CenterfocusWiki - New pages [en]

Steiner abstract

2011-01-21T09:26:00Z

CfWikiSysop:

=="Examining components of the Poincaré center variety with interpolation"==
====Johannes Steiner, Bachelor thesis 2011====

<br>
We use lifts calculated with centerfocus to identify components of the center-variety in degree 3 over <math>F_{29}</math>. <br>
Then we analyze the components by checking if the differential forms are Darboux-integrable and comparing them to the families of {Zoladek's} lists. <br>
Finally we make an attempt to parametrize the componoents of codimension 11.

Centerfocus findings

2011-01-17T10:34:33Z

CfWikiSysop:

==Preprints==
* [http://arxiv.org/pdf/1012.3402 A survey of the Poincaré Center Problem in degree 3 using finite field heuristics ], [http://www.crcg.de/wiki/User:Bothmer Hans-Christian Graf v. Bothmer], [http://www.crcg.de/wiki/User:Kroeker Jakob Kröker]

==Publications==
* [http://arxiv.org/abs/math/0505547 Experimental results for the Poincaré center problem], [http://www.crcg.de/wiki/User:Bothmer Hans-Christian Graf v. Bothmer] (including an appendix with M. Cremer), NoDEA, Volume 14, Numbers 5-6 / December 2007.

All computations can be performed with our most recent program available at [http://sourceforge.net/projects/centerfocus/files/ sourceforge]. [ [[cf:strudel_programs.html | code at submission time]] ]

* "Focal values of plane cubic centers", [http://www.crcg.de/wiki/User:Bothmer Hans-Christian Graf v. Bothmer], [http://www.crcg.de/wiki/User:Kroeker Jakob Kröker], Qualitative Theory of Dynamical Systems, Volume 9, Number 1-2 (2010): 319-324, DOI [http://dx.doi.org/10.1007/s12346-010-0031-8 10.1007/s12346-010-0031-8], [http://arxiv.org/abs/0902.3415 arXiv:0902.3415v1].

==Theses==

'''Diploma  thesis''' [http://ulrich.rhein.name/diplom/da.pdf Das Poincarésche Zentrumsproblem: Komponenten der Zentrumsvarietät ], Ulrich Rhein, 2008 [ [http://ulrich.rhein.name/diplom/ Abstract ] ]   [ [http://sourceforge.net/projects/poincareisecs/ source code] ]

'''Bachelor thesis''' [http://www.centerfocus.de/theses/js.pdf Untersuchung der Komponenten der Zentrumsvarietät mit Interpolation], Johannes Steiner, 2011 [ [[Steiner_abstract | Abstract]] ] <br>
[ source code ('''macaulay-packages/LiftpointInterpolation.m2''') is currently available from the svn trunk of the centerfocus project, checkout with<br>
svn co https://centerfocus.svn.sourceforge.net/svnroot/centerfocus/strudel/trunk/
]

Centerfocus main

2011-01-12T10:33:52Z

CfWikiSysop:

__NOTOC__
=   About <tt>centerfocus.de</tt> =

On this website we present our research on the ''Poincaré center problem'' in degree 3 obtained by applying a [http://www.math.nyu.edu/~tschinke/books/finite-fields/final/01_bothmer.pdf heuristic method] of Frank-Olaf Schreyer.

The Poincaré center problem is derived from the question about the stability of our solar system. For background you might want to
read the [http://books.google.com/books?id=f_oOSGY2rtoC&printsec=frontcover&dq=siegel+mechanics&source=bl&ots=oCxX6eqpFp&sig=M2hqYK-dEDODV7TeNBqFSKwQd24&hl=en&ei=WtLrTOLrHNCXOsLP3YcB&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBkQ6AEwAA#v=onepage&q&f=false Lectures on Celestial Mechanics] by Siegel, Moser and Kalme. The ''Poincaré center problem'' is treated in [http://books.google.com/books?id=f_oOSGY2rtoC&lpg=PP1&ots=oCxX6eqpFp&dq=siegel%20mechanics&pg=PA198#v=onepage&q&f=false § 27]

== The Poincaré center problem ==

In 1885 Poincaré asked when the solutions of the plane autonomous system<br>
 x' = y + q(x,y) = Q(x,y)<br>
 y' = -x - p(x,y) = -P(x,y)<br><math></math>
are stable in the neighbourhood of the equilibrium (x,y)=(0,0).

If the solutions are
stable (i.e. closed curves around the orgin) one says that the system has a ''center'' at the origin.

If the solutions are not stable (i.e spirals) the system has a ''focus''.

== Some history ==

Poincaré showed in 1885, that there exists an infinite number of so called focal polynomials <math>s_i</math> in the coefficients of P and Q that vanish if and only if the system above has a center.
The values of these polynomials are called focal values. If the first n-1 focal values vanish, but the n'th doesn't, the system has a n-th order focus.

