关于几类系统混沌性的研究
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摘要
混沌学是近四十年发展起来的一个跨学科的学术分支,其本质在于研究确定的非线性系统中的不确定性,尤其关注的是不确定性中所蕴含的各种规律,从而达到对系统加以控制的目的。随着基础科学和应用科学的发展,混沌理论和应用研究已经成为非线性科学中的重要课题之一。
本文在简述混沌理论发展的基础上,首先较系统地回顾了常见的混沌定义,然后在五个系统中集中讨论连续自映射的混沌性质,尤其是Li-Yorke混沌、Devaney混沌、spatio-temporal混沌、(F_1,F_2)-混沌、分布混沌、稠混沌、对初值的敏感依赖和Li-Yorke敏感的一些特征。获得如下六个方面的结果:
1、在拓扑空间上证明了ω-混沌和四种Devaney混沌(DevC, EDevC, MDevC,WMDevC)在拓扑共轭下是保持的。从而在一般度量空间上这些混沌也在拓扑共轭下保持。通过举反例得出:在一般度量空间中拓扑共轭不保持Li-Yorke混沌。研究发现:“紧空间+拓扑共轭”或“一致共轭”的条件才能保持Li-Yorke混沌。此外,还证明了在一般度量空间上拓扑一致共轭保持Auslander-Yorke混沌、敏感性、分布混沌、序列分布混沌、稠混沌和稠δ-混沌。
2、通过对线性序拓扑系统上连续自映射的周期点的研究,得出:如果马蹄存在,则周期点存在;如果奇周期点存在,则马蹄存在。通过对周期点的不稳定流形的研究,得出:相邻不动点构成的区间含于其中一个不动点的单侧不稳定流形之中;若周期点集有限,则不动点p的不稳定流形被p分成两个区间,分别是p的左、右侧不稳定流形。研究稠密轨道的性质,得出:若点轨道稠密,则拓扑传递;若偶次迭代点集稠密,则k (mod s)次迭代点集稠密。
3、对与Belousov-Zhabotinsky振荡反应相关的一类耦合映象格子,在文中所定义的度量下,系统具有(F_1,F_2)-混沌性(或Li-Yorke混沌性、分布混沌性)的一个充分条件是原始映射具有该种混沌性。若将诱导映射限制在空间的对角线上,那么系统的稠混沌性、稠δ-混沌性、spatio-temporal混沌性、敏感性或Li-Yorke敏感性也有类似的充分条件。但如果改变空间度量,原始映射的混沌性不一定能保证耦合系统的混沌性。通过对系统拓扑熵的研究,得出:系统的拓扑熵不小于原始映射的拓扑熵;若原始映射的拓扑熵大于0,则系统是ω-混沌的。
4、研究了一类权移位算子的分布混沌性,得到:权移位算子是分布ε-混沌的(ε取大于0小于空间直径的任意值),也是一致分布混沌的,并且这些性质在乘积运算下保持;然后计算出该权移位算子的准测度为1。
5、在非自治系统中证明了映射序列f_1,∞的混沌性与fn,∞的混沌性之间是充要条件的关系。还证明了如果映射序列f_1,∞具有P-混沌性质,那么乘积映射f[m]
1,∞(m为一个正整数)也具有P-混沌性质。其中P-混沌是指Li-Yorke混沌,分布混沌,敏感, Li-Yorke敏感,或稠Li-Yorke敏感。并且,当映射序列f_1,∞一致收敛时,逆命题也成立。
6、研究了双寡头博弈系统中的Cournot映射,得到: Cournot映射在全空间上的Li-Yorke混沌性(或分布混沌性,序列分布混沌性)和限制在MPE-集上的Li-Yorke混沌性(或分布混沌性,序列分布混沌性)等价。并举例说明了对敏感性和Li-Yorke敏感性而言,这个结论不成立。
最后,本文对所做工作进行了系统的总结,对所研究课题中还需要深入研究的地方进行了展望,为将来的研究奠定了一定的基础。
Chaos is an interdisciplinary academic branch which is developed for nearly fourdecades. Its essence is to study the uncertainty in nonlinear system, especially theregularities inherent in the uncertainty. These properties can be used to control thesystems. With the development of basic science and applied science, the research onchaos theory and applied has become one of the main topics of nonlinear science.
On the basis of the discussion about the development of chaos, this dissertationfirst reviews the common definitions of chaos. Then the dissertation focus on thechaotic properties of continuous self-maps in five systems, especially the characteristicsof Li-Yorke chaos, Devaney chaos, spatio-temporal chaos,(F_1,F_2)-chaos, distributionalchaos, dense chaotic, sensitivity, and Li-Yorke sensitivity. The following six aspects ofresults are obtained:
1. On topological spaces, ω-chaos and four Devaney chaos (DevC, EDevC,MDevC, WMDevC) are preserved under topological conjugation. So as in generalmetric spaces. While, on general metric spaces, topological conjugate do not keepLi-Yorke chaos. It needs conditions ‘compact space+topological conjugate’ or‘uniformly conjugation’. Additionally, it is proved that, on general metric spaces, theAuslander-Yorke chaos, sensitivity, distributional chaos, distributional chaos in asequence, dense chaos, and dense δ-chaos are all maintained under uniformlyconjugation.
