Morse Decomposition of Semiflows on Topological Spaces

This paper studies Morse decompositions of discrete and continuous-time semiflows on compact Hausdorff topological spaces. We extend two classical results which are well-known facts for flows on compact metric spaces: the characterization of the Morse decompositions through increasing sequences of attractors and the existence of Lyapunov functions.     

Principal Lyapunov Exponents and Principal Floquet Spaces of Positive Random Dynamical Systems. III. Parabolic Equations and Delay Systems

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors’ paper Mierczyński and Shen (Trans Am Math Soc 365(10):5329–5365, 2013), to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents.     

Random attractors for asymptotically upper semicompact multivalue random semiflows

The present paper studied the dynamics of some multivalued random semi- flow.The corresponding concept of random attractor for this case was introduced to study asymptotic behavior.The existence of random attractor of multivalued random semiflow was proved under the assumption of pullback asymptotically upper semicompact,and this random attractor is random compact and invariant.Furthermore,if the system has ergodicity,then this random attractor is the limit set of a deterministic bounded set.     

Minimal Spaces with Cyclic Group of Homeomorphisms

There are two main subjects in this paper. (1) For a topological dynamical system \((X,T)\) we study the topological entropy of its “functional envelopes” (the action of \(T\) by left composition on the space of all continuous self-maps or on the space of all self-homeomorphisms of \(X\)). In particular we prove that for zero-dimensional spaces \(X\) both entropies are infinite except when \(T\) is equicontinuous (then both equal zero). (2) We call Slovak space any compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Using Slovak spaces we provide examples of (minimal) systems \((X,T)\) with positive entropy, yet, whose functional envelope on homeomorphisms has entropy zero (answering a question posed by Kolyada and Semikina). Finally, also using Slovak spaces, we resolve a long standing open problem whether the circle is a unique non-degenerate continuum admitting minimal continuous transformations but only invertible: No, some Slovak spaces are such, as well.     

FIXED POINT THEOREMS FOR MAPPINGS SATISFYING IMPLICIT RELATION ON TWO COMPLETE AND COMPACT METRIC SPACES

First, the implicit relations were given. A common fixed point theorem was proved for two mappings satisfying implicit relation functions. A further fixed point theorem was proved for mappings satisfying implicit relation functions on two compact metric spaces.     

Affine Transformation in Random Iterated Function Systems

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The embedded theorem for family of quasi metric spaces in Menger space and its application

In this paper the notion of embedding for family of quasi metric spaces in Menger spaces is introduced and its properties are investigated. A common fixed point theorem for sequence of continuous mappings in Menger spaces is proved. These mappings are assumed to satisfy some generalizations of the contraction condition. The proving technique herein seems to be new even for mappings in Menger spaces.     

Dissipation and Compact Attractors

We present an approach to the study of the qualitative theory of infinite dimensional dynamical systems. In finite dimensions, most of the success has been with the discussion of dynamics on sets which are invariant and compact. In the infinite dimensional case, the appropriate setting is to consider the dynamics on the maximal compact invariant set. In dissipative systems, this corresponds to the compact global attractor. Most of the time is devoted to necessary and sufficient conditons for the existence of the compact global attractor. Several important applications are given as well as important results on the qualitative properties of the flow on the attractor.     

Global Solutions of Systems of Nonlinear Impulsive Volterra Integral Equations in Banach Spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｔｈｅｏｒｙｏｆｉｍｐｕｌｓｉｖｅｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｈａｓｂｅｅｎｅｍｅｒｇｉｎｇａｓａｎｉｍｐｏｒｔａｎｔａｒｅａｏｆｉｎｖｅｓｔｉｇａｔｉｏｎｓｉｎｒｅｃｅｎｔｙｅａｒｓ (ｓｅｅ [1 ] ) .Ｉｎｐａｐｅｒ [2 ] ,ＧｕｏＤａ＿ｊｕｎｅｓｔａｂｌｉｓｈｅｄｓｏｍｅｅｘｉｓｔｅｎｃｅｔｈｅｏｒｅｍｓｏｆｅｘｔｒｅｍａｌｓｏｌｕｔｉｏｎｓｆｏｒｎｏｎｌｉｎｅａｒｉｍｐｕｌｓｉｖｅＶｏｌｔｅｒｒａｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎｓｏｎａｆｉｎｉｔｅｉｎ…     

Analyticity of Homogenized Coefficients Under Bernoulli Perturbations and the Clausius–Mossotti Formulas

This paper is concerned with the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a Bernoulli perturbation. Introducing a new family of energy estimates that combine probability and physical spaces, we prove the analyticity of the perturbed homogenized coefficients with respect to the Bernoulli parameter. Our approach holds under the minimal assumptions of stationarity and ergodicity, both in the scalar and vector cases, and gives analytical formulas for each derivative that essentially coincide with the so-called cluster expansion used by physicists. In particular, the first term yields the celebrated (electric and elastic) Clausius–Mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium. This work constitutes the first general proof of these formulas in the case of random inclusions.     

D/A Converters and Iterated Function Systems

This paper discusses A/D and D/A converters as applications of iterated function systems. We describe operation of basic converters in terms of iterate function systems and introduce typical examples for binary and Gray encodings/decodings. We then comment that converters in the field of analog signal processing have different character from iterated function systems in the field of nonlinear dynamical systems. Simple implementation circuits for the binary and Gray D/A converters are also shown.     

