Author : Alexey V. Myshev (Obninsk Institute for Nuclear Power Engineering, National Research Nuclear University MEPhI)

Title: Mathematical Modeling of the Dynamical Evolution of Small Solar System Bodies under Uncertainty

Abstracts:

The results of the doctoral dissertation entitled “Mathematical Modeling of the Dynamical Evolution of Small Solar System Bodies under Uncertainty” are presented. The relevance of this research is multifaceted and stems from several factors. Current trends in scientific progress and its future evolution are associated both with ensuring the safety of space exploration, the ecology of near-Earth space, its “colonization,” and other related issues, as well as with addressing the challenges involved in spacecraft operation and space exploration

The feasibility of practically addressing these problems largely depends on resolving theoretical and applied issues related to mathematical modeling technologies for the dynamical evolution of small bodies under uncertainty, including:

1. uncertainty in the initial and boundary conditions and parameters of the system being modeled;
2. factors determining the theoretical and practical limits of predictability;
3. the inability of classical theories and models to describe and explain many phenomena associated with the dynamical evolution of small bodies;
4. the inadequacy of the criteria used to represent the investigated reality and assess the modeling results.

In view of the above, the main objectives of the research are:

1. to develop and establish a new integrative concept and methodology for investigating and analyzing the dynamical evolution of small Solar System bodies under uncertainty;
2. to construct a theory of mathematical modeling of the dynamical evolution of small bodies under uncertainty, based on new principles and approaches to computational experiments, as well as methods for the analysis, verification, validation, and identification of models and modeling results.

To achieve these objectives, the dissertation develops and implements a formalism for the mathematical modeling of the dynamical evolution of small bodies under uncertainty, in which such phenomena as singular phase transitions, energy and information exchange, capture and ejection, trajectory scattering and focusing, intermittency, fractality, and other “phenomena” arise.

A theory of the virtual perspective method has been constructed for the development of mathematical modeling technologies under multifactor uncertainty and computational-environment constraints. A probabilistic map method has been developed to implement technologies for constructing solutions to the modeled problems using Monte Carlo algorithms based on the Bernoulli scheme.

To model the dynamical evolution of small bodies within the framework of fuzzy problems, a numerical theory for constructing solutions to such problems on virtual lattices has been developed as a new paradigm of neural-network modeling technologies. A synergetic approach involving the application of fractal methods to the processing and analysis of modeling results and observational data has also been proposed

The application of the developed theories, methods, and technologies for modeling, processing, and analyzing the results has made it possible, for the first time, to:

1. demonstrate that the dynamical evolution of small bodies is governed by such mechanisms as intermittency in a fuzzy environment and a non-averaged description of stochastic processes;
2. identify the mechanism responsible for the transition from regular dynamics with a complex structure to irregular dynamics;
3. describe the fractal properties arising during close encounters;
4. demonstrate that the developed theory and methods make it possible to determine the conditions and constraints under which the modeling process remains controllable and manageable, thereby ensuring an adequate representation of the actual dynamical evolution being modeled, as well as to determine the prediction horizon under uncertainty and obtain other previously unknown results.

The uniqueness of the results determines the scientific novelty of the research. A new theory of mathematical modeling of the dynamical evolution of small bodies under uncertainty has been constructed. Methods and technologies for modeling fuzzy problems under uncertainty and computational-environment constraints have been developed and methodologically substantiated. Unique and previously unknown solutions to astronomical problems within the framework of the restricted (N)-body problem have been obtained

The practical significance of the research is confirmed by patents for inventions issued by Rospatent for software-based and virtual mathematical modeling systems. The research results have been reported in publications in journals included in the lists of the Higher Attestation Commission and Scopus, as well as in presentations delivered at Russian and international conferences.