scholarly journals Lower bounds for the Hilbert number of polynomial systems

2012 ◽  
Vol 252 (4) ◽  
pp. 3278-3304 ◽  
Author(s):  
Maoan Han ◽  
Jibin Li
Keyword(s):  
Lower Bounds ◽  
Polynomial Systems ◽  
Hilbert Number

ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES IN A QUINTIC PLANAR VECTOR FIELD

2008 ◽  
Vol 18 (07) ◽  
pp. 1939-1955 ◽  
Author(s):  
YUHAI WU ◽  
YONGXI GAO ◽  
MAOAN HAN
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Hilbert Problem ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Analysis Method ◽  
Quintic Polynomial ◽  
Hilbert Number ◽  
Planar Vector ◽  
16Th Hilbert Problem

This paper is concerned with the number and distributions of limit cycles in a Z2-equivariant quintic planar vector field. By applying qualitative analysis method of differential equation, we find that 28 limit cycles with four different configurations appear in this special planar polynomial system. It is concluded that H(5) ≥ 28 = 52+ 3, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to the study of the second part of 16th Hilbert problem.

scholarly journals Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tao Li ◽  
Jaume Llibre
Keyword(s):  
Periodic Orbits ◽  
Lower Bounds ◽  
Limit Cycles ◽  
Averaging Method ◽  
Polynomial Systems ◽  
Upper Bounds ◽  
Piecewise Polynomial ◽  
Differential Systems ◽  
Discontinuity Line ◽  
Polynomial Differential Systems

<p style='text-indent:20px;'>In this paper we study the maximum number of limit cycles bifurcating from the periodic orbits of the center <inline-formula><tex-math id="M1">\begin{document}$ \dot x = -y((x^2+y^2)/2)^m, \dot y = x((x^2+y^2)/2)^m $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ m\ge0 $\end{document}</tex-math></inline-formula> under discontinuous piecewise polynomial (resp. polynomial Hamiltonian) perturbations of degree <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula> with the discontinuity set <inline-formula><tex-math id="M4">\begin{document}$ \{(x, y)\in\mathbb{R}^2: xy = 0\} $\end{document}</tex-math></inline-formula>. Using the averaging theory up to any order <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, we give upper bounds for the maximum number of limit cycles in the function of <inline-formula><tex-math id="M6">\begin{document}$ m, n, N $\end{document}</tex-math></inline-formula>. More importantly, employing the higher order averaging method we provide new lower bounds of the maximum number of limit cycles for several types of piecewise polynomial systems, which improve the results of the previous works. Besides, we explore the effect of 4-star-symmetry on the maximum number of limit cycles bifurcating from the unperturbed periodic orbits. Our result implies that 4-star-symmetry almost halves the maximum number.</p>

Two Lower Bounds for Shortest Double-Base Number System

2015 ◽  
Vol E98.A (6) ◽  
pp. 1310-1312 ◽  
Author(s):  
Parinya CHALERMSOOK ◽  
Hiroshi IMAI ◽  
Vorapong SUPPAKITPAISARN
Keyword(s):  
Lower Bounds ◽  
Number System ◽  
Base Number ◽  
Double Base

Lower bounds on the dimension of the Rauzy gasket

2020 ◽  
Vol 148 (2) ◽  
pp. 321-327
Author(s):  
Rodolfo Gutiérrez-Romo ◽  
Carlos Matheus
Keyword(s):  
Lower Bounds
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