In 1934 Max Frommer presented an algorithm to compute the focal polynomials and analysed the degree 2 case. We use his algorithm to calculate focal polynomials and focal values.

From then on many authors worked on the degree 2 case, which is completely understood today. A good starting point for this story is [http://www.ams.org/journals/tran/1993-338-02/S0002-9947-1993-1106193-6/S0002-9947-1993-1106193-6.pdf D. Schlomiuks 1993 article] and the references therein.

In 1996 Zoladek gave a list of 52 families of degree 3 differential forms known to have a center. The complete classification of degree 3 centers remains unsolved.

== Our work ==

Let <math>Z_i</math> be the locus where the first <math>i</math> focal values vanish. We obtain heuristic information about the variety <math>Z_{13}</math> and <math>Z_{14}</math> respectively by evaluating its defining equations at random points over a small finite field. For this we calculate the focal values of a plane autonomous system using an C++ implementation of a modified algorithm from Kay Moritzen (see his [http://www.uni-bayreuth.de/departments/math/org/mathe6/publ/da/moritzen/diplom.html diploma thesis], in german) who in turn build on the work of Frommer 1934. To speedup the calculation we parametrized the first three focal values.

The interested reader will find more details in our [[centerfocus_findings | findings ]].

The statistics from our experiments and some of the analyzed points are stored in a public accessible [[cf:QueryDatabase.html | database]].<br>

Our program centerfocus is freely available at [http://sourceforge.net/projects/centerfocus/files/ sourceforge].

Centerfocus contact

2011-01-12T09:50:05Z

Bothmer:

__NOTOC__
<table border=0 width=600>
<tr>
<td valign=baseline >[[Image:H-Chr_v_Bothmer.png|H-Chr_v_Bothmer.png]]</td>
<td>

<table width=400>
<TR valign=baseline>
<TH align=left valign="top"></TH>
<TD> <h4>Hans-Christian v. Bothmer</h4>
</TD>
</TR>

<TR valign=baseline>
<TH align=left valign="top">Address: </TH>
<TD>
[https://www.math.uni-hamburg.de/home/bothmer/ Universität Hamburg]<BR>
Fachbereich Mathematik<BR>
Bundesstraße 55 (Geomatikum)<BR>
D-20146 Hamburg Germany
</TD>
</TR>
<TR>
<TH align=left>Room: </TH>
<TD align=left>218</TD>
<TD></TD>
</TR>
<TR>
<TH align=left>Tel.: </TH>
<TD align=left>+49 40 42838-5149 </TD>
<TD></TD>
</TR>
<TR>
<TH align=left valign="top">e-mail: </TH>
<TD align=left> hcvbothmer at gmail.com
</TD>
</TR>
</table>
</td>
</tr>
</table>
<br>
<br>
<table border=0>
<tr>
<td valign=baseline >[[Image:Kroeker_2010_125x154a.png| ]]</td>
<td>
<table>
<TR valign=baseline>
<TH align=left valign="top"></TH>
<TD> <h4>Jakob Kröker </h4>
</TD>
</TR>

<TR>
<TH ALIGN=left valign="top">e-mail: </TH>
<TD ALIGN=left> jakobkroeker.academic at spaceship-earth.net
</TD>
</TR>
</table>
</td>
</tr>
</table>

Centerfocus publications

2011-01-12T09:49:18Z

CfWikiSysop: New page: * [http://arxiv.org/abs/math/0505547 Experimental results for the Poincaré center problem], [http://www.crcg.de/wiki/User:Bothmer Hans-Christian Graf v. Bothmer] (including an appendix wi...

* [http://arxiv.org/abs/math/0505547 Experimental results for the Poincaré center problem], [http://www.crcg.de/wiki/User:Bothmer Hans-Christian Graf v. Bothmer] (including an appendix with M. Cremer), NoDEA, Volume 14, Numbers 5-6 / December 2007.

All computations can be performed with our most recent program available at [http://sourceforge.net/projects/centerfocus/files/ sourceforge]. [ [[cf:strudel_programs.html | code at submission time]] ]

* "Focal values of plane cubic centers", [http://www.crcg.de/wiki/User:Bothmer Hans-Christian Graf v. Bothmer], [http://www.crcg.de/wiki/User:Kroeker Jakob Kröker], Qualitative Theory of Dynamical Systems, Volume 9, Number 1-2 (2010): 319-324, DOI [http://dx.doi.org/10.1007/s12346-010-0031-8 10.1007/s12346-010-0031-8], [http://arxiv.org/abs/0902.3415 arXiv:0902.3415v1].