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Entropy of nonautonomous dynamical systems

非自主动力系统的熵

基本信息

批准号：
241933299
负责人：
Privatdozent Dr. Christoph Kawan
金额：
--
依托单位：
Courant Institute of Mathematical Sciences
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2013
资助国家：
德国
起止时间：
2012-12-31 至 2014-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/241933299?language=en
关键词：
Entropy nonautonomous dynamical systems

项目摘要

The aim of this research project is the advancement of the entropy theory of nonautonomous deterministic dynamical systems. In the 1990s, the notion of topological entropy for nonautonomous systems has been introduced by S.Kolyada and L. Snoha, and in the subsequent years it has been investigated by several authors. This notion generalizes the classical notion of topological entropy in the theory of autonomous dynamical systems, which, as arguably the most important invariant of such systems, has been investigated thoroughly since the 1960s and nowadays is considered as well-understood. The topological entropy of a system is a real number which can be regarded as a measure for the global exponential complexity of the orbit structure. The more complicated or chaotic the trajectories depend on their initial values, the higher is the value of the entropy. The quantity introduced by Kolyada and Snoha is a similar measure for systems with time-dependent dynamics, which, for example, are generated by differential equations with explicitly time-dependent right-hand sides. Beyond that, it generalizes several other concepts of entropy, in particular topological sequence entropy, topological entropy of systems with non-compact state space and topological entropy of random dynamical systems. Finally, there is also a relation to control-theoretic notions of entropy which are measures for minimal bit rates necessary to accomplish control tasks. The first objective of the research project is to establish the measure-theoretic counterpart of topological entropy for nonautonomous systems, the metric entropy, and to relate it to the topological entropy as this is accomplished for the corresponding notions in the autonomous theory via the variational principle. This principle, which asserts that the topological entropy is equal to the supremum of the metric entropies with respect to all invariant measures of the system, is the basis for most nontrivial results about topological entropy. Further topics to be treated are: The characterization of systems with vanishing entropy via their nonwandering sets, the entropy of hyperbolic systems, and relations between escape rates and entropy. All of these topics are also related to control-theoretic problems.

这个研究项目的目的是推进非自治确定性动力系统的熵理论。20世纪90年代，S.Kolyada和L. Snoha，并在随后的几年里，它已被调查的几个作者。这个概念概括了自治动力系统理论中的拓扑熵的经典概念，可以说是此类系统中最重要的不变量，自20世纪60年代以来已经被彻底研究，现在被认为是很好理解的。系统的拓扑熵是一个真实的数，它可以被看作是轨道结构的全局指数复杂性的度量。轨迹对初始值的依赖越复杂或越混乱，熵值就越高。Kolyada和Snoha引入的量是具有时间依赖动态的系统的类似度量，例如，由具有显式时间依赖右侧的微分方程产生。除此之外，还推广了熵的其它几个概念，特别是拓扑序列熵、非紧状态空间系统的拓扑熵和随机动力系统的拓扑熵。最后，也有一个关系到控制理论的熵概念，这是必要的最小比特率来完成控制任务的措施。该研究项目的第一个目标是建立非自治系统拓扑熵的测度论对应物，度量熵，并将其与拓扑熵联系起来，因为这是通过变分原理在自治理论中实现的相应概念。这个原则，它断言拓扑熵等于度量熵的上确界关于所有不变的措施的系统，是大多数非平凡的结果拓扑熵的基础。进一步的主题是：系统的特点消失熵通过其非游荡集，熵的双曲系统，以及逃逸率和熵之间的关系。所有这些主题也都与控制理论问题有关。