel version of artificial rabbits optimization (ARO) called the Nonlinear based chaotic artificial rabbits optimization (NCARO) and utilizes NCARO for task scheduling in the fog computing environment (TSNCARO). The NCARO optimizes ARO by using chaotic and nonlinear control parameters. In the proposed method, chaotic maps are used to improve the exploratory behavior of ARO. ARO's exploratory and exploitative behaviors are also adjusted by means of a nonlinear control parameter. Three objectives are considered: service time, cost, and energy consumption. It improves performance by prioritizing tasks according to deadlines. An extensive scenario compares NCARO and TSNCARO algorithms with other algorithms. On the basis of the comparison results, the proposed algorithms achieved the best results in terms of makespan, service time, total cost, energy consumption, carbon dioxide emission rate, and percentage of deadline satisfaction. [ABSTRACT FROM AUTHOR]

This paper proposes a metaheuristic algorithm, called advanced differential evolution (ADE), by improving the DE algorithm. The ADE algorithm was developed with the goal of creating an optimization framework that addresses the challenges of exploration and exploitation balance, avoiding local minima, utilizing chaos theory for diverse initialization, and improving solution quality and convergence speed. By incorporating these features, ADE aims to enhance the effectiveness of optimization processes. The proposed algorithm utilizes chaos theory to generate the initial population, which is subsequently divided into two sub-populations with adaptive sizes. The size of each sub-population is determined using a formula based on the number of iterations during the algorithm's execution. The first sub-population has a larger size in the beginning and the second one has a smaller size, but the total size of these two populations is always constant. The main contribution of this paper is the proposal of two novel improved differential evolution algorithms, namely MDE1 and MDE2, which are utilized for exploration within these sub-populations. The proposed ADE is tested on 29 well-known benchmarks and six engineering problems, and the results are compared with seven other algorithms. Various statistical experiments are carried out showing that the proposed algorithm provides significant superiority over other well-known algorithms. [ABSTRACT FROM AUTHOR]

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Introduction: Existing studies suggest that the number of night shifts may impact the occurrence of adverse events. However, while this relationship is well-documented, previous research has not thoroughly examined the non-linear associations between night shifts and adverse events among nursing staff, which remains a gap in our understanding. Methods: Participants were 1,774 Chinese nurse staff. Psychosocial characteristics were screened by The Chinese version of the multidimensional scale of perceived social support (MSPSS) for social support, the 9-item Patient Health Questionnaire (PHQ-9) for depressive symptoms, the Generalized Anxiety Disorder-7 (GAD-7) for anxiety symptoms. Binary logistic regression and restricted cubic splines were applied to analyze the data. The statistical software used were R version 3.6.2 and SPSS version 22.0. Results: Over the past year, 325 cases (18.3%) were classified as adverse events. Logistic regression unveiled that social support played a protective role against adverse events, with an OR of 0.991 (95% CI: 0.983, 0.999). Furthermore, night shifts continued to surface as a substantial risk factor for adverse events, with an OR to 1.300 (95% CI: 1.181, 1.431). The restricted cubic spline regression model highlights a nonlinear relationship between night shifts and adverse events (P for non-liner < 0.001). The probability of adverse events increases with the number of night shifts, but compared to individuals working 3–4 night shifts per month, those working 5–6 night shifts per month have a lower probability of adverse events. Conclusion: Our findings indicate a non-linear relationship between the frequency of night shifts and adverse events, suggesting a complex interplay of factors. This highlights the need for nursing practice and policy to consider the intricacies of night shift scheduling and explore more reasonable rostering strategies to mitigate the probability of adverse events. [ABSTRACT FROM AUTHOR]

The Maximum Lyapunov Exponent(MLE) measures the sensitivity of a chaotic system to its initial conditions. In chaos theory, systems with positive MLE values are considered chaotic, and larger MLE values generally indicate more robust chaos. A higher MLE suggests a more chaotic and complex behavior, which could be beneficial for encryption as it might provide enhanced security due to increased sensitivity to initial conditions and resistance to attacks. This paper searches for whether a 1D seed chaotic map or a 1D integrated chaotic map is preferred for image encryption. Does the MLE have any role? Firstly, three 1D integrated chaotic maps were designed: Sine-Cubic Integrated Map(SCIM), Tent-Logistic Integrated Map(TLIM), and Sine-Logistic Integrated Map(SLIM). These integrated chaotic maps are designed using the four available seed maps: sine, logistics, cubic, and tent. Thus, we have considered seven 1D chaotic maps to analyse and answer the question. Secondly, image encryption and decryption are performed using the considered seven 1D chaotic maps, one after the other, and the security measures of the encrypted image are analysed using various available tools. The image encryption is performed using block shuffling as diffusion and bit-Xor operation as the confusion process. A comparative analysis is performed using the six quantitative security analysis tools. According to the encryption correlation coefficient value of 0.0017, the Pick Signal-To-Noise Ratio(PSNR) value of 9.204, the Mean Square Error(MSE) value of 7809.1, the Number of Pixel Change Rate(NPCR) value of 95.3903, the Unified Average Change Intensity(UACI) value of 33.3676, and the information entropy value of 7.9635, the sine map is ranked first in security. The comparative analysis result reveals that seed maps give better encrypted image security than integrated chaotic maps. Therefore, integrating 1D chaotic maps is not guaranteed to get a better-secured encrypted image. Further analysis is made to understand if the MLE directly impacts the security of the encrypted process. It is found that the integrated chaotic maps provide a higher MLE. However, in this analysis, we couldn't observe the direct relationship between the MLE and the security of the encrypted image. This suggests that other factors beyond just MLE contribute to the security of the encryption process. [ABSTRACT FROM AUTHOR]

