Victor Edneral CASC 2007. Bonn, September 16-20 1 On algorithm of the normal form building Victor Edneral Skobeltsyn Institute of Nuclear Physics of Moscow

Victor Edneral CASC 2007. Bonn, September 16-201

On algorithm of the normal form building

Victor EdneralSkobeltsyn Institute of Nuclear Physics

of Moscow State UniversityLeninskie Gory, Moscow, 119991, Russia

[email protected]


Introduction

The normal form method is based on a transformation of an ODEs system to a simpler set called the normal form. The importance of this method for an analyzing of ODEs near stationary point has been recognized for a long time.

We will speak here about the resonant normal form.

• Poincare (1875)• Dulac (1912)• Bruno (1964)


Since the system MAO [Rom, A., Mechanized Algebraic Operations (MAO). Celestial Mechanics,1 (1970) 301–319] by which was checked Delaney’s theory of a motion of the Moon there were created many programs for creating normal forms and corresponding transformations.

For example:

K. Godziewski & A.J. Maciejewski(1990)I.I. Shevchenko & A.G. Sokolsky (1993)J. Mikram & F. Zinoun (2001)L. Vallier (1993)V.F. Zhuravlev & A.G. Petrov (2005)


The discussed algorithm was mainly implemented in 1985:

Edneral V.F., Khrustalev O.A., Normalizing transformation for systems of nonlinear ordinary differential equations. International Conference on Computer Algebra and its Applications in Theoretical Physics (Dubna, September 1985), ed. by Rostovtsev V.A. JINR D11–85–791, Dubna, 1986, pp. 219–224. In Russian.

The implementation above was written for the REDUCE system on the STANDARD LISP language.


Later this algorithm was rewritten for the MATHEMATICA system.

Edneral V.F., Khanin R., (2002) Multivariate Power Series and Nor-mal Form Calculation in Mathematica. Proceed. of the Fifth Workshop on Computer Algebra in Scientific Computing (CASC 2002, Big Yalta, Ukraine , September, 2002), ed.by Ganzha et al., Tech.Univ.Munchen, Munich, 2002, pp. 63–70.

Edneral V.F., Khanin R., (2003) Application of the resonant normal form to high order nonlinear ODEs using MATHEMATICA; Nuclear Inst. and Methods in Physics Research, A, 502/2-3, pp. 643 – 645.


Problem Formulation





For this paper, we assume that system (2) satisfies thefollowing assumptions:

• the system is autonomous and has polynomial nonlinearities;• 0 is a stationary point and the system will be studied near y = 0;• the linear part of the right hand side is diagonal and not all eigenvalues are zero, i.e. Λ≠0.

Remark that the last assumption is a restriction of a current implementation rather the approach itself.


The Normal Form Method

,n}




Main Algorithm




Main Ideas of the Implementation

• We should have an effective package for a truncated formal power series treatment;

• This package should have an internal representation with splitting terms to groups which are homogeneous in common powers of variables;

• Summation in (7) should been made without an enumeration of all possible values of the summation parameters.


We group terms of series in homogeneous sums in variable order and we store the value of this order with the corresponding sum. For example if we have a truncated series:

It is obviously that this form is very convenient for a summation. And objects in this representation can be very effective multiplied in the sense of truncated series – for excluding from results negligible for corresponding order of truncation terms, it is enough to eliminate from the multiplied groups the terms with common orders which are over the negligibleone. For example if we wish to calculate a square of the series above till the 5th order we need to square only sum of the first two homogeneous groups above (with 2 and 3 common orders), not more.


Computer Algebra Implementation of the Normal Form Method

The calculation of the coefficients of the normal form (5) and corresponding transformation (4) with respect of (7) and (8) was implemented as the NORT package. Earlier attempts of the author to compute sufficiently high orders of the normal form using REDUCE language internal representation of polynomials were not successful. Because of this, the NORT package was created. The NORT is written in Standard LISP and contains about 2000 operators. The NORT is a package of procedures to treat truncated multivariate formal power series in arbitrary dimensions.


In addition to procedures for arithmetic operations with series, there are special procedures for the creation of normal forms and procedures for substitutions, for calculations of some elementary functions (when it is possible), for differentiating, for printing and for inverting multivariate power series, etc. It contains also special procedures for a calculation of Lyapunov’s values. The NORT can be used as a separate program or as a REDUCE package.Besides series, expressions in NORT can contain also non-negligible variables (parameters). There is implemented multivariate series-polynomial arithmetic. The complex-valued numerical coefficients of the truncated power series-polynomials may be treated in three different arithmetics: rational, modular, floating point and approximate rational. There are also several options for the output form of these numbers, the output is in a REDUCE readable form.


The program uses an internal recurrence representation for its objects. Remark that a garbage collection time for different examples was smaller than 3% of evaluation time. This can characterize the NORT package as a program with a good enough internal organization. Many important results were obtained by a computer with 1 Mbyte RAM only.

In 1993 the normal form till 12th order for the Henon─Heiles system took ~2 hours on a HP workstation.


Unfortunately at this moment the NORT package has no friendly user interface yet. So we create a package for usage with MATHEMATICA package. This package works with truncated multivariate formal power series. The PolynomialSeries package can be accessed at www.mathsource.com site. The existing version is enough for a support of a normal form method. The comparison of MATHEMATICA package with an earlier version of normal form package NORT written in LISP demonstrates that the calculations within the MATHEMATICA system are strong more flexible and convenient but are considerably slower than under the LISP.


A cost of the algorithm

A cost of the algorithm above is low in comparison with a cost of evaluation of the right hand side of the nonlinear system. Under such circumstances it is very important to calculate the right-hand sides very economically, using so much as possible the fact that we need to calculate at each step of (ii) the homogeneous terms of order k only and all terms of lower orders are not changed during the later operations. During 24 hours you can calculate with 3 Ghz processor the normal form for 6 dimensional system till 8 order and for 2 dimensional system till 80 order.


Ideas for Future Implementations

• Optimization of a calculation of right hand side by storage of all preliminary calculated productions in RHS;

• Usage symmetries for a simplification of the calculations;

• Implementations for systems which are under Gnu Public License (GPL), such that MACSYMA and AXIOM.

Documents

Victor Edneral CASC 2007. Bonn, September 16-20 1 On algorithm of the normal form building Victor Edneral Skobeltsyn Institute of Nuclear Physics of Moscow