Some conditions for a center of autonomous and nonautonomous systems (Q1324498)

The author considers systems of the form \(x'= y+ p(x,y)\), \(y'= -x+ q(x,y)\) in which \(p\) and \(q\) are homogeneous polynomials of degree \(n\) and gives sufficient conditions for the origin to be a center. These conditions have a rather complicated form and generalize previous work of the author and \textit{N. G. Lloyd} [Proc. R. Soc. Edinburgh, Sect. A 105, 129-152 (1987; Zbl 0618.34026)]. They are obtained by first transforming to polar coordinates \((r,\theta)\) and letting \(s= r^{n-1}[1- r^{n-1} g(\theta)]^{-1}\), where \(g(\theta)= \cos \theta q(\cos \theta,\sin\theta)- \sin \theta p(\cos \theta,\sin\theta)\), to transform the original system to the form \({ds\over d\theta}= A(\theta) s^ 3+ B(\theta) s^ 2\). The conditions are requirements on the functions \(A\) and \(B\), obtained from a study of the differential equation \({dz\over dt}= A(t) z^ 3+ B(t) z^ 2\), with \(t\) real, \(A\) and \(B\) real-valued, and \(z\) complex.