CA Seminar, 21 February, 2001

[Alexander Kryukov]

Dear Colleagues,

Computer Algebra seminar will at 17:00, 21 February, 2001 in NII Yadernoj
MGU, korpus Fiziki vysokih energij, room 2-15. 

               About Normal Form Method
                 Victor F. Edneral

                     Abstract

In this report we will use the algorithm based on the approach that was
develope d by A.D. Bruno for resonant normal form. The important advantage
of this approach is a possibil ity to investigate a wide class of
autonomous systems in united, easily algorithmized frame. In parti cular
it provides a constructive way for obtaining the approximations of local
families of periodic and conditionally periodic solutions in the form of
power/Fourier series for real families and in a form of power series in
time dependent exponents for complex ones. For this paper it is especially
im portant that the problem of convergence of used transformations is
investigated. This circumstanc e allows us to hope that approximations of
frequencies and corresponding periodic solutions fam ilies near stationary
points by finite formulas can be done with acceptable precision. Exce pt
solutions themselves we can find also approximations of origin conditions,
which initiate such periodic solutions. I.e. we can produce some elements
of a phase analyses.  It is possibl e to use the method for bifurcation
analysis of ODEs.

Another advantage of the used approach is an algorithmic simplicity of the
creat ion of the normal form and the corresponding transformations. We
have a direct recurrence f ormula for this procedure. The usage does not
demand keeping of some large intermediate results as it is in other
algorithms. The approach is free from a necessity to solve intermediate
systems of linear equations and from any restrictions on low resonance
cases.
 It is also possible to approximate by the proposed method the
non-periodic fami lies of solutions ("crude" case). The results are close
to the results of the Carleman linearizati on method. For periodic and
conditionally periodic cases the method is a generalization of the
Poincare-Lindstedt approach.

Below we describe the creation of the normal form and the application for
buildi ng of periodic solutions of well-known equations.

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More details see: http://theory.npi.msu.su/CA

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         Alexander Kryukov         Nuclear Physics Institute
                                   Moscow State University
                                   119899, Moscow
                                   Russia
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