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| On the Number of Limit Cycles in Small Perturbations of a PiecewiseLinear Hamiltonian System with a Heteroclinic Loop |
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Citation： |
Feng LIANG,Maoan HAN.On the Number of Limit Cycles in Small Perturbations of a PiecewiseLinear Hamiltonian System with a Heteroclinic Loop[J].Chinese Annals of Mathematics B,2016,