scholarly journals The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line

2021 ◽  
Vol 31 (4) ◽  
pp. 043112
Author(s):  
M. Esteban ◽  
J. Llibre ◽  
C. Valls
Keyword(s):  
Hilbert Problem ◽  
Straight Line ◽  
16Th Hilbert Problem ◽  
Isochronous Centers

scholarly journals Limit Cycles Coming from Some Uniform Isochronous Centers

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
Haihua Liang ◽  
Jaume Llibre ◽  
Joan Torregrosa
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Hilbert Problem ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
16Th Hilbert Problem ◽  
Isochronous Centers

AbstractThis article is about the weak 16th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centersof degree

The Extended 16th Hilbert Problem for Discontinuous Piecewise Linear Centers Separated by a Nonregular Line

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Marina Esteban ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Hilbert Problem ◽  
Piecewise Linear ◽  
Upper Bounds ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential ◽  
16Th Hilbert Problem ◽  
To Receive

The study of the piecewise linear differential systems goes back to Andronov, Vitt and Khaikin in 1920’s, and nowadays such systems still continue to receive the attention of many researchers mainly due to their applications. We study the discontinuous piecewise differential systems formed by two linear centers separated by a nonregular straight line. We provide upper bounds for the maximum number of limit cycles that these discontinuous piecewise differential systems can exhibit and we show that these upper bounds are reached. Hence, we solve the extended 16th Hilbert problem for this class of piecewise differential systems.

scholarly journals On the 16th Hilbert problem for algebraic limit cycles

2010 ◽  
Vol 248 (6) ◽  
pp. 1401-1409 ◽  
Author(s):  
Jaume Llibre ◽  
Rafael Ramírez ◽  
Natalia Sadovskaia
Keyword(s):  
Limit Cycles ◽  
Hilbert Problem ◽  
16Th Hilbert Problem ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

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.

On the Limit Cycles of a Perturbed Z3-Equivariant Planar Quintic Vector Field

2015 ◽  
Vol 25 (05) ◽  
pp. 1550073
Author(s):  
Yunlei Ma ◽  
Yuhai Wu
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Hilbert Problem ◽  
Polynomial System ◽  
Heteroclinic Loop ◽  
Planar Vector ◽  
16Th Hilbert Problem

In this paper, the number and distributions of limit cycles in a Z3-equivariant quintic planar polynomial system are studied. 24 limit cycles with two different configurations are shown in this quintic planar vector field by combining the methods of double homoclinic loops bifurcation, heteroclinic loop bifurcation and Poincaré–Bendixson Theorem. The results obtained are useful to the study of weakened 16th Hilbert problem.

scholarly journals On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

2011 ◽  
Vol 250 (2) ◽  
pp. 983-999 ◽  
Author(s):  
Jaume Llibre ◽  
Rafael Ramírez ◽  
Natalia Sadovskaia
Keyword(s):  
Limit Cycles ◽  
Algebraic Curves ◽  
Hilbert Problem ◽  
16Th Hilbert Problem
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