全局性N元F-强混沌系统的一个判据

符和满

摘要: 设(X,f)是一个动力系统, 其中X是一个含至少2个点的完备度量空间,f是X上的一个连续自映射. 对给定的 Furstenberg 族F与整数

$N\geq2$ ,将F-混沌推广到N元F-混沌. 为此, 对于X的2个非空子集A,B, 借助集对(A,B)的F-往复点来引入F-攀援串的概念, 进而定义N元 F-混沌以及讨论N元F-混沌的一些性质. 最后以 Furstenberg 族理论为主要工具, 给出一个动力系统是全局性N元F-强混沌的一个判据, 并通过例子来阐述它在动力系统中的应用.

关键词:

$N$ 元 $\mathcal{F}$ -攀援集

Abstract: Chaoticity is a vital property of a dynamical system. For a given Furstenberg family

$\mathcal{F}$ , the notion of

$\mathcal{F}$ -chaos has been introduced in a dynamical system recently. In order to extend it to

$\mathcal{F}$ -

$N$ -chaos,

$\mathcal{F}$ -scrambled tuples are defined by means of

$\mathcal{F}$ -reciprocating points for a pair

$(A,B)$ of sets. Hence, some properties of

$\mathcal{F}$ -

$N$ -chaos are considered similarly. A criterion for generically strongly

$\mathcal{F}$ -

$N$ -chaotic systems is obtained by heavy use of the theory of Furstenberg families, and it applications in dynamical systems are given with two examples.

Keywords:

$\mathcal{F}$ - $N$ -scrambled set

[1]Akin E.Recurrence in topological dynamics: Furstenberg families and Ellis actions[M], New York: Plenum Press, 1997. [2]Fu H M, Xiong J C, Tan F.On distributionally chaotic and null systems[J].J Math Anal Appl, 2011, 375(1):166-173 [3]Huang W, Ye X D.Homeoporphisms with the whole compacta being scrambled sets[J].Ergodic Theory and Dynamical Systems, 2001, 21(1):77-91 [4]Li J.Chaos and Entropy for Interval Maps[J].Journal of Dynamics and Differential Equations, 2011, 23(2):333-352 [5]吕杰, 熊金城, 谭枫.周期吸附系统的分布混沌[J].数学学报, 2008, 51(6):1109-1114 [6]Tan F, Fu H M.On distributional

$n$ -chaos[J].Acta Mathematica Scientia, 2014, 34(5):1473-1480 [7]Xiong J C, L\"{u} J, Tan F.Furstenberg Family and Chaos[J].Science in China Series A: Mathematics, 2007, 50(9):1325-1333 [8]Xiong J C, Fu H M, Wang H Y.A class of Furstenberg families and their applications to chaotic dynamics[J].Sci China Math, 2014, 57(4):823-836

[1]Akin E.Recurrence in topological dynamics: Furstenberg families and Ellis actions[M], New York: Plenum Press, 1997. [2]Fu H M, Xiong J C, Tan F.On distributionally chaotic and null systems[J].J Math Anal Appl, 2011, 375(1):166-173 [3]Huang W, Ye X D.Homeoporphisms with the whole compacta being scrambled sets[J].Ergodic Theory and Dynamical Systems, 2001, 21(1):77-91 [4]Li J.Chaos and Entropy for Interval Maps[J].Journal of Dynamics and Differential Equations, 2011, 23(2):333-352 [5]吕杰, 熊金城, 谭枫.周期吸附系统的分布混沌[J].数学学报, 2008, 51(6):1109-1114 [6]Tan F, Fu H M.On distributional $n$ -chaos[J].Acta Mathematica Scientia, 2014, 34(5):1473-1480 [7]Xiong J C, L\"{u} J, Tan F.Furstenberg Family and Chaos[J].Science in China Series A: Mathematics, 2007, 50(9):1325-1333 [8]Xiong J C, Fu H M, Wang H Y.A class of Furstenberg families and their applications to chaotic dynamics[J].Sci China Math, 2014, 57(4):823-836

全局性N元F-强混沌系统的一个判据

通讯作者:
符和满

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A Criterion for Generically Strongly $\mathcal{F}$ - $N$ -chaotic Systems

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目录

全局性N元F-强混沌系统的一个判据

通讯作者: 符和满

计量

出版历程

A Criterion for Generically Strongly F\mathcal{F}-NN-chaotic Systems

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出版历程

目录

通讯作者:
符和满

A Criterion for Generically Strongly $\mathcal{F}$ - $N$ -chaotic Systems