This article investigates a class of Duffing nonlinear dynamic systems with fractional‐order dry friction and conducts in‐depth research on the stability, chaotic characteristics, and erosion of the safety basin of this system; the results are verified through numerical simulation. First, the average method is used to approximate the amplitude–frequency relationship of the system, and the accuracy of the analytical results is verified through numerical experiments. Second, the Melnikov method is used to obtain the conditions for the system to enter chaos in the Smale horseshoe sense, and the Melnikov curve is drawn for further verification. Then, bifurcation diagrams are drawn for the changes in various parameters in the system, with a focus on analyzing the influence of friction factors on chaotic bifurcation. By applying the definition and calculation principle of the maximum Lyapunov exponent, and drawing and utilizing the maximum Lyapunov exponent graph, the chaotic state that the system enters under different parameters is more clearly defined. Finally, the evolution law of the safety basin under various parameter changes, especially dry friction changes, is analyzed, and the erosion and bifurcation mechanism of the safety basin is studied. Comparing with the bifurcation diagram, it reveals that chaos primarily contributes to the erosion of the safety basin. [ABSTRACT FROM AUTHOR]

Background: Primary care is often described as slow to change. But conceptualized through complexity theory, primary care is continually changing in unpredictable, non-linear ways through self-organization processes. Self-organization has proven hard to study directly. We aimed to develop a methodology to study self-organization and describe how a primary care clinic self-organizes over time. Methodology: We completed a virtual case study of an urban primary care clinic from May-Nov 2021, applying methodological insights from actor-network theory to examine the complexity theory concept of self-organization. We chose to focus our attention on self-organization activities that alter organizational routines. Data included fieldnotes of observed team meetings, document collection, interviews with clinic members, and notes from brief weekly discussions to detect actions to change clinical and administrative routines. Adapting schema analysis, we described changes to different organizational routines chronologically, then explored intersecting changes. We sought feedback on results from the participating clinic. Findings: Re-establishing equilibrium remained challenging well into the COVID-19 pandemic. The primary care clinic continued to self-organize in response to changing health policies, unintended consequences of earlier adaptations, staff changes, and clinical care initiatives. Physical space, technologies, external and internal policies, guidelines, and clinic members all influenced self-organization. Changing one created ripple effects, sometimes generating new, unanticipated problems. Member checking confirmed we captured most of the changes to organizational routines during the case study period. Conclusions: Through insights from actor-network theory, applied to studying actions taken that alter organizational routines, it is possible to operationalize the theoretical construct of self-organization. Our methodology illuminates the primary care clinic as a continually changing entity with co-existing and intersecting processes of self-organization in response to varied change pressures. [ABSTRACT FROM AUTHOR]

This interdisciplinary study critically analyzes current research, establishing a profound connection between sea water, sea ice, sea temperature, and surface temperature through a 4D hyperchaotic Caputo fractional differential equation. Emphasizing the collective impact on climate, focusing on challenges from anthropogenic global warming, the study scrutinizes theoretical aspects, including existence and uniqueness. Two sliding mode controllers manage chaos in this 4D fractional system, assessed amid uncertainties and disruptions. The global stability of these controlled systems is also confirmed, considering both commensurate and non-commensurate 4D fractional order. To demonstrate the intricate chaotic motion within the system, we employ the Lyapunov exponent and Poincare sections. Numerical simulations are conducted by using the predictor-corrector method. The effects of surface temperature on chaotic dynamics are discussed. The crucial role of sea ice reflection in climate stability is highlighted in two scenarios. Correlation graphs, comparing model and observational data using the predictor-corrector method, enhance the proposed 4D hyperchaotic model's credibility. Subsequently, numerical simulations validate theoretical assertions about the controllers' influence. These controllers indicate which variable significantly contributes to controlling the chaos. [ABSTRACT FROM AUTHOR]

Encryption techniques have been proposed to improve security using chaos theory, mathematical computations, DNA computations, substitution-masking operations, and neural networks. However, existing encryption techniques have focused on intra-block variations to improve security and largely ignored the impact of inter-block variations. Inter-block variations are critical because they ensure that the encryption of a block is affected by its internal variation and also by the variations of all its previous blocks. This can significantly increase the confusion and the technique's ability to resist security attacks. This paper proposes a hybrid technique that addresses the weaknesses of existing techniques by combining both intra- and inter-block variations to greatly improve security. The proposed technique uses chaotic systems to increase confusion and conceal the key. It also incorporates fuzzy encoding and distortion methods that use both inter- and intra-block variations to further strengthen security. The technique was rigorously evaluated using state-of-the-art testing tools. The evaluation showed that the technique is effective because it passed the randomness tests with a high rate that exceeds the threshold for effective techniques, and it has an execution time that is better than state-of-the-art techniques. [ABSTRACT FROM AUTHOR]

