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The dynamo action, which is of importance in the study of the geomagnetism mechanism, is considered to be caused by the convection structure formed inside a rotating spherical shell. This convection structure elongated in the rotation axis is generated by the action of both heat and rotation on the fluid inside a spherical shell. In this study, we analyzed thermal convection in such a rotating spherical shell and attempted to understand the phenomenon of this convective structure. It is known that each value of the Prandtl number, the Ekman number and the Rayleigh number and their balance are important for the generation of such convective structure. We fixed these three parameters and considered the effect of centrifugal buoyancy as the Froude number additionally. To investigate how the effects of centrifugal buoyancy affect the convective structure, we carried out both three-dimensional numerical simulations and linear stability analyses. In particular, we focused on the transition from axisymmetric flow to non-axisymmetric flow having wavenumbers in the toroidal direction and investigated both growth rate and phase velocity of the disturbance. It was found that axisymmetric flow tends to be maintained as the effect of centrifugal buoyancy increases.

The present study deals with the effects of non-uniform inclined thermal gradient and internal heat source effect on the stability of buoyant flows in a fluid-saturated horizontal porous layer. Both linear and non-linear stability analyses have been performed. The non-linear stability analysis has been carried out by using the energy functional. The eigenvalue problems in both cases are solved numerically by shooting and Runga-Kutta methods. It is observed that the preferred mode at the onset of convection is longitudinal stationary mode. Comparison between linear and energy thresholds is given and found that the flow is stabilized at higher horizontal Rayleigh number for linear and non-linear cases irrespective of heat source. For fixed horizontal thermal Rayleigh number, increasing the value of internal heat source parameter destabilizes the system and favors the convection to commence.

This paper examines a new vibrating dynamical motion of a novel auto-parametric system with three degrees of freedom. It consists of a damped Duffing oscillator as a primary system attached to a damped spring pendulum as a secondary system. Lagrange’s equations are utilized to acquire the equations of motion according to the number of the system’s generalized coordinates. The perturbation technique of multiple scales is applied to provide the solutions to these equations up to a higher order of approximations, with the aim of obtaining more accurate novel results. The categorizations of resonance cases are presented, in which the case of primary external resonance is examined to demonstrate the conditions of solvability of the steady-state solutions and the equations of modulation. The time histories of the achieved solutions, the resonance curves in terms of the modified amplitudes and phases, and the regions of stability are outlined for various parameters of the considered system. The non-linear stability, in view of both the attained stable fixed points and the criterion of Routh–Hurwitz, is investigated. The results of this paper will be of interest for specialized research that deals with the vibration of swaying buildings and the reduction in the vibration of rotor dynamics, as well as studies in the fields of mechanics and space engineering.

This study examines the magnetic effect on Darcy Brinkman convection in a Bidispersive horizontal porous layer, considering the importance of convective motions of electrically conducting porous media accompanying a magnetic field in real-life applications such as geophysics, metallurgical field and solidification structures. In order to conduct a thorough study, the boundaries are classified as free-free, rigid-free, and rigid-rigid. The fluid motion is described using the Brinkman-Darcy equation with a single temperature in the macropores and micropores. The eigenvalue problem is solved analytically for the free-free case by employing linear stability theory. A non-linear analysis using the energy method is undertaken to prove that linear instability and global non-linear stability thresholds are the same. The eigenvalue problem for rigid-free and rigid-rigid boundaries is numerically solved with the bvp4c routine in MATLAB R2020 with the Rayleigh number as the eigenvalue. It is found that the Hartmann number M2, Darcy number Da, permeability ratio κr, and momentum transfer coefficient γ stabilize the system. Rigid-rigid boundaries are found to be the most stable ones, followed by rigid-free and free-free, which are the least stable boundaries.

The common intuition among the ecologists of the midtwentieth century was that large ecosystems should bemore stable than those with a smaller number of species.This view was challenged by Robert May, who found a stability bound for randomly assembled ecosystems; they become unstable for a sufficiently large number of species. In the present work, we show that May’s bound greatly changes when the past population densities of a species affect its own current density. This is a common feature in real systems, where the effects of species’ interactions may appear after a time lag rather than instantaneously.The local stability of these models with self-interaction is described by bounds, which we characterize in the parameter space.We find a critical delay curve that separates the region of stability fromthat of instability, and correspondingly, we identify a critical frequency curve that provides the characteristic frequencies of a system at the instability threshold. Finally, we calculate analytically the distributions of eigenvalues that generalizeWigner’s aswell asGirko’s laws. Interestingly,we find that, for sufficiently large delays, the eigenvalues of a randomly coupled system are complex even when the interactions are symmetric.

