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Lie algebras and invariants for differential systems with projections on some mathematical model


Author: Neagu Natalia
Degree:doctor of
Speciality: 01.01.02 - Differential equations
Year:2018
Scientific advisers: Mihail Popa
doctor habilitat, professor, Institute of Mathematics and Computer Science of the ASM
Dumitru Cozma
doctor habilitat, associate professor (docent)
Institution: Institute of Mathematics and Computer Science of the ASM

Status

Term of presenting of the thesis 22 June, 2018

Abstract

Adobe PDF document0.21 Mb / in romanian

Thesis

CZU 517.925

Adobe PDF document 3.10 Mb / in romanian
125 pages


Keywords

polynomial differential system, ternary differential system of Darboux (Lyapunov-Darboux) type, stability of unperturbed motion, Lie algebra, invariant, comitant

Summary

ANNOTATION of the thesis "Lie algebras and invariants for differential systems with projections on some mathematical models", presented by Neagu Natalia for obtaining the Doctor degree in Mathematics, specialty 111.02 – differential equations. The thesis was elaborated at Tiraspol State University (Chișinău) and presented for defense in 2017. The language of the thesis is Romanian. It comprises 125 pages and has the following structure: Introduction, 4 Chapters, General Conclusions and Recommendations, Bibliography with 73 References. Research outcomes were reflected in 14 scientific works. Keywords: polynomial differential system, ternary differential system of Darboux (Lyapunov-Darboux) type, stability of unperturbed motion, Lie algebra, invariant, comitant. Field of study of the thesis: qualitative theory of dynamical systems, integrability of polynomial differential systems, stability of unperturbed motion. The purpose and objectives of the thesis are: to determinate the centro-affine invariant conditions of stability of unperturbed motion described by two-dimensional and ternary differential systems with polynomial nonlinearities; to investigate the critical and noncritical cases for such systems; to study the integrability of ternary differential systems of Darboux type and of Lyapunov-Darboux type. Novelty and scientific originality. For the first time there were used the methods of Lie algebras and of theory of algebraic invariants and comitants in study of stability of unperturbed motion described by two-dimensional and ternary differential systems with polynomial nonlinearities. The main scientific problem solved consists in approaching of some differential systems by Lie algebras and algebras of invariants, which contributed to obtain the centro-affine invariant conditions of stability of unperturbed motion described by two-dimensional and ternary differential systems with polynomial nonlinearities. This made possible to apply the obtained results in future investigation of concrete mathematical models. The significance of theoretical and practical values of the work. The results obtained in thesis are new and represent a new approach on the use of Lie algebras and theory of algebraic invariants and comitants in study of stability of unperturbed motion described by two-dimensional and ternary differential systems with polynomial nonlinearities, the integrability of ternary differential systems on some invariant varieties. Implementation of the scientific results. The obtained results can: be used in further investigation of the theory of stability of unperturbed motion described by multidimensional differential systems with polynomial nonlinearities using Lie algebras and the theory of invariants; be used in the study of some mathematical models describing processes from physics, medicine, biology, chemistry, economy, etc; serve as support for Master thesis and for teaching courses at the university level.