有关映射的动力性状的研究-Devaney混沌
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摘要
本论文研究了紧致系统（X，f)的Devaney混沌性状。将对Devaney混沌的三个条件进行改变，进而得到了不同的Devaney混沌。因为Devaney混沌的三个条件中，第一个条件表示系统是不可以分解的，即混Devaney沌系统不可以分解成两个子系统的和，第二个条件说明没有周期点的系统不是Devaney混沌的，第三个条件表示系统不可以预测的，初值的很小的改变导致迭代的结果有很大的差别。
本文的第二章讨论了Devaney混沌及其修改意义下的Devaney混沌。先构造了符号空间，证明它是紧致的度量空间，利用已经学过的知识及其符号空间的某些性质证明这个符号空间是Devaney混沌。集值动力系统的研究实际上是这集合的子集的运动性状的研究，在人口统计学，吸引子等得很多领域中，知道子集的运动性状非常重要的，因此集值动力系统的研究显得很重要。因为符号空间的周期稠密蕴含它的集值动力系统的周期稠密，拓扑传递蕴含集值动力系统的拓扑传递，则符号空间的集值动力系统是Devaney混沌的。由于拓扑半共轭保持拓扑混合性不变拓扑混合性蕴含传递性和初值敏感依赖，则拓扑半共轭保持修改意义下的Devaney混沌不变。
第三章讨论了修改的弱Devaney混沌。将初值敏感依赖改为初值弱敏感依赖，则修改的Devaney混沌改为修改的弱Devaney混沌。主要讨论了紧致的系统的逆极限空间，因为拓扑传递蕴含逆极限空间的拓扑传递，拓扑混合蕴含初值敏感，也是弱初值敏感，则用所学的知识可以以证明逆极限空间也是弱初值敏感的，即使修改的弱Devaney混沌。因为讨论Devaney混沌时拓扑传递是比不可以少得，则拓扑传递系统成为一个重要的研究对象。由于任何一个拓扑动力系统不一定是拓扑传递的，但是一定存在拓扑传递的子系统。
第四章在符号空间上构造了新的集合，构造了一个新的系统是强的Devaney混沌的。
本文讨论了Devaney混沌的三个条件在相应的改变下得到了不同的Devaney混沌，使得我们对映射的动力性状Devaney混沌的研究更加的方便，更加彻底。
This paper studies the compact system (X, f) of Devaney chaotic characters.three conditions of Devaney chaos are changed, and then we get different of Devaneychaos. Because in the three conditions of Devaney chaos, the first condition can not bedecomposed, namely the Devaney chaos chaotic system can be decomposed into twosubsystems,the second conditions describes the system without periodic points is notDevaney chaos system, the third conditions describes the system cannot be predicted,the initial value of small change leads that the iterative results have varied considerably.
In the first chapter of this paper discusses Devaney chaos and its modificationof Devaney chaos. To construct the symbol space, it is proved that it is a compactmetric space, using the learned knowledge and some nature of symbolic space,we proofthe symbol space is Devaney chaos.The studing of Set valued dynamical system isactually the movement characteristics of the subset,in the demography, attractor, andmany fields, knowing the movement characters of the subset is very important, soresearching of the set valued dynamical system is very important. Because the symbolspace periodic density contains its set value dynamic system of dense, topologicaltransitivity contains topology delivery of set valued dynamical system, then the setvalued dynamical system of symbol space is Devaney chaos. Due to the topologicalSemi-conjugacy maintain topological mixing invariant, topological mixing containstransitivity and sensitive dependence on initial conditions, then the topologicalSemi-conjugacy maintain modification of Devaney chaotic invariant.
The second chapter discusses the modification of weak Devaney chaos. Whenthe sensitive dependence on initial conditions changes weakly sensitive dependence on initial value, a modification of the Devaney chaos changs to modify the weakDevaney chaos. We mainly discuss inverse limit space of the compact system, becausethe topological transitive contains topological transitive of inverse limit space,topologically mixing contains sensitive dependence on initial conditions, is weaklysensitive dependence on initial conditions, using the learned knowledge, we show thatthe inverse limit space is weakly sensitive dependence on initial conditions, even if themodifications of the weak Devaney chaos. As when we discuss Devaney chaos,topological transitivity is less than not, so topological transitivity system has become animportant research object. Because topological dynamical system is not to betopologically transitive, but there must be topologically transitive subsystem.
In the third chapter, we construct a new collection of the symbol space, constructa new system that is a strong Devaney chaos.
This paper discusses the corresponding change of the three conditions ofDevaney chaos,we get different Devaney chaos. That allows us to research the map ofthe dynamic properties of Devaney chaos more convenientiy, more thorough.
本文的第二章讨论了Devaney混沌及其修改意义下的Devaney混沌。先构造了符号空间，证明它是紧致的度量空间，利用已经学过的知识及其符号空间的某些性质证明这个符号空间是Devaney混沌。集值动力系统的研究实际上是这集合的子集的运动性状的研究，在人口统计学，吸引子等得很多领域中，知道子集的运动性状非常重要的，因此集值动力系统的研究显得很重要。因为符号空间的周期稠密蕴含它的集值动力系统的周期稠密，拓扑传递蕴含集值动力系统的拓扑传递，则符号空间的集值动力系统是Devaney混沌的。由于拓扑半共轭保持拓扑混合性不变拓扑混合性蕴含传递性和初值敏感依赖，则拓扑半共轭保持修改意义下的Devaney混沌不变。
第三章讨论了修改的弱Devaney混沌。将初值敏感依赖改为初值弱敏感依赖，则修改的Devaney混沌改为修改的弱Devaney混沌。主要讨论了紧致的系统的逆极限空间，因为拓扑传递蕴含逆极限空间的拓扑传递，拓扑混合蕴含初值敏感，也是弱初值敏感，则用所学的知识可以以证明逆极限空间也是弱初值敏感的，即使修改的弱Devaney混沌。因为讨论Devaney混沌时拓扑传递是比不可以少得，则拓扑传递系统成为一个重要的研究对象。由于任何一个拓扑动力系统不一定是拓扑传递的，但是一定存在拓扑传递的子系统。
第四章在符号空间上构造了新的集合，构造了一个新的系统是强的Devaney混沌的。
本文讨论了Devaney混沌的三个条件在相应的改变下得到了不同的Devaney混沌，使得我们对映射的动力性状Devaney混沌的研究更加的方便，更加彻底。
This paper studies the compact system (X, f) of Devaney chaotic characters.three conditions of Devaney chaos are changed, and then we get different of Devaneychaos. Because in the three conditions of Devaney chaos, the first condition can not bedecomposed, namely the Devaney chaos chaotic system can be decomposed into twosubsystems,the second conditions describes the system without periodic points is notDevaney chaos system, the third conditions describes the system cannot be predicted,the initial value of small change leads that the iterative results have varied considerably.
In the first chapter of this paper discusses Devaney chaos and its modificationof Devaney chaos. To construct the symbol space, it is proved that it is a compactmetric space, using the learned knowledge and some nature of symbolic space,we proofthe symbol space is Devaney chaos.The studing of Set valued dynamical system isactually the movement characteristics of the subset,in the demography, attractor, andmany fields, knowing the movement characters of the subset is very important, soresearching of the set valued dynamical system is very important. Because the symbolspace periodic density contains its set value dynamic system of dense, topologicaltransitivity contains topology delivery of set valued dynamical system, then the setvalued dynamical system of symbol space is Devaney chaos. Due to the topologicalSemi-conjugacy maintain topological mixing invariant, topological mixing containstransitivity and sensitive dependence on initial conditions, then the topologicalSemi-conjugacy maintain modification of Devaney chaotic invariant.
The second chapter discusses the modification of weak Devaney chaos. Whenthe sensitive dependence on initial conditions changes weakly sensitive dependence on initial value, a modification of the Devaney chaos changs to modify the weakDevaney chaos. We mainly discuss inverse limit space of the compact system, becausethe topological transitive contains topological transitive of inverse limit space,topologically mixing contains sensitive dependence on initial conditions, is weaklysensitive dependence on initial conditions, using the learned knowledge, we show thatthe inverse limit space is weakly sensitive dependence on initial conditions, even if themodifications of the weak Devaney chaos. As when we discuss Devaney chaos,topological transitivity is less than not, so topological transitivity system has become animportant research object. Because topological dynamical system is not to betopologically transitive, but there must be topologically transitive subsystem.
