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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 ' =-y+P n (x,y),y ' =x+Q n (x,y),

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.

34C23Bifurcation (ODE)
34C07Theory of limit cycles of polynomial and analytic vector fields
37C10Vector fields, flows, ordinary differential equations