Search Results (2,849)

By applying the Non-Perturbative Approach (NPA), the corresponding linear differential equation is obtained. Aimed at organizational investigation, the resulting linear equation is used. Strong agreement between numerical calculations and the precise frequency is demonstrated, and the reliability of the results acquired is established by the correlation with the numerical solution. Additionally, this study explores a new control process to affect the stability and behavior of dynamic motorcycle systems that vibrate nonlinearly. A multiple time-scale method (MTSM) is applied to examine the analytical solution of the nonlinear differential equations describing the aforementioned system. Every instance of resonance was taken out of the second-order approximations. The simultaneous primary and 1:1 internal resonance case ($\mathsf{\Omega }\approx {\omega }_{eq},$${\omega }_2\approx {\omega }_{eq}$) is recorded as the worst resonance case caused while working on the model. We investigated stability with frequency–response equations and bifurcation. Numerical solutions for the system are covered. The effects of the majority of the system parameters were examined. In order to mitigate harmful vibrations, the controller under investigation uses (PD) proportional derivatives with (PPF) positive position feedback as a new control technique. This creates a new active control technique called PDPPF. A comparison between the PD, PPF, and PDPPF controllers demonstrates the effectiveness of the PDPPF controller in reducing amplitude and suppressing vibrations. Unwanted consequences like chaotic dynamics, limit cycles, or loss of stability can result from bifurcation, which is the abrupt qualitative change in a system’s behavior as a parameter. The outcomes showed how effective the suggested controller is at reducing vibrations. According to the findings, bifurcation analysis and a control are crucial for designing vibrating dynamic motorcycle systems for a range of engineering applications. The MATLAB software is utilized to match the analytical and numerical solutions at time–history and frequency–response curves (FRCs) to confirm their comparability. Additionally, case studies and numerical simulations are presented to show how well these strategies work to control bifurcations and guarantee the desired system behaviors. An analytical and numerical solution comparison was prepared. Full article

We consider first-order impulses for impulsive stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $H\in ({\displaystyle \frac{1}{2}},1)$ involving a nonlinear $\varphi$-Laplacian operator. The system incorporates both state and derivative impulses at fixed time instants. First, we establish the existence of at least one mild solution under appropriate conditions in terms of nonlinearities, impulses, and diffusion coefficients. We achieve this by applying a nonlinear alternative of the Leray–Schauder fixed-point theorem in a generalized Banach space setting. The topological structure of the solution set is established, showing that the set of all solutions is compact, closed, and convex in the function space considered. Our results extend existing impulsive differential equation frameworks to include fractional stochastic perturbations (via fBm) and general $\varphi$-Laplacian dynamics, which have not been addressed previously in tandem. These contributions provide a new existence framework for impulsive systems with memory and hereditary properties, modeled in stochastic environments with long-range dependence. Full article

Endocrine axes are pathways of interactions involved in various aspects of the organism’s functioning, also implicated in deviations from physiological states leading to pathological conditions. The hypothalamic–pituitary–adrenal (HPA) axis releases corticosteroid hormones promoting adaptation to environmental stimuli (acute stress) or inducing altered conditions due to long-term noxious solicitations (chronic stress). The HP–gonadal (HPG) axis regulates reproductive activities by releasing gonadal steroids. These axes have been shown to engage in reciprocal inhibition under certain conditions, particularly when they rise beyond normal ultradian and circadian fluctuations. Based on the literature data, we reconstructed a neuroendocrine network responsible for this type of interaction. Thereafter, we developed a model of the HPA-HPG inhibition based on a series of nonlinear interactions represented by a system of differential equations in the Matlab environment. The quantitative analysis of the system’s behavior revealed the occurrence of bifurcations leading to bistable behavior, allowing us to detect bifurcation parameters. Bifurcation arises as the system’s components increase hypersensitivity and sustained activity in response to activating inputs. This involves transition from a single low-activity attractor to two distinct attractors, with a new high-activity state representing a breakdown of homeostasis. These results provide insights into the potential involvement of the HPA-HPG interaction in neuroendocrine disorders, and the identification of therapeutic targets from bifurcation parameters. Full article

