非自治动力系统中周期跟踪和极限跟踪的研究
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摘要
根据自治动力系统中周期跟踪性和极限跟踪性的定义,将其引入到非自治动力系统。研究了非自治动力系统中周期跟踪性和极限跟踪性的动力学性质,得到:(1)若F={f_i}_(i=0)~∞拓扑共轭于G={g_i}_(i=0)~∞,则F具有周期跟踪性当且仅当G具有周期跟踪性;(2)若F={f_i}_(i=0)~∞拓扑共轭于G={g_i}_(i=0)~∞,则F具有极限跟踪性当且仅当G具有极限跟踪性;(3)若乘积系统(X×Y,F×G)具有周期跟踪性,则(X,F)和(Y,G)具有周期跟踪性。以上结论对非自治动力系统中跟踪性的发展有一定的促进作用。
According to the definition of the periodic shadowing property and the limit shadowing property in autonomous dynamical systems, this paper introduces the concept of periodic shadowing property and limit shadowing property in nonautonomous dynamical systems,and studies the dynamical properties of both shadowing properties and limit shadowing property in nonautonomous dynamical systems. The following results are obtained:(1) If F = { f_i}_(∞= 0)~i and G = { g_i} _(∞= 0)~i are topologically conjugate, then F has periodic shadowing property if and only if G has periodic shadowing property;(2) If F = { f_i}_(∞= 0)~i and G = { g_i} _(∞= 0)~i are topologically conjugate, then F has limit shadowing property if and only if G has limit shadowing property;(3) If the product system( X × Y,F × G) has periodic shadowing property, then( X,F) and(Y,G) have periodic shadowing property. The above results have the positive effect on the development of the shadowing property in autonomous dynamical systems.
According to the definition of the periodic shadowing property and the limit shadowing property in autonomous dynamical systems, this paper introduces the concept of periodic shadowing property and limit shadowing property in nonautonomous dynamical systems,and studies the dynamical properties of both shadowing properties and limit shadowing property in nonautonomous dynamical systems. The following results are obtained:(1) If F = { f_i}_(∞= 0)~i and G = { g_i} _(∞= 0)~i are topologically conjugate, then F has periodic shadowing property if and only if G has periodic shadowing property;(2) If F = { f_i}_(∞= 0)~i and G = { g_i} _(∞= 0)~i are topologically conjugate, then F has limit shadowing property if and only if G has limit shadowing property;(3) If the product system( X × Y,F × G) has periodic shadowing property, then( X,F) and(Y,G) have periodic shadowing property. The above results have the positive effect on the development of the shadowing property in autonomous dynamical systems.
引文
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[3]李明军,曾凡平.序列伪轨跟踪性与拓扑可迁[J].广西大学学报(自然科学版),2000,25(2):137-140.LI M J,ZENG F P.Sequence pseudo-orbit-tracing property and topological transitivity[J].Journal of Guangxi University(Natural Science Edition),2000,25(2):137-140.
[4]冀占江.乘积空间与拓扑群作用下逆极限空间的动力学性质[D].南宁:广西大学,2014.JI Z J.Dynamical Property of Product Space and the Inverse Limit Space of A Topological Group Action[D].Nanning:Guangxi University,2014.
[5]GU R B,SHENG Y Q.Adopt for the inverse limit spaces[J].Applied Mathematics Journal,2006,21(4):473-478.
[6]赵俊玲.强链回归集与跟踪性[J].数学研究,2004,37(3):286-291.ZHAO J L.Strong chain recurrent sets and strong shadowing property[J].Journal of Mathematical Study,2004,37(3):286-291.
[7]牛应轩.具有渐进平均跟踪性质的系统[J].高校应用数学学报(A辑),2007,22(4):462-468.NIU Y X.Dynamical systems with the asymptotic average shadowing property[J].Applied Mathematics AJournal of Chinese Universities,2007,22(4):462-468.
[8]朱玉峻,何连法.线性系统的极限跟踪性[J].数学物理学报,2007,27(2):314-321.ZHU Y J,HE L F.Limit shadowing property of linear systems[J].Acta Mathematica Scientia,2007,27(2):314-321.
[9]SAHAI K.Various shadowing properties for positively expansive maps[J].Topology and Its Applications,2003,131(1):15-31.
