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
1
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)
Victor Edneral
CASC 2007. Bonn, September 16-20
2
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)
Victor Edneral
CASC 2007. Bonn, September 16-20
3
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.
Victor Edneral
CASC 2007. Bonn, September 16-20
4
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.
Victor Edneral
CASC 2007. Bonn, September 16-20
5
Problem Formulation
Victor Edneral
CASC 2007. Bonn, September 16-20
6
Victor Edneral
CASC 2007. Bonn, September 16-20
7
Victor Edneral
CASC 2007. Bonn, September 16-20
8
Victor Edneral
CASC 2007. Bonn, September 16-20
9
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 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.
Victor Edneral
CASC 2007. Bonn, September 16-20
10
The Normal Form Method
,n}
Victor Edneral
CASC 2007. Bonn, September 16-20
11
Victor Edneral
CASC 2007. Bonn, September 16-20
12
Victor Edneral
CASC 2007. Bonn, September 16-20
13
Main Algorithm
Victor Edneral
CASC 2007. Bonn, September 16-20
14
Victor Edneral
CASC 2007. Bonn, September 16-20
15
Victor Edneral
CASC 2007. Bonn, September 16-20
16
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
17
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.
Victor Edneral
CASC 2007. Bonn, September 16-20
18
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.
Victor Edneral
CASC 2007. Bonn, September 16-20
19
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 nonnegligible 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.
Victor Edneral
CASC 2007. Bonn, September 16-20
20
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.
Victor Edneral
CASC 2007. Bonn, September 16-20
21
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.
Victor Edneral
CASC 2007. Bonn, September 16-20
22
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.
Victor Edneral
CASC 2007. Bonn, September 16-20
23
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
24