This paper aims to refine and expand the theoretical and application framework of higher-dimensional digital chaotic system (HDDCS). Topological mixing for HDDCS is strictly proved theoretically at first. Topological mixing implies Devaney's definition of chaos in a compact space, but not vice versa. Therefore, the proof of topological mixing promotes the theoretical research of HDDCS. Then, a general design method for constructing HDDCS via loop-state contraction algorithm is given. The construction of the iterative function uncontrolled by random sequences (hereafter called iterative function) is the starting point of this research. On this basis, this paper put forward a general design method to solve the construction problem of HDDCS, and several examples illustrate the effectiveness and feasibility of this method. The adjacency matrix corresponding to the designed HDDCS is used to construct the chaotic Echo State Network (ESN) for predicting Mackey-Glass time series. Compared with other ESNs, the chaotic ESN has better prediction performance and is able to accurately predict a much longer period of time.

中文翻译：

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通过循环状态收缩算法构建高维数字混沌系统


本文旨在完善和扩展高维数字混沌系统（HDDCS）的理论和应用框架。首先对HDDCS的拓扑混合进行了严格的理论证明。拓扑混合暗示了德瓦尼对紧凑空间中混沌的定义，但反之则不然。因此，拓扑混合的证明促进了HDDCS的理论研究。然后，给出了通过循环状态收缩算法构建HDDCS的通用设计方法。构造不受随机序列控制的迭代函数（以下简称迭代函数）是本研究的出发点。在此基础上，提出了解决HDDCS构建问题的通用设计方法，并通过几个算例说明了该方法的有效性和可行性。与设计的HDDCS对应的邻接矩阵用于构建混沌回波状态网络（ESN）来预测Mackey-Glass时间序列。与其他ESN相比，混沌ESN具有更好的预测性能，并且能够准确预测更长的时间段。