Symbolic Approximation of Periodic Solutions and
Trajectories of Henon--Heiles' System by the Normal Form
Method
Date: July 19th (Friday)
Time: 17:20-17:30
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).
