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

scholarly journals The 16th Hilbert problem restricted to circular algebraic limit cycles

2016 ◽  
Vol 260 (7) ◽  
pp. 5726-5760 ◽  
Author(s):  
Jaume Llibre ◽  
Rafael Ramírez ◽  
Valentín Ramírez ◽  
Natalia Sadovskaia
Keyword(s):  
Limit Cycles ◽  
Hilbert Problem ◽  
16Th Hilbert Problem ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

scholarly journals The Cubic Polynomial Differential Systems with two Circles as Algebraic Limit Cycles

2018 ◽  
Vol 18 (1) ◽  
pp. 183-193 ◽  
Author(s):  
Jaume Giné ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Cubic Polynomial ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractIn this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.

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.

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