Qualitative Theory of Dynamical Systems最新文献

In this paper we study the generalized Lagrangian system with a small perturbation. We assume the main term in the system to have a maximum, but do not suppose any condition for perturbation term. Then we prove the existence of a periodic solution via Ekeland’s principle. Moreover, we prove a convergence theorem for periodic solutions of perturbed systems.

The aim of this paper is to construct the general solution to a nonlocal linear differential equation of first-order, either homogeneous or inhomogeneous, together with its stability analysis. The success lies in decomposing functions into their even and odd parts, which presents an innovative approach to nonlocal equations. Our analysis also exhibits an unusual solution phenomenon occurring in nonlocal models.

This article primarily focuses on the single-valued and multi-valued cases of the class of nonlinear Caputo two-term fractional differential equation with three-point integral boundary conditions. In the single-valued case, we employ Schaefer’s fixed point theorem and the Banach fixed point theorem to establish results regarding the existence and uniqueness of solutions, using linear growth and Lipschitz conditions. Furthermore, we delve into the stability analysis of the single-valued problem using Ulam–Hyers and Ulam–Hyers–Rassias stabilities. In addition to the above, we address the multi-valued scenario and provide results on the existence of solutions. This is achieved by employing the Covitz–Nadler FPT and the nonlinear alternative for contractive maps. As an application of our fundamental findings, we present illustrative examples that validate our results. These examples have been implemented using MATLAB.

Traveling wave solutions are a class of invariant solutions which are critical for shallow water wave equations. In this paper, traveling wave solutions for two perturbed KP-MEW equations with a local delay convolution kernel are examined. The model equation is reduced to a planar near-Hamiltonian system via geometric singular perturbation theorem, and the qualitative properties of the corresponding unperturbed system are analyzed by using dynamical system approach. The persistence of the bounded traveling wave solutions for the perturbed KP-MEW equations with delay is investigated. By using a criterion for the monotonicity of ratio of two Abelian integrals and Melnikov’s method, the existence of kink (anti-kink) wave solutions and periodic wave solutions of the model equation are established. The result shows that the delayed KP-MEW equations with positive perturbation and the one with negative perturbation exhibit completely diverse dynamical properties. These new findings greatly enrich the understanding of dynamical properties of the traveling wave solutions of perturbed nonlinear wave equations with local delay convolution kernel. Numerical experiments further confirm and illustrate the theoretical results.

In this work, the Riemann–Hilbert (RH) problem is employed to study the Lakshmanan–Porsezian–Daniel (LPD) equation with arbitrary-order pole points under finite density initial data condition. By performing spectral analysis on Lax pairs, a suitable matrix RH problem is established. Through the residue theorem, the explicit expression of simple pole solutions is obtained by Binet–Cauchy theorem. In addition, utilizing the Wronskian form of scattering data (s_{11}(mu )) which degenerates to zero at high-order zero points and the Taylor expansion of oscillation index (e^{2itheta }), the expression of the high-order pole solutions is constructed. Moreover, the detailed analysis is conducted on the dynamic behaviors of special soliton solutions, and some interesting soliton phenomena are presented by taking the influence of various parameters into consideration.

This paper studies topological entropy and pseudo-entropy of iterated set-valued function systems. Firstly, the notions of topological entropy defined by separating and spanning sets and by open covers are introduced respectively, and they are proved equivalent, then a formula is obtained for the topological entropy of an iterated set-valued function system concerning the corresponding skew product system, and topological entropy of iterated set-valued function systems is a topological conjugacy invariant. Finally, the notions of pseudo-entropy of set-valued function systems and iterated set-valued function systems are introduced and it is proved that the pseudo-entropy is equal to the topological entropy of iterated set-valued function systems.

This paper investigates the existence, uniqueness, and continuous dependence of solutions to fractional order nonlocal thermistor two-point boundary value problems on time scales. We employ the Schauder fixed point theorem to establish the existence of solutions, and the contraction principle to prove uniqueness. We also obtain a result on the continuous dependence of solutions. Finally, we present several examples to illustrate our findings. This work is the first to study a fractional model of thermistor on Department of Medical Research,time scales, and it makes a significant contribution to the field of modeling on time scales. The results of this paper can be used to develop new and improved mathematical models for thermistors, which can be used to design more efficient and reliable thermistor-based devices.

In this article, a classical predator–prey system with linear cross-diffusion and Holling-II type functional response and subject to homogeneous Neuamnn boundary condition is considered. The spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of the constant coexistence equilibrium are established with the help of the linearized analysis. When the bifurcation parameters are restricted to the Turing instability region and near the Turing bifurcation curve, the associated amplitude equations of the original system near the constant coexistence equilibrium are obtained by means of multiple-scale time perturbation analysis. According to the obtained amplitude equations, the stability and classification of spatiotemporal patterns of the original system near the constant coexistence equilibrium are determined. It is shown that the cross-diffusion in the classical predator–prey system plays an important role in formation of spatiotemporal patterns. Also, the theoretical results are verified numerically.

Combining the spectral radius of matrix with the generalized Banach fixed point theory and some properties of exponential contraction, we prove periodic solution and almost periodic solution of a neutral delay multispecies logarithmic population model with piecewise constant argument is existent and unique in appropriate conditions. The results have generalized and improved some results of literature on logarithmic population model. Finally, one example is given to illustrate our results.

A hierarchy of nonlocal nonlinear Schrödinger equations is derived by using the Lenard gradients and the zero-curvature equation. According to the Lax matrix of the nonlocal nonlinear Schrödinger equations, we introduce a hyperelliptic Riemann surface ({mathcal {K}}_{n}) of genus n, from which Dubrovin-type equations, meromorphic function, and Baker–Akhiezer function are established. By the theory of algebraic curves, the corresponding flows are straightened by resorting to the Abel–Jacobi coordinates. Finally, we obtain the explicit Riemann theta function representations of the Baker–Akhiezer function, specifically, that of solutions for the hierarchy of nonlocal nonlinear Schrödinger equations in regard to the asymptotic properties of the Baker–Akhiezer function.