Chaos of discrete dynamical systems in complete metric spaces

ZHAI Hongli, LIN Yajing, SUN Bo

?

Chaos of discrete dynamical systems in complete metric spaces

ZHAI Hong-li, LIN Ya-jing, SUN Bo

(College of Mathematics and Computers, Changsha University of Science and Technology, Changsha 410114, China)

Chaos of discrete dynamical systems in metric spaces was discussed. Two existing criteria for chaos was improved, and it is proven that a system was topologically conjugate to a symbolic dynamical system if it has a regular degenerate snap-back repeller.

Metric space; discrete dynamical system; chaos; snap-back repeller

Consider the following discrete dynamical system:

Chaos for interval maps or one dimensional discrete dynamical systems have been studied by T.Y. Li, J. A. Yorke, Goong Chen and other authors[1-4]. Chaos for n-dimensional discrete dynamical systems was first studied by F. R. Marotto, then by Li, Chen, Hsu S and Zhou[5-7]. In 2004, Y. Shi and G. Chen studied system (1) for general metric spaces, and obtained results as follows.

In this paper, we aim to modify Shi's work and deduce chaos of discrete dynamical systems in complete metric spaces under fewer conditions. We modify Proposition 1, and obtain the same chaos results under condition (a) and (b). Then we modify Proposition 2, and prove that the discrete dynamical system is chaotic under condition (a). Finally, we simplify Proposition 3 and get a better one.

This paper is organized as follows: In section 1 we recall some preliminary definitions and lemmas; In section 2 we state our main results and prove them.

1 Definitions and lemmas

For convenience, we recall some definitions and lemmas as follows[3,8]:

2 Main results

Proof The proof is similar to that of Yuming Shi (Ref. [8], Theorem 1). So we recall the main steps of Yuming Shi, and modify a key step.

The proof is divided into three steps:

Combining Proposition 6 and the arguments on Theorem 1, we have:

Acknowledgements

The research for this work was supported, in part, by the Natural Sciences Council of China.

[1] Roger Temam. Infinte-Dimensional Dynamical Systems in Mechanics and Physics[M]. New York: Springer-Verlag, 1998.

[2] Li T Y, Yorke J A. Period three implies chaos[J]. American Mathematical Monthly, 1975, 82: 985-992,.

[3] Chen G, Huang T, Huang Y. Chaotic behavior of interval maps and total variations of iterates[J]. Int J Bifur chaos, 2004, 14: 2161-2186.

[4] Zhou Z L. Symbolic Dynamical Systems[M]. Shanghai: Shanghai Science and Technology Press, 1997.

[5] Sun B, Xiao H, Zhang X. A note on chaotic behavior of interval maps[J]. J Physics: Conference Series, 2008, 96: 1-4.

[6] Frederick R. Marotto. Snap-back repellers imply chaos inR[J]. J Math Anal Appl, 1978, 63: 199-223.

[7] Chen G, Hsu S, Zhou J. Snap-back repellers as a cause of chaotic vibration of the wave equation with a Van der Pol boundary condition and energy injection at the middle of the span[J]. J Math Phys, 1998, 39: 6459-6489.

[8] Li C, Chen G. An improved version of the Marotto Theorem[J]. Chaos, Solitons and Fractals, 2003, 18: 969-977.

[9] Shi Yuming, Chen Guanrong. Chaos of discrete dynamical systems in complete metric spaces[J]. Chaos, solitons and fractals, 2004, 22: 555-571.

完備度量空間中離散動(dòng)力系統(tǒng)的混沌

翟紅利,林亞靜,孫 波

(長(zhǎng)沙理工大學(xué) 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院, 湖南 長(zhǎng)沙, 410114)

考慮度量空間中離散動(dòng)力系統(tǒng)的混沌, 改進(jìn)了兩條現(xiàn)有判據(jù), 證明了當(dāng)一個(gè)系統(tǒng)有正則退化snap-back repeller時(shí)拓?fù)涔曹椨诜?hào)動(dòng)力系統(tǒng).

度量空間; 離散動(dòng)力系統(tǒng); 混沌; snap-back repeller

O 193

1672-6146(2012)01-0001-04

10.3969/j.issn.1672-6146.2012.01.001

2011-10-24

翟紅利(1987-), 女, 碩士研究生, 研究方向?yàn)閯?dòng)力系統(tǒng)與控制理論.

孫波(1965-), 男, 博士, 教授, 主要研究方向?yàn)槠⒎址匠?、?dòng)力系統(tǒng)與控制理論. E-mail:sunbo52002@ yahoo.com.cn

(責(zé)任編校:劉曉霞)