To research the vibration problem caused by bearing assembly errors in machine tool spindles, a special test rig for rolling bearing misalignment is designed and built. The dynamic model of the spindle-bearing-pedestal system of the test rig is established, and the characterization method for bearing parallel and angular misalignments is given through the Hertz contact theory. The effect rule of the misalignment level on spindle vibration response is studied by simulation and experiment, and the system dynamic model and bearing misalignment model are verified. On this basis, the vibration characteristics of the spindle system under the joint effect of raceway misalignment and waviness are further analyzed. The numerical and experimental results show that when the bearing is in the misalignment state, the excitation of bearing variable compliance increases. The assembly errors of the bearing raceway lead to the deviation of the centroid motion trajectory of the spindle tool end, which seriously affects the machining accuracy of the machine tool. In the case of severe angular misalignment, the system shows chaotic motion, indicating that the instability of the system is enhanced. The impact of angular misalignment on system vibration is greater than parallel misalignment, so bearing ring tilt should be avoided as far as possible in practice. In addition, it is also found that the variation of excitation amplitude of bearing waviness caused by raceway tilt is related to the relationship between the number of waviness and ball number. [ABSTRACT FROM AUTHOR]

The front facia gives the car vivid features; accordingly, and an appropriate product personalisation degree (PD) can improve product branding, related to the appearance design, while avoiding excessive design features. This study investigates the effects of the car front facia PD on user perceptions and proposes design guidelines for vehicle designers and manufacturers. An open-ended experiment was conducted with 30 participants using 60 car fronts to study user emotions and preferences related to product personality based on questionnaire and physiological electrical signal evaluation results. The features were extracted using the chaos theory and cvxEDA algorithm. The obtained results indicate that the PD was consistent with user emotions. Furthermore, when the consistency between PD and user emotions is high, a significant variation in user preferences can be observed. Therefore, when designing high-PD products, the sub-item product personalisation degree (sub-item PD) should be carefully managed, considering the overall logic and coherence. Highlights Product personalisation degree is positively correlated with the subjective and objective emotions of users. The impact of product personalisation degree on user preferences and subjective emotions is not always the same, particularly pleasure and arousal. When product personalisation degree is highly correlated with the subjective and objective emotions of users, the user preferences are highly differentiated. [ABSTRACT FROM AUTHOR]

BI 187004, a selective small‐molecule inhibitor of 11β‐hydroxysteroid dehydrogenase‐1 (11β‐HSD1), displayed complex nonlinear pharmacokinetics (PK) in humans. Following nine single oral doses, BI 187004 exhibited nonlinear PK at low doses and linear PK at higher doses. Notably, substantial hepatic 11β‐HSD1 inhibition (50%) was detected in a very low‐dose group, achieving a consistent 70% hepatic enzyme inhibition in subsequent ascending doses without any dose‐dependent effects. The unusual PK and PD profiles of BI 187004 suggest the presence of pharmacological target‐mediated drug disposition (TMDD), arising from the saturable binding of BI 187004 compound to its high‐affinity and low‐capacity target 11β‐HSD1. The non‐intuitive dose, exposure, and response relationship for BI 187004 pose a significant challenge in rational dose selection. This study aimed to construct a TMDD model to explain the complex nonlinear PK behavior and underscore the importance of recognizing TMDD in this small‐molecule compound. Among the various models explored, the best model was a two‐compartment TMDD model with three transit absorption components. The final model provides insights into 11β‐HSD1 binding‐related parameters for BI 187004, including the total amount of 11β‐HSD1 in the liver (estimated to be 8000 nmol), the second order association rate constant (estimated to be 0.102 nM−1h−1), and the first‐order dissociation rate constant (estimated to be 0.11 h−1). Our final population PK model successfully characterized the intricate nonlinear PK of BI 187004 across a wide dose range. This modeling work serves as a valuable reference for the rational selection of the dose regimens for BI 187004's future clinical trials. [ABSTRACT FROM AUTHOR]

Atherosclerosis, a leading cause of cardiovascular disease, necessitates advanced and innovative modeling techniques to better understand and predict plaque dynamics. The present work presents two distinct hypothetical models inspired by different research fields: the logistic map from chaos theory and Markov models from stochastic processes. The logistic map effectively models the nonlinear progression and sudden changes in plaque stability, reflecting the chaotic nature of atherosclerotic events. In contrast, Markov models, including traditional Markov chains, spatial Markov models, and Markov random fields, provide a probabilistic framework to assess plaque stability and transitions. Spatial Markov models, visualized through heatmaps, highlight the spatial distribution of transition probabilities, emphasizing local interactions and dependencies. Markov random fields incorporate complex spatial interactions, inspired by advances in physics and computational biology, but present challenges in parameter estimation and computational complexity. While these hypothetical models offer promising insights, they require rigorous validation with real-world data to confirm their accuracy and applicability. This study underscores the importance of interdisciplinary approaches in developing theoretical models for atherosclerotic plaques. [ABSTRACT FROM AUTHOR]

This paper investigates the Li–Yorke chaos in linear systems with weak topology on Hilbert spaces. A weak topology induced by bounded linear functionals is first constructed. Under this weak topology, it is shown that the weak Li–Yorke chaos can be equivalently measured by an irregular or a semi-irregular vector, which are utilized to establish criteria for the weak Li–Yorke chaos of diagonalizable operators, Jordan blocks, and upper triangular operators. In particular, for a linear operator that can be decomposed into a direct sum of finite-dimensional Jordan blocks, it is Li–Yorke chaotic in weak topology if its point spectrum contains a pair of real opposite eigenvalues with absolute values not less than 1, or a pair of complex conjugate eigenvalues with moduli not less than 1. Interestingly, as a specific example of upper triangular operator, the existence of Li–Yorke chaos in weak topology can be derived for a class of linear operators expressed as the direct sum of finite-dimensional Jordan blocks and a strongly irreducible operator. [ABSTRACT FROM AUTHOR]

The objective of this work is to investigate the complex dynamics of a discrete predator–prey system using the method of piecewise constant argument for discretization. An analysis is conducted to examine the presence and stability of fixed points. Furthermore, the system is shown to undergo period-doubling (PD) and Neimark–Sacker (NS) bifurcations by the use of center manifold and bifurcation theories. The feedback and hybrid control strategies are used to regulate the system's bifurcating and chaotic behaviors. Both strategies seem to be effective in managing bifurcation and chaos inside the system. Finally, the main results are validated by numerical evidence. Parameters of the system are varied to produce time graphs, phase portraits, bifurcation diagrams, and maximum Lyapunov exponent (MLE) graphs. The discrete model displays rich dynamics, as seen in the numerical simulations and graphs, indicating a complex and chaotic system. [ABSTRACT FROM AUTHOR]

This research investigates the interaction between a generalist predator and a prey, where the predator exhibits cooperative behavior during hunting, inducing fear into the prey population. Additionally, both the prey and predator populations are subject to harvesting. The study establishes the positivity and boundedness of the model's solutions, ensuring the existence of the population. Analyzing the system, we explore its feasible steady states and their stability, along with various types of bifurcations, including Hopf with direction of stability, Saddle-node, Transcritical, Homoclinic, Bogdanov–Takens, and Cusp bifurcation. We also demonstrate the stability and bifurcation behavior of a delayed system. These findings are verified through one-parameter and two-parameter bifurcation structures, complemented by respective phase portraits. Notably, the system displays transition between different equilibria and bistability. Furthermore, numerical investigations reveal the impact of gestation delay, indicating chaotic behavior in the system due to this time delay. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

In this paper, we present a new class of extended oscillators in light of chaos theory. It is based on dynamical complex systems built on the concept of self-describing with a stopping criterion process. We offer an effective studying approach with a specific focus on learning, provoking students' thinking through the triad of enigmatics–creativity–acmeology. Dynamic processes are the basis of mathematical modeling; thus, we can reach the goal of the above-mentioned triad by the proposed differential systems. The results we derive strongly confirm the presence of symmetry in the outcomes of the proposed models. We suggest a stochastic approach to structuring the proposed dynamical systems by modeling the coefficients that drive them by some discrete probability distribution that exhibits symmetry or asymmetry. We propose specific tools for researching the behavior of these systems. [ABSTRACT FROM AUTHOR]

It is widely acknowledged that fluidelastic instability (FEI), among other mechanisms, is of the greatest concern in the flow-induced vibration (FIV) of tube bundles in steam generators and heat exchangers. A range of theoretical models have been developed for FEI analysis, and, in addition to the earliest semi-empirical Connors' model, the unsteady model, the quasi-steady model and the semi-analytical model are believed to be three advanced models predominant in the literature. The unsteady and the quasi-static models share the merits of having explicit fluid force expressions and ease of being implemented but require more experimental inputs, whereas the semi-analytical model requires fewer parameters due to its analytical nature but is hard, if not prohibitive, to derive explicit fluid force expressions. Since the fluid force formulations set in the core of development of FEI models, the understanding and in particular the implementation of the semi-analytical model has been impaired by the nonexistence of explicit fluid force expression. This issue becomes more profound in time-domain analysis whereby the simple harmonic assumption is discarded. Here we report a new semi-analytical time-domain (SATD) FEI model with explicit fluid force expressions. The new model allows a consistent frequency-domain stability analysis and more importantly a truly time-domain response analysis. The theory was validated by calculating linear stability thresholds of two typical tube array patterns and comparing against reported experimental data. We then present a nonlinear time-domain analysis of a single loosely-supported tube with piece-wise linear stiffness. The nonlinear and nonsmooth dynamics was probed in details by utilizing various techniques, playing an emphasis on characterizing and distinguishing the chaotic vibration. We found that the system follows a quasi-periodic route to chaos. Such an in-depth study of the nonlinear dynamics of tubes in crossflow has never been reported in the context of SATD model. Our results enrich the theory and provide a different approach for linear and nonlinear dynamics of tube bundles, which are essential for the subsequent fretting wear analysis. • A new formulation on semi-analytical time-domain model for fluidelastic instability of tube bundles. • The formulation has explicit fluid force expressions. • Linear frequency- and time-domain analysis are validated against available experimental and theoretical results. • Nonlinear dynamics behavior of a loosely-supported tube is probed based on the new formulation by using various techniques. • The route to chaos of the system is found through quasi-periodic to chaos. [ABSTRACT FROM AUTHOR]

