## The Cyclicity Problem for Two-dimensional Polynomial Systems

### Author(s):

**V. G. Romanovski**

Center for Applied Mathematics

and Theoretical Physics, University of Maribor},

CAMTP, Krekova, 2,

Maribor, SI-2000,

Slovenia.

valery.romanovsky@uni-mb.si

**A. S. Jarrah**

New Mexico State University,

Department of Mathematical Sciences,

New Mexico State University,

Las Cruces, NM 88003,

USA

ajarrah@nmsu.edu

**R. Laubenbacher**

Virginia Bioinformatics Institute,

1880 Pratt Drive Blacksburg,

VA 24061, USA

reinhard@almaren.bioinformatics.vt.edu

### Abstract:

The problem of small limit cycles bifurcations
(the cyclicity problem) is considered for the system with homogeneous
cubic nonlinearities

and for the cubic system

where akj are complex parameters and x=u+iv.
Considering as an example the first system we show that using
algorithms of computational commutative algebra one can easily solve the
cyclicity problem in the case when the ideal of focus quantities is a
radical ideal.

In the case of the second system it appears the ideal of focus quantities
is not a radical one. Nevertheless using the monoid structure of the focus
quantities we are able to find a basis of the ideal and to solve the cyclicity problem for "almost
all" values of parameters of the system.