This paper presents a detailed study of the $(3+1)$-dimensional Zakharov–Kuznetsov–Burgers equation to investigate shock-wave phenomena in dusty plasmas with quantum effects. The model provides significant physical insight into nonlinear dispersive and dissipative structures arising in charged-dust–ion environments, corresponding to both laboratory and astrophysical plasmas. We then perform a qualitative, numerically assisted dynamical analysis using bifurcation diagrams, multistability checks, return maps, Poincaré sections, and phase portraits. For both the unperturbed and a perturbed system, we identify chaotic, quasi-periodic, and periodic regimes from these numerical diagnostics; accordingly, our dynamical conclusions are qualitative. We also examine frequency-response and time-delay sensitivity, providing a qualitative classification of nonlinear behavior across a broad parameter range. After establishing the global dynamical picture, traveling-wave solutions are obtained using the Paul–Painlevé approach. These solutions represent shock and solitary structures in the plasma system, thereby bridging the analytical and dynamical perspectives. The significance of this study lies in combining a detailed dynamical framework with exact traveling-wave solutions, allowing a deeper understanding of nonlinear shock dynamics in quantum dusty plasmas. These results not only advance theoretical plasma modeling but also hold potential applications in plasma-based devices, wave propagation in optical fibers, and astrophysical plasma environments. Full article

This work investigates the coupled Ivancevic option-pricing model, a nonlinear wave alternative to the Black–Scholes model. By utilizing the recently developed Kumar-Malik method, modified Sardar sub-equation method and the generalized Arnous method, the substantial results of this research are the successful derivation of novel exact soliton solutions, including bright, singular, dark, combined dark–bright, singular-periodic, complex solitons, exponential and Jacobi elliptic functions. A detailed analysis of option price wave functions and modulation instability analysis is conducted, with the conditions for valid solutions outlined. Additionally, a mathematical framework is established to capture market price fluctuations. Numerical simulations, illustrated through 2D, 3D and contour graphs, highlight the effects of parameter variations. Our findings demonstrate the effectiveness of the coupled Ivancevic model as a fractional nonlinear wave system, providing valuable insights into stock volatility and returns. This study contributes to creating new option-pricing models, which affect financial market analysis and risk management. Full article

This paper addresses the trajectory tracking control problem of Autonomous Underwater Vehicles (AUVs). A comprehensive mathematical model is first established based on Newtonian mechanics, incorporating both kinematic and dynamic equations. By reasonably neglecting the minor influence of roll motion, a five-degree-of-freedom (5-DOF) underactuated AUV model is derived. Considering the strong nonlinearities, high coupling, and time-varying hydrodynamic parameters typical of underwater environments, a fuzzy adaptive PID controller is proposed. This controller combines the adaptability of fuzzy logic with the structural simplicity and reliability of PID control, making it well-suited to the demanding requirements of AUV motion control. Extensive simulation experiments are conducted to evaluate the controller’s performance under various operating conditions. The results show that the fuzzy adaptive PID controller significantly outperforms conventional PID and standalone fuzzy logic controllers in terms of convergence speed and oscillation suppression. Furthermore, a theoretical stability analysis is provided to ensure that the proposed control system remains stable under time-varying fuzzy gain scheduling, confirming its effectiveness and potential for practical application in underwater vehicle control. Full article

This work presents a systematic study of nonlinear differential equations within Sobolev spaces, focusing on mild solutions and their qualitative properties. An iterative reconstruction method is developed to obtain uniform a priori bounds, which ensure both the existence and tightness of invariant measures. Furthermore, uniqueness of these measures is established under appropriate structural conditions. The results provide a rigorous foundation for analyzing the asymptotic behavior of nonlinear dynamical systems. Full article

Acid stimulation is a widely used technique for enhancing hydrocarbon recovery from carbonate reservoir formations. In this study, a mathematical model is developed to describe acidizing-induced pressure behavior in carbonate rocks, based on fluid dynamics and acid transport equations. The model is discretized using the finite volume method, resulting in a numerical framework suitable for simulating acidizing processes in carbonate reservoirs. Due to the model’s inherent characteristics—strong nonlinearity and the presence of high-dimensional sparse systems of equations—a sparse quasi-Newton method is proposed to efficiently solve the resulting system. Numerical experiments confirm the practicality and effectiveness of the proposed approach. Full article

