In this study, we investigate the bifurcation of limit cycles from a class of uniform isochronous cubic centers in the plane under small continuous perturbations. Such systems provide classic yet challenging examples in nonlinear differential equations, offering valuable insight into the dynamics of planar polynomial systems and the emergence of small-amplitude periodic solutions. Understanding these bifurcations is fundamental for advancing the qualitative theory of planar systems and their applications in modeling oscillatory phenomena.

The differential system under consideration is given by:

\dot{x}=-y+x^{2}y+xy^{2}+\sum\limits_{i=1}^{6}\epsilon^{i}P_{i}(x,y),\quad

\dot{y}=x+xy^{2}-x^{2}y+\sum\limits_{i=1}^{6}\epsilon^{i}Q_{i}(x,y),​

where P_{i}(x,y) and Q_{i}(x,y) are cubic polynomials, and ε is a small perturbation parameter. Introducing polar coordinates, the system is transformed into a form suitable for the averaging method. Applying the sixth-order averaging theory, we derive sufficient conditions for the existence of limit cycles through the analysis of the averaged functions. This higher-order approach enables the detection of subtle bifurcation phenomena that may be missed in lower-order analyses.

Our analysis demonstrates that at most three small-amplitude limit cycles can bifurcate from the uniform isochronous cubic center under small continuous perturbations. The use of higher-order averaging reveals complex dynamics and interactions between the perturbation terms that are otherwise undetectable.

These findings enrich the qualitative theory of planar polynomial differential systems and showcase the effectiveness of sixth-order averaging methods in studying limit cycle bifurcations. The results highlight the intricate behavior of cubic systems and provide a robust framework for future investigations in nonlinear dynamics, with potential applications for modeling oscillatory behaviors in applied mathematics, physics, and engineering.

The Schrödinger equation with a unidimensional Gaussian potential barrier admits a finite number of complex eigenvalues associated with purely outgoing wavefunctions, called resonances.

We compute the resonances for this problem with varying barrier strength parameter by means of the well-known Rayleigh–Ritz method (RR) with complex rotation, using a harmonic oscillator basis set, and the Riccati–Padé method (RPM), which is based on the application of a Padé approximant to the Riccati equation for the regularized logarithmic derivative of the wavefunction. The latter method does not need to perform a complex rotation explicitly. We also show that a second set of complex eigenvalues exists, which is obtainable by both methods; in the case of the RR method, the computation of either of them requires different choices of the rotation parameter, setting it to be either below or above a certain threshold, respectively, whereas the RPM yields both sets of resonances. The lowest-lying eigenvalues of the second set are close to the resonances of the first set.

We perform a simple asymptotic analysis of the eigenfunctions, which allows us to determine the threshold value of the complex rotation parameter, as well as analyze the characteristics of both sets of eigenfunctions.
Finally, we draw parallels with similar results obtained in previous examples that present similar behavior, such as the radial exponential potential barrier and a well-known barrier potential that presents pre-dissociating resonances akin to those found in diatomic molecules.

This poster presents an analysis of an even Hermite–Sobolev system associated with the Gaussian weight rho(x) = exp(−x²) on the real line. The considered system is generated from normalized even Hermite functions and naturally arises in functional analysis, spectral theory, and the asymptotic study of generalized orthogonal systems.

Let r ≥ 1 be a fixed integer. The family of functions is defined by polynomial expressions for the initial indices and by integral representations involving even Hermite functions for higher orders. A fundamental property of this construction is that the r-th derivative of the system coincides with the classical even Hermite functions. We first prove that this family forms an orthogonal system with respect to a Sobolev-type inner product restricted to the space of even functions. This orthogonality reflects the intrinsic differential structure of the system and highlights its close connection with Hermite analysis.

The main objective of this work is to investigate the asymptotic behavior of the system as the index tends to infinity. By combining the classical Plancherel–Rotach asymptotic formulas for Hermite functions with the steepest descent method applied to the associated integral representations, we identify three distinct asymptotic regimes: the oscillatory (bulk) region, the transition region, and the exponential (outer) region. In particular, we show that the asymptotic behavior in the transition region is governed by the Airy function, revealing a universal feature characteristic of Gaussian-based orthogonal systems. Finally, numerical illustrations are provided to demonstrate the consistency between the asymptotic results and computational observations.