This paper proposes a hybrid Modified Coronavirus Herd Immunity Aquila Optimization Algorithm (MCHIAO) that compiles the Enhanced Coronavirus Herd Immunity Optimizer (ECHIO) algorithm and Aquila Optimizer (AO). As one of the competitive human-based optimization algorithms, the Coronavirus Herd Immunity Optimizer (CHIO) exceeds some other biological-inspired algorithms. Compared to other optimization algorithms, CHIO showed good results. However, CHIO gets confined to local optima, and the accuracy of large-scale global optimization problems is decreased. On the other hand, although AO has significant local exploitation capabilities, its global exploration capabilities are insufficient. Subsequently, a novel metaheuristic optimizer, Modified Coronavirus Herd Immunity Aquila Optimizer (MCHIAO), is presented to overcome these restrictions and adapt it to solve feature selection challenges. In this paper, MCHIAO is proposed with three main enhancements to overcome these issues and reach higher optimal results which are cases categorizing, enhancing the new genes' value equation using the chaotic system as inspired by the chaotic behavior of the coronavirus and generating a new formula to switch between expanded and narrowed exploitation. MCHIAO demonstrates it's worth contra ten well-known state-of-the-art optimization algorithms (GOA, MFO, MPA, GWO, HHO, SSA, WOA, IAO, NOA, NGO) in addition to AO and CHIO. Friedman average rank and Wilcoxon statistical analysis (p-value) are conducted on all state-of-the-art algorithms testing 23 benchmark functions. Wilcoxon test and Friedman are conducted as well on the 29 CEC2017 functions. Moreover, some statistical tests are conducted on the 10 CEC2019 benchmark functions. Six real-world problems are used to validate the proposed MCHIAO against the same twelve state-of-the-art algorithms. On classical functions, including 24 unimodal and 44 multimodal functions, respectively, the exploitative and explorative behavior of the hybrid algorithm MCHIAO is evaluated. The statistical significance of the proposed technique for all functions is demonstrated by the p-values calculated using the Wilcoxon rank-sum test, as these p-values are found to be less than 0.05. [ABSTRACT FROM AUTHOR]

This paper is dedicated to exterminate the ( 2 + 1 )-dimensional integro-differential Jaulent–Miodek equation, a prominent model linked to energy-dependent Schrödinger potential. This equation is employed in a wide array of disciplines, including fluid dynamics, condensed matter physics, optics, and various engineering systems. First, we are given to derive exact wave solutions for the ( 2 + 1 )-dimensional integro-differential Jaulent–Miodek equation using an innovative approach known as the new modified unified auxiliary equation method. We offer a comprehensive visual representation and some exact solutions propagate of these solutions in 2D and 3D plots, using various parametric values for a comprehensive analysis. In addition, we employed the planar dynamical system method to study phase portraits and chaotic behavior of the governing equation. Through the analysis of phase portraits, the sensitivity of dynamic system is examined. The chaotic behavior of dynamic system is examined by time series, Poincaré section, and 2D, 3D phase portraits. [ABSTRACT FROM AUTHOR]

The focus of this paper is the design of open-loop control profiles for linear differentially flat systems whose model parameters are uncertain and are represented probabilistically. To account for the time-invariant model parameter uncertainties, polynomial chaos is used to derive a surrogate model which permits easy evaluation of the mean and variance of the uncertain states. A chance constrained optimisation problem is posed to minimise the terminal error of the stochastic states for a prescribed risk of not satisfying the terminal state bounds. To permit posing a convex optimisation problem, the cost function which is the residual energy is approximated by a hypercube circumscribing the hypersphere which bounds the terminal residual error. This relaxation permits posing a convex optimisation problem to arrive at the robust control profile. The proposed technique is illustrated on two examples including the benchmark spring-mass-dashpot system undergoing a rest-to-rest maneuver and a UAV undergoing a translation in a prescribed time. Parametric studies are conducted where the impact of varying the order of the polynomial chaos expansion and the degree of the parameterised control profiles are used to conclude that a 2–3 degree polynomial chaos expansion is sufficient to capture the time evolution of the mean and variance of the states which are required for the chance constrained optimisation problems. As expected, the design results in improved performance as the number of free variables used to parameterise the control profiles increases. [ABSTRACT FROM AUTHOR]

This article takes a class of strongly nonlinear Duffing systems with fractional-order terms as the research object, studying their main resonance and conditions for chaos under single-frequency excitation. The approximate analytical solution of the main resonance of the system under single-frequency excitation is obtained by the multi-scale method. The approximate analytical solution is utilized to construct the steady-state motion's amplitude–frequency response equation, and Lyapunov's first method is used to determine the constant solution's stability condition, which is then used to analyze the steady-state motion's stability. The system is examined by using the Melnikov method to identify the circumstances that would appear to result in a transverse intersection of heterodox orbits and the onset of chaos in the system. The amplitude–frequency characteristics of the total response of the system at various excitation frequencies are investigated in numerical simulations using analytical and simulation methods, respectively, and the system makes a comparison of amplitude–frequency curves, confirming that the numerical results and results achieve a consistent trend. The effects of nonlinear stiffness coefficients, excitation amplitude, fractional-order differential term order and fractional-order differential term coefficients on the system's amplitude–frequency response are each examined in turn. A numerical investigation of the effects of system parameters on chaotic motion is then conducted using many kinds of diagrams. [ABSTRACT FROM AUTHOR]