The friction-induced vibration of a novel 5-DoF (degree-of-freedom) mass-on-oscillating-belt model considering multiple types of nonlinearities is studied. The first type of nonlinearity in the system is the nonlinear contact stiffness, the second is the non-smooth behaviour including stick, slip and separation, and the third is the geometrical nonlinearity brought about by the moving-load feature of the mass slider on the rigid belt. Both the linear stability of the system and the nonlinear steady-state responses are investigated, and rich dynamic behaviours of the system are revealed. The results of numerical study indicate the necessity of the transient dynamic analysis in the study of friction-induced-vibration problems as the linear stability analysis fails to detect the occurrence of self-excited vibration when two stable solutions coexist in the system. The bifurcation behaviour of the steady-state responses of the system versus some parameters is determined. Additionally, the significant effects of each type of nonlinearity on the linear stability and nonlinear steady-state responses of the system are discovered, which underlie the necessity to take multiple types of nonlinearities into account in the research of friction-induced vibration and noise.

Alluvial bars in straight channels generally migrate downstream or can be non‐migrating but rarely migrate upstream. On the other hand, a recent long‐term field observation suggested a possibility of the upstream migration of bars. To understand this rare morphodynamic feature of alluvial bars, we herein perform non‐linear numerical simulations of alluvial bar formation and development in a straight channel. More specifically, we simulate alluvial bar dynamics under steady discharge and unsteady discharge (repeated hydrographs), wherein the bar regime retains super‐resonance condition. The result shows that initially formed, short downstream‐migrating free bars elongate, becoming much longer upstream migrating bars. The former bar elongation can be understood as a well‐observed non‐linear response of the bars, but much longer bar formation might originate from different morphodynamic features. The theoretical interpretation for this result using temporal and spatial mode linear analysis suggests a sort of forced bar formation in the upstream channel originating from nearly non‐migrating, slow‐movement bars. In other words, nearly non‐migration bars act as forcing factors, similar to an asymmetric obstacle in the channel, leading to an upstream morphodynamic influence observed under super‐resonance conditions. Subsequently, an interaction/coexistence of free‐migrating and forced‐steady bar responses may contribute to the formation of upstream‐migrating bars. These results provide new insights into the migration features of alluvial bars, but they still require a comprehensive theoretical framework to explain them. Additionally, observing upstream‐migrating bars will be an important future research, although it will be difficult due to the considerably slow migration speed predicted by the present numerical simulation.

This work investigates dispersive optical solitons governed by a perturbed cubic–quartic nonlinear Schrödinger equation with parabolic self-phase modulation, a model of direct relevance to high-capacity fiber-optic systems where simultaneous higher-order dispersion and nonlinear perturbations shape pulse dynamics. The model is physically motivated by fibers with intensity-dependent refractive index profiles, where the interplay between fourth-order chromatic dispersion and parabolic (cubic–quintic) nonlinearity generates wave structures that the standard Kerr approximation cannot capture. To extract exact traveling-wave solutions, we employ the improved modified extended tanh-function method (IMETFM), which is selected for its ability to handle multi-parameter auxiliary equations and yield a wider diversity of solution families than classical expansion methods such as the tanh-function or -expansion approaches, without requiring the integrability of the underlying system. Our analysis produces five families of exact solutions: bright solitons, dark solitons, exponential-type solutions, singular periodic waves, and solutions expressed in terms of Weierstrass elliptic functions. For each family, explicit existence conditions and free-parameter restrictions are stated. The parametric constraints governing solution validity are derived and physically interpreted in terms of the dispersion, nonlinearity, and perturbation coefficients. Graphical representations of the spatial and temporal profiles illustrate the distinct propagation features of each solution type. A linear stability analysis, conducted via perturbation theory, yields an explicit eigenvalue dispersion relation and identifies a critical wavenumber threshold at which modulational instability sets in. The stability criteria provide actionable guidelines for maintaining soliton integrity under weak disturbances in practical optical environments. The results have direct implications for optical fiber communications, ultrafast signal processing, and dispersion-engineered photonic waveguides. The novelty lies in the simultaneous treatment of the parabolic law nonlinearity, fourth-order dispersion, and perturbative effects within a unified algebraic framework, yielding solution families including Weierstrass elliptic solutions that have not previously been reported for this model. Future work will address numerical validation, extension to stochastic and variable-coefficient models, and higher-dimensional soliton dynamics.

In this work, the wave solutions of the generalized (3+1)-dimensional P-type equation, a significant model for describing the evolution of waves in plasma physics, are investigated. The modified extended mapping method (MEMM) is applied as an effective analytical tool to get these solutions. Through the use of this method, a large variety of exact solutions is successfully derived, including Jacobi elliptic function solutions, bright and dark solitons, singular solitons, exponential forms, and singular periodic waveforms solutions. These solutions provide additional insight into the complex dynamics of the used equation. Furthermore, a linear stability analysis is performed to examine the stability of the steady-state solutions. The dispersion relation shows that the perturbation growth rate is purely imaginary for generic parameters, indicating neutral stability and the absence of modulation instability. Moreover, graphical representations of some of the solutions are given in order to disclose their physical behavior and better understand the corresponding wave phenomena.