This paper presents an analytical study of the use of mathematical modelling, such as the iterated function system (IFS), for patch fractal antenna design. Current, modern research in the field of communications focuses on the realization of compact communication systems. Therefore, they concentrate on reducing the size of individual devices and the constitution of elements, particularly antennas. Fractal geometries are generated by recursive transformations, which provide a rigorous mathematical framework for constructing self-similar sets through affine contraction mappings using an iterative procedure for compact, miniaturized self-similar structures. With each iteration n, the effective electrical length of the antenna increases. Finite element simulations validate the theoretical predictions, demonstrating close agreement between calculated and simulated resonant frequencies so that additional resonant frequencies appear, which enable wideband and multiband operations. The results confirm that the iteration order can be used as a systematic design parameter to achieve target frequency bands without relying just on numerical optimization. This work illustrates how mathematical concepts such as IFS and recursion can guide the analytical design of compact, multiband, or wideband fractal antennas for wireless communication systems and aims to demonstrate the role of applied mathematics in the systematic design and miniaturization of fractal antenna structures.

We investigate the Green’s function associated with a one-dimensional fractional viscoelastic wave equation driven by a Dirac delta source. The starting point of the formulation is the classical one-dimensional viscoelastic wave equation posed with a Dirac delta driving term, which defines the fundamental solution of the corresponding wave operator. Fractional effects are incorporated directly at the level of this forced equation by generalizing the time and space operators: the second-order time derivative is replaced by the Caputo fractional derivative of order α∈(1,2], while the spatial Laplacian is replaced by the Riesz fractional operator of order β∈(1,2]. This construction leads to a fractional wave equation with memory in time and nonlocality in space, formulated explicitly as a Green’s function problem.

The Green’s function is obtained by applying the Laplace transform in time and the Fourier transform in space to the fractional wave equation, reducing the problem to an algebraic equation in the transform domain. The inverse transforms yield an explicit representation of the fundamental solution in terms of Fox H-function series, which naturally arise from the combined action of the fractional temporal and spatial operators. The resulting expression provides an explicit form of the Green’s function for the fractional viscoelastic wave equation with singular forcing.

This paper introduces a three-variable mathematical model that integrates susceptible biomass (\(S\)), infected biomass (\(I\)), and environmental inoculum (\(E\)), offering a novel framework that extends beyond classical SIR-type models by explicitly incorporating inoculum ecology and climate-driven dynamics. The model captures nonlinear infection saturation, logistic host growth, seasonal forcing, and a control intervention representing management practices such as fungicide application or crop rotation. Analytical exploration of the system reveals the existence of both disease-free and endemic equilibria, with stability determined by the basic reproduction number, \(R_0\). When \(R_0 < 1\), the disease-free equilibrium is stable, leading to the eradication of infection and recovery of host biomass, whereas \(R_0 > 1\) results in endemic persistence sustained by environmental inoculum reservoirs. Possible outcomes include stable disease-free states, endemic equilibria, seasonal oscillations driven by climatic variability, and shifts toward eradication under effective control strategies. Sensitivity analysis highlights that \(R_0\) is strongly influenced by inoculum survival rates, host growth capacity, and seasonal forcing intensity, underscoring the ecological complexity of plant disease dynamics. Numerical simulations substantiate these theoretical findings, demonstrating that strong seasonal forcing can destabilize equilibria and generate recurrent epidemic waves, while optimal control interventions can suppress infection prevalence by reducing \(R_0\) below unity. The results emphasize the critical role of environmental reservoirs in sustaining epidemics and provide a robust theoretical foundation for designing sustainable agroecological forecasting and management strategies under climate variability.

The heat transfer in channels that have varying cross sections is fundamentally not the same as that in channels of the same cross section. The heat transfer mechanism is enhanced in comparison with those channels with the channels of constant cross section is based on increased flow intensity. This paper is a numerical exploration of the laminar flow and thermal transport properties of a viscoplastic fluid. The Casson model is a model of viscosity that varies with temperature. The equations of governance include mixed convection driven by buoyancy, with the plastic viscosity being an exponential function of temperature. The proximity of the behavior to the yield surface requires the use of the Casson–Papanastasiou regularization, which guarantees a smooth transition in stress between the yielded zones and the unyielded zones. The finite volume numerical simulations method displays detailed flow fields, temperature fields, and yield surface dynamics depending on the values of Reynolds and Casson numbers. The major finding of this work is the major role contributed by the half-channel geometry to the improvement of flow development and heat transfer. The lopsided form encourages enhanced convectional circulation and speeds up the destruction of unyielded areas with symmetric enclosures, enhancing mixing and thermal diffusion. Findings indicate that the fluid mobility is inhibited by a rise in the Casson number by increasing the area of unyielded flow, whereas increased Reynolds numbers lower the resistance of the flow and pressure drop. The interaction between temperature-dependent viscosity and enclosure geometry provides more insight into the mechanisms of flow enhancement of non-Newtonian systems. These findings can be useful for optimizing industrial and biomedical practices, such as exposure of yield-stress fluids to confined geometries.