The Internet of Multimedia Things (IoMT) refers to a network of interconnected multimedia devices that communicate with each other over the Internet. Recently, smart healthcare has emerged as a significant application of the IoMT, particularly in the context of knowledge‐based learning systems. Smart healthcare systems leverage knowledge‐based learning to become more context‐aware, adaptable, and auditable while maintaining the ability to learn from historical data. In smart healthcare systems, devices capture images, such as X‐rays, Magnetic Resonance Imaging. The security and integrity of these images are crucial for the databases used in knowledge‐based learning systems to foster structured decision‐making and enhance the learning abilities of AI. Moreover, in knowledge‐driven systems, the storage and transmission of HD medical images exert a burden on the limited bandwidth of the communication channel, leading to data transmission delays. To address the security and latency concerns, this paper presents a lightweight medical image encryption scheme utilising bit‐plane decomposition and chaos theory. The results of the experiment yield entropy, energy, and correlation values of 7.999, 0.0156, and 0.0001, respectively. This validates the effectiveness of the encryption system proposed in this paper, which offers high‐quality encryption, a large key space, key sensitivity, and resistance to statistical attacks. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a discrete-time prey-predator model. The model is formulated by using the piecewise constant argument method for differential equations and taking into account Holling type III. The existence and local behavior of equilibria are studied. We established that the system experienced both Neimark-Sacker and perioddoubling bifurcations analytically by using bifurcation theory and the center manifold theorem. In order to control chaos and bifurcations, the state feedback method is implemented. Numerical simulations are also provided for the theoretical discussion. [ABSTRACT FROM AUTHOR]

The users of 5G and others greatly benefit from Device-to-device (D2D) communication technologies. It improves spectral efficiency, extends coverage, reduces offload traffic and lowers the latency rate in a cellular network. The resource-sharing method utilizing D2D communication must be appropriately designed to minimize interference while cellular user (CU) resources are reused. This paper presents a game theory algorithm to maximize the overall network throughput capacity. At the same time the quality of services for both CU and D2D users (DU) shall be guaranteed. The proposed new method enables the DU to reuse available uplink or downlink resources of the CU. In addition, an optimization problem is framed based on the Chaos Game Theory Algorithm. To reduce the complexity of the objective function, the optimization problem can be divided into two-stage subproblems, i.e., the subchannel assignment (SC) stage, followed by the power allocation (PA) stage as they are NP-hard in Nature. It converges as an optimal solution in convex polynomial time. The proposed approach considered network throughput performance related to the number of iterations, D2D link distance, number of active CUs and maximum transmission power of CU, D2D users, and BS as the parameter metrics. Then they are compared with the existing resource allocation schemes. The numerical results infer that the proposed game theory approach improves the network throughput more than the Hungarian joint resource allocation scheme. [ABSTRACT FROM AUTHOR]

Chaotic time series prediction is a prediction method based on chaos theory, and has important theoretical and application value. At present, most prediction methods only pursue digital fitting and do not consider the directional trend. In addition, using the single model will not achieve better prediction results. Therefore, a chaotic time series combined prediction model for improving trend lagging (ITL) is proposed. An improved dual‐stage attention‐based long short‐term memory model with the improved training objective fuction is designed to solve the trend lagging problem. Then, an auto regressive moving average model with the sliding window is established to mine other characteristics of the time series except nonlinear characteristic. Finally, the idea of optimization algorithm is introduced to construct a time series combined prediction model with high accuracy based on the above two models, so as to perform the chaotic time series prediction from multiple perspectives. Multiple datasets are selected as experimental datasets, and the proposed method is compared with common prediction methods. The results show that the proposed method can achieve single‐step prediction with high accuracy and effectively improve the lagging of chaotic time series prediction. This research can provide theoretical support for the complex chaotic time series prediction. [ABSTRACT FROM AUTHOR]

The design of chaotic systems with complex dynamic behaviors has always been a key aspect of chaos theory in engineering applications. This study introduces a novel fractional-order system characterized by hidden dynamics, hyperchaotic behavior, and multi-scroll attractors. By employing fractional calculus, the system's order is extended beyond integer values, providing a richer dynamic behavior. The system's hidden dynamics are revealed through detailed numerical simulations and theoretical analysis, demonstrating complex attractors and bifurcations. The hyperchaotic nature of the system is verified through Lyapunov exponents and phase portraits, showing multiple positive exponents that indicate a higher degree of unpredictability and complexity. Additionally, the system's multi-scroll attractors are analyzed, showcasing their potential for secure communication and encryption applications. The fractional-order approach enhances the system's flexibility and adaptability, making it suitable for a wide range of practical uses, including secure data transmission, image encryption, and complex signal processing. Finally, based on the proposed fractional-order system, we designed a simple and efficient medical image encryption scheme and analyzed its security performance. Experimental results validate the theoretical findings, confirming the system's robustness and effectiveness in generating complex chaotic behaviors. [ABSTRACT FROM AUTHOR]

