Search Results (360)

This paper investigates the nonlinear dynamics of the wheel-side planetary reducer, considering the tooth wear effect. The tooth wear model based on the Archard adhesion wear theory is established, and the impact of tooth wear on meshing stiffness and piecewise-linear backlash of the planetary gear system is discussed. Then, the torsional vibration model and dimensionless differential equations considering tooth wear for the wheel-side planetary reducer are established, in which meshing excitations include time-varying mesh stiffness (TVMS), piecewise-linear backlash, and transmission error. The dynamic responses are numerically solved using the fourth-order Runge–Kutta method. On this basis, the nonlinear dynamics, such as the bifurcation and chaos properties of the wheel-side planetary reducer with tooth wear, are analyzed. Simulation results demonstrate that the existence of tooth wear reduces meshing stiffness and increases backlash. The reduction in the meshing stiffness changes the bifurcation path and chaotic amplitude of the system, inducing chaotic phenomena more easily. The increase in the gear backlash causes a higher amplitude of the relative displacement and more severe vibration. Full article

This study investigates the dynamic behavior of a discrete epidemic model as affected by media coverage through integrated analytical and numerical methods. The main objective is to quantitatively assess the impact of media coverage on disease outbreak models through mathematical modeling. We use the central manifold theorem and bifurcation theory to perform a rigorous analysis of the periodic solutions, focusing on the coefficients and conditions governing the flip bifurcation. On this basis, state feedback and hybrid control are utilized to control the system chaotically. Under certain conditions, the chaos and bifurcation of the system can be stabilized by the control strategy. Numerical simulations further reveal the bifurcation dynamics, chaotic behavior, and control techniques. Our results show that media coverage is a key factor in regulating the intensity and chaos of disease transmission. Control techniques can effectively prevent large-scale outbreaks of epidemics. Notably, enhanced media coverage can effectively increase public awareness and defensive behaviors, thus contributing to mitigating disease spread. Full article

This study examines a discrete-time predator–prey model constructed via piecewise constant discretization of its continuous counterpart. Through comprehensive qualitative and dynamical analyses, we reveal a rich set of nonlinear phenomena, encompassing Neimark–Sacker bifurcation, flip bifurcation, and codimension-two bifurcations corresponding to 1:2, 1:3, and 1:4 resonances. Rigorous analysis of these bifurcation scenarios, conducted via center manifold theory and bifurcation methods, establishes a robust mathematical framework for their characterization. Numerical simulations corroborate the theoretical predictions, exposing intricate dynamical phenomena such as quasiperiodic oscillations and chaotic attractors. Our results demonstrate that resonance-driven bifurcations are potent drivers of ecological complexity in discrete systems, acting as key determinants that orchestrate the emergent dynamics of populations—a finding with profound implications for interpreting patterns in real-world ecosystems subject to discrete generations or seasonal pulses. Full article

This study investigates the dynamic behaviors of the discrete epidemic model influenced by media coverage through integrated analytical and numerical approaches. The primary objective is to quantitatively assess the impact of media coverage on disease outbreak patterns using mathematical modeling. Firstly, the Euler method is used to discretize the model (2), and the periodic solution is strictly analyzed. Secondly, the coefficients and conditions of restricted flip and Neimark–Sacker bifurcation are studied by using the center manifold theorem and bifurcation theory. By calculating the largest Lyapunov exponent near the critical bifurcation point, the occurrence of chaos and limit cycles is proved. On this basis, the chaotic control of the system is carried out by using state feedback and hybrid control. Under certain conditions, the chaos and bifurcation of the system can be stabilized by control strategies. Numerical simulations further reveal bifurcation dynamics, chaotic behaviors, and control technologies. Our results show that media coverage is a key factor in regulating the intensity of disease transmission and chaos. The control technology can effectively prevent the large-scale outbreak of epidemic diseases. Importantly, enhanced media coverage can effectively promote public awareness and defensive behaviors, thereby contributing to the mitigation of disease transmission. Full article