This article proposes a three-step strategy for improving fuzzy systems using simple models and high-fidelity data. In the first step, data are generated from a basic model (e.g., CART tree), simplified, and converted into fuzzy sets using a GMM that defines membership functions. The process is a generic, open process. Then, in the second step, a refinement with high-precision data leads to a neutrosophic extension: new points are fed into a high-resolution model; for each rule, the degrees of truth (T), falsehood (F), and indeterminacy (I) are calculated. Finally, the last step is sending feedback to the initial model: the refined rules are reinterpreted and the fuzzy set is regenerated and merged again with the knowledge acquired in step 2, obtaining an updated FIS that preserves the computational lightness of step 1. Overall, the strategy combines automatic rule generation, neutrosophic refinement, and iterative reuse to improve accuracy without losing efficiency. The originality of this paper is that it closes the loop. There are studies on the first two steps but here we have included refinement and feedback by means of the neutrosophic paradigm. This work also outlines a numerical procedure for analyzing neutrosophic refinement convergence through a so-called Indeterminacy-First Aggregation Operator (IFAO). As a test case, we applied the proposed scheme to a Physics-Guided Unification of Scattering Regimes for Fractal Aggregates. The example was considered due to its clear physical regimes, explicit mathematical forms, and hierarchical structure, which provide an ideal testbed where we can demonstrate how to i) extract rules (the distinct power laws and Guinier law from data); ii) assemble them into a unified model; and iii) refine that model with a full physical theory.

Understanding pedestrian dynamics within social groups is a key issue in improving crowd safety, particularly in emergency evacuation scenarios. Pedestrians often move in families or friendship groups, and social interactions within these groups strongly influence collective movement patterns, especially under conditions of heightened stress and anxiety. Accurately capturing these effects is therefore essential for realistic modeling and reliable evacuation assessment.
In this study, a kinetic modeling framework is proposed to describe the dynamics of pedestrian social groups in evacuation situations, with a particular emphasis on the role of anxiety. Anxiety is introduced into the model through modifications of pedestrians’ preferred velocity, allowing stress-induced behavioral changes such as increased urgency or loss of coordination to be represented at the microscopic level. The resulting kinetic equations govern the time evolution of the pedestrian distribution and reflect both individual motion and group interactions.
To solve the proposed model, a numerical scheme based on the Monte Carlo particle method is developed, enabling efficient simulation of large crowds with complex interactions. A series of numerical experiments is performed to investigate the influence of different anxiety levels on evacuation performance. The simulation results indicate that moderate levels of anxiety may enhance evacuation efficiency by increasing walking speed and coordination within groups. In contrast, excessive anxiety can trigger disorganized movement, local congestion, and inefficient use of space, ultimately leading to longer evacuation times. These findings highlight the nontrivial role of anxiety in crowd dynamics and provide insights for safer evacuation planning and design.

This study presents a numerical investigation of natural convection and heat transfer in a square cavity subjected to non-uniform wall heating. A spatially varying heat source is imposed on the left vertical wall, while the right vertical wall is maintained at a uniform cold temperature, and the horizontal walls are assumed to be adiabatic. The main objective is to analyze the effects of heater non-uniformity and position on flow structure, fluid mixing, and convective heat transfer characteristics within the enclosure.

The governing equations, expressed in terms of stream function, vorticity, and temperature, are solved using the Dual Reciprocity Boundary Element Method (DRBEM). In this framework, the fundamental solution of the Laplace equation is employed for the stream function equation, whereas the vorticity transport and energy equations are transformed into modified Helmholtz form and solved using the corresponding fundamental solution. This transformation is achieved by introducing a relaxation parameter into the Laplacian terms, while time derivatives are discretized using a forward difference scheme. The adopted approach eliminates the need for an additional time integration procedure and provides enhanced numerical stability, in addition to the computational efficiency offered by the boundary-only discretization.

Numerical simulations are carried out for Rayleigh numbers ranging from 10³ to 10⁶ and for various values of the non-uniformity parameter β in the interval [−2, 2]. The results demonstrate that both the flow intensity and the average Nusselt number increase with increasing Rayleigh number and β, indicating a significant enhancement in convective heat transfer. Furthermore, the heater distribution is shown to have a pronounced influence on the thermal and hydrodynamic fields. These findings highlight the critical role of non-uniform heating patterns in optimizing heat transfer performance in natural convection systems.