This article introduces and formalizes a novel stochastic method that combines inverse simulation with the theory of generalized Polynomial Chaos (gPC) to solve and study inverse problems under uncertainty in energy system design applications. The method is particularly relevant to design tasks where only a deterministic forward model of a physical system is available, in which a target design quantity is an input to the model that cannot be obtained directly, but can be quantified reversely via the outputs of the model. In this scenario, the proposed method offers an analytical and direct approach to invert such system models. The method puts emphasis on user-friendliness, as it enables its users to conduct the inverse simulation under uncertainty directly in the gPC domain by redefining basic algebra operations for computations. Moreover, the method incorporates an optimization-based approach to integrate supplementary constraints on stochastic quantities. This feature enables the solution of inverse problems bounding the statistical moments of stochastic system variables. The authors exemplify the application of the proposed method with proof-of-concept tests in energy system design, specifically performing uncertainty quantification and sensitivity analysis for a Multi-Energy System (MES). The findings demonstrate the high accuracy of the method as well as clear advantages over conventional sampling-based methods when dealing with a small number of stochastic variables in a system or model. However, the case studies also highlight the current limitations of the proposed method such as slow execution speed due to the optimization-based approach and the challenges associated with, for example, the curse of dimensionality in gPC. [ABSTRACT FROM AUTHOR]

In this paper, the Caputo fractional derivative is assumed to be the prey–predator model. In order to create Caputo fractional differential equations for the prey–predator model, a discretization process is first used. The fixed points of the model are categorized topologically. We identify requirements for the fixed points of the suggested prey–predator model's local asymptotic stability. We demonstrate analytically that, under specific parametric conditions, a fractional order prey–predator model supports both a Neimark–Sacker (NS) bifurcation and a Flip bifurcation. We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey–predator model. As the bifurcation parameter is increased, the system displays chaotic behavior. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. The suggested prey–predator dynamical system's chaotic behavior will be controlled by the OGY and hybrid control methodology, which will also visualize the chaotic state for various biological parameters. [ABSTRACT FROM AUTHOR]

Dynamic systems theory and complexity theory (DST/CT) is a framework explaining how complex systems change and adapt over time. In psychotherapy, DST/CT can be used to understand how a person's mental and emotional state changes during therapy incorporating higher levels of complexity. This study aimed to systematically review the variability of DST/CT methods applied in psychotherapy research. A primary studies search was conducted in the EBSCO and Web of Knowledge databases, extracting information about the analyzed DST/CT phenomena, employed mathematical methods to investigate these phenomena, descriptions of specified dynamic models, psychotherapy phenomena, and other information regarding studies with empirical data (e.g., measurement granularity). After screening 38,216 abstracts and 4,194 full texts, N = 41 studies published from 1990 to 2021 were identified. The employed methods typically included measures of dynamic complexity or chaoticity. Computational and simulation studies most often employed first-order ordinary differential equations and typically focused on describing the time evolution of client-therapist dyadic influences. Eligible studies with empirical data were usually based on case studies and focused on data with high time intensity of within-session dynamics. This review provides a descriptive synthesis of the current state of the proliferation of DST/CT methods in the psychotherapy research field. [ABSTRACT FROM AUTHOR]

The present work presents the investigation of the dynamics and influence of chaotic behavior on energy capture for a U-shaped structure (portal frame) that contains shape memory alloy (SMA), piezoelectric material (PZT), a nonlinear energy sink (NES) and a non-ideal excitation source represented by an unbalanced electric motor coupled to the U-structure. The mathematical model presents nonlinearities arising from the nonlinear stiffness of the U-structure, the NES system, the SMA, and the PZT material. Chaotic behavior is assessed through time history, bifurcation diagrams, phase diagrams, and the 0–1 test. Energy capture is carried out through a piezoelectric material (PZT), represented by a non-linear electromechanical coupling model, and electromagnetic induction generated by the non-linear electromagnetic energy sink coupled to the structure (NES). Dynamic analysis is performed through parametric analysis of parameters related to piezoelectric coupling and NES parameters. Numerical simulations demonstrate that the system has chaotic behavior for specific parameters and that its energy capture is influenced by parametric variation. It is shown numerically that the parameters of the SMA material, the PZT material, and the NES significantly influence the chaotic behavior and energy capture of the investigated electromechanical system. [ABSTRACT FROM AUTHOR]

In this paper, we explore the rich dynamics in the Chen system using Runge–Kutta method and Deep Neural Networks (DNNs). Compared with the Lorenz system, we find that the first return map P of the Chen system exhibits a more complex structure of continuous regions in the Poincaré section. Moreover, the existence of six crossing blocks with respect to the second return map implies that there is a closed invariant set Λ in the Poincaré section such that P 2 | Λ is semi-conjugate to a 6-shift map, thus demonstrating remarkably chaos in the Chen system. [ABSTRACT FROM AUTHOR]