Background/Objectives: Synaptic plasticity is fundamental to learning and memory, yet classical models such as Hebbian learning and spike-timing-dependent plasticity often overlook the distributed and wave-like nature of neural activity. We present a computational framework grounded in Scale Relativity Theory (SRT), which describes neural propagation along fractal geodesics in a non-differentiable space-time. The objective is to link nonlinear wave dynamics with the emergence of structured memory representations in a biologically plausible manner. Methods: Neural activity was modeled using nonlinear Schrödinger-type equations derived from SRT, yielding complex wave solutions. Synaptic plasticity was coupled through a reaction–diffusion rule driven by local activity intensity. Simulations were performed in one- and two-dimensional domains using finite difference schemes. Analyses included spectral entropy, cross-correlation, and Fourier methods to evaluate the organization and complexity of the resulting synaptic fields. Results: The model reproduced core neurobiological features: localized potentiation resembling CA1 place fields, periodic plasticity akin to entorhinal grid cells, and modular tiling patterns consistent with V1 orientation maps. Interacting waveforms generated interference-dependent plasticity, modeling memory competition and contextual modulation. The system displayed robustness to noise, gradual potentiation with saturation, and hysteresis under reversal, reflecting empirical learning and reconsolidation dynamics. Cross-frequency coupling of theta and gamma inputs further enriched trace complexity, yielding multi-scale memory structures. Conclusions: Wave-driven dynamics in fractal space-time provide a hypothesis-generating framework for distributed memory formation. The current approach is theoretical and simulation-based, relying on a simplified plasticity rule that omits neuromodulatory and glial influences. While encouraging in its ability to reproduce biological motifs, the framework remains preliminary; future work must benchmark against established models such as STDP and attractor networks and propose empirical tests to validate or falsify its predictions. Full article

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. Full article

As a pivotal carrier of emerging human–computer interaction technologies, artificial intelligence (AI) conversational agents (CAs) hold critical significance for research on the mechanisms of users’ continuance usage behaviour, which is essential for technological optimization and commercial transformation. However, the differential impact pathways of multidimensional perceived interactivity on continuance usage intention, particularly the synergistic mechanisms between technical and affective dual-path dimensions, remain unclear. This study investigates the personalized AI-based CAs project “Dialogue with Great Souls,” launched on a Chinese social platform, using survey data from 305 users. A hybrid approach combining partial least squares structural equation modelling (PLS-SEM) and artificial neural networks (ANN) was employed for empirical analysis. The results indicate that technical dimensions, such as control and responsiveness, are key factors influencing trust, while affective interactive dimensions, including communication, personalization, and playfulness, significantly affect social presence, thereby shaping users’ continuance usage intention. ANN results corroborated most PLS-SEM findings but revealed inconsistencies in the predictive importance of personalization and communication on social presence, highlighting the complementary nature of linear and nonlinear interaction mechanisms. By expanding the interactivity model and adopting a hybrid methodology, this study constructs a novel framework for AI CAs. The empirical findings suggest that developers should strengthen socio-emotional bonds in anthropomorphic interactions while ensuring technical credibility to enhance users’ continuance usage intention. This research not only advances theoretical perspectives on the integration of technical and affective dimensions in agent systems but also provides practical recommendations for optimizing the design and development of AI CAs. Full article

