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In 1885 Poincaré asked when the solutions of the plane autonomous system<br> | In 1885 Poincaré asked when the solutions of the plane autonomous system<br> | ||
x' = y + q(x,y) = Q(x,y)<br> | x' = y + q(x,y) = Q(x,y)<br> | ||
| - | y' = -x - p(x,y) = -P(x,y)<br><math> | + | y' = -x - p(x,y) = -P(x,y)<br><math></math> |
are stable in the neighbourhood of the equilibrium (x,y)=(0,0). | are stable in the neighbourhood of the equilibrium (x,y)=(0,0). | ||
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== Some history == | == 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. | + | 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. | 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. | ||
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== Our work == | == 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. | + | 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 [[ | + | 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> | 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]. | Our program centerfocus is freely available at [http://sourceforge.net/projects/centerfocus/files/ sourceforge]. | ||
Current revision
About centerfocus.de
On this website we present our research on the Poincaré center problem in degree 3 obtained by applying a 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 Lectures on Celestial Mechanics by Siegel, Moser and Kalme. The Poincaré center problem is treated in § 27
The Poincaré center problem
In 1885 Poincaré asked when the solutions of the plane autonomous system
x' = y + q(x,y) = Q(x,y)
y' = -x - p(x,y) = -P(x,y)
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 si 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 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 Zi be the locus where the first 
focal values vanish. We obtain heuristic information about the variety Z13 and Z14 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 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 findings .
The statistics from our experiments and some of the analyzed points are stored in a public accessible database.
Our program centerfocus is freely available at sourceforge.
