時變參數動力系統的兩種跟蹤性質

宋曉倩,王良偉,馮玉明

(重慶三峽學院數學與統計學院,重慶 404100)

時變參數動力系統的兩種跟蹤性質

宋曉倩,王良偉,馮玉明

(重慶三峽學院數學與統計學院,重慶 404100)

時變參數動力系統;鏈傳遞;偽軌;漸進偽軌

1 引言

經典動力系統是研究某一個映射迭代(也稱為自治動力系統)所產生的動力性狀,目前關于經典動力系統的研究成果已經很多,研究的主要問題圍繞軌道的各種性狀[1-3].如果在動力系統中,迭代的映射不是唯一的,而是隨著時間而變化的一序列映射,則構成非自治動力系統,也稱為時變參數動力系統.非自治動力系統的概念打破了經典自治動力系統的限制,拓展了動力系統的研究范圍.由于非自治動力系統能更靈活方便的描述現實世界的各種動態和動力學行為,因此具有重要的實際應用價值.對此類動力系統的研究已逐年成為熱點.1996年,文獻[4]首次提出了非自治動力系統的在開覆蓋意義下的拓撲熵和類似于Bowen的拓撲熵.2006年,文獻[5]提出時變參數動力系統的周期點、回復性、傳遞性、擴張性、一致拓撲共軛等概念,并且給出了時變參數Devaney混沌系統的一個簡單構造方法.2007年,文獻[6]研究了線段非自治動力系統的向前熵和向后熵與逆極限空間之間的關系.2008年,文獻[7]研究了非自治動力系統的預像熵.2012年,文獻[8]研究了非自治動力系統的弱混合性和混沌性質.同年文獻[9]研究了非自治動力系統的測度熵和拓撲熵.

偽軌跟蹤性是動力系統中的一個重要概念,它與系統的穩定性密切相關,而且在數值計算中也有廣泛應用.偽軌不是真正的軌道,而是一種近似軌道.關于偽軌跟蹤的研究已經很多,參見文獻[10-11]等.隨著研究的深入,各種跟蹤性質層出不窮.例如逐點偽軌跟蹤[12]、周期偽軌跟蹤[13]、漸近偽軌跟蹤[10]、漸近平均偽軌跟蹤[14]、Lipschitz跟蹤及強跟蹤性[15].其中漸進偽軌跟蹤概念于1996年被文獻[10]提出,他們研究了映射被半流漸進偽軌跟蹤的動力學性質,2003年,文獻[16]證明了漸近偽軌跟蹤是拓撲共軛下的不變量,并且討論了有限乘積系統的漸近偽軌跟蹤性質,2006年,文獻[17]研究了漸近偽軌跟蹤和拓撲傳遞之間的關系,證明了系統滿足漸近偽軌跟蹤性質時,其拓撲傳遞和鏈傳遞是等價的.這些成果都是在自治動力系統,也就是固定參數動力系統中得出的.目前對于時變參數動力系統(非自治動力系統)的偽軌跟蹤還是個未知領域,于是,下面的問題是自然的:

問題 時變參數動力系統是否也有各種偽軌跟蹤性質?若有,它們的具體性質如何？

2 預備知識

2.1 偽軌概念

2.2 漸進偽軌概念

3 主要結果

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Two tracing properties of timevarying discrete dynamical system

Song Xiaoqian,Wang Liangwei,Feng Yuming
(College of Mathematics and Statistics,Chongqing Three Gorges University,Chongqing 404100,China)

The purpose of this paper is to introduce chain transitivity,pseudo-orbit and asymptotic pseudoorbit for time-varying discrete dynamical system.And through these new conceptions,the pseudo-orbit tracing property and asymptotic pseudo-orbit tracing property of time-varying discrete dynamical system are studied. We proved that expansive timevarying discrete dynamical system with pseudo-orbit tracing property implies it has asymptotic pseudo-orbit tracing property.We investigate the tracing property between the product system and subsystem.It is showed that the product system has the pseudo-orbit tracing property and asymptotic pseudo-orbit tracing property if and only if each subsystem has the same property.Finally,we construct an example,which is chain transitivity and has asymptotic pseudo-orbit tracing property.

time-varying discrete dynamical system,chain transitivity,pseudo-orbit,asymptotic pseudo-orbit

O189.1

A

1008-5513(2012)05-0641-08

2012-04-07.

重慶市教委資助項目(KJ091104,KJ121105);國家自然科學基金(11126212).

宋曉倩(1985-),碩士,助教,研究方向:拓撲學動力系統.

2010 MSC:54A10