For a non-conservative nonlinear oscillator (NCNO) having a periodic solution, the existence of the first integral is a certain symmetry of the nonlinear dynamical system, which signifies the balance of kinetic energy and potential energy. A first-order nonlinear ordinary differential equation (ODE) is used to derive the first integral, which, equipped with a right-end boundary condition, can determine an implicit potential function for computing the period by an exact integral formula. However, the integrand is singular, which renders a less accurate value of the period. A generalized integral conservation law endowed with a weight function is constructed, which is proved to be equivalent to the exact integral formula. Minimizing the error to satisfy the periodicity conditions, the optimal initial value of the weight function is determined. Two non-iterative methods are developed by integrating three first-order ODEs or two first-order ODEs to compute the period. Very accurate value of the period can be observed upon testing five examples. For the NCNO without having the first integral, the integral-type period formula is derived. Four examples belong to the Liénard equation, involving the van der Pol equation, are evaluated by the proposed iterative method to determine the oscillatory amplitude and period. For the case with one or more limit cycles, the amplitude and period can be estimated very accurately. For the NCNO of a broad type with or without having the first integral, the present paper features a solid theoretical foundation and contributes integral-type formulations for the determination of the oscillatory period. The development of new numerical algorithms and extensive validation across a diverse set of examples is given. Full article

A class of boundary value problems for fractional differential systems involving variable-order derivatives is considered. Such problems can be transformed into some boundary value problems for nonlinear Caputo fractional differential systems. Here, the relations between linear Caputo fractional differential equations and their corresponding linear integral equations are investigated, and the results demonstrate that a proper Lipschitz-type condition is needed for studying nonlinear Caputo fractional differential equations. Then, an existence and uniqueness result is established in some vector subspaces by Banach’s fixed-point theorem and ${\parallel ·\parallel }_e$ norm. In addition, two examples are presented to illustrate the theoretical conclusions. Full article

With the rapid development of deep-sea resource exploration and marine scientific research, wheeled remotely operated vehicles (ROVs) have become crucial for seabed operations. However, under complex seabed conditions, traditional ROV control systems suffer from insufficient trajectory tracking accuracy, poor disturbance rejection capability, and low dynamic torque distribution efficiency. These issues lead to poor motion stability and high energy consumption on sloped terrains and soft substrates, which limits the effectiveness of deep-sea engineering. To address this, we proposed a comprehensive motion control solution for deep-sea wheeled ROVs. To improve modeling accuracy, a coupled kinematic and dynamic model was developed, together with a body-to-terrain coordinate frame transformation. Based on rigid-body kinematics, three-degree-of-freedom kinematic equations incorporating the slip ratio and sideslip angle were derived. By integrating hydrodynamic effects, seabed reaction forces, the Janosi soil model, and the impact of subsidence depth, a dynamic model that reflects nonlinear wheel–seabed interactions was established. For optimizing disturbance rejection and trajectory tracking, a hierarchical control method was designed. At the kinematic level, an improved model predictive control framework with terminal constraints and quadratic programming was adopted. At the dynamic level, non-singular fast terminal sliding mode control combined with a fixed-time nonlinear observer enabled rapid disturbance estimation. Additionally, a dynamic torque distribution algorithm enhanced traction performance and trajectory tracking accuracy. Full article

This study introduces a novel numerical approach to analyze the axisymmetric bending behavior of functionally graded (FG) graphene platelet (GPL)-reinforced annular sandwich nanoplates featuring a porous core. The nanostructures are exposed to coupled magnetic and hygrothermal environments. The porosity distribution and GPL weight fraction are modeled as nonlinear functions through the thickness, capturing realistic gradation effects. The governing equations are derived using the virtual displacement principle, taking into account the Lorentz force and the interaction with an elastic foundation. To address the size-dependent behavior and thickness-stretching effects, the model employs the nonlocal strain gradient theory (NSGT) integrated with a modified version of Shimpi’s quasi-3D higher-order shear deformation theory (Q3HSDT). The differential quadrature method (DQM) is applied to obtain numerical solutions for the displacement and stress fields. A detailed parametric study is conducted to investigate the influence of various physical and geometric parameters, including the nonlocal parameter, strain gradient length scale, magnetic field strength, thermal effects, foundation stiffness, core thickness, and radius-to-thickness ratio. The findings support the development of smart, lightweight, and thermally adaptive nano-electromechanical systems (NEMS) and provide valuable insights into the mechanical performance of FG-GPL sandwich nanoplates. These findings have potential applications in transducers, nanosensors, and stealth technologies designed for ultrasound and radar detection. Full article