Mathematical Methods in the Applied Sciences最新文献

This article investigates the chaotic analysis and attractive domain of a fractional-order $�{\Phi }^6$-van der Pol with time delay velocity under harmonic excitation. Firstly, eight different types of bifurcation states of the system under different parameters are calculated by using the undisturbed system. Secondly, the Melnikov method is used to explore the effect of time delay velocity on the threshold of chaos in the Smale horseshoe sense under the double-well potential and three-well potential of the system. Finally, through numerical analysis of the phase diagram, bifurcation diagram, and maximum Lyapunov exponent, the influence of time delay velocity on system chaos is studied. The results indicate that an increase in the delay velocity coefficient will lead to the system transitioning from a chaotic state to a periodic state, while an increase in the delay velocity term will lead to the system transitioning from a periodic state to a chaotic state. In the study of system bifurcation, it is found that the position of the equilibrium points of the system changes during periodic motion. Therefore, cell mapping is used to draw the attractive domain of the system is studying the influence of initial conditions on the equilibrium point of the system and the results show that there is a close relationship between the attraction domain and the process of chaos occurrence.

In this paper, we study the boundedness, persistence, and global stability of the equilibrium point and rate of convergence of positive solutions of the exponential form of a system of difference equations

In this study, we employ new results on the semi-inner product, the adjoint operator, and the Schatten class of operators on a separable Banach space, which allow us to investigate a differential operator with unbounded operator-valued coefficients. Specifically, we derive an asymptotic formula for the second regularized trace based on the asymptotic and spectral properties of the extended operator. For this purpose, we use the theory of continuous dense embeddings and known results regarding the regularized trace in Hilbert space. We conclude our paper by providing examples that support our findings.

The Boltzmann equation is a typical example of partially dissipative equations, where the linearized collision operator is positive definite with respect to the microscopic part and the dissipation of the hydrodynamic part is discovered from the coupling structure between the transport operator and the linearized collision operator. Guo and Wang (Comm. Partial Differential Equations, 37, 2012) developed a general energy method for proving the optimal time decay rates of the solution to such type of equations in the whole space; however, the decay rate of the highest order spatial derivatives of the solution is not optimal. In this paper, by incorporating the high-low frequency decomposition in the energy estimates, both linearly and nonlinearly, we prove the optimal decay rates of any high order spatial derivatives of the low frequency part of the solution to the Boltzmann equation and the almost exponential decay rate of the high frequency part, which imply in particular the optimal decay rate of the highest order spatial derivatives of the solution. Moreover, the velocity-weighted assumption of the initial data required in Guo and Wang (Comm. Partial Differential Equations, 37, 2012) is removed by capturing the time-weighted dissipation estimates via the time-weighted energy method. The method can be applied to the compressible Navier–Stokes equations and many partially dissipative equations in kinetic theory and fluid dynamics.

By constructing various Pohozaev identities, we study the uniqueness of minimizers for $��{�L�}^{�2�}�$-subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contains $��x�=�0�$ as a singular point.

This paper is concerned with the asymptotic behavior of the solution to the Cauchy problem for the three-dimensional bipolar Euler–Poisson system with time-dependent over-damping. We prove that the Cauchy problem admits a unique global-in-time smooth solution that tends to the planar diffusion wave as time goes to infinity provided that the initial perturbation is sufficiently small. Moreover, we also obtain the convergence rate of the solution toward the planar diffusion wave. The proof is completed by the time-weighted energy method.

In this paper, the imperfect and Bogdanov–Takens bifurcations are investigated in a Leslie–Gower predator–prey model with constant stocking and harvesting. It is shown that the introduction of the stocking of predators can prompt the extinction of prey; with the harvesting of predators, the model will experience the imperfect bifurcation and the Bogdanov–Takens bifurcation; meanwhile, no such bifurcations occur in the model with the stocking. The results imply that human intervention may lead to complex dynamical phenomena among species. Numerical simulations are presented to illustrate the theoretical results.

In this paper, we introduce some explicit and recursive formulas for the $��n�$th $��q�$-derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of $��q�$-differential operator. One of the formulas involves the determinant of the functions and their $��q�$-derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting $��q�$-identities.

In this article, the inverse problems for the wave equation in a medium in which multiple types of cavities and inclusion exist in a mixture are considered. From the point of view of the indicator function of the enclosure method, there are two types of heterogeneous parts: “minus group” and “plus group.” For example, cavities with the Dirichlet boundary condition belong to the minus group, while inclusions with smaller propagation velocity belong to the plus group. The heterogeneous part of the minus group gives a negative sign to the indicator function, and the heterogeneous part of the plus group gives a positive sign. In general, the presence of many types of heterogeneous parts causes cancelation of the sign of the indicator function. Such cases are referred to as “mixed cases.” Here, we consider the case that the shortest length obtained from the indicator function is attained only by heterogeneous parts of the same group. This case is called the “mixed but separated case,” and it is shown that the method of elliptic estimates developed by Ikehata works well. We also show that the case of a two-layered background medium with a flat layer can be considered in the same way as the case of a homogeneous background medium.

In this paper, we investigate a chemotaxis system under homogeneous Neumann boundary conditions within a bounded domain with a smooth boundary. The system describes the movement of cells in response to two chemical signal substances: one acts as a chemoattractant, while the other serves as a chemorepellent, both produced by the cells. The system takes into account chemotactic sensitivity in the reaction movement when detecting these chemicals. Under certain assumptions, we demonstrate the existence of a unique global bounded classical solution for the proposed problem. To further understand the time evolution of the system's solutions, we conduct numerical experiments and analyze the dynamic properties of the $��{�L�}^{�\infty �}�\left(�\Omega �\right)�$ norm of the solutions with respect to variations in chemical production rates.