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Almost automorphically-forced flows on S^(1) or R in one-dimensional almost periodic semilinear heat equations

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Science China Mathematics 2022年第9期65卷 1875-1894页

作者： Wenxian Shen Yi Wang Dun Zhou Department of Mathematics and Statistics Auburn UniversityAuburnAL 36849USA School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026China School of Mathematics and Statistics Nanjing University of Science and TechnologyNanjing 210094China

In this paper,we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost periodically-forced scalar reaction-diffusion equation:u_(t)=u_(xx)+f(t,u,u_(x)),t>0,0sets of the skew-product *** is proved that any compact minimal invariant set(throughout this paper,we refer to it as a minimal set)can be residually embedded into an invariant set of some almost automorphically-forced flow on a circle S^(1)=R/LZ(see Definition 2.4 for“residually embedded”).Particularly,if f(t,u,p)=f(t,u,-p),then the flow on a minimal set can be embedded into an almost periodically-forced minimal flow on R(see Definition 2.4 for“embedded”).Moreover,it is proved that the ω-limit set of any bounded orbit contains at most two minimal sets that cannot be obtained from each other by phase *** addition,we further consider the asymptotic dynamics of the skew-product semiflow generated by(0.1)with the Neumann boundary condition u_(x)(t,0)=u_(x)(t,L)=0 or the Dirichlet boundary condition u(t,0)=u(t,L)=*** such a system,it has been known that theω-limit set of any bounded orbit contains at most two minimal *** applying the new results for(0.1)+(0.2),under certain direct assumptions on f,we prove in this paper that the flow on any minimal set of(0.1)with the Neumann boundary condition or the Dirichlet boundary condition can be embedded into an almost periodically-forced minimal flow on ***,a counterexample is given to show that even for quasi-periodically-forced equations,the results we obtain here cannot be further improved in general.

关键词： non-autonomous parabolic equation almost automorphically-forced circle flow function of the number of zeros minimal set ω-limit set

minimal subshifts which display Schweizer-Smítal chaos and have zero topological entropy

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Science China Mathematics 1998年第1期41卷 33-38页

作者：廖公夫范钦杰 Department of Mathematics Jilin University Changchun China Department of Mathematics Siping Teachers College Siping China

A class of minimal subshifts which display Schweizer-Smital chaos and have zero topotogical entropy is constructed, and it is proved that for a compact system, the positive topological entropy is not generally equival... 详细信息

关键词： compact system subshlft Schweizer-Smital chaos topological entropy minimal set

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具有伪轨跟踪性质的逐点回归动力系统

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Northeastern Mathematical Journal 1996年第1期12卷 73-78页

作者：麦结华

Let X be a metric space,and F:X×R→X α C0-flow on *** is proved in this peper that if every point in X is a positively or negatively recurrent(Posisson stable) point of F,and F has the pseudo-orbit tracing property,then every chain-connected component of X is a minimal set of *** follows from this that each pointwise recurrent C0-flow on a Chain-connected space having the pseudoorbit tracing property must be a minimal flow,i.e.,the whole space must be a (unique) minimal set of *** the discrete semi-flow generated by a continuous map f:X→*** results are also given.

关键词： Recurrent Point Pseudo-orbit Tracing Property minimal set Recurrent Orbit.Chain-connected Space Almost Periodic Point