Differential equations have demonstrated significant practical effectiveness across diverse fields, including physics, chemistry, biological engineering, computer science, electrical power systems, and security cryptography. This study investigates the dynamics of a Caputo fractional discrete-time modified Brusselator model. Conditions for the existence and local stability of the fixed point $FP$ are established. Using bifurcation theory, the occurrence of both period-doubling and Neimark–-Sacker bifurcations is analyzed, revealing that these bifurcations occur at specific values of the bifurcation parameter. Maximum Lyapunov characteristic exponents are computed to assess system sensitivity. Two-dimensional diagrams are presented to illustrate phase portraits, local stability regions, closed invariant curves, bifurcation types, and Lyapunov exponents, while the 0-1 test confirms the presence of chaos in the model. Finally, MATLAB simulations validate the theoretical results. Full article

To model the response of neural networks to electromagnetic radiation in real-world environments, this study proposes a memristive dual-wing fractional-order Hopfield neural network (MDW-FOMHNN) model, utilizing a fractional-order memristor to simulate neuronal responses to electromagnetic radiation, thereby achieving complex chaotic dynamics. Analysis reveals that within specific ranges of the coupling strength, the MDW-FOMHNN lacks equilibrium points and exhibits hidden chaotic attractors. Numerical solutions are obtained using the Adomian Decomposition Method (ADM), and the system’s chaotic behavior is confirmed through Lyapunov exponent spectra, bifurcation diagrams, phase portraits, and time series. The study further demonstrates that the coupling strength and fractional order significantly modulate attractor morphologies, revealing diverse attractor structures and their coexistence. The complexity of the MDW-FOMHNN output sequence is quantified using spectral entropy, highlighting the system’s potential for applications in cryptography and related fields. Based on the polynomial form derived from ADM, a field programmable gate array (FPGA) implementation scheme is developed, and the expected chaotic attractors are successfully generated on an oscilloscope, thereby validating the consistency between theoretical analysis and numerical simulations. Finally, to link theory with practice, a simple and efficient MDW-FOMHNN-based encryption/decryption scheme is presented. Full article

This paper addresses the challenge of determining optimal parameters in chaotic systems used for image encryption algorithms based on chaos theory. A baseline algorithm employing a third-order Lorenz chaotic system is examined, incorporating core procedures such as permutation (shuffling) and diffusion. Graphical results are presented to illustrate the variation of image entropy in relation to changes in system parameters. The analysis reveals a distinct region in the parameter space where entropy reaches its highest values. Based on these observations, an optimality criterion is formulated, defining an objective function that captures the entropy’s sensitivity to two key system parameters, including the bifurcation parameter. A complex objective function is derived, and the optimization problem is solved using a modified version of the Price algorithm enhanced with artificial intelligence techniques. The proposed modification demonstrates superior performance in locating the global extremum of the objective function, resulting in enhanced security of the encrypted image. Numerical and graphical results for various images are provided, along with a comparative analysis between the standard and the modified Price method. Full article

This paper investigates the stability and Hopf bifurcation problems of fractional-order quaternion-valued neural networks (FOQVNNs) with leakage delay and communication delay. Utilizing the Hamilton rule of quaternions, the fractional-order quaternion-valued time-delay neural network model is transformed into an equivalent fractional-order real-valued time-delay neural network system. Then, employing the stability theory and bifurcation theory of fractional-order dynamical systems, novel sufficient criteria are derived to ensure system stability and to induce Hopf bifurcation, respectively, using the leakage delay and the communication delay as bifurcation parameters. Furthermore, the influences of both delay types on the bifurcation behavior of FOQVNNs are analyzed in depth. To verify the correctness of the theoretical results, bifurcation diagrams and simulation results generated using MATLAB are presented. The theoretical results established in this paper provide a significant theoretical basis for the analysis and design of FOQVNNs. Full article

