分段常数变元的中立型方程的振动性

给出了一类含连续和分段常数变元的中立型方程振动的两个充分条件,改进和推广了已知文献中的一些结果.     

Stability and bifurcations analysis of a competition model with piecewise constant arguments

In this paper, we investigate local and global asymptotic stability of a positive equilibrium point of system of differential equations where t ≥ 0, the parameters r₁, k₁, α₁, α₂, r₂, k₂, and d₁ are positive, and [t] denotes the integer part of t ∈ [0, ∞ ). x(t) and y(t) represent population density for related species. Sufficient conditions are obtained for the local and global stability of the positive equilibrium point of the corresponding difference system. We show through numerical simulations that periodic solutions arise through Neimark–Sacker bifurcation. Copyright © 2014 John Wiley & Sons, Ltd.     

广义M—J集的界与J—集Hausdorff维数的估计

解析出给出了广义M－J集的界,其中某界在某种意义上是最佳的。解决了应用逃兔时间等算法计算机构造其混沌分形图的首要问题,并在此基础上通过线性逼近的方法给出了某些情况下J－集Hausdorff维数的近似估计。     

On the Classification of Laminations Associated to Quadratic Polynomials

Given any rational map f, there is a lamination by Riemann surfaces associated to f. Such laminations were constructed, in general, by Lyubich and Minsky. In this article, we classify laminations associated to quadratic polynomials with periodic critical point. In particular, we prove that the topology of such laminations determines the combinatorics of the parameter. We also describe the topology of laminations associated to other types of quadratic polynomials.      

A class of boundary value problems for first-order impulsive integro-differential equations with deviating arguments

This paper is concerned with a class of boundary value problems for nonlinear mixed impulsive integro-differential equations with deviating arguments. We establish a new comparison principle and use the method of upper and lower solutions together with the monotone iterative technique. Under suitable conditions, we obtain the existence results of extremal solutions for the problems. An example is also given to illustrate our results.     

Connectivity of the Mandelbrot set for the family of renormalization transformations

We show that the Mandelbrot set for the family of renormalization transformations of 2-dimensional diamond-like hierachical Potts models in statistical mechanics is connected. We also give an upper bound for the Hausdorff dimension of Julia set when it is a quasi-circle.     

On a topological closeness of perturbed Mandelbrot sets

In the present work we propose a numerical and visual tool for the study of the deformation of the Mandelbrot sets of perturbed Mandelbrot maps by noise in comparison with the original Mandelbrot set. Further, by employing these numerical tools, we support the invariance of the Mandelbrot set of a noise-perturbed Mandelbrot map under different noise realizations. Finally, we provide evidence for the non-fractal structure of the Mandelbrot set of a noise-perturbed Mandelbrot map.     

Stability analysis of parabolic partial differential equations with piecewise continuous arguments

This article deals with the analytic and numerical stability of numerical methods for a parabolic partial differential equation with piecewise continuous arguments of alternately retarded and advanced type. First, application of the theory of separation of variables in matrix form and the Fourier method, the necessary and sufficient condition under which the analytic solution is asymptotically stable is derived. Then, the θ‐methods are applied to solve the corresponding initial value problem, the sufficient conditions for the asymptotic stability of numerical methods are obtained. Finally, several numerical examples are presented to support the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 531–545, 2017     

Almost periodic solutions of second-order neutral equations with piecewise constant arguments

In this paper, we present a theorem on the almost periodic solutions of second-order neutral equations with piecewise constant arguments of the form

(x(t)+px(t−1))^″=qx([t])+f(t),${(x\left(t\right)+px(t-1))}^{″}=qx\left(\left[t\right]\right)+f\left(t\right)\text{,}$

Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions

We use a fixed point theorem due to Avery and Peterson to establish the existence of at least three non-negative solutions of some nonlocal boundary value problems to third order differential equations with advanced arguments. An example is given to illustrate the main results.     

Global attractivity of periodic solution for neutral functional differential system with multiple deviating arguments

In this paper, a kind of neutral functional differential system with multiple deviating arguments is considered. By means of Mawhin's coincidence degree theory and Lyapunov method, a sufficient condition is obtained for guaranteeing the existence and global attractivity of periodic solution for the system. It is interesting that the result is related to the multiple deviating arguments τ_i(i = 1, 2, …, m). An example is given to show the feasibility of the result in the last section. Copyright © 2011 John Wiley & Sons, Ltd.     

First order differential equations with piecewise constant arguments and nonlinear boundary value conditions

This paper is devoted to the study of differential equations with piecewise constant arguments coupled with nonlinear boundary value conditions. Under suitable assumptions on the data of the equation, by means of the method of (weakly coupled) lower and upper solutions, we derive the existence of extremal solutions and extremal quasi-solutions. Moreover some results are given concerning the uniqueness of solutions. Furthermore, we deduce some maximum principles related to the linear equation which allow us to develop the monotone iterative method. Some illustrative examples are also presented.     

A general view to relaxation methods in control theory1

A general method of construction of relaxed optimal control problems is proposed. Two standard relaxed problem in optimal control of ordinary differential equation and relation between them are thus obtained as special cases, and besides these relaxed problems are justified in accord with their extremal properties.     

Analytical and numerical stability of partial differential equations with piecewise constant arguments

This article is concerned with the stability analysis of the analytic and numerical solutions of a partial differential equation with piecewise constant arguments of mixed type. First, by means of the similar technique in Wiener and Debnath [Int J Math Math Sci 15 (1992), 781–788], the sufficient conditions under which the analytic solutions asymptotically stable are obtained. Then, the θ‐methods are used to solve the above‐mentioned equation, the sufficient conditions for the asymptotic stability of numerical methods are derived. Finally, some numerical experiments are given to demonstrate the conclusions.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1‐16, 2014     

具连续偏差变元的中立型向量抛物偏微分方程的H-振动性

讨论一类具连续偏差变元的中立型向量抛物偏泛函微分方程的H-振动性,利用内积降维的方法和Green公式,得到了该类方程在Robin边值条件下所有解Hm-振动的若干充分判据,这里H是Rm中的单位向量.     

Multiple fixed-sign solutions for a system of generalized right focal problems with deviating arguments

We consider the following system of generalized right focal boundary value problems

     

Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions

In this paper we investigate integral boundary value problems for fourth order differential equations with deviating arguments. We discuss our problem both for advanced or delayed arguments. We establish sufficient conditions under which such problems have positive solutions. To obtain the existence of multiple (at least three) positive solutions, we use a fixed point theorem due to Avery and Peterson. An example is also included to illustrate that corresponding assumptions are satisfied. The results are new.     

On a topological closeness of perturbed Julia sets

In the present work we expand our previous work in [1] by introducing the Julia Deviation Distance and the Julia Deviation Plot in order to study the stability of the Julia sets of noise-perturbed Mandelbrot maps. We observe a power-law behaviour of the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelbrot maps from the Julia set of the Mandelbrot map as a function of the noise level. Additionally, using the above tools, we support the invariance of the Julia set of a noise-perturbed Mandelbrot map under different noise realizations.     

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy.