Construct Loop Node Shrinkage Algorithm For High-dimensional Digital Chaotic Systems

In recent years,many theoretical and practical advances have been made in the field of high-dimensional digital chaotic systems on finite state machines that can generate real provable chaos,and this thesis is part of that progress to refine and extend the theoretical framework of high-dimensional digital chaotic systems as well as to provide more examples of applications,providing topological mixing proofs for the literature on the theoretical side.Topological mixing implies Devaney chaos in tight spaces,but the converse is not necessarily true.Thus,the topological mixing proof completes and extends the theoretical framework for high-dimensional numerical chaotic systems.This paper gives a general design method for constructing high-dimensional digital chaotic systems based on loop node shrinkage algorithm,in which the construction of iterative functions not controlled by random sequences(hereafter referred to as iterative functions)is the entry point of this study.The design of iterative functions is used as the entry point of the study,and then simple per-bit operations and random sequences are used to create iterative functions,such that the corresponding states in the high-dimensional digital chaotic systems The connectivity of the transformation graph do not change with finite precision.Noting that the strong connectivity of the state transition diagram in high-dimensional digital chaotic systems is the path to digital chaos,the iterative function is constructed according to the loop node shrinkage algorithm so that the state transition diagram of a medium-to the high-dimensional digital chaotic system is strongly connected.A loop is a non-empty directed path in the state transition graph where the first node and the last node are the same.A set of nodes in the loop in the state transition diagram is replaced with a single new node(and all edges of the nodes in the set that are connected to other points are also connected to this new node),and all subsequent self-loops(edges connected to themselves)and polygons are removed.New state Each edge in the transformed graph is identified with its counterpart in the original graph,or in the case of polygons,a single remaining edge is identified with any corresponding edge in the original graph.The process is repeated until no loop is found,and only the strongly connected graph eventually shrinks to a single node.This is the algorithm for strong connectivity discrimination of state transition graphs based on loop node shrinkage,based on which a general design method is proposed to solve the construction problem of high-dimensional digital chaotic systems,and several examples are used to illustrate the effectiveness and feasibility of the method.In terms of application,the echo state network,also known as reserve pool computation,uses a reserve pool consisting of randomly sparsely connected neurons as the hidden layer,so the state transition graph topology of the high-dimensional digital chaotic system constructed in this thesis is used to replace the topology structure of the hidden layer of the echo state network to obtain the chaotic echo state network,and then the Mackey-Glass time The prediction accuracy increases with the increase of the network size and the number of dimensions of the high-dimensional digital chaos system,indicating that the high-dimensional digital chaos system has better prediction performance than the low-dimensional digital chaos system.