离散与间断动力系统的Conley指标理论及应用
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摘要
Conley指标由Charles C.Conley在处理天体力学中的微分方程时首先提出。它是定义在任意孤立不变集上的同伦不变量,是Morse指标的推广。Conley指标既可证明不变集的存在性,又包含稳定性,同时还具有对微小扰动保持不变的连续不变性。基于Morse分解的Conley指标可以刻划不变集的诸多复杂动力学行为。这些良好的性质使Conley指标从诞生之日起就得到了充分的重视和广泛的应用。目前在动力系统的各领域几乎都已建立了Conley指标理论,惟独没有间断动力系统的Conley指标。本文解决了这一难题,给出了某一类间断动力系统(有限分片连续映射)的Conley指标。此外,我们还研究了离散Conley指标与分支点的关系,以及用Conley指标对Ikeda映射的混沌性作计算机严格辅助证明。
具体来说,论文首先将流上的素孤立不变集推广到离散系统的情形,证明了素孤立不变集的不交性;给出了利用Conley指标判断离散动力系统分支点存在的充分条件。我们还首次从范畴角度研究了动力系统族范畴中,怎样的态射有相同分支点的问题,得到了两族系统在某一条件下有一致C-分支点的充分条件。这一条件比两族系统完全拓扑共轭弱,因此是有理论意义的,可以使我们从相对较简单的或已知分支点状况的系统来推知未知系统的分支点。接着,我们把编码映射的图的概念从分片等距映射扩展至一般的有限分片连续映射,并以此为工具定义出了有限分片连续映射的Conley指标。详细论述了不连续性对定义指标的困难,并设计了一系列的方法加以解决。最终得到的指标仍然有Wazewski性质,虽弱于局部紧空间的连续映射的Conley指标,但比非局部紧空间上的连续映射的Conley指标要强,因此是有意义的.我们所给的指标是连续映射指标的真推广,可以用来证明不变集的存在性。最后,我们用区间算法、Conley指标和计算同调等工具,研究了在某一参数下的Ikeda映射的混沌性。用计算机严格辅助的方法,证明了给定参数下的Ikeda映射含有极小周期分别为2和3的周期点。利用连接此2,3周期点的轨道,构造了相应的指标偶,并最终证明了Ikeda映射在某一不变集上的限制与一有限型子移位拓扑半共轭。这一有限型子移位有正的拓扑熵,因此原系统混沌。
The Conley index, introduced by Charles C. Conley in his work in dealing with the differential equations of celestial mechanics, is a homotopy invariant defined on any isolated invariant sets. It is a generalization of the traditional Morse index. Conley index can be used to prove both the existence and the stability of the invariant sets, and also has the continuity property which makes the index unchanged by small disturbation. Assisted by the Morse decompositions, some complex dynamical properties can be characterized by the Conley index. All of the above perfect properties make the Conley index attract many attentions and cover extensive applications since its appearance. It is well known that the Conley index theory has been established in almost every field of dynamical systems. However, so far as we know, there is still no Conley index theory in discontinuous dynamical systems. In this thesis, we aim to tackle this hard problem and we give the definition of the Conley index for a kind of discontinuous dynamical systems, i.e., finite piecewise continuous maps. Other works include the study of the relation between discrete Conley index and bifurcation points, and the rigorous computer assisted proof of the chaotic property of the Ikeda map by Conley index. In details, the main contents of this thesis are as follows.
We extend the definition of prime isolated invariant sets from flow to discrete dynamical systems, and show that any two prime isolated invariant sets do not intersect; we present a sufficient condition to detect bifurcation points by the Conley index. We originally use the viewpoint of categories to study those morphisms in the categories of the family of dynamical systems, which may have corresponding bifurcation points. We obtain a sufficient condition for two systems to have the same C-detectable bifurcation points. This condition is weaker than the conditon for complete topological conjugacy of two families of systems, so it is significant and useful for deducing from relatively simple systems or systems with known bifurcation points to the systems with unknown bifurcation points.
We extend the graph of coding map from piecewise isometries to general finite piecewise continuous maps, by which we give the definition of the Conley index for finite piecewise continuous maps. We specifically discuss the difficulties caused by the discontinuity while defining the Conley index, and we design a series of methods to overcome these problems. The index we obtained still possess the Wazewski property, though weaker than the general Conley index of continuous maps, it is stronger than the index defined on continuous maps without compactness. This implies that our definition of index is meaningful. The index we designed is a true extension of the index of continuous maps and it can be utilized to prove the existence of invariant sets.
