Development of a self-adjusting method for calculating recurrent diagrams in a space with a scalar product

Abstract

A self-adjusting method for calculating recurrence diagrams has been developed. The proposed method is aimed at overcoming the metric-threshold uncertainty inherent in the known methods for calculating recurrence diagrams. The method provides invariance to the nature of the measured data, and also allows to display the recurrence of states, adequate to real systems of various fields. A new scientific result consists in the theoretical justification of the method for calculating recurrence diagrams, which is capable of overcoming the existing metric-threshold uncertainty of known methods on the basis of self-adjusting by measurements by improving the topology of the metric space. The topology is improved due to the additional introduction of the scalar product of state vectors into the operation space. This allowed to develop a self-adjusting method for calculating recurrence diagrams with increased accuracy and adequacy of the display of recurrence states of real systems. Moreover, the method has a relatively low computational complexity, providing invariance with respect to the nature of the irregularity of measurements.

Verification of the proposed method was carried out on the basis of experimental measurements of concentrations of gas pollutants of atmospheric air for a typical industrial city. The main gas pollutants of the atmosphere are formaldehyde, ammonia and nitrogen dioxide, caused by stationary and mobile sources of urban pollution. The obtained experimental verification results confirm the increased accuracy and adequacy of the display of the recurrence of atmospheric pollution states, as well as the invariance of the method with respect to the nature of the irregularity of measurements. It has been established that the accuracy of the method is influenced by the a priori boundary angular dimensions of the recurrence cone. It was shown that with a decrease in the boundary angular dimensions of the recurrence cone, the accuracy of the recurrence mapping of the real states of dynamical systems in the calculated diagrams increases. It was experimentally established that the accuracy and adequacy of the mapping of the recurrence states of real dynamical systems acceptable for applications is provided for a boundary angular size of the recurrence cone of 10° or less.