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Symbolic Approximation of Periodic Solutions and
 Trajectories of Henon--Heiles' System by the Normal Form
                             Method
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Victor Edneral

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Date:  July 19th (Friday)\\
Time:  17:20-17:30\\

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Abstract
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This report describes the computer algebra application of the normal form method for building
analytical approximations for all (including complex) periodic orbits of Henon--Heiles' system in
a neighbourhood of the zero stationary point. The solutions are represented as truncated Fourier
series in approximated frequencies and the corresponding trajectories are described by
intersections of hypersurfaces which are defined by pieces of multivariate power series in phase
variables of the system. A comparison numerical values created by a tabulation of the
approximated solutions above with results of a numerical integration Henon--Heiles' system
displays a good agreement which is enough for a usage these approximate solutions for engineering
applications. Such approximations can be useful for a phase analysis of wide class autonomous
nonlinear systems with smooth enough right sides near an equilibrium point. The method also
provides a new convenient graphic representation for the phase portrait of such systems. If the
system has polynomial partial integrals they can be evaluated in described approach in a finite
form. The algorithm also is suitable for a search of all local conditionally periodic solutions
(invariant tori). 