We study the chaotic properties of the Ikeda map by tools of Interval Algorithm, Conley Index and Computational Homology. We prove that the Ikeda map under the given parameters contains minimal periodic points with periods 2 and 3. By the orbits connecting the period 2, 3 points, we construct the corresponding index pair, and prove that there exists a semi-conjugacy between some invariant set of Ikeda map and a shift of finite type. The positive entropy of the shift of finite type implies that the Ikeda map is chaotic under the given parameters.
具体来说,论文首先将流上的素孤立不变集推广到离散系统的情形,证明了素孤立不变集的不交性;给出了利用Conley指标判断离散动力系统分支点存在的充分条件。我们还首次从范畴角度研究了动力系统族范畴中,怎样的态射有相同分支点的问题,得到了两族系统在某一条件下有一致C-分支点的充分条件。这一条件比两族系统完全拓扑共轭弱,因此是有理论意义的,可以使我们从相对较简单的或已知分支点状况的系统来推知未知系统的分支点。接着,我们把编码映射的图的概念从分片等距映射扩展至一般的有限分片连续映射,并以此为工具定义出了有限分片连续映射的Conley指标。详细论述了不连续性对定义指标的困难,并设计了一系列的方法加以解决。最终得到的指标仍然有Wazewski性质,虽弱于局部紧空间的连续映射的Conley指标,但比非局部紧空间上的连续映射的Conley指标要强,因此是有意义的.我们所给的指标是连续映射指标的真推广,可以用来证明不变集的存在性。最后,我们用区间算法、Conley指标和计算同调等工具,研究了在某一参数下的Ikeda映射的混沌性。用计算机严格辅助的方法,证明了给定参数下的Ikeda映射含有极小周期分别为2和3的周期点。利用连接此2,3周期点的轨道,构造了相应的指标偶,并最终证明了Ikeda映射在某一不变集上的限制与一有限型子移位拓扑半共轭。这一有限型子移位有正的拓扑熵,因此原系统混沌。
The Conley index, introduced by Charles C. Conley in his work in dealing with the differential equations of celestial mechanics, is a homotopy invariant defined on any isolated invariant sets. It is a generalization of the traditional Morse index. Conley index can be used to prove both the existence and the stability of the invariant sets, and also has the continuity property which makes the index unchanged by small disturbation. Assisted by the Morse decompositions, some complex dynamical properties can be characterized by the Conley index. All of the above perfect properties make the Conley index attract many attentions and cover extensive applications since its appearance. It is well known that the Conley index theory has been established in almost every field of dynamical systems. However, so far as we know, there is still no Conley index theory in discontinuous dynamical systems. In this thesis, we aim to tackle this hard problem and we give the definition of the Conley index for a kind of discontinuous dynamical systems, i.e., finite piecewise continuous maps. Other works include the study of the relation between discrete Conley index and bifurcation points, and the rigorous computer assisted proof of the chaotic property of the Ikeda map by Conley index. In details, the main contents of this thesis are as follows.
We extend the definition of prime isolated invariant sets from flow to discrete dynamical systems, and show that any two prime isolated invariant sets do not intersect; we present a sufficient condition to detect bifurcation points by the Conley index. We originally use the viewpoint of categories to study those morphisms in the categories of the family of dynamical systems, which may have corresponding bifurcation points. We obtain a sufficient condition for two systems to have the same C-detectable bifurcation points. This condition is weaker than the conditon for complete topological conjugacy of two families of systems, so it is significant and useful for deducing from relatively simple systems or systems with known bifurcation points to the systems with unknown bifurcation points.
We extend the graph of coding map from piecewise isometries to general finite piecewise continuous maps, by which we give the definition of the Conley index for finite piecewise continuous maps. We specifically discuss the difficulties caused by the discontinuity while defining the Conley index, and we design a series of methods to overcome these problems. The index we obtained still possess the Wazewski property, though weaker than the general Conley index of continuous maps, it is stronger than the index defined on continuous maps without compactness. This implies that our definition of index is meaningful. The index we designed is a true extension of the index of continuous maps and it can be utilized to prove the existence of invariant sets.
We study the chaotic properties of the Ikeda map by tools of Interval Algorithm, Conley Index and Computational Homology. We prove that the Ikeda map under the given parameters contains minimal periodic points with periods 2 and 3. By the orbits connecting the period 2, 3 points, we construct the corresponding index pair, and prove that there exists a semi-conjugacy between some invariant set of Ikeda map and a shift of finite type. The positive entropy of the shift of finite type implies that the Ikeda map is chaotic under the given parameters.
引文
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