[10]KEONHE L,KAZUHIRO S.Various shadowing properties and their equivalence[J].Discrete and Continuous Dynamical Systems,2005,13(2):533-540.
[11]张金莲,朱玉峻,何连法.非自治动力系统的原像熵[J].数学学报(中文版),2005,48(4):693-702.ZHANG J L,ZHU Y J,HE L F.Preimage entropies of nonautonomous dynamical systems[J].Acta Mathematica Sinica(Chinese Series),2005,48(4):693-702.
[12]KOLYADA S,MISIUREWICZ M,SNOHA L.Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval[J].Fundamenta Mathematicae,1999,160(2):161-181.
[13]TIAN C J,CHEN G R.Chaos of sequence of maps in a metric space[J].Chaos Solitons and Fractals,2006,28(4):1067-1075.
[14]HOSSEIN R.On the shadowing property of nonautonomous discrete systems[J].International Journal of Nonlinear Analysis and Applications,2016,7(1):271-277.
[15]HOSSEIN R,REZA M.On the relation of shadowing and expansivity in nonautonomous discrete systems[J].Analysis in Theory and Applications,2017,33(1):11-19.
[2]赵俊玲,张莉.周期伪轨跟踪性和伪轨跟踪性的关系[J].浙江大学学报(理学版),2010,37(5):505-507.ZHAO J L,ZHANG L.Relationship between the periodic pseudo orbit tracing property and the pseudo orbit tracing property[J].Journal of Zhejiang University(Science Edition),2010,37(5):505-507.
[3]李明军,曾凡平.序列伪轨跟踪性与拓扑可迁[J].广西大学学报(自然科学版),2000,25(2):137-140.LI M J,ZENG F P.Sequence pseudo-orbit-tracing property and topological transitivity[J].Journal of Guangxi University(Natural Science Edition),2000,25(2):137-140.
[4]冀占江.乘积空间与拓扑群作用下逆极限空间的动力学性质[D].南宁:广西大学,2014.JI Z J.Dynamical Property of Product Space and the Inverse Limit Space of A Topological Group Action[D].Nanning:Guangxi University,2014.
[5]GU R B,SHENG Y Q.Adopt for the inverse limit spaces[J].Applied Mathematics Journal,2006,21(4):473-478.
[6]赵俊玲.强链回归集与跟踪性[J].数学研究,2004,37(3):286-291.ZHAO J L.Strong chain recurrent sets and strong shadowing property[J].Journal of Mathematical Study,2004,37(3):286-291.
[7]牛应轩.具有渐进平均跟踪性质的系统[J].高校应用数学学报(A辑),2007,22(4):462-468.NIU Y X.Dynamical systems with the asymptotic average shadowing property[J].Applied Mathematics AJournal of Chinese Universities,2007,22(4):462-468.
[8]朱玉峻,何连法.线性系统的极限跟踪性[J].数学物理学报,2007,27(2):314-321.ZHU Y J,HE L F.Limit shadowing property of linear systems[J].Acta Mathematica Scientia,2007,27(2):314-321.
[9]SAHAI K.Various shadowing properties for positively expansive maps[J].Topology and Its Applications,2003,131(1):15-31.
[10]KEONHE L,KAZUHIRO S.Various shadowing properties and their equivalence[J].Discrete and Continuous Dynamical Systems,2005,13(2):533-540.
[11]张金莲,朱玉峻,何连法.非自治动力系统的原像熵[J].数学学报(中文版),2005,48(4):693-702.ZHANG J L,ZHU Y J,HE L F.Preimage entropies of nonautonomous dynamical systems[J].Acta Mathematica Sinica(Chinese Series),2005,48(4):693-702.
[12]KOLYADA S,MISIUREWICZ M,SNOHA L.Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval[J].Fundamenta Mathematicae,1999,160(2):161-181.
[13]TIAN C J,CHEN G R.Chaos of sequence of maps in a metric space[J].Chaos Solitons and Fractals,2006,28(4):1067-1075.
[14]HOSSEIN R.On the shadowing property of nonautonomous discrete systems[J].International Journal of Nonlinear Analysis and Applications,2016,7(1):271-277.
[15]HOSSEIN R,REZA M.On the relation of shadowing and expansivity in nonautonomous discrete systems[J].Analysis in Theory and Applications,2017,33(1):11-19.