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topological stability and Entropy for Certain Set-valued Maps

Acta Mathematica Sinica,English Series 2024年第4期40卷 962-984页

作者： Yu ZHANG Yu Jun ZHU School of Mathematical Sciences Xiamen UniversityXiamen 361005P.R.China

In this paper,the dynamics(including shadowing property,expansiveness,topological stability and entropy)of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view.It is shown that(1)if f is a hyperbolic endomorphism then for eachε>0 there exists a C^(1)-neighborhood U of f such that the induced set-valued map F_(f,U)has theε-shadowing property,and moreover,if f is an expanding endomorphism then there exists a C^(1)-neighborhood U of f such that the induced set-valued map F_(f,U)has the Lipschitz shadowing property;(2)when a set-valued map F is generated by finite expanding endomorphisms,it has the shadowing property,and moreover,if the collection of the generators has no coincidence point then F is expansive and hence is topologically stable;(3)if f is an expanding endomorphism then for eachε>0 there exists a C^(1)-neighborhood U of f such that h(F_(f,U,ε))=h(f);(4)when F is generated by finite expanding endomorphisms with no coincidence point,the entropy formula of F is given.Furthermore,the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.

关键词： Set-valued map orbit space hyperbolic endomorphism perturbation shadowing expansiveness topological stability entropy

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topological stability and Pseudo-orbit Tracing Property for Homeomorphisms on Uniform Spaces

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Acta Mathematica Sinica,English Series 2022年第2期38卷 431-442页

作者： Ke Song YAN Fan Ping ZENG School of Information and Statistics Guangxi University of Finance and EconomicsNanning 530003P.R.China

In this paper,we investigate the topological stability and pseudo-orbit tracing property for homeomorphisms on uniform spaces.We introduce the concept of topological stability for homeomorphisms on compact uniform spaces and prove that if a homeomorphism on a compact uniform space is expansive and has pseudo-orbit tracing property,then it is topologically stable.Moreover,we discuss the topological stability for homeomorphisms on uniform spaces from the view of localization.We introduce definitions of topologically stable point and shadowable point for homeomorphisms on uniform spaces and show that every shadowable point of an expansive homeomorphism on a compact uniform space is topologically stable.

关键词： topological stability pseudo-orbit tracing property expansiveness uniform space

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SYMMETRIC VORTICAL STRUCTURE AND ITS topological stability

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Journal of Hydrodynamics 2005年第5期17卷 552-557页

作者： WANG Gang LIANG Xin-gang GANG Ming-li Department of Engineering Mechanics Tsinghua University Beijing 100084 China Exploration and Development Research Institute PetroChina Daqing Oilfield Company Ltd. Daqing 163712 China

Using time dependent compressible laminar Navier-Stokes equations with a finite volume method incorporating a third-order-accurate discretization scheme, the flow structures around a slender at certain incidences are numerical simulated and typical crossflow patterns are presented. At incidence 10°, these vortical configurations are different at dissimilar axial locations though they are symmetric. At 35°, the symmetric vortical structures still maintain over the slender, yet their interaction at afterbody is intense than that at the forehody since the two vortices have fully developed downstream. The unstable topological structure of trajectory of saddle-to-saddle points and multiple limit cycle are further discussed in topological stability theory. These structures easily produce bifurcation with perturbation. The results support the view of hydrodynamic instability of vortices flow field.

关键词： symmetric vortices slender body flow pattern topological stability flow mechanism

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