A symbolic approximation of periodic solutions of the Henon–Heiles system by the normal form method

This article describes the computer algebra application of the normal form method for building analytic approximations for all (including complex) local families of periodic solutions in a neighborhood of the stationary point to the Henon–Heiles system. The families of 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 of the 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 of autonomous nonlinear systems with smooth enough right sides near a stationary point. The method also provides a new convenient graphic representation for the phase portrait of such systems. There is possibility for searching polynomial integral manifolds of the system. For the given example they can be evaluated in a finite form.