This work investigates a fractional-order multi-wing chaotic system for detecting weak signals. The influence of the order of fractional calculus on chaotic systems' dynamical behavior is examined using phase diagrams, bifurcation diagrams, and SE complexity diagrams. Then, the principles and methods for determining the frequencies and amplitudes of weak signals are examined utilizing fractional-order multi-wing chaotic systems. The findings indicate that the lowest order at which this kind of fractional-order multi-wing chaotic system appears chaotic is 2.625 at a = 4 , b = 8 , and c = 1 , and that this value decreases as the driving force increases. The four-wing and double-wing change dynamics phenomenon will manifest in a fractional-order chaotic system when the order exceeds the lowest order. This phenomenon can be utilized to detect weak signal amplitudes and frequencies because the system parameters control it. A detection array is built to determine the amplitude using the noise-resistant properties of both four-wing and double-wing chaotic states. Deep learning images are then used to identify the change in the array's wing count, which can be used to determine the test signal's amplitude. When frequencies detection is required, the MUSIC method estimates the frequencies using chaotic synchronization to transform the weak signal's frequencies to the synchronization error's frequencies. This solution adds to the contact between fractional-order calculus and chaos theory. It offers suggestions for practically implementing the chaotic weak signal detection theory in conjunction with deep learning. [ABSTRACT FROM AUTHOR]

The dynamical behavior of the double-chain deoxyribonucleic acid (DNA) system holds significant implications for advancing the understanding of DNA transmission laws in the realms of biology and medicine. This study delves into the investigation of chaos patterns and solitary wave solutions for the (2+1) Beta-fractional double-chain DNA system, employing the theory of planar dynamical systems and the method of complete discrimination system for polynomials (CDSP). The results demonstrate a diverse spectrum of solitary wave solutions, sensitivity to perturbations, and manifestations of chaotic behavior within the system. Through the utilization of the complete discrimination system for polynomials, a multitude of novel solitary wave solutions, encompassing periodic, solitary wave, and Jacobian elliptic function solutions, were systematically constructed. The influence of Beta derivatives on the solutions was elucidated through parameter comparison analysis, emphasizing the innovative nature of this study. These findings underscore the potential of this system in unraveling various biologically significant DNA transmission mechanisms. [ABSTRACT FROM AUTHOR]

The symmetric multi-scroll strange attractor has shown great potential in chaos-based applications due to its high complexity in phase space. Here, the approach of symmetrization is employed for attractor doubling to generate pseudo-multi-scroll attractors in a discrete map, where a carefully selected offset constant is the key to organizing coexisting attractors. By choosing the Hénon map to generate the pseudo-multi-scroll attractor and implementing the digital circuit on a microcontroller, this study fills a significant gap in the research on discrete chaotic systems. The complexity performance is further validated using a pseudo-random number generator, demonstrating substantial academic contributions to the field of chaos theory. Additionally, a pseudo-multi-scroll attractor-based squirrel search algorithm is first developed, showcasing its practical application in mobile robot path planning. This work not only advances the theoretical understanding of chaotic systems but also provides practical methods for implementation in digital systems, offering valuable insights for policy-making in advanced robotic systems and intelligent manufacturing. [ABSTRACT FROM AUTHOR]

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C 0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map's states in commensurate and incommensurate cases. [ABSTRACT FROM AUTHOR]

The Internet of Things (IoT) is made up of billions of interconnected devices that can transmit and receive data over the Internet. IoT devices have many vulnerabilities that attackers could use to compromise their security because of the heterogeneity of device connectivity. Distributed denial-of-service (DDoS) attacks against those applications become more common as IoT applications continue to expand and devolve. Identifying DDoS attacks is a difficult process due to the variety of IoT devices connected. The present article proposed a new method to detect DDoS attacks using an optimized Elman recurrent neural network (ERNN) based on chaotic bacterial colony optimization (CBCO) called CBCO-ERNN. The proposed method uses CBCO for obtaining optimal parameters (weights and biases) and structure (number of hidden neurons) of ERNN architecture. The chaos theory is applied to improve BCO's exploration and exploitation capabilities by initializing the bacterial population and selecting the appropriate chemotaxis step size value. The CBCO approach is used to train the ERNN model to avoid local optima and enhance the convergence rate. The performance of the CBCO-ERNN is tested and evaluated using four benchmark attack datasets such as the BoT-IoT, CIC-IDS2017, CIC-DDoS2019, and IoTID20 datasets, and five performance metrics are considered: accuracy, sensitivity, specificity, precision, and F-Score. According to the experimental results, the CBCO-ERNN method provides a high detection and a faster convergence rate when compared to earlier algorithms. [ABSTRACT FROM AUTHOR]

Investigating the chaotic characteristics of classical autonomous systems necessitates prioritizing the dynamic behavior of simple system models, which plays a crucial role in advancing research in this field. This study examines the influence of exponential variations in state variables on the dynamic behavior of chaotic systems, using a classical chaotic system as the main research case. Using classical autonomous systems as the examples, three possible scenarios of exponential change in state variables are listed and the chaotic characteristics of the system under these conditions are demonstrated. The system model exhibits a wide range of chaotic characteristics under different conditions. Additionally, the exponential change of state variables is applied to a simplified chaotic system to illustrate the universality of this phenomenon. Finally, circuit simulation experiments confirm the feasibility of circuit simulations for chaotic system models with exponential change characteristics of state variables, providing theoretical support for the application of such chaotic systems. [ABSTRACT FROM AUTHOR]