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Communications in Theoretical Physics 2008年第5期49卷 1119-1124页

作者： QU Gai-Zhu ZHANG Shun-Li ZHU Chun-Rong Center for Nonlinear Studies Department of Mathematics Northwest University Xi＇an 710069 China Center of Nonlinear Science Ningbo University Ningbo 315211 China Department of Mathematics Anhui Normal University Wuhu 241000 China

The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = ... 详细信息

关键词： wave equation invariant set exact solution

invariant sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

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Communications in Theoretical Physics 2008年第10期50卷 821-826页

作者： JIA Hua-Bing XU Wei Department of Applied Mathematics Northwestern Polytechnical University Xi＇an 710072 China Department of Matheinatics Baoji University of Arts and Sciences Baoji 721013 China

In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.

关键词： invariant set exact solution nonlinear diffusion equations rotation group scaling group

invariant sets and solutions to the generalized thin film equation

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Science China Mathematics 2007年第6期50卷 875-886页

作者： Chang-zheng QU & Chun-rong ZHU Center for Nonlinear Studies, Northwest University, Xi’an 710069, China Department of Mathematics, Northwest University, Xi’an 710069, China Department of Mathematics, Anhui Normal University, Wuhu 241000, China Center for Nonlinear Studies Northwest University Xi’an China Department of Mathematics Northwest University Xi’an China Department of Mathematics Anhui Normal University Wuhu China

The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set $$E_0 = \{ u:u_x = v_x F(u),u_y = v_y F(u)\} ,$$ where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the 1+1-dimensional nonlinear evolution equations.

关键词： thin film equation invariant set invariant solution rotation group scaling group 35K55 35K40 82C26

Exact Solutions and invariant sets to General Reaction-Diffusion Equation

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Communications in Theoretical Physics 2008年第6期49卷 1389-1392页

作者： JIA Hua-Bing XU Wei Department of Applied Mathematics Northwestern Polytechnical University Xi＇an 710072 China

In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations ... 详细信息

关键词： invariant set exact solution general reaction-diffusion equation rotation group scaling group

Regularity of invariant sets in Variable Internal Damped Wave Equations

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Acta Mathematicae Applicatae Sinica 2020年第4期36卷 952-974页

作者： Gao-cheng YUE Yu-xin LIANG Jia-jia YANG Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China

In this paper we prove that every compact invariant subset■associated with the semigroup{Sn,k(t)}t≥0 generated by wave equations with variable damping,either in the interior or on the boundary of the domainΩwhereΩ■R^3 is a smooth bounded domain,in H^10(Ω)×L^2(Ω)is in fact bounded in D(B0)×H^10(Ω)As an application of our results,we obtain the upper-semicontinuity for global attractor of the weakly damped semilinear wave equation in the norm of H^1(Ω)×L^2(Ω)when the interior variable damping converges to theboundary damping in the sense of distributions.

关键词： Damped wave equation boundary damping invariant set regularity

CONSTRUCTIONS OF THE invariant setS AND invariant MEASURES

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Applied Mathematics(A Journal of Chinese Universities) 1997年第3期12卷 363-370页

作者： FENGJINGHAI Departmentofappliedmathematics zhejianguniversityhangzhou310027

In this paper, we will discuss the constructiOn problems about the invariant sets and invariant measures of continues maps~ which map complexes into themselves, using simplical approximation and Markov cbeirs. In [7], the author defined a matrix by using r-normal subdivi of the w,dimensional unit cube, considered it a Markov matrix, and constructed the invariantset and invafiant measure, In fact, the matrix he defined is not Markov matrix generally. So wewill give [7] and amendment in the last pert of this paper. We also construct an invariant set thatis the chain-recurrent set of the map by means of a non-negative matrix which only depends on themap. At hst, we will prove the higher dimension?Banach variation formuls that can simplify thetransition matrix.

关键词： Markov chain invariant set invariant measure recurrent chain Banach formula

The Box-counting Dimension of invariant sets for Iterated Function Systems

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Communications in Mathematical Research 1999年第1期 59-64页

作者： Geng Xiangyi (Department of Computer Sciences, Dalian Railway College, Dalian, 116028)

Ｕｎｄｅｒｔｈｅｏｐｅｎｓｅｔｃｏｎｄｉｔｉｏｎ，ａｎｉｎｅｑｕａｌｉｔｙｏｎｔｈｅｂｏｘ－ｃｏｕｎｔｉｎｇｄｉｍｅｎｓｉｏｎｏｆｉｎｖａｒｉ－ａｎｔｓｅｔｓｆｏｒｃｏｎｔｒａｃｔｉｏｎｉｔｅｒａｔｅｄｆｕｎｃｔｉｏ... 详细信息

Ｕｎｄｅｒｔｈｅｏｐｅｎｓｅｔｃｏｎｄｉｔｉｏｎ，ａｎｉｎｅｑｕａｌｉｔｙｏｎｔｈｅｂｏｘ－ｃｏｕｎｔｉｎｇｄｉｍｅｎｓｉｏｎｏｆｉｎｖａｒｉ－ａｎｔｓｅｔｓｆｏｒｃｏｎｔｒａｃｔｉｏｎｉｔｅｒａｔｅｄｆｕｎｃｔｉｏｎｓｙｓｔｅｍｓｗｅｒｅｇｉｖｅｎ...

关键词： invariant set Box counting Dimension Iterated Function System

Reconfigurability evaluation method for input-constrained control systems

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Journal of Systems Engineering and Electronics 2021年第5期32卷 1023-1030页

作者： TU Yuanyuan WANG Dayi LI Wenbo Beijing Institute of Control Engineering China Academy of Space TechnologyBeijing 100190China Beijing Institute of Spacecraft System Engineering China Academy of Space TechnologyBeijing 100190China

This paper proposes a quantitative reconfigurability evaluation method for control systems with actuator saturation and additive faults from the perspective of system *** the saturated feedback law in the convex hull of a group of auxiliary linear controls,the sufficient reconfigurability conditions for the system under additive faults are derived using invariant *** conditions are then expressed as linear matrix inequalities(LMIs)and applied to quantify the degree of reconfigurability for the fault *** largest fault magnitude for which the system can be stabilized,the largest initial state domain from which all the trajectories are convergent,and the minimum final state domain to which the trajectories will converge are *** effectiveness of the proposed method is illustrated through an application example.

关键词： control reconfigurability additive fault actuator saturation invariant set fault-tolerant control

A Note on invariant sets

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Northeastern Mathematical Journal 2000年第4期16卷 451-453页

作者：瞿成勤苏维宜许勇 Zhongshan University Nanjing University South China University of Technology

The lower bound of Hausdorff dimension of the invariant set is given.

关键词： invariant set Hausdorff dimension iterated function