This study presents a comprehensive dynamical analysis of a ratio-dependent reaction–diffusion prey–predator model, with an emphasis on nonlinear oscillatory behavior and the evolution of spatiotemporal patterns. The governing system of partial differential equations is investigated through local and global stability analysis to characterize the response of equilibrium states to small perturbations and to establish the conditions for long-term dynamic stability. Particular attention is devoted to Hopf-bifurcation analysis, which identifies the critical parameter values that govern the onset of self-sustained oscillations and periodic dynamics in the interacting populations. The emergence of diffusion-driven instability is further examined to reveal the formation of spatially distributed oscillatory patterns arising from the coupling between reaction kinetics and diffusion. Numerical simulations are performed to validate the theoretical findings and to illustrate the transition between stable, periodic, and spatially structured dynamic regimes. A coupled amplification-matrix analysis of the non-standard finite difference (NSFD) scheme, mesh-refinement tests, and a direct comparison with a classical explicit finite-difference method are used to assess discrete stability, consistency, positivity preservation, and small-step accuracy. The results demonstrate the proposed model’s ability to capture oscillatory responses and complex spatiotemporal behaviors governed by nonlinear interactions. The obtained results provide a comprehensive understanding of stability, bifurcation, and diffusion-driven pattern formation in the reaction-diffusion prey-predator model and may serve as a useful mathematical framework for future studies on nonlinear distributed dynamical systems.
This paper investigates the in-plane coupled vibration characteristics of a rotating Euler–Bernoulli beam with a non-uniform cross-section. It is assumed that the cross-sectional radius of the beam varies exponentially along its longitudinal direction. Based on Hamilton’s principle, the governing equations for the in-plane coupled vibrations of the rotating variable-cross-section beam are established. The partial differential equations of motion are discretized numerically using the Galerkin method, and the complex frequencies of the system are determined. The effects of key parameters, such as hub radius, taper ratio, slenderness ratio, and rotational speed, on the frequency characteristics of the beam are further examined. Moreover, the complex modal motions of the rotating beam are investigated, with a detailed analysis of how these key parameters influence the complex modal behavior. The results indicate that the in-plane coupled vibration characteristics of the rotating variable-cross-section beam depend on both the structural parameters and the rotational speed. As the taper ratio and slenderness ratio increase, the natural frequencies and vibration amplitudes of the beam decrease. The effects of rotational speed and hub radius on axial and transverse vibrations are different, which shows distinct dynamic behaviors at different modal orders.
In this work, we address the stochastic–fractional Drinfel’d–Sokolov–Wilson (SFDSW) system, a generalized nonlinear evolution model that incorporates fractional-order dynamics and stochastic perturbations to provide a more realistic framework for modeling complex physical processes. For the purpose of obtaining analytical solutions, a generalized Riccati equation mapping method is elaborated and applied to the fractional and stochastic structure of the stochastic–fractional Drinfel′d–Sokolov–Wilson model. Numerical solutions are created by using the differential transform method that allows for reliable approximation of the system. A comparative analysis is conducted with the purpose of examining the impact of fractional derivatives on the behavior of waves, stability, and dissipation, along with sensitivity to different noise intensities. It is determined that analytical mappings and numerical solutions exhibit strong agreement, confirming the efficiency of the approach. Moreover, this work reveals how the fractional order and stochastic forcing jointly shape the qualitative structure of solutions, providing further insight into the dynamics of nonlinear systems of the stochastic–fractional Drinfel′d–Sokolov–Wilson type.
This work explores the propagation of photo-thermoelastic disturbances in an infinite and isotropic semiconductor containing a spherical cavity. The analysis utilizes a fully coupled approach that integrates carrier (plasma) transport, heat conduction, and elastic deformation, effectively portraying the interplay among electronic excitation, increase in temperature, and mechanical deformation. The cavity's inner boundary excited by a surface heat flux that decays exponentially in time, representing a short pulse. An analytical framework relating to pulsed photo-thermal loading in semiconductors containing internal voids is developed and presented in this research. The governing equations are solved, and boundary conditions are applied to derive closed-form expressions for the physical fields, and the transformed solutions are inverted using eigenvalue decomposition to retrieve the fields in space-time. Numerical illustrations are reported for a silicon-like material to provide quantitative insight into the evolution of temperature, displacement, stress, and carrier density, as well as the associated electrochemical potential energy. The graphical results elucidate how transient behavior depends on radial position, and they highlight the role of coupling among the plasma, thermal, and elastic subsystems in shaping peak magnitudes, decay rates, and spatial attenuation.
The conventional filtered-x least mean square (FxLMS) algorithm for active vibration control (AVC) is limited by the trade-off between convergence rate and steady-state error and by its sensitivity to abrupt input changes. To address these issues, a nonlinear step-size adjustment mechanism combining a bounded hyperbolic tangent mapping with a normalized variable step-size strategy is developed, based on which a variable step-size FxLMS algorithm, termed VSS-FxtanhLMS, is proposed. The convergence properties of the algorithm are analyzed in the mean and mean-square senses. For AVC of a flexible cantilever beam, the proposed algorithm is evaluated on a co-simulation platform established using ADAMS and Simulink under single-frequency, dual-frequency, and amplitude-step excitations, and is further validated on an experimental AVC platform. Co-simulation results show that VSS-FxtanhLMS achieves the smallest steady-state error, the lowest mean squared error, and the shortest convergence time under both single-frequency and dual-frequency excitations, while remaining convergent under amplitude-step excitation. In experiments, compared with FxasinhLMS, the strongest competing method, VSS-FxtanhLMS reduces the steady-state error by 17.48%, shortens the convergence time by 33.33%, and reduces the power spectral density by 1.72 dB/Hz. These results indicate that VSS-FxtanhLMS achieves improved steady-state accuracy, convergence behavior, and robustness under both simulation and practical conditions.
