# Lopatin A. K.

### Normalization and averaging on compact lie groups in nonlinear mechanics

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 47–67

We consider the method of normal forms, the Bogolyubov averaging method, and the method of asymptotic decomposition proposed by Yu. A. Mitropol’skii and the author of this paper. Under certain assumptions about group-theoretic properties of a system of zero approximation, the results obtained by the method of asymptotic decomposition coincide with the results obtained by the method of normal forms or the Bogolyubov averaging method. We develop a new algorithm of asymptotic decomposition by a part of the variables and its partial case — the algorithm of averaging on a compact Lie group. For the first time, it became possible to consider asymptotic expansions of solutions of differential equations on noncommutative compact groups.

### Bogolyubov averaging and normalization procedures in nonlinear mechanics. IV

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1044–1068

In this paper, we apply the theory developed in parts I-III [*Ukr. Math. Zh.*,**46**, No. 9, 1171–1188; No. 11, 1509–1526; No. 12, 1627–1646 (1994)] to some classes of problems. We consider linear systems in zero approximation and investigate the problem of invariance of integral manifolds under perturbations. Unlike nonlinear systems, linear ones have centralized systems, which are always decomposable. Moreover, restrictions connected with the impossibility of diagonalization of the coefficient matrix in zero approximation are removed. In conclusion, we apply the method of local asymptotic decomposition to some mechanical problems.

### Bogolyubov averaging and normalization procedures in nonlinear mechanics. III

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1627–1646

We describe the technique of normalization based on the method of asymptotic decomposition in the space of representation of a finite-dimensional Lie group. The main topics of the theory necessary for understanding the method are outlined. Models based on the Van der Pol equation are investigated by the method of asymptotic decomposition in the space of homogeneous polynomials (the space of representation of a general linear group in a plane) and in the space of representation of a rotation group on a plane (ordinary Fourier series). The comparison made shows a dramatic decrease in the necessary algebraic manipulations in the second case. We also discuss other details of the technique of normalization based on the method of asymptotic decomposition.

### Bogolyubov averaging and normalization procedures in nonlinear mechanics. II

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1509–1526

By using a new method suggested in the first part of the present work, we study systems which become linear in the zero approximation and have perturbations in the form of polynomials. This class of systems has numerous applications. The following fact is even more important: Our technique demonstrates how to generalize the classical method of Poincaré-Birkhoff normal forms and obtain new results by using group-theoretic methods. After a short exposition of the general theory of the method of asymptotic decomposition, we illustrate the new normalization technique as applied to models based on the Lotka-Volterra equations.

### Bogolyubov averaging and normalization procedures in nonlinear mechanics. I

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1171–1188

We suggest a new method for asymptotic analysis of nonlinear dynamical systems based on group-the-oretic methods. On the basis of the Bogolyubov averaging method, we develop a new normalization procedure — “asymptotic decomposition.” We clarify the contribution of this procedure to the interpretation and development of the averaging method for systems in the standard form and systems with several fast variables. According to this method, the centralized system is regarded as a direct analog of the system averaged in Bogolyubov's sense. The operation of averaging is interpreted as the Bogolyubov projector, i.e., the operation of projection of an operator onto the algebra of centralizer.

### Asymptotic decomposition of completely integrable pfaffian systems with small parameter

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1988. - 40, № 3. - pp. 349-356

### Basis of the algorithm of asymptotic decomposition for a finite number of approximations

Ukr. Mat. Zh. - 1987. - 39, № 6. - pp. 732-737

### Asymptotic decomposition of differential systems with a small parameter

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 194-204

### Asymptotic decomposition of differential systems with a small parameter in the representation space of a finite-dimensional Lie group

Ukr. Mat. Zh. - 1987. - 39, № 1. - pp. 56–64

### Asymptotic decomposition of systems of nonlinear ordinary differential equations

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1984. - 36, № 1. - pp. 35 - 45