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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.

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