渐近紧随机动力系统及其极限集的性质

doi:10.12341/jssms10220

系统科学与数学 ›› 2008, Vol. 28 ›› Issue (6): 669-678.DOI: 10.12341/jssms10220

渐近紧随机动力系统及其极限集的性质

黎育红, 王乘, 周建中

华中科技大学水电与数字化工程学院, 武汉 430074

收稿日期:2006-04-10 修回日期:2007-04-09 出版日期:2008-06-25 发布日期:2008-06-25

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黎育红;王乘;周建中. 渐近紧随机动力系统及其极限集的性质[J]. 系统科学与数学, 2008, 28(6): 669-678.

LI Yuhong;WANG Cheng;ZHOU Jianzhong. Asymptotically Compact Random Dynamical Systems and its Limit Set[J]. Journal of Systems Science and Mathematical Sciences, 2008, 28(6): 669-678.

Asymptotically Compact Random Dynamical Systems and its Limit Set

LI Yuhong, WANG Cheng, ZHOU Jianzhong

Huazhong University of Science and Technology, Wuhan 430074

Received:2006-04-10 Revised:2007-04-09 Online:2008-06-25 Published:2008-06-25

通过对随机动力系统极限行为的研究,推广了传统动力系统的相关定义和理论.由于在无界区域上,Sobolev紧嵌入的缺乏,Crauel, Debussche及Flandoli 等人在研究有界区域上的随机演化系统时所引入的渐近紧的概念不再适用.通过对Ladyzhenskaya,~Rosa等人在对确定性系统的研究中所提出的渐近紧概念的推广,引入了随机动力系统渐近紧的概念,并以相应的示例及严格的理论推导证明了此概念的合理性和必要性.最后,作为这一新的概念的应用,证明了渐近紧随机动力系统极限集的性质.

关键词： 随机动力系统, 渐近紧, 极限集, 演化系统, 耗散.

In order to study the asymptotic behaviour of random dynamical systems, corresponding notions and theories from the traditional dynamical systems are generalized. Due to the absence of the Sobolev embedding compactness, the idea of asymptotical compactness for random evolution systems introduced by
Crauel, Debussche and Flandoli will be no longer suitable for random dynamical systems on unbounded domain. A new definition of asymptotic compactness for random dynamical systems is introduced. The theoretical analysis and some examples show the reasonability and applicability of the new notion. Finally, the ${\it \Omega}$-limit property for the asymptotically compact RDS is proved.

Key words: Random dynamical systems, asymptotical compactness, ${\Omega}$-limit set, evolution system, dissipativity.

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