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Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials. (English) Zbl 1238.34074

Summary: Using the algorithm presented in [J. Giné and X. Santallusia, Appl. Math. 28, 17, No. 1, 17–20 (200; Zbl 1022.34028)] the Poincaré-Liapunov constants are calculated for polynomial systems of the form

${x}^{\text{'}}=-y+{P}_{n}\left(x,y\right),\phantom{\rule{3.33333pt}{0ex}}{y}^{\text{'}}=x+{Q}_{n}\left(x,y\right),$

where ${P}_{n}$ and ${Q}_{n}$ are homogeneous polynomials of degree n. The objective of this work is to calculate the minimum number of ideal generators i.e., the number of functionally independent Poincaré-Liapunov constants, through the study of the highest fine focus order for $n=4$ and $n=5$ and compare it with the results that give the conjecture presented in [J. Giné, Appl. Math. Comput. 188, No. 2, 1870–1877 (2007; Zbl 1124.34018)]. Moreover, the computational problems which appear in the computation of the Poincaré-Liapunov constants and the determination of the number of functionally independent ones are also discussed.

##### MSC:
 34C23 Bifurcation (ODE) 34C07 Theory of limit cycles of polynomial and analytic vector fields 37C10 Vector fields, flows, ordinary differential equations