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= &nbsp;&nbsp;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>
&nbsp;x' = y + q(x,y) = Q(x,y)<br>
&nbsp;y' = -x - p(x,y) = -P(x,y)<br><math>a+b=d</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 s_i 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 Z_i be the locus  where the first i focal values vanish. We obtain heuristic information about the variety Z_{13} and Z_{14} 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_publications | publications]].

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].