Construction of methods for computing recurrence plots in space with a scalar product

Abstract

The methods for calculation of recurrence plots in space with a scalar product were developed. They enable examining the properties and features of the vector of states of dynamical systems of various complexity in natural and social spheres. A new scientific result is the development of scientific-methodical apparatus to calculate recurrence plots of vectors of states of systems in metric spaces expanded on the basis of scalar product. Two methods for calculation of recurrence plots for vectors of states of complex dynamic systems, which are highly informative, moderately complex and universal in dimensionality of the studied space of states, were proposed. In practice, the proposed methods can be used to calculate and compare recurrence plots of states of the studied systems in metric spaces of different dimensionality without additional normalizing. The functionality of the proposed methods was verified based on experimental observations of concentrations of formaldehyde, ammonia and carbon monoxide in the atmosphere of an industrial city. It was established that at the values of angular size of the area of 10° and 30°, the proposed method for calculation of recurrence plots is more informative, less complex and invariant relative to dimensionality of the space of states. It was shown that the methods for calculation of recurrence plots in space with scalar product make it possible to use them if there are short-time intervals of the absence of observations. It was experimentally determined that in some cases of parameters, the results of computation of recurrence plots based on the developed methods coincide with the results obtained when using the known methods. This indicates a more general nature of the proposed methods.