In the third chapter, we construct a new collection of the symbol space, constructa new system that is a strong Devaney chaos.
This paper discusses the corresponding change of the three conditions ofDevaney chaos,we get different Devaney chaos. That allows us to research the map ofthe dynamic properties of Devaney chaos more convenientiy, more thorough.
引文
[1]周作领．符号动力系统[M]．上海：上海科技教育出版社，1997：1-15.
[2] J.bank,J.Brooks,G.Gairns,G.Davis and D.stacey.on Devaney’s Definition ofchaos.The Amer Math.Moonthly,99:4.
[3] KleinE,ThompsonA.Theory of correspondences[M].New York:wiley-intersince,1984.
[4] FedeliA.on chaotic set-valued discrete dynamical systems chaos,solitons andFractls[J].2005,23:1381-1384.
[5]张景中，熊金城.函数迭代与一维动力系统[M].成都：四川教育出版社，1992：15-27.
[6]段晓君，张增辉.符号动力系统中的相关结论和证明[J].湖南工业大学学报，2010,24（2）：32-33.
[7]熊金城．点集拓扑讲义[M]．北京：高等教育出版社，1998：107-108.
[8]廖公夫.一类集值映射的传递性，混合性和混沌[J].中国科学（A辑），2005，35（10）：1155-1161.
[9]刘洪刚，朱培勇.Devaney混沌的等价刻画[J].西南民族大学学报，2009,35（1）：49-53.
[10]牛应轩.强初值敏感性的一些性质及充分条件[J].皖西学院学报，2010,26（2）:1-6.
[11] KolyadaS.Snohal.Nonivertible Minimal Maps[J].FundMath.2001,168(2):141-163.
[12]顾荣宝.逆极限空间的转移映射的拓扑熵与混沌[J].武汉大学学报，1995,41（1）：22-26.
[13] TIEN-YIENLI,JAMS YORKE.period three implies chaos[J].Amer.Math.Monthly,1975,82:985-992.
[14] BANKS ETALJ.on Devaney Definition of chaos Amer[J].Math Mon,1992,99(4):334-334.
[15] AkinE. kolyadaS.Li-York sensitivity[J].Nonlinearity,2003,16:1421-1433.
[16]熊金城．拓扑传递系统中的混沌[J]．中国科学(A)，2005，35（1）：302-311.
[17] ChristopheA,GerardB,BenoitC.chaotic properties on a probapility space[J].Math,Anal.Appl,2002,266(2):420-431.
[18] EAkin,JAuslander and KBerg,when is a Transitive map chaotic?[A].V.Bergelson,P.March,JRosenblatt.convergence in Ergodic theory and probability(columabus,oh,1993)(ohio coniversity Math,Res.inst.pub.5)[c],Berin,NewYork:de Gruter,1996.
[19] GlasnerE,weissB,sensitive dependence on initial conditiona[J]. Nonlinearity.1993,9(6):1067-1075.
[20] HuangW,YeX.Anexplicit scattering,Non-weakly mixing example and weak disjointness [J].Nonlinearity,2002,15:1-14.
[21] BlockL.Diffeomorphisms obtained From endomrphisms.Thans Amer Math soc,1975,214:403-413.
[22] LiSH.Dynamical properties of the shift maps on the inverse limt space,Ergodthe dynamsys,1992,12:95-108.
[23] ChenL,LiSH.shadowing property for inverse limt space.proc Amer Mathsoc,1992,115:573-581.
[24]耿祥义.Li-York混沌的充要条件[J].数学学报，2001,929-932.
[25]刘增荣.混沌研究中的解析方法[M].上海：上海教育出版社，2002:1-5.
[26]李继彬.混沌与Melnikov方法[M].重庆：重庆大学出版社，1989：7-11.
[27]张伟年.动力系统基础[M].北京：高等教育出版社，2001：52-71.
[28]张筑生.微分动力系统原理[M]．北京:科学出版社，1987：11-15.