2. Considering of periodic points of continuous self-maps on linear orderedtopological system, it is pointed that there exist periodic points if horseshoes existed,and odd periodic points imply the existence of horseshoes. According to the study of theunstable manifolds, we prove that the interval with endpoints of two adjacent fixedpoints is contained in the unilateral unstable manifold of one of the endpoints. If thecontinuous self-map has finitely many periodic points, then the unstable manifold of afixed point p is divided into two zones, namely, the left and right unilateral unstablemanifolds of p. Through the research of dense orbits, we point that dense orbits implytopological transitivity. And if the set of even iterate points is dense, then the set ofk (mod s)iterate points is dense too.
3. On a class of coupled map lattice related to the Belousov-Zhabotinsky oscillating reaction, with the metric defined in the fourth chapter, a sufficient conditionfor the system is(F_1, F_2)-Chaos (Li-Yorke chaos, or distributional chaos) is obtained.If the induced mapping is limited on the diagonal of the space, a similar sufficientcondition will be obtained for the system is dense chaos, dense δ-chaos,spatio-temporal chaos, sensitivity, or Li-Yorke sensitivity. However, these sufficientconditions do not always established if the metric changes. Chaotic original maps cannot guarantee the chaoticity of the system. The dissertation also points out that thetopological entropy of the system is not less than topological entropy of the originalmapping. If the topological entropy of original mapping is greater than0, then thesystem is ω-chaotic.
4. According to study distributional chaotic of a weighted shifts operator. Weproved that the weighted shifts operator is distributional ε-chaos and uniformlydistributional chaos (where ε is any value greater than0and less than diameter of thespace). And these chaotic properties are preserved under product operation. Then, it isproved that the principal measure of this weighted shifts operator is1.
5. In non-autonomous system, it is proved that the chaotic off_1,∞is a necessaryand sufficient condition for the chaotic offn,∞. If the map sequencef_1,∞is P-chaos,then the product mapping sequencef[m]
1,∞(m is a positive integer) is P-chaos too.There P-chaos denote Li-Yorke chaos, distributional chaos, sensitivity, Li-Yorkesensitivity, or Li-Yorke sensitivity. And iff_1,∞converges uniformly, then the converseof the above conclusion is true.
6. The Li-Yorke chaotic (distributional chaotic, or distributional chaotic in asequence) on the whole space of Cournot mapping is equivalent to the one limited onMPE-set. And an example is illustrated to show that this conclusion does not hold forsensitivity or Li-Yorke sensitivity.
Finally, the dissertation summarized the work, prospected the work which isrequired in-depth study. Thus, the dissertation laid some foundation for the futureresearch.
本文在简述混沌理论发展的基础上,首先较系统地回顾了常见的混沌定义,然后在五个系统中集中讨论连续自映射的混沌性质,尤其是Li-Yorke混沌、Devaney混沌、spatio-temporal混沌、(F_1,F_2)-混沌、分布混沌、稠混沌、对初值的敏感依赖和Li-Yorke敏感的一些特征。获得如下六个方面的结果:
1、在拓扑空间上证明了ω-混沌和四种Devaney混沌(DevC, EDevC, MDevC,WMDevC)在拓扑共轭下是保持的。从而在一般度量空间上这些混沌也在拓扑共轭下保持。通过举反例得出:在一般度量空间中拓扑共轭不保持Li-Yorke混沌。研究发现:“紧空间+拓扑共轭”或“一致共轭”的条件才能保持Li-Yorke混沌。此外,还证明了在一般度量空间上拓扑一致共轭保持Auslander-Yorke混沌、敏感性、分布混沌、序列分布混沌、稠混沌和稠δ-混沌。
2、通过对线性序拓扑系统上连续自映射的周期点的研究,得出:如果马蹄存在,则周期点存在;如果奇周期点存在,则马蹄存在。通过对周期点的不稳定流形的研究,得出:相邻不动点构成的区间含于其中一个不动点的单侧不稳定流形之中;若周期点集有限,则不动点p的不稳定流形被p分成两个区间,分别是p的左、右侧不稳定流形。研究稠密轨道的性质,得出:若点轨道稠密,则拓扑传递;若偶次迭代点集稠密,则k (mod s)次迭代点集稠密。
3、对与Belousov-Zhabotinsky振荡反应相关的一类耦合映象格子,在文中所定义的度量下,系统具有(F_1,F_2)-混沌性(或Li-Yorke混沌性、分布混沌性)的一个充分条件是原始映射具有该种混沌性。若将诱导映射限制在空间的对角线上,那么系统的稠混沌性、稠δ-混沌性、spatio-temporal混沌性、敏感性或Li-Yorke敏感性也有类似的充分条件。但如果改变空间度量,原始映射的混沌性不一定能保证耦合系统的混沌性。通过对系统拓扑熵的研究,得出:系统的拓扑熵不小于原始映射的拓扑熵;若原始映射的拓扑熵大于0,则系统是ω-混沌的。
4、研究了一类权移位算子的分布混沌性,得到:权移位算子是分布ε-混沌的(ε取大于0小于空间直径的任意值),也是一致分布混沌的,并且这些性质在乘积运算下保持;然后计算出该权移位算子的准测度为1。
5、在非自治系统中证明了映射序列f_1,∞的混沌性与fn,∞的混沌性之间是充要条件的关系。还证明了如果映射序列f_1,∞具有P-混沌性质,那么乘积映射f[m]
1,∞(m为一个正整数)也具有P-混沌性质。其中P-混沌是指Li-Yorke混沌,分布混沌,敏感, Li-Yorke敏感,或稠Li-Yorke敏感。并且,当映射序列f_1,∞一致收敛时,逆命题也成立。
6、研究了双寡头博弈系统中的Cournot映射,得到: Cournot映射在全空间上的Li-Yorke混沌性(或分布混沌性,序列分布混沌性)和限制在MPE-集上的Li-Yorke混沌性(或分布混沌性,序列分布混沌性)等价。并举例说明了对敏感性和Li-Yorke敏感性而言,这个结论不成立。
最后,本文对所做工作进行了系统的总结,对所研究课题中还需要深入研究的地方进行了展望,为将来的研究奠定了一定的基础。
Chaos is an interdisciplinary academic branch which is developed for nearly fourdecades. Its essence is to study the uncertainty in nonlinear system, especially theregularities inherent in the uncertainty. These properties can be used to control thesystems. With the development of basic science and applied science, the research onchaos theory and applied has become one of the main topics of nonlinear science.
