Search Results (141)

This study investigates the dynamics of dust acoustic periodic waves in a three-component, unmagnetized dusty plasma system using generalized $(r^{★},q^{★})$ 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

This paper investigates the periodic boundary value problem for impulsive evolution equation in ordered Banach space. By applying the Poincaré mapping and monotone iterative method, we obtain the existence results of mild solutions and positive mild solutions for impulsive evolution equation. Further, we obtain the uniqueness of mild solution. Full article

This paper presents a novel four-dimensional autonomous conservative model characterized by an infinite set of equilibrium points and an unusual algebraic structure in which all eigenvalues of the Jacobian matrix are zero. The linearization of the proposed model implies that classical stability analysis is inadequate, as only the center manifolds are obtained. Consequently, the stability of the system is investigated through both analytical and numerical methods using Lyapunov functions and numerical simulations. The proposed model exhibits rich dynamics, including hyperchaotic behavior, which is characterized using the Lyapunov exponents, bifurcation diagrams, sensitivity analysis, attractor projections, and Poincaré map. Moreover, in this paper, we explore the model with fractional-order derivatives, demonstrating that the fractional dynamics fundamentally change the geometrical structure of the attractors and significantly change the system stability. The Grünwald–Letnikov formulation is used for modeling, while numerical integration is performed using the Caputo operator to capture the memory effects inherent in fractional models. Finally, an analog electronic circuit realization is provided to experimentally validate the theoretical and numerical findings. Full article

Fin-stabilized guided rockets exhibit ballistic characteristics such as low initial velocity, high flight altitude, and long flight duration, which render their impact point accuracy and flight stability highly susceptible to the influence of wind. In this paper, the four-dimensional nonlinear angular motion equations describing the changes in attack angle and the law of axis swing of a dual-spin rocket are established, and the phase trajectory and equilibrium point stability characteristics of the nonlinear angular motion system under windy conditions are analyzed. Aiming at the problem that the equilibrium point of the angular motion system cannot be solved analytically with the change in wind speed, a phase trajectory projection sequence method based on the Poincaré cross-section and stroboscopic mapping is proposed to analyze the effect of wind on the angular motion bifurcation characteristics of a dual-spin rocket. The possible instability of angular motion caused by nonlinear aerodynamics under strong wind conditions is explored. This study is of reference significance for the launch control and aerodynamic design of guided rockets in complex environments. Full article

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

This study investigates the nonlinear aeroelastic behavior and energy harvesting performance of a two-degrees-of-freedom NACA 0012 airfoil under varying reduced velocities and electrical load resistances. The system exhibits a range of dynamic responses, including periodic and chaotic states, governed by strong fluid–structure interactions. Nonlinear oscillations first appear near the critical reduced velocity $U_r^{*}=6$, with large-amplitude limit-cycle oscillations emerging around $U_r^{*}=8$ in the absence of the electrical loading. As the load resistance increases, this transition shifts to higher $U_r^{*}$, reflecting the damping effect of the electrical load. Fourier spectra reveal the presence of odd and even superharmonics in the lift coefficient, indicating nonlinearities induced by fluid–structure coupling, which diminishes at higher resistances. Phase portraits and Poincaré maps capture transitions across dynamical regimes, from periodic to chaotic behavior, particularly at a low resistance. The voltage output correlates with variations in the lift force, reaching its maximum at an intermediate resistance before declining due to a suppressing nonlinearity. Flow visualizations identify various vortex shedding patterns, including single (S), paired (P), triplet (T), multiple-pair (mP) and pair with single (P + S) that weaken at higher resistances and reduced velocities. The results demonstrate that nonlinearity plays a critical role in efficient voltage generation but remains effective only within specific parameter ranges. Full article

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. Full article

This study investigates the dynamic behavior and reliability of planar multi-link linkages with clearance in translational pairs. Using the Lagrange multiplier method, a dynamic model that accounts for clearance effects is developed. Furthermore, a reliability model is established by combining the first-order second-moment method with the stress-strength interference theory. Numerical simulations were performed to evaluate the impact of varying clearance sizes and driving speeds on motion errors and system reliability. This study also explores the nonlinear dynamics of the end-effector. The results indicate that increased clearance and higher driving speeds lead to certain changes in motion errors and operational reliability. Phase diagrams and Poincaré maps reveal directional differences in dynamic stability: chaotic motion along the X-direction and periodic oscillations along the Y-direction. These findings provide valuable insights for optimizing mechanism design and enhancing operational reliability. Full article

The dynamic response of symmetrical press mechanisms is severely affected by revolute clearance, translational clearance, and the elasticity of the components. Therefore, the coupling effects of disturbance factors were studied in this paper, including revolute clearance, translational clearance, and component elastic deformation; the influence of their coupling effects on the dynamic chaos characteristic are also discussed. A dynamic model of a rigid–flexible coupling mechanism with revolute clearance and translational clearance was established. Using MATLAB R2024a to solve the model, chaos identification was researched through phase diagrams, Poincaré maps, and maximum Lyapunov exponents. Under the parameters studied in this paper, the maximum Lyapunov exponents at revolute clearance A (X direction and Y direction) and translational clearance B (X direction and Y direction) were 0.0521, 0.0573, 0.3915, and −0.0287, respectively. The motion state of revolute pair A (X direction and Y direction) and translational pair B (X direction) were more prone to chaotic states; translational pair B (Y direction) was more prone to periodic motion. The influence of various factors on the dynamic response were analyzed. With the increase in driving speed and clearance value, as well as the decrease in friction coefficient, the stability of the mechanism weakened, and the vibration of the mechanism’s dynamic response intensified. This paper provides theoretical support for the establishment of precise dynamic models for multi-link symmetrical structure press mechanisms. Full article

Revolute and prismatic pair clearances are common in various mechanisms, and their motion state seriously affects the accuracy of the mechanism. Adding lubricant to a kinematic pair can effectively counteract the adverse influence of a collision force. Thus, this work introduces an advanced modeling method that considers the combined effects of a lubricated revolute and prismatic clearance, as well as component flexibility, and studies the influence of their coupling effect on the dynamic response and nonlinear characteristic of mechanisms. The specific content of this paper is as follows: Firstly, revolute lubrication clearance and prismatic pair clearance models are established. Secondly, rigid components and flexible components are described based on the reference point coordinate method and absolute nodal coordinate formulation. Then, based on the Lagrange multiplier method, a rigid–flexible coupling dynamics model with revolute lubrication clearance and prismatic clearance is established. Finally, the dynamic responses of the mechanism are analyzed, including the displacement, velocity, and acceleration of the slider, the driving torque of the crank, and the center trajectories of the revolute clearance and prismatic clearance. Qualitative research is conducted on the nonlinear characteristics of the system through a phase diagram and Poincaré map. This quantitative study is conducted on the nonlinear characteristics of a system using the maximum Lyapunov exponent. The influences of different parameters on the dynamic response and nonlinear characteristic of the mechanism are analyzed. The results indicate that lubrication effectively reduces the influence of the clearance on the dynamic response and nonlinear characteristic of the mechanism, resulting in a decrease in the peak dynamic response and a weakening of the chaotic phenomenon. Further, as the driving speed increases, the dynamic viscosity decreases the clearance value increases, and the stability of the mechanism decreases. Full article

As an inherent property of polyvinyl chloride (PVC) gel material, viscoelasticity is closely related to the deformation of the material, which will affect its dynamic behavior. However, the existing theoretical model does not consider the influence of time-varying damping on its nonlinear vibration, which leads to the unclear nonlinear dynamic behavior of the material under the dual influence of viscoelasticity and electromechanical parameters and limits the further application of the material. Therefore, in this study, the standard linear solid (SLS) model was used to describe the time-varying dynamic change of viscoelasticity of PVC gel, and the electromechanical coupling second-order nonlinear constitutive equation of PVC gel actuator was established by combining the Gent free energy theory model. The harmonic resonance, stability and periodicity of PVC gel actuator under different loading conditions were investigated by using dynamic analysis methods such as phase path, Poincaré map, bifurcation diagram, and Lyapunov exponent. Through the systematic research in this study, the deformation law of PVC gel with time-varying damping under different electromechanical parameters was revealed, and the parameter control strategy of deformation stability and chaos was obtained, which provided the design method and theoretical basis for the further application of the material. Full article