[29]徐园芬，戴振祥．一类描述五分Cantor集的拟移位映射[J]．浙江师范大学学报（自然科学版），2010，33(4)：387-391.
[30] Coiiet.P,Eckmam,J.P,Iterated maps on the Interval as Dynamics systems,Birkhauser,Boston,1980.
[31] Li,T.Yorke,J.A, Reriod3implies chaos, Amer Math Monthly.1975,82:985-992.
[32] J.Milnor,W.Thurston.Dynamical System[J].Lect,Notesin,math.1988,1342:469.
[33] S.-L.Peng and L.-m.Du，Dualstar products and symbolic dymamics of Lorenz mapswith the same entropy [J]. Phys.Lete.A.1999，261：63.
[34]庄云标.符号空间上左拟移位映射[J].台州学院学报，2003，25（6）：37-39.
[35] Z-X.Chen,K-F Cao and S-L.Peng,symbolic dynamics anaiysis of topologicalentropy and its multifractal structure[J].Phys.Rev.E,1995,51:1985.
[36] MinQ.Journal of MengZi Tenchers’College,2001,23(24):39.
[37]陈凤娟.双边拟移位映射的混沌性[J].浙江师范大学学报，2002,25(3):221-223.
[38] BargeM,MartinJ.chaos,periodicity and snakelike continua.Trans Amer Math soc,1985,289:355-365.
[39]熊金城，陈二才.强混合的保测度变换引起的混沌.中国科学（A），1996,26（1）：961-967.
[40]汪火云，熊金城．拓扑遍历映射的一些性质[J]．数学学报，2004，4795):859－866.
[41]廖公夫，范钦杰．拓扑熵为零且schweizer－smital混沌的极小子转移[J]．中国科学，1997,27(9):770－774．
[42]夏道行．实变函数论与泛函分析[M]．北京：高等教育出版社，1979．
[43]廖公夫，范钦杰．混沌与ss混沌不等价[J]．数学年刊，A辑，6（2000），749－754．
[44]王立冬，陈知之．符号空间一类极小子转移的混沌性[J]．大连民族学院学报，20035（2）：63－65．
[45]孙丽华，赵恩良，刘丹．非周期回复点生成子转移的混沌性质[J]．沈阳建筑大学学报，2009,25（5）：1019－1020．
[46]王立冬，廖公夫．Σ上非弱几乎周期的回复点集和ss混沌集[J]．数学物理学报，2006,26A（5）：778－784．
[2] J.bank,J.Brooks,G.Gairns,G.Davis and D.stacey.on Devaney’s Definition ofchaos.The Amer Math.Moonthly,99:4.
[3] KleinE,ThompsonA.Theory of correspondences[M].New York:wiley-intersince,1984.
[4] FedeliA.on chaotic set-valued discrete dynamical systems chaos,solitons andFractls[J].2005,23:1381-1384.
[5]张景中，熊金城.函数迭代与一维动力系统[M].成都：四川教育出版社，1992：15-27.
[6]段晓君，张增辉.符号动力系统中的相关结论和证明[J].湖南工业大学学报，2010,24（2）：32-33.
[7]熊金城．点集拓扑讲义[M]．北京：高等教育出版社，1998：107-108.
[8]廖公夫.一类集值映射的传递性，混合性和混沌[J].中国科学（A辑），2005，35（10）：1155-1161.
[9]刘洪刚，朱培勇.Devaney混沌的等价刻画[J].西南民族大学学报，2009,35（1）：49-53.
[10]牛应轩.强初值敏感性的一些性质及充分条件[J].皖西学院学报，2010,26（2）:1-6.
[11] KolyadaS.Snohal.Nonivertible Minimal Maps[J].FundMath.2001,168(2):141-163.
[12]顾荣宝.逆极限空间的转移映射的拓扑熵与混沌[J].武汉大学学报，1995,41（1）：22-26.
[13] TIEN-YIENLI,JAMS YORKE.period three implies chaos[J].Amer.Math.Monthly,1975,82:985-992.
[14] BANKS ETALJ.on Devaney Definition of chaos Amer[J].Math Mon,1992,99(4):334-334.
[15] AkinE. kolyadaS.Li-York sensitivity[J].Nonlinearity,2003,16:1421-1433.
