Foundations of Computational Mathematics an Algorithmic Approach to Chain Recurrence Dedicated to Steve Smale on His 75th Birthday
Foundations of Computational Mathematics an Algorithmic Approach to Chain Recurrence Dedicated to Steve Smale on His 75th Birthday
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通讯作者：
W. Kalies;K. Mischaikow;R. Vandervorst
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作者：
W. Kalies;K. Mischaikow;R. Vandervorst
本文利用有限空间离散化给出了连续映射链回归集的一个新定义。这种方法允许隔离块的莫尔斯分解的组件，近似的链经常性设置任意接近以及离散近似的康利的李雅普诺夫函数的算法，rithmic建设。这是一个自然的框架，在其中开发计算技术的定性动力学分析，包括严格的计算机辅助证明。Conley基本分解定理及其对莫尔斯分解的推广是动力系统理论中的一个强有力的工具。然而，标准理论所建立的框架并不自然地导致用于近似链递归集的算法或计算方法，即，生成的莫尔斯分解或近似的一个李雅普诺夫函数的梯度状的系统的一部分。对于有限ε > 0，可以用ε-链递归集来近似链递归集，但是没有算法或计算技术来直接计算这个集。在本文中，我们提出了一种替代方法的基础上有限离散化和组合多值映射。这种方法有几个优点。这个理论的基本要素可以用简单明了的方式加以证明。此外，这些方法本质上是组合的，因此是算法的。该框架自然导致计算技术分析定性动力学，包括严格的计算机辅助证明，见，e。
In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algo-rithmic construction of isolating blocks for the components of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley's Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs. Conley's Fundamental Decomposition Theorem and its extension to Morse decom-positions is a powerful tool in dynamical systems theory. However, the framework on which the standard theory is built does not lead naturally to an algorithmic or computational approach for the approximation of the chain recurrent set, i.e., generation of Morse decompositions or the approximation of a Lyapunov function for the gradient-like part of the system. One can approximate the chain recurrent set by the ε-chain recurrent set for finite ε > 0, but there are no algorithmic or computational techniques for computing this set directly. In this paper, we present an alternative approach based on finite discretizations and combinatorial multivalued maps. This approach has several advantages. The basic elements of the theory can be proved in a straightforward manner. Moreover, the methods are inherently combinatorial and hence algorithmic. The framework leads naturally to computational techniques for analyzing qualitative dynamics including rigorous computer-assisted proofs, see, e.