Limit cycle bifurcations from a non-degenerate center

摘要

In this work we discuss the computational problems which appear in the computation of the Poincaré–Liapunov constants and the determination of their functionally independent number. Moreover, we calculate the minimum number of Bautin ideal generators which give the number of small limit cycles under certain hypothesis about the generators. In particular, we consider polynomial systems of the form x˙=-y+Pn(x,y),y˙=x+Qn(x,y), where Pn and Qn are a homogeneous polynomial of degree n. We use center bifurcation rather than multiple Hopf bifurcations, used a previous work [19], to estimate the cyclicity of a unique singular point of focus–center type for n = 4, 5, 6, 7 and compare with the results given by the conjecture presented in [18].