On the basis of the discussion about the development of chaos, this dissertationfirst reviews the common definitions of chaos. Then the dissertation focus on thechaotic properties of continuous self-maps in five systems, especially the characteristicsof Li-Yorke chaos, Devaney chaos, spatio-temporal chaos,(F_1,F_2)-chaos, distributionalchaos, dense chaotic, sensitivity, and Li-Yorke sensitivity. The following six aspects ofresults are obtained:
1. On topological spaces, ω-chaos and four Devaney chaos (DevC, EDevC,MDevC, WMDevC) are preserved under topological conjugation. So as in generalmetric spaces. While, on general metric spaces, topological conjugate do not keepLi-Yorke chaos. It needs conditions ‘compact space+topological conjugate’ or‘uniformly conjugation’. Additionally, it is proved that, on general metric spaces, theAuslander-Yorke chaos, sensitivity, distributional chaos, distributional chaos in asequence, dense chaos, and dense δ-chaos are all maintained under uniformlyconjugation.
2. Considering of periodic points of continuous self-maps on linear orderedtopological system, it is pointed that there exist periodic points if horseshoes existed,and odd periodic points imply the existence of horseshoes. According to the study of theunstable manifolds, we prove that the interval with endpoints of two adjacent fixedpoints is contained in the unilateral unstable manifold of one of the endpoints. If thecontinuous self-map has finitely many periodic points, then the unstable manifold of afixed point p is divided into two zones, namely, the left and right unilateral unstablemanifolds of p. Through the research of dense orbits, we point that dense orbits implytopological transitivity. And if the set of even iterate points is dense, then the set ofk (mod s)iterate points is dense too.
3. On a class of coupled map lattice related to the Belousov-Zhabotinsky oscillating reaction, with the metric defined in the fourth chapter, a sufficient conditionfor the system is(F_1, F_2)-Chaos (Li-Yorke chaos, or distributional chaos) is obtained.If the induced mapping is limited on the diagonal of the space, a similar sufficientcondition will be obtained for the system is dense chaos, dense δ-chaos,spatio-temporal chaos, sensitivity, or Li-Yorke sensitivity. However, these sufficientconditions do not always established if the metric changes. Chaotic original maps cannot guarantee the chaoticity of the system. The dissertation also points out that thetopological entropy of the system is not less than topological entropy of the originalmapping. If the topological entropy of original mapping is greater than0, then thesystem is ω-chaotic.
4. According to study distributional chaotic of a weighted shifts operator. Weproved that the weighted shifts operator is distributional ε-chaos and uniformlydistributional chaos (where ε is any value greater than0and less than diameter of thespace). And these chaotic properties are preserved under product operation. Then, it isproved that the principal measure of this weighted shifts operator is1.
5. In non-autonomous system, it is proved that the chaotic off_1,∞is a necessaryand sufficient condition for the chaotic offn,∞. If the map sequencef_1,∞is P-chaos,then the product mapping sequencef[m]
1,∞(m is a positive integer) is P-chaos too.There P-chaos denote Li-Yorke chaos, distributional chaos, sensitivity, Li-Yorkesensitivity, or Li-Yorke sensitivity. And iff_1,∞converges uniformly, then the converseof the above conclusion is true.
6. The Li-Yorke chaotic (distributional chaotic, or distributional chaotic in asequence) on the whole space of Cournot mapping is equivalent to the one limited onMPE-set. And an example is illustrated to show that this conclusion does not hold forsensitivity or Li-Yorke sensitivity.
Finally, the dissertation summarized the work, prospected the work which isrequired in-depth study. Thus, the dissertation laid some foundation for the futureresearch.
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