scholarly journals The number of limit cycles in perturbations of polynomial systems with multiple circles of critical points

2016 ◽  
Vol 440 (1) ◽  
pp. 220-239 ◽  
Author(s):  
Yanqin Xiong
Keyword(s):  
Critical Points ◽  
Limit Cycles ◽  
Polynomial Systems

Differential Equations Defined by the Sum of two Quasi-Homogeneous Vector Fields

1997 ◽  
Vol 49 (2) ◽  
pp. 212-231 ◽  
Author(s):  
B. Coll ◽  
A. Gasull ◽  
R. Prohens
Keyword(s):  
Differential Equation ◽  
Critical Points ◽  
Limit Cycles ◽  
Vector Fields ◽  
Polynomial Systems ◽  
Polar Coordinates ◽  
The Stability ◽  
Derivatives Of ◽  
Homogeneous Vector Fields ◽  
Homogeneous Vector

AbstractIn this paper we prove, that under certain hypotheses, the planar differential equation: ˙x = X1(x, y) + X2(x, y), ˙y = Y1(x, y) + Y2(x, y), where (Xi, Yi), i = 1, 2, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar´e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

2021 ◽  
Vol 31 (09) ◽  
pp. 2150123
Author(s):  
Xiaoyan Chen ◽  
Maoan Han
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

Bifurcation Analysis in a Class of Piecewise Nonlinear Systems with a Nonsmooth Heteroclinic Loop

2018 ◽  
Vol 28 (02) ◽  
pp. 1850026
Author(s):  
Yuanyuan Liu ◽  
Feng Li ◽  
Pei Dang
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
Heteroclinic Loop ◽  
First Order ◽  
Nilpotent Critical Point

We consider the bifurcation in a class of piecewise polynomial systems with piecewise polynomial perturbations. The corresponding unperturbed system is supposed to possess an elementary or nilpotent critical point. First, we present 17 cases of possible phase portraits and conditions with at least one nonsmooth periodic orbit for the unperturbed system. Then we focus on the two specific cases with two heteroclinic orbits and investigate the number of limit cycles near the loop by means of the first-order Melnikov function, respectively. Finally, we take a quartic piecewise system with quintic piecewise polynomial perturbation as an example and obtain that there can exist ten limit cycles near the heteroclinic loop.

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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