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# Algorithms of the Asymptotic Nonlinear Analysis

• Alexander D. Bruno
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 5)

## Abstract

All local and asymptotic first approximations of a polynomial, of a differential polynomial and of a system of such polynomials may be selected algorithmically. Here the first approximation of a solution of the system of equations is a solution of the corresponding first approximation of the system of equations. The power transformations induce linear transformations of vector exponents and commute with the operation of selecting first approximations. In a first approximation of a system of equations they allow to reduce number of parameters and to reduce the presence of some variables to the form of derivatives of their logarithms. If the first approximation is the linear system, then in many cases the system of equations can be transformed into the normal form by means of the formal change of coordinates. The normal form is reduced to the problem of smaller dimension by means of the power transformation. Combining these algorithms, in many problems we can resolve a singularity, find parameters determining properties of solutions and obtain the asymptotic expansions of solutions. Some applications from Mechanics, Celestial Mechanics and Hydrodynamics are indicated.

## Keywords

Periodic Solution Normal Form Normal Cone Power Transformation Center Manifold
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media Dordrecht 2000

## Authors and Affiliations

• Alexander D. Bruno
• 1
1. 1.Department of MathematicsKeldysh Institute of Applied MathematicsMoscowRussia