Mathematical Methods in the Applied Sciences最新文献

This paper is devoted to a numerical resolution of spectral shape optimization problems governed by the Robin $��p�$-Laplacian operator under a volume constraint. We deal notably with the numerical computation of optimal shapes and corresponding higher eigenvalues of the problem governed by the $��p�$-Laplacian operator with Robin boundary conditions, in two and three dimensions. In this regard, it is worth mentioning that to our knowledge, no existing work has yet addressed this numerical investigation. So, we establish first the existence of the shape derivative for simple eigenvalues, providing both volume and boundary shape derivative formulas. Then we develop a numerical approach using gradient descent methods to approximate minimizers of higher Robin eigenvalues. This is based on a finite element discretization, combined with a Picard iteration scheme, designed to compute higher order Robin eigenvalues for various values of $��p�$. Finally, several numerical experiments in 2D and 3D are presented to demonstrate the effectiveness and robustness of the proposed approaches, and new conjectures are stated, based on these numerical simulations.

This research investigates the discrete-time dynamics of a predator–prey model governed by a ratio-dependent Ivlev functional response incorporating harvesting on the predator population. Through a detailed algebraic analysis, we show that the system experiences both period-doubling (PD) and Neimark–Sacker (NS) bifurcations within the positive quadrant of the phase space. Using the center manifold theorem and bifurcation theory, we provide a theoretical understanding of these bifurcations. To support our theoretical results, numerical simulations are performed, revealing chaotic behavior such as phase portraits, period-12 orbits, invariant closed curves, and attractor chaotic sets. Additionally, we compute the Lyapunov exponents to confirm the chaotic nature of the system. The results demonstrate that parameter values play a crucial role in shaping the model's dynamic behavior. Finally, we showcase the practical application of chaos control by employing state feedback and the OGY method to stabilize chaotic trajectories around an unstable equilibrium point. This study deepens our understanding of complex predator–prey dynamics and highlights the potential for managing chaos in ecological systems. Further, bifurcations are explored in a discrete predator–prey model within a coupled network, with numerical simulations showing that chaotic behavior emerges in such networks when the coupling strength reaches a critical threshold. Furthermore, we implemented the Euler–Maruyama method for stochastic simulations to analyze our system in the context of environmental uncertainties. We examined various cases to investigate different environmental scenarios. All theoretical results on stability, bifurcations, and chaotic transitions in the coupled network are validated through numerical simulations.

The $��q�$-Dunkl-classical symmetric orthogonal $��q�$-polynomials provide a unification and generalization of the $��{�q�}^{�2�}�$-analogue of the generalized Hermite polynomials and the $��{�q�}^{�2�}�$-analogue of the generalized Gegenbauer polynomials. In this paper, we characterize these polynomials by means of the so-called structure relation. The paper concludes with an open problem based upon the findings of this paper.

We first prove the sharp Hausdorff–Young inequality for the right-sided continuous quaternion Fourier transform (RCQFT), and the Hausdorff–Young inequalities for the discrete quaternion Fourier transform (DQFT) and for the quaternion Fourier series (QFS), respectively. Next, on the basis of the measure of uncertainty for the Rényi entropy in the quantum world, the uncertainty principle (UP) of the Rényi entropy is established for position and momentum. Moreover, similar uncertainty relations are also derived for the $��N�$-level systems and for the angle and angular momentum, respectively. Finally, all uncertainty relations are shown more attractive while they are expressed in terms of the symmetrized Rényi entropy.

In many applications of interest, the dynamics of systems described by universal differential operators are widely known. Is it possible to control the non-asymptotic or unstable dynamics of such-type systems? We consider nonlinear $��\psi �$-Caputo fractional order systems and design a linear state feedback control technique to address a novel concept, $��\psi �$-Mittag-Leffler asymptotic stabilization. By using a quadratic positive function and Lyapunov function inequality, we establish local and global conditions for $��\psi �$-Mittag-Leffler asymptotic stability and stabilization under Lipschitz perturbation and a norm bound nonlinearity. The interesting aspects of our proposed results have been applied to five non-trivial systems to show the intuitive understanding of systems. We discover that two systems give rise to memory chaos, whereas it is shown that under the action of an implemented linear state feedback controller, it can be possible to control the memory chaotic dynamics to some control objective.

This study presents an analysis of a novel fractional two prey–one predator model incorporating predator-taxis sensitivity. We conduct a comprehensive stability analysis, explore the model's chaotic nature through period-doubling bifurcations, and also show the existence of limit cycles through fractional Hopf bifurcation. It is observed that the fractional-order parameter brings in a stabilizing effect and, simultaneously, a shift of the Hopf bifurcation point. At the Hopf bifurcation point, the system moves from stable equilibria to sustained oscillations. In addition, regardless of initial conditions, the system approaches a stable limit cycle, showing the robustness of the method. We also demonstrate the effectiveness of the active control method to eliminate the periodicity of the fractional system and also unravel the decelerating influence of the fractional-order parameter on the convergence time to equilibrium. These results provide valuable insights into the stabilization of ecosystem dynamics and contribute more broadly to our understanding of population dynamics in ecological systems.

We present the novel Hamiltonian descriptions of some three-dimensional systems including two well-known systems describing the three-wave interaction problem and some well-known chaotic systems, namely, the Chen, Lü, and Qi systems. We show that all of these systems can be described in a Hamiltonian framework in which the Poisson matrix $��𝒥�$ is supplemented by a resistance matrix $��ℛ�$. While such resistive-Hamiltonian systems are manifestly nonconservative, we construct higher degree Poisson matrices via the Jordan product as $��𝒩�=�𝒥�ℛ�+�ℛ�𝒥�$, thereby leading to new bi-Hamiltonian systems. Finally, we discuss conformal Hamiltonian dynamics on Poisson manifolds and demonstrate that by appropriately choosing the underlying parameters, the reduced three-wave interaction model as well as the Chen and Lü systems can be described in this manner where the concomitant nonconservative part of the dynamics is described with the aid of the Euler vector field.

This paper is devoted to the study of a two-point impulsive boundary value problem for a system of nonlinear first-order ordinary differential equations involving an unknown parameter appearing in both the differential equation and the boundary condition. The main objective is to establish sufficient conditions for the existence of an isolated solution within some set. To achieve this, we apply the Dzhumabaev parametrization method. Based on this approach, we develop a constructive algorithm for finding the solution. The method effectively addresses the discontinuities introduced by the impulses and ensures solvability under clearly defined conditions. A numerical example is presented to illustrate the efficiency and practical applicability of the proposed algorithm. The results demonstrate the method's potential for solving a wide range of nonlinear impulsive boundary value problems with parameters.