Computation of focus quantities of three-dimensional polynomial systems

• Main Authors: Valery G. Romanovski, Douglas S. Shafer
• Format: Article
• Language: English
• Published: Academic Journals Center of Shanghai Normal University 2014-10-01
• Series: Journal of Shanghai Normal University (Natural Sciences)
• Subjects: Integrability; center conditions; focus quantities
• Online Access: http://qktg.shnu.edu.cn/zrb/shsfqkszrb/ch/reader/create_pdf.aspx?file_no=201405010&flag=1&year_id=2014&quarter_id=5

Fix a collection of polynomial vector fields on <i>R</i>³ with a singularity at the origin,for every one of which the linear part at the origin has two pure imaginary and one non-zero eigenvalue.Some such systems admit a local analytic first integral,which then defines a local center manifold of the system.Conditions for existence of a first integral are given by the vanishing certain polynomial or rational functions in the coefficients of the system called focus quantities.In this paper we prove that the focus quantities have a structure analogous to that in the two-dimensional case and use it to formulate an efficient algorithm for computing them.