SYMBOLIC APPROXIMATION OF PERIODIC SOLUTIONS AND TRAJECTORIES OF HENON--HEILES' SYSTEM BY THE NORMAL FORM METHOD

Victor Edneral

Date: July 19th (Friday)
Time: ??.??
Abstract
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).

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