Open AccessThe application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems
Author(s) -
В. И. Никонов
Publication year - 2018
Publication title -
žurnal srednevolžskogo matematičeskogo obŝestva
Language(s) - English
Resource type - Journals
eISSN - 2587-7496
pISSN - 2079-6900
DOI - 10.15507/2079-6900.20.201803.295-303
Subject(s) - mathematics , lie group , partial differential equation , lie algebra , semi elliptic operator , pure mathematics , invariant (physics) , representation of a lie group , stability (learning theory) , linear map , nonlinear system , algebra over a field , differential operator , mathematical analysis , mathematical physics , computer science , physics , quantum mechanics , machine learning
The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.