Abstract
Paper contains the results of the analysis of the laws of functioning of discrete dynamical systems, as mathematical models of which, using the apparatus of geometric images of automatons, are used numerical sequences which interpreted as sequences of second coordinates of points of geometric images of automatons. The geometric images of the laws of the functioning of the automaton are reduced to numerical sequences and numerical graphs. The problem of constructing an estimate of the complexity of the structures of such sequences is considered. To analyze the structure of sequences, recurrence forms are used that give characteristics of the relative positions of elements in the sequence. The parameters of recurrent forms are considered, which characterize the lengths of the initial segments of sequences determined by recurrence forms of fixed orders, the number of changes of recurrent forms required to determine the entire sequence, the place of change of recurrence forms, etc. All these parameters are systematized into the special spectrum of dynamic parameters used for the recurrent determination of sequences, which is used as a means of constructing estimates of the complexity of sequences. In this paper, it also analyzes return sequences (for example, Fibonacci numbers), for the analysis of the properties of which characteristic sequences are used. The properties of sequences defining approximations of fundamental mathematical constants (number e, pi number, golden ratio, Euler constant, Catalan constant, values of Riemann zeta function, etc.) are studied. Complexity estimates are constructed for characteristic sequences that distinguish numbers with specific properties in a natural series, as well as for characteristic sequences that reflect combinations of the properties of numbers.
Highlights
The study of the functioning of discrete systems is the subject of research by thousands of scientists around the world
The traditional used mathematical models of discrete systems are defined by symbolic structures: tables, graphs, matrices, logical equations, etc
We study the properties of fundamental mathematical quantities using a spectrum of dynamic parameters, determine the specificity of the structure of the arrangement of elements in each sequence
Summary
The study of the functioning of discrete systems is the subject of research by thousands of scientists around the world. The traditional used mathematical models of discrete systems are defined by symbolic structures: tables, graphs, matrices, logical equations, etc These models are not suitable for use in the control and diagnostics of large-scale and complex discrete systems (see, for example [9,10]), due to the enormous dimension. In [1, 2], a new approach based on numerical structures by professor Tverdokhlebov V.A. was proposed and developed to set the laws of functioning of control objects and technical diagnostics, which are complex systems. This approach allows you to use powerful idealizations of classical continuous mathematics: infinitesimal, actual infinity, summation of infinite series, passage to the limit, etc
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