Anholonomic configuration spaces and metric tensors in finite elastoplasticity

Deformation mappings are considered that correspond to the motions of lattice defects, elastic stretch and rotation of the lattice, and initial defect distributions. Intermediate (i.e., relaxed) configuration spaces associated with these deformation maps are identified and then classified from the differential-geometric point of view. A fundamental issue is the proper selection of coordinate systems and metric tensors in these configurations when such configurations are classified as anholonomic. The particular choice of a global, external Cartesian coordinate system and corresponding covariant identity tensor as a metric on an intermediate configuration space is shown to be a constitutive assumption often made regardless of the existence of geometrically necessary crystal defects associated with the anholonomicity (i.e., the non-Euclidean nature) of the space. Since the metric tensor on the anholonomic configuration emerges necessarily in the definitions of scalar products, certain transpose maps, tensorial symmetry operations, and Jacobian invariants, its selection should not be trivialized. Several alternative (i.e., non-Euclidean) representations proposed in the literature for the metric tensor on anholonomic spaces are critically examined.     

Solutions for a system of nonlinear random integral and differential equations under weak topology

I.IntroductionTheexistenceandcomparisonresultsofsolutionsfornonlinearVolterraintegralequationsinstrongtopologyofBanachspaceshavebeenobtainedbyVaughnI"'"],LakshmikanthamIl3]andLakshmikantham-Leela114l.TheexistenceresultsofweaksolutionsfortheCauchyproblemof…     

Invariant Measures for Dissipative Systems and Generalised Banach Limits

Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier–Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521–540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on metric spaces that have compact attracting sets, taking limits evaluated along individual trajectories. We also show that if the space is a reflexive separable Banach space, or if the dynamical system has a compact absorbing set, then rather than taking limits evaluated along individual trajectories, we can take an ensemble of initial conditions: the generalised Banach limit can be used to construct an invariant measure based on an arbitrary initial probability measure, and any invariant measure can be obtained in this way. We thus propose an alternative to the classical Krylov–Bogoliubov construction, which we show is also applicable in this situation.     

Multi-type Entire Solutions in a Nonlocal Dispersal Epidemic Model

This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space.     

基于Copula理论的模糊可靠度隶属函数求解的自适应截断抽样法

针对工程中同时存在随机和模糊基本变量且随机概率信息不全的可靠性问题,利用Copula理论逼近随机基本变量的联合分布函数和联合概率密度函数,进而建立了Copula逼近基础上的模糊可靠度隶属函数求解的自适应截断抽样法。所建模型在Copula逼近基础上采用优化建模和迭代策略,交替考虑模糊不确定性和随机不确定性对功能函数的影响...     

Existence and stability of common fixed points for systems of mappings

In this paper,we establish some common fixed point theorems and stability theorems of the sets of common fixed points for the systems of set-valued and single-valued nonlinear contractive type mappings in a finite Cartesian product of metric spaces.     

Wiener-Hopf analysis of an acoustic plane wave in a trifurcated waveguide

In this paper, we have made Wiener-Hopf analysis of an acoustic plane wave by a semi-infinite hard duct that is placed symmetrically inside an infinite soft/hard duct. The method of solution is integral transform and Wiener-Hopf technique. The imposition of boundary conditions result in a 2 × 2 matrix Wiener-Hopf equation associated with a new canonical scattering problem which is solved by using the pole removal technique. In the solution, two infinite sets of unknown coefficients are involved that satisfy two infinite systems of linear algebraic equations. These systems of linear algebraic equations are solved numerically. The graphs are plotted for sundry parameters of interest. Kernel functions are also factorized.     

哈密顿系统正则变换在时变最优控制中的应用

利用哈密顿系统正则变换和生成函数理论求解线性时变最优控制问题,构造了新的最优控制律形式并提出了控制增益计算的保结构算法. 利用生成函数求解最优控制导出的哈密顿系统两端边值问题,并构造线性时变系统的最优控制律,由第2类生成函数所构造的最优控制律避免了末端时刻出现无穷大反馈增益. 控制系统设计中需求解生成函数满足的时变矩阵微分方程组. 根据生成函数与哈密顿系统状态转移矩阵之间的关系,从正则变换的辛矩阵描述出发,导出了求解这组微分方程组的保结构递推算法.为了保持递推计算中的辛矩阵结构,哈密顿系统状态转移矩阵的计算中利用了Magnus级数.      

A Uniform Exponential Spectrum for Linear Flows on Vector Bundles

For linear flows on vector bundles we define a uniform exponential spectrum. For a compact invariant set for the projected flow we obtain this spectrum by taking all accumulation points for the time tending to infinity of the union over the finite time exponential growth rates for all initial values in this set. Using direct arguments we show that for a connected compact invariant set this spectrum is a closed interval whose boundary points are Lyapunov exponents. For a compact invariant set on which the flow is chain transitive we show that this spectrum coincides with the Morse spectrum. In particular, this approach admits a straightforward analytic proof for the regularity and continuity properties of the Morse spectrum without using cohomology or ergodicity results.