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be not only non-diagonal but also non-square, without the conservative requirements in plenty of the existing literature. Next, an equation transformation and an inequality estimate for the anisotropic diffusion term are established, which are fundamental for analyzing the diffusion phenomenon in network dynamics. Subsequently, three control laws are devised to offer a detailed discussion for HSF control law’s outstanding performances, including the swifter convergence rate, the tighter bound of the settling time and the suppression of chattering. Following this, by a designed chaotic system with multi-scroll chaotic attractors tested with bifurcation diagrams, Poincaré map, and Turing pattern, several simulations are pvorided to attest the correctness of our developed findings. Finally, a formulated image encryption algorithm, which is evaluated through imperative security tests, reveals the effectiveness and superiority of the obtained results. Full article

The isolated boost full-bridge converter (IBFBC) is a DC–DC conversion topology that achieves a high boost ratio and provides electrical isolation, making it suitable for applications requiring both. Its operational dynamics are often intricate due to its inherent characteristics and the prevalent usage of nonlinear switching elements, leading to bifurcation and chaos. Chaos theory was employed to investigate the impact of changes in the voltage feedback coefficient K and input voltage E on the dynamic behavior of the IBFBC when operating in discontinuous conduction mode (DCM). Based on an analysis of its operating principles, a discrete iterative mapping model of the system in DCM is constructed using the stroboscopic mapping method. The effects of two control parameters, namely the proportional coefficient and input voltage, on system performance are studied using bifurcation diagrams, fold diagrams, Poincaré sections, and Lyapunov exponents. Simulation experiments are conducted using time-domain and waveform diagrams to validate the discrete mapping model and confirm the correctness of the theoretical analysis. The results indicate that when the IBFBC operates in DCM, its operating state is influenced by the voltage feedback coefficient $K$ and input voltage $E$. Under varying values of $K$ and $E$, the system may operate in a single-period stable state, multi-period bifurcation, or chaotic state, exhibiting typical nonlinear behavior. Full article

An examination was previously derived to conclude the understanding of the response of a cantilever beam with a tip mass (CBTM) that is stimulated by a parameter to undergo small changes in flexibility (stiffness) and tip mass. The study of this problem is essential in structural and mechanical engineering, particularly for evaluating dynamic performance and maintaining stability in engineering systems. The existing work aims to study the same problem but in different situations. He’s frequency formula (HFF) is utilized with the non-perturbative approach (NPA) to transform the nonlinear governing ordinary differential equation (ODE) into a linear form. Mathematica Software 12.0.0.0 (MS) is employed to confirm the high accuracy between the nonlinear and the linear ODE. Actually, the NPA is completely distinct from any traditional perturbation technique. It simply inspects the stability criteria in both the theoretical and numerical calculations. Temporal histories of the obtained results, in addition to the corresponding phase plane curves, are graphed to explore the influence of various parameters on the examined system’s behavior. It is found that the NPA is simple, attractive, promising, and powerful; it can be adopted for the highly nonlinear ODEs in different classes in dynamical systems in addition to fluid mechanics. Bifurcation diagrams, phase portraits, and Poincaré maps are used to study the chaotic behavior of the model, revealing various types of motion, including periodic and chaotic behavior. Full article

Hierarchical clustering analysis (HCA) is a widely used unsupervised learning method. Limitations of HCA, however, include imposing an artificial hierarchy onto non-hierarchical data and fixed two-way mergers at every level. To address this, the current work describes a novel rootlets hierarchical principal component analysis (hPCA). This method extends typical hPCA using multivariate statistics to construct adaptive multiway mergers and Riemannian geometry to visualize nested dependencies. The rootlets hPCA algorithm and its projection onto the Poincaré disk are presented as examples of this extended framework. The algorithm constructs high-dimensional mergers using a single parameter, interpreted as a p-value. It decomposes a similarity matrix from GL(m, ℝ) using a sequence of rotations from SO(k), k << m. Analysis shows that the rootlets algorithm limits the number of distinct eigenvalues for any merger. Nested clusters of arbitrary size but equal correlations are constructed and merged using their leading principal components. The visualization method then maps elements of SO(k) onto a low-dimensional hyperbolic manifold, the Poincaré disk. Rootlets hPCA was validated using simulated datasets with known hierarchical structure, and a neuroimaging dataset with an unknown hierarchy. Experiments demonstrate that rootlets hPCA accurately reconstructs known hierarchies and, unlike HCA, does not impose a hierarchy on data. Full article