Search Results (598)

Building on the first paper in this series (Paper I), a unified perspective on Poincaré and Galilei physics in a 5-dimensional spacetime setting is further pursued through a consideration of the kinematics of general relativity, with the gravitational dynamics to be addressed separately. The metric of the 5-dimensional affine spacetimes governed by the Bargmann groups considered in Paper I (central extensions of the Poincaré and Galilei groups) is generalized to curved spacetime by extending the usual 1 + 3 (traditionally `3 + 1’) formalism of general relativity on 4-dimensional spacetime to a 1 + 3 + 1 formalism, whose spacetime kinematics is shown to be consistent with that of the usual 1 + 3 formalism. Spacetime tensor laws governing the motion of an elementary classical material particle and the dynamics of a simple fluid are presented, along with their 1 + 3 + 1 decompositions; these reference the foliation of spacetime in a manner that partially reverts the Einstein perspective (accelerated fiducial observers, and geodesic material particles and fluid elements) to a Newton-like perspective (geodesic fiducial observers, and accelerated material particles and fluid elements subject to a gravitational force). These spacetime laws of motion for particles and fluids also suggest that a strong-field Galilei general relativity would involve a limit in which not only $c\to \infty$ but also $G\to \infty$, such that $G/c^2$ remains constant. 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

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

There is limited published guidance available to help less experienced practitioners assess and manage shoulder conditions, including spasticity, after acquired central nervous system (CNS) lesions. To address this gap, 11 spasticity and dystonia experts convened in a 2023 meeting to build on existing guidance, provide consensus on best treatment practice, and develop expert recommendations to guide the diagnosis and treatment of complications of shoulder conditions following CNS lesions. Presentations by each expert on diagnosis and management were followed by discussion; consensus on assessment and treatment practices was identified and recommendations developed. The expert panel recommended an assessment approach structured using the following components: patient history, including interpretation of reported symptoms; observation of postures and pain responses; clinical examination with targeted tests for specific signs; diagnostic tests; and assessment of upper limb impairment, activity limitations, and participation restrictions. This assessment process and the recommended measures recognize the importance of identifying shoulder involvement in upper limb spasticity as part of the diagnostic process in shoulder conditions following CNS lesions. These recommendations provide a practical approach to diagnosis and treatment for clinicians who are less experienced in evaluating and treating such conditions, simplifying otherwise complicated clinical scenarios. 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

In this paper, an eco-epidemiological model with a weak Allee effect and prey disease dynamics is discussed. Mathematical features such as non-negativity, boundedness of solutions, and local stability of the feasible equilibria are discussed. Additionally, the transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation are proven using Sotomayor’s theorem and Poincare–Andronov–Hopf theorems. In addition, the correctness of the theoretical analysis is verified by numerical simulation. The numerical simulation results show that the eco-epidemiological model with a weak Allee effect has complex dynamics. If the prey population is not affected by disease, the predator becomes extinct due to a lack of food. Under low infection rates, all populations are maintained in a coexistent state. The Allee effect does not influence this coexistence. At high infection rates, if the prey population is not affected by the Allee effect, the infected prey is found to coexist in an oscillatory state. The predator population and the susceptible prey population will be extinct. If the prey population is affected by the Allee effect, all species will be extinct. Full article

In a large deep lake, the generation of internal Kelvin waves and internal Poincaré waves due to wind stress on the lake surface is a significant phenomenon. These internal waves play a crucial role in material transport within the lake and have profound effects on its ecosystem and environment. Our study, which investigated the modes of internal waves in Lake Biwa using the vertical temperature distribution from field observations, has yielded important findings. We have demonstrated the applicability of the frequency equation solutions, considering the Coriolis force. The period of the internal Poincaré waves, as observed in the field, was found to match the solutions of the frequency equation. For example, observational data collected in late October revealed excellent agreement with the theoretical solutions derived from the frequency equation, showing periods of 14.7 h, 11.8 h, 8.2 h, and 6.3 h compared to the theoretical values of 14.4 h, 11.7 h, 8.5 h, and 6.1 h, respectively. However, the periods of the internal Kelvin waves in the field observation results were longer than those of the theoretical solutions. The Modified Mathew function uses a series expansion around $q_i=0$, making it difficult to estimate the periods of internal Kelvin waves under conditions where $\left|q_i\right|>1.0$. Furthermore, in lakes with an elliptical shape, such as Lake Biwa, the elliptical cylinder showed better reproducibility than the circular cylinder. These findings have significant implications for the rapid estimation of internal wave periods using the frequency equation. Full article

This investigation is centered on the analysis of frequency response characteristics of a p-i-n photodiode using InxGa_1−xAs/InP. The InGaAs/InP can be developed under three conditions: compression, tensile strain, and lattice matching. Initially, we performed calculations on strain, bandgap energy (E_g), and absorption coefficient. We then optimized the influence of indium concentration (x) on stability, critical thickness, bandgap energy, and absorption coefficient. The effects of temperature and deformation on E_g were also studied. Finally, we optimized the cutoff frequency (f_c), capacitive effects, and response frequency by considering the impact of x, active layer thickness (d), and surface area (S). For our future endeavors, we intend to explore additional parameters that may affect the p-i-n response. In future work, we can add transparent double layers in the i. InGaAs layer to reduce the transit time, leading to the development of an ultrafast photodiode. 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 paper elucidates the Lorentz group, a fundamental subgroup of the Poincaré group. The orbits and little groups associated with the Lorentz group are described in detail, along with their corresponding properties. The Poincaré group is presented. Another fundamental aspect of the Poincaré group is Wigner’s little groups obtained from this group. An in-depth discussion on the cases of both massive and massless relativistic particles within the context of little groups is given. Our examination extends to the properties of various special groups associated with the Poincaré group. Applications of these groups are elaborated by physical examples taken from high-energy physics and optics from both classical and quantum domains. Specifically, covariant harmonic oscillators including entangled states, proton form factors, and the parton picture as proposed by Feynman are discussed. In this context, laser cavities and shear states are also addressed. We lay out the underlying mathematics that connects these apparently disparate realms of physics. Full article

After linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been studied by many authors. Such quadratic polynomial differential systems have been divided into ten families. Here, for two of these families, we classify all topologically distinct phase portraits in the Poincaré disc. These two families have already been studied previously, but several mistakes made there are repaired here thanks to the use of a more powerful technique. This new technique uses the invariant theory developed by the Sibirskii School, applied to differential systems, which allows to determine all the algebraic bifurcations in a relatively easy way. Even though the goal of obtaining all the phase portraits of quadratic systems for each of the ten families is not achievable using only this method, the coordination of different approaches may help us reach this goal. Full article

For a massive scalar field in a fixed Schwarzschild background, the radial wave equation obeyed by Fourier modes is first studied. After reducing such a radial wave equation to its normal form, we first study approximate solutions in the neighborhood of the origin, horizon and point at infinity, and then we relate the radial with the Heun equation, obtaining local solutions at the regular singular points. Moreover, we obtain the full asymptotic expansion of the local solution in the neighborhood of the irregular singular point at infinity. We also obtain and study the associated integral representation of the massive scalar field. Eventually, the technique developed for the irregular singular point is applied to the homogeneous equation associated with the inhomogeneous Zerilli equation for gravitational perturbations in a Schwarzschild background. Full article