Limit cycle bifurcations from a non-degenerate center. (English) Zbl 1391.34073
Summary: 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 \(\dot x = -y +P_n(x,y), \dot y=x+Q_n(x,y)\), where \(P_{n}\) and \(Q_{n}\) are a homogeneous polynomial of degree n. We use center bifurcation rather than multiple Hopf bifurcations, used a previous work [the author and J. Mallol, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, e132–e137 (2009; Zbl 1238.34074)], 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 [the author, Appl. Math. Comput. 188, No. 2, 1870–1877 (2007; Zbl 1124.34018)].
MSC:
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
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