Victor F. Edneral

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.

Computer evaluation of cyclicity in planar cubic system

The paper describes an evaluation of c~clicity for a planar system of ODF, with a cubic polynomial right side. This problem is the important, as it is a suhproblem of the famous XIrI-th Hilbert problem. Estimations of cyclicity and c’onditious for a center art! calculated by direct computer alge}]ra methods for the cubic homogeneous and some extrnsirmsof thrhomogmrmuscase. In particular the 5 (complex) parametm case is investigated for the first time. For a lligllor{ter Lya~}llllov focllsclllantitiesc alculation a modular arithmetic is used, The NORT and GROEBNER packages under REDUCE-3.6 systcm wet-eapplied.

An algorithm for construction of normal forms

The normal form method is widely used in the theory of nonlinear ordinary differential equations (ODEs). But in practice it is impossible to evaluate the corresponding transformations without computer algebra packages. Here we describe an algorithm for normalization of nonlinear autonomous ODEs. Some implementations of these algorithms are also discussed.

Computer generation of normalizing transformation for systems of nonlinear ODE

This article describes the Standard LISP program for building a normal form and a corresponding normalizing transformation of a system of ordinary differential equations (ODE) in Bruno’s notation [1] up to the specified order. This program includes also a complete set of procedures of arithmetic for the truncated power series and input/output services. This gives us an opportunity to continue a treatment of obtained results autonomically or in a REDUCE environment. The program can work in a rational arithmetic or in an approximate rational arithmetic, or in a float point arithmetic. The program usage is illustrated by treating systems of weakly nonlinear ODES in the language of the truncated series. The approximate solution is produced from the normal form calculated up to enough high order and from the corresponding normalizing transformation. This method demonstrates rather good agreement with numerical solutions of some well known equations.

Investigation of the Double Pendulum System by the Normal Form Method in MATHEMATICA

The paper is devoted to the application of the normal form method to the investigation of the double pendulum system with the use of the MATHEMATICA computer algebra system.

Looking for Periodic Solutions of ODE Systems by the Normal Form Method

We describe usage of the normal form method and corresponding computer algebra packages for building an approximation of local periodic solutions of nonlinear autonomous ordinary differential equations (ODEs). For illustration a number of systems are treated.

Bifurcation Analysis of Low Resonant Case of the Generalized Henon - Heiles System

The paper describes the computer algebra application of the normal form method to bifurcation analysis of a low resonant case of the generalized Henon - Heiles system. A behavior of all local families of periodic solutions in system parameters is determined. Corresponding approximated solutions were checked by a comparison with the numerical solutions of the system.

About Normal Form Method

The paper describes usage of a normal form method for building an approximation of families of periodic solutions of nonlinear autonomous ordinary differential equations (ODEs). For illustration a center case and a limit circle axe chosen.

On Integrability of a Planar ODE System Near a Degenerate Stationary Point

We consider an autonomous system of ordinary differential equations, which is solved with respect to derivatives. To study local integrability of the system near a degenerate stationary point, we use an approach based on Power Geometry method and on the computation of the resonant normal form. For a planar 5-parametric example of such system, we found the complete set of necessary and sufficient conditions on parameters of the system for which the system is locally integrable near a degenerate stationary point.

On New Integrals of the Algaba-Gamero-Garcia System

We study local integrability of a plane autonomous polynomial system of ODEs depending on five parameters with a degenerate singular point at the origin. The approach is based on making use of the Power Geometry Method and the computation of normal forms. We look for the complete set of necessary conditions on parameters of the system under which the system is locally integrable near the degenerate stationary point. We found earlier that the sets of parameters satisfying these conditions consist of four two-parameter subsets in the full five-parameter co-space. Now we consider the special subcase of the case \(b^2 = 2/3\) and separate subsubcases when additional first integrals can exist. Here we have found two such integrals.