  In recent years, the generation of rigorously provable chaos in finite
precision digital domain has made a lot of progress in theory and practice,
this article is a part of it. It aims to improve and expand the theoretical and
application framework of higher-dimensional digital chaotic system (HDDCS). In
this study, topological mixing for HDDCS is strictly proved theoretically at
first. Topological mixing implies Devaney's definition of chaos in compact
space, but not vise versa. Therefore, the proof of topological mixing improves
and expands the theoretical framework 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 article put forward a general design method to solve the
problem of HDDCS construction, and a number of 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 the Mackey-Glass time series prediction. The prediction
accuracy is improved with the increase of the size of the reservoir and the
dimension of HDDCS, indicating that higher-dimensional systems have better
prediction performance than lower-dimensional systems.