[16]熊金城．拓扑传递系统中的混沌[J]．中国科学(A)，2005，35（1）：302-311.
[17] ChristopheA,GerardB,BenoitC.chaotic properties on a probapility space[J].Math,Anal.Appl,2002,266(2):420-431.
[18] EAkin,JAuslander and KBerg,when is a Transitive map chaotic?[A].V.Bergelson,P.March,JRosenblatt.convergence in Ergodic theory and probability(columabus,oh,1993)(ohio coniversity Math,Res.inst.pub.5)[c],Berin,NewYork:de Gruter,1996.
[19] GlasnerE,weissB,sensitive dependence on initial conditiona[J]. Nonlinearity.1993,9(6):1067-1075.
[20] HuangW,YeX.Anexplicit scattering,Non-weakly mixing example and weak disjointness [J].Nonlinearity,2002,15:1-14.
[21] BlockL.Diffeomorphisms obtained From endomrphisms.Thans Amer Math soc,1975,214:403-413.
[22] LiSH.Dynamical properties of the shift maps on the inverse limt space,Ergodthe dynamsys,1992,12:95-108.
[23] ChenL,LiSH.shadowing property for inverse limt space.proc Amer Mathsoc,1992,115:573-581.
[24]耿祥义.Li-York混沌的充要条件[J].数学学报，2001,929-932.
[25]刘增荣.混沌研究中的解析方法[M].上海：上海教育出版社，2002:1-5.
[26]李继彬.混沌与Melnikov方法[M].重庆：重庆大学出版社，1989：7-11.
[27]张伟年.动力系统基础[M].北京：高等教育出版社，2001：52-71.
[28]张筑生.微分动力系统原理[M]．北京:科学出版社，1987：11-15.
[29]徐园芬，戴振祥．一类描述五分Cantor集的拟移位映射[J]．浙江师范大学学报（自然科学版），2010，33(4)：387-391.
[30] Coiiet.P,Eckmam,J.P,Iterated maps on the Interval as Dynamics systems,Birkhauser,Boston,1980.
[31] Li,T.Yorke,J.A, Reriod3implies chaos, Amer Math Monthly.1975,82:985-992.
[32] J.Milnor,W.Thurston.Dynamical System[J].Lect,Notesin,math.1988,1342:469.
[33] S.-L.Peng and L.-m.Du，Dualstar products and symbolic dymamics of Lorenz mapswith the same entropy [J]. Phys.Lete.A.1999，261：63.
[34]庄云标.符号空间上左拟移位映射[J].台州学院学报，2003，25（6）：37-39.
[35] Z-X.Chen,K-F Cao and S-L.Peng,symbolic dynamics anaiysis of topologicalentropy and its multifractal structure[J].Phys.Rev.E,1995,51:1985.
[36] MinQ.Journal of MengZi Tenchers’College,2001,23(24):39.
[37]陈凤娟.双边拟移位映射的混沌性[J].浙江师范大学学报，2002,25(3):221-223.
[38] BargeM,MartinJ.chaos,periodicity and snakelike continua.Trans Amer Math soc,1985,289:355-365.
[39]熊金城，陈二才.强混合的保测度变换引起的混沌.中国科学（A），1996,26（1）：961-967.
[40]汪火云，熊金城．拓扑遍历映射的一些性质[J]．数学学报，2004，4795):859－866.
[41]廖公夫，范钦杰．拓扑熵为零且schweizer－smital混沌的极小子转移[J]．中国科学，1997,27(9):770－774．
[42]夏道行．实变函数论与泛函分析[M]．北京：高等教育出版社，1979．
[43]廖公夫，范钦杰．混沌与ss混沌不等价[J]．数学年刊，A辑，6（2000），749－754．
[44]王立冬，陈知之．符号空间一类极小子转移的混沌性[J]．大连民族学院学报，20035（2）：63－65．
[45]孙丽华，赵恩良，刘丹．非周期回复点生成子转移的混沌性质[J]．沈阳建筑大学学报，2009,25（5）：1019－1020．
[46]王立冬，廖公夫．Σ上非弱几乎周期的回复点集和ss混沌集[J]．数学物理学报，2006,26A（5）：778－784．