In this paper, a predator–prey model with hunting cooperation and maturation delay is studied. Through theoretical analysis, we investigate the existence of multiple stability switches of the positive equilibrium. By applying Hopf bifurcation theory, the conditions for Hopf bifurcation are derived, indicating the emergence of periodic solutions as the maturation delay passes through critical values. Utilizing center manifold theory and normal form analysis, we determine the stability and direction of the bifurcating orbits. Numerical simulations are performed to validate the theoretical results. Furthermore, the simulations vividly demonstrate the appearance of period-doubling bifurcations, which is the onset of chaotic behavior. Bifurcation diagrams and phase portraits are employed to precisely characterize the transition processes from a stable equilibrium to periodic, period-doubling solutions and chaotic states under different maturation delay values. The study reveals the significant influence of maturation delay on the stability and complex dynamics of predator–prey systems with hunting cooperation. Full article

This study examines the Enron collapse through an integrated theoretical framework combining the financial saturation paradox with the dynamics of a naturally occurring Ponzi process. The central objective is to evaluate whether endogenous market mechanisms—beyond managerial misconduct—played a decisive role in the emergence and breakdown of the Enron stock bubble. A logistic-growth-based saturation model is formulated, incorporating positive feedback effects and bifurcation thresholds, and applied to Enron’s stock price data from 1996 to 2001. The computations were performed using LogletLab 4 (version 4.1, 2017) and Microsoft^® Excel^® 2016 MSO (version 2507). The model estimates market saturation ratios (P/Pp) and logistic growth rate (r), treating market potential, initial price, and time as constants. The results indicate that Enron’s share price approached a saturation level of approximately 0.9, signaling a hyper-accelerated, unsustainable growth phase consistent with systemic overheating. This finding supports the hypothesis that a naturally occurring Ponzi dynamic was underway before the firm’s collapse. The analysis further suggests a progression from market-driven expansion to intentional manipulation as the bubble matured, linking theoretical saturation stages with observed price behavior. By integrating behavioral–financial insights with saturation theory and Natural Ponzi dynamics, this work offers an alternative interpretation of the Enron case and provides a conceptual basis for future empirical validation and comparative market studies. Full article

This study investigates the dynamics of dust acoustic periodic waves in a three-component, unmagnetized dusty plasma system using generalized $(r^{\star },q^{\star })$ distributions. First, boundary conditions are applied to reduce the model to a second-order nonlinear ordinary differential equation. The Galilean transformation is subsequently applied to reformulate the second-order ordinary differential equation into an unperturbed dynamical system. Next, phase portraits of the system are examined under all possible conditions of the discriminant of the associated cubic polynomial, identifying regions of stability and instability. The Runge–Kutta method is employed to construct the phase portraits of the system. The Hamiltonian function of the unperturbed system is subsequently derived and used to analyze energy levels and verify the phase portraits. Under the influence of an external periodic perturbation, the quasi-periodic and chaotic dynamics of dust ion acoustic waves are explored. Chaos detection tools confirm the presence of quasi-periodic and chaotic patterns using Basin of attraction, Lyapunov exponents, Fractal Dimension, Bifurcation diagram, Poincaré map, Time analysis, Multi-stability analysis, Chaotic attractor, Return map, Power spectrum, and 3D and 2D phase portraits. In addition, the model’s response to different initial conditions was examined through sensitivity analysis. Full article

In this paper, we propose a diffusion-type predator–prey interaction model with harvest and disease in prey, and conduct stability analysis and pattern formation analysis on the model. For the temporal model, the asymptotic stability of each equilibrium is analyzed using the linear stability method, and the conditions for Hopf bifurcation to occur near the positive equilibrium are investigated. The simulation results indicate that an increase in infection force might disrupt the stability of the model, while an increase in harvesting intensity would make the model stable. For the spatiotemporal model, a priori estimate for the positive steady state is obtained for the non-existence of the non-constant positive solution using maximum principle and Harnack inequality. The Leray–Schauder degree theory is used to study the sufficient conditions for the existence of non-constant positive steady states of the model, and pattern formation are achieved through numerical simulations. This indicates that the movement of prey and predators plays an important role in pattern formation, and different diffusions of these species may play essentially different effects. Full article

