宋曉倩,王良偉,馮玉明

(重慶三峽學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,重慶 404100)

時(shí)變參數(shù)動(dòng)力系統(tǒng)的兩種跟蹤性質(zhì)

宋曉倩,王良偉,馮玉明

(重慶三峽學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,重慶 404100)

時(shí)變參數(shù)動(dòng)力系統(tǒng);鏈傳遞;偽軌;漸進(jìn)偽軌

1 引言

經(jīng)典動(dòng)力系統(tǒng)是研究某一個(gè)映射迭代(也稱為自治動(dòng)力系統(tǒng))所產(chǎn)生的動(dòng)力性狀,目前關(guān)于經(jīng)典動(dòng)力系統(tǒng)的研究成果已經(jīng)很多,研究的主要問題圍繞軌道的各種性狀[1-3].如果在動(dòng)力系統(tǒng)中,迭代的映射不是唯一的,而是隨著時(shí)間而變化的一序列映射,則構(gòu)成非自治動(dòng)力系統(tǒng),也稱為時(shí)變參數(shù)動(dòng)力系統(tǒng).非自治動(dòng)力系統(tǒng)的概念打破了經(jīng)典自治動(dòng)力系統(tǒng)的限制,拓展了動(dòng)力系統(tǒng)的研究范圍.由于非自治動(dòng)力系統(tǒng)能更靈活方便的描述現(xiàn)實(shí)世界的各種動(dòng)態(tài)和動(dòng)力學(xué)行為,因此具有重要的實(shí)際應(yīng)用價(jià)值.對(duì)此類動(dòng)力系統(tǒng)的研究已逐年成為熱點(diǎn).1996年,文獻(xiàn)[4]首次提出了非自治動(dòng)力系統(tǒng)的在開覆蓋意義下的拓?fù)潇睾皖愃朴贐owen的拓?fù)潇?2006年,文獻(xiàn)[5]提出時(shí)變參數(shù)動(dòng)力系統(tǒng)的周期點(diǎn)、回復(fù)性、傳遞性、擴(kuò)張性、一致拓?fù)涔曹椀雀拍?并且給出了時(shí)變參數(shù)Devaney混沌系統(tǒng)的一個(gè)簡(jiǎn)單構(gòu)造方法.2007年,文獻(xiàn)[6]研究了線段非自治動(dòng)力系統(tǒng)的向前熵和向后熵與逆極限空間之間的關(guān)系.2008年,文獻(xiàn)[7]研究了非自治動(dòng)力系統(tǒng)的預(yù)像熵.2012年,文獻(xiàn)[8]研究了非自治動(dòng)力系統(tǒng)的弱混合性和混沌性質(zhì).同年文獻(xiàn)[9]研究了非自治動(dòng)力系統(tǒng)的測(cè)度熵和拓?fù)潇?

偽軌跟蹤性是動(dòng)力系統(tǒng)中的一個(gè)重要概念,它與系統(tǒng)的穩(wěn)定性密切相關(guān),而且在數(shù)值計(jì)算中也有廣泛應(yīng)用.偽軌不是真正的軌道,而是一種近似軌道.關(guān)于偽軌跟蹤的研究已經(jīng)很多,參見文獻(xiàn)[10-11]等.隨著研究的深入,各種跟蹤性質(zhì)層出不窮.例如逐點(diǎn)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(qiáng)跟蹤性[15].其中漸進(jìn)偽軌跟蹤概念于1996年被文獻(xiàn)[10]提出,他們研究了映射被半流漸進(jìn)偽軌跟蹤的動(dòng)力學(xué)性質(zhì),2003年,文獻(xiàn)[16]證明了漸近偽軌跟蹤是拓?fù)涔曹椣碌牟蛔兞?并且討論了有限乘積系統(tǒng)的漸近偽軌跟蹤性質(zhì),2006年,文獻(xiàn)[17]研究了漸近偽軌跟蹤和拓?fù)鋫鬟f之間的關(guān)系,證明了系統(tǒng)滿足漸近偽軌跟蹤性質(zhì)時(shí),其拓?fù)鋫鬟f和鏈傳遞是等價(jià)的.這些成果都是在自治動(dòng)力系統(tǒng),也就是固定參數(shù)動(dòng)力系統(tǒng)中得出的.目前對(duì)于時(shí)變參數(shù)動(dòng)力系統(tǒng)(非自治動(dòng)力系統(tǒng))的偽軌跟蹤還是個(gè)未知領(lǐng)域,于是,下面的問題是自然的:

問題 時(shí)變參數(shù)動(dòng)力系統(tǒng)是否也有各種偽軌跟蹤性質(zhì)?若有,它們的具體性質(zhì)如何？

2 預(yù)備知識(shí)

2.1 偽軌概念

2.2 漸進(jìn)偽軌概念

3 主要結(jié)果

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

[2]Bowen R.Entropy for group endomorphisms and homogeneous spaces[J].Transactions of the American Mathematical Society,1971,153:401-414.

[3]Vellekoop M,Berglund R.On intervals,Transitivity=chaos[J].American Mathematical Monthly,1994,101:353-355.

[4]Kolyada S,Snoha L.Topological entropy of non-autonomous dynamical systems[J].Random and Computational Dynamics,1996,4,205-233.

[5]Tian C J,Chen G R.Chaos of a sequence of maps in a metric space[J].Chaos Solitons and Fractals, 2006,28:1067-1075.

[6]Mouron C.Positive entropy on non-autonomous interval maps and the topology of the inverse limit space[J]. Topology and its Applications,2007,154:894-907.

[7]Huang X J,Wen X,Zeng F P.Pre-image entropy of nonautonomous dynamical systems[J].Journal of Systems Science and Complexity,2008,3:441-445.

[8]Balibrea F,Oprocha P.Weak mixing and chaos in nonautonomous discrete systems[J].Applied Mathematics Letters,2012,25:1135-1141.

[9]Zhu Y J,Liu Z F,Xu X L,et al.Entropy of nonautonomous dynamical systems[J].Journal of the Korean Mathematical Society,2012,49:165-185.

[10]BenaiM,Hirsch M W.Asymptotic pseudo trajectories and chain recurrent fl ows with applications[J].Journal of Dynamics and Di ff erential Equations,1996,8:141-176.

[11]楊潤(rùn)生.偽軌跟蹤與混沌[J].數(shù)學(xué)學(xué)報(bào),1996,39:382-386.

[12]Li M J.Pointwise pseudo-orbit tracing property and its application[J].Journal of Mathematics Research and Exposition,2005,25:23-30.

[13]Koscielniak P.On genericity of shadowing and periodic shadowing property[J].Journal of Mathematical Analysis and Applications,2005,310:188-196.

[14]Gu R B.The asymptotic average shadowing property and transitivity[J].Nonlinear Analysis,2007,67:1680-1689.

[15]Sakai K.Various shadowing properties for positively expansive maps[J].Topology and its Applications, 2003,131:15-31.

[16]顧榮寶,盛業(yè)青.關(guān)于漸近的偽軌跟蹤性質(zhì)[J].安徽大學(xué)學(xué)報(bào):自然科學(xué)版,2003,27:1-5.

[17]Gu R B.Recurrence and the asymptotic pseudo-orbit tracing property[J].Nonlinear Analysis,2007,66:1698-1706.

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(xiàng)目(KJ091104,KJ121105);國(guó)家自然科學(xué)基金(11126212).

宋曉倩(1985-),碩士,助教,研究方向:拓?fù)鋵W(xué)動(dòng)力系統(tǒng).

2010 MSC:54A10