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On algorithm of the normal form building

On algorithm of the normal form building. Victor Edneral Skobeltsyn Institute of Nuclear Physics of Moscow State University Leninskie Gory, Moscow, 11999 1 , Russia edneral@theory.sinp.msu.ru. Introduction

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On algorithm of the normal form building

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  1. On algorithm of the normal form building Victor Edneral Skobeltsyn Institute of Nuclear Physics of Moscow State University Leninskie Gory, Moscow, 119991, Russia edneral@theory.sinp.msu.ru Victor Edneral CASC 2007. Bonn, September 16-20

  2. Introduction • The normal form method is based on a transformation ofan ODEs system to a simpler set called the normal form.The importance of this method for an analyzing of ODEsnear stationary point has been recognized for a long time. • We will speak here about the resonant normal form. • Poincare (1875) • Dulac (1912) • Bruno (1964) Victor Edneral CASC 2007. Bonn, September 16-20

  3. Since the system MAO [Rom, A., Mechanized Algebraic Operations (MAO).Celestial Mechanics,1 (1970) 301–319]by which was checked Delaney’stheory of a motion of the Moon there were createdmany programs for creatingnormal 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) Victor Edneral CASC 2007. Bonn, September 16-20

  4. The discussed algorithm was mainly implemented in 1985: Edneral V.F., Khrustalev O.A., Normalizing transformation for systemsof nonlinear ordinary differential equations. International Conference onComputer 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. Victor Edneral CASC 2007. Bonn, September 16-20

  5. Laterthis 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 Workshopon 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 formto high order nonlinear ODEs using MATHEMATICA; Nuclear Inst. andMethods in Physics Research, A, 502/2-3, pp. 643 – 645. Victor Edneral CASC 2007. Bonn, September 16-20

  6. Problem Formulation Victor Edneral CASC 2007. Bonn, September 16-20

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  10. For this paper, we assume that system (2) satisfies the • following assumptions: • the system is autonomous and has polynomial nonlinearities; • 0 is a stationary point and the system will be studiedneary = 0; • the linear part of the right hand side is diagonal andnot all eigenvalues are zero, i.e. Λ≠0. • Remark that the last assumption is a restriction of a currentimplementation rather the approach itself. Victor Edneral CASC 2007. Bonn, September 16-20

  11. The Normal Form Method ,n} Victor Edneral CASC 2007. Bonn, September 16-20

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  14. Main Algorithm Victor Edneral CASC 2007. Bonn, September 16-20

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  17. 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. Victor Edneral CASC 2007. Bonn, September 16-20

  18. We group terms of series in homogeneoussums in variable order and we store the value of this order with the corresponding sum. Forexample if we have a truncated series: It is obviously that this form is very convenient for a summation. And objects in thisrepresentation can be very effective multiplied in the sense of truncated series – for excludingfrom results negligible for corresponding order of truncation terms, it is enough to eliminatefrom 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 weneed to square only sum of the first two homogeneous groups above (with 2 and 3 commonorders), not more. Victor Edneral CASC 2007. Bonn, September 16-20

  19. Computer Algebra Implementation of the Normal Form Method The calculation of the coefficients of the normal form (5) andcorresponding transformation(4) with respect of (7) and (8) was implemented as the NORT package. Earlier attemptsof 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 inarbitrary dimensions. Victor Edneral CASC 2007. Bonn, September 16-20

  20. In addition to procedures for arithmetic operations with series, thereare special procedures for the creation of normal forms and procedures for substitutions,for calculations of some elementary functions (when it is possible), for differentiating, forprinting and for inverting multivariate power series, etc. It contains also special proceduresfor a calculation of Lyapunov’s values. The NORT can be used as a separate program oras a REDUCE package. Besides series, expressions in NORT can contain also non-negligible variables (parameters).There is implemented multivariate series-polynomial arithmetic. The complex-valuednumerical coefficients of the truncated power series-polynomials may be treated in three differentarithmetics: rational, modular, floating point and approximate rational. There are alsoseveral options for the output form of these numbers, the output is in a REDUCE readableform. Victor Edneral CASC 2007. Bonn, September 16-20

  21. The program uses an internal recurrence representation for its objects. Remark thata garbage collection time for different examples was smaller than 3% of evaluation time. Thiscan characterize the NORT package as a program with a good enough internal organization.Many important results were obtained by a computer with 1 MbyteRAM only. In 1993 the normal form till 12th order for the Henon─Heiles system took ~2 hours on a HP workstation. Victor Edneral CASC 2007. Bonn, September 16-20

  22. Unfortunately at this moment the NORT package has no friendly user interface yet. Sowe create a package for usage with MATHEMATICA package. This package works withtruncated multivariate formal power series. ThePolynomialSeries package can be accessedat www.mathsource.comsite. The existing version is enough for a support of a normal formmethod. The comparison of MATHEMATICA package with an earlier version of normal formpackage NORT written in LISP demonstrates that the calculations within the MATHEMATICAsystem are strong more flexible and convenient but are considerably slower than under theLISP. Victor Edneral CASC 2007. Bonn, September 16-20

  23. A cost of the algorithm A cost of the algorithm above is low in comparison with a cost of evaluation of the righthandside of the nonlinear system. Under such circumstances it is very important to calculatethe right-hand sides very economically, using so much as possible the fact that we need tocalculate at each step of (ii) the homogeneous termsof order k only and all terms oflower 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. Victor Edneral CASC 2007. Bonn, September 16-20

  24. 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. Victor Edneral CASC 2007. Bonn, September 16-20

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