Based on the theory of optical sensing, we propose a high-precision plasmonic refractive index nanosensor, which consists of a symmetric rectangular waveguide and a circular ring containing a rectangular cavity. The designed novel tunable micro-resonant circular cavity filter based on surface plasmon excitations is able to confine light to sub-wavelength dimensions. The data show that different geometrical factors have different effects on sensing, with the geometry of the rectangular cavity and the radius of the circular ring being the key factors affecting the Fano resonance. Furthermore, the resonance bifurcation enables the structure to achieve a tunable dual Fano resonance system. The structure was tuned to obtain optimal sensitivity (S) and figure of merit values up to 3066 nm/RIU and 78. The designed structure has excellent sensing performance with sensitivities of 0.4767 nm·(mg/d${\mathrm{L}}^{-1}$) and 0.6 nm·(mg/d${\mathrm{L}}^{-1}$) in detecting Na⁺ and K⁺ concentrations in the electrolyte solution, respectively, and can be easily achieved by the spectrometer. The wavelength accuracy of 0.001 nm can be easily achieved by a spectrum analyzer, which has a broad application prospect in the field of optical integration. Full article

In this article, we propose a novel nonlinear cross-diffusion framework to model the distribution of susceptible and infected individuals within their habitat using a reduced SIR model that incorporates saturated incidence and treatment rates. The study investigates solution boundedness through the theory of parabolic partial differential equations, thereby validating the proposed spatio-temporal model. Through the implementation of the suggested cross-diffusion mechanism, the model reveals at least one non-constant positive equilibrium state within the susceptible–infected (SI) system. This work demonstrates the potential coexistence of susceptible and infected populations through cross-diffusion and unveils Turing instability within the system. By analyzing codimension-2 Turing–Hopf bifurcation, the study identifies the Turing space within the spatial context. In addition, we explore the results for Turing–Bogdanov–Takens bifurcation. To account for seasonal disease variations, novel perturbations are introduced. Comprehensive numerical simulations illustrate diverse emerging patterns in the Turing space, including holes, strips, and their mixtures. Additionally, the study identifies non-Turing and Turing–Bogdanov–Takens patterns for specific parameter selections. Spatial series and surfaces are graphed to enhance the clarity of the pattern results. This research provides theoretical insights into the implications of cross-diffusion in epidemic modeling, particularly in contexts characterized by localized mobility, clinically evident infections, and community-driven isolation behaviors. Full article

This study develops a dynamic IS-LM macroeconomic model that incorporates delayed taxation and a memory-dependent income effect, and calibrates it to quarterly data for Romania (2000–2023). Within this framework, fiscal policy lags are modelled using a “memory” income variable that weights past incomes, an approach grounded in distributed lag theory to capture how historical economic conditions influence current dynamics. The model is analysed both analytically and through numerical simulations. We derive stability conditions and employ bifurcation analysis to explore how the timing of taxation influences macroeconomic equilibrium. The findings reveal that an immediate taxation regime yields a stable adjustment toward a unique equilibrium, consistent with classical IS-LM expectations. In contrast, delayed taxation, where tax revenue depends on past income, can destabilise the system, giving rise to cycles and even chaotic fluctuations for parameter values that would be stable under immediate collection. In particular, delays act as a destabilising force, lowering the threshold of the output-adjustment speed at which oscillations emerge. These results highlight the critical importance of policy timing: prompt fiscal feedback tends to stabilise the economy, whereas lags in fiscal intervention can induce endogenous cycles. The analysis offers policy-relevant insights, suggesting that reducing fiscal response delays or counteracting them with other stabilisation tools is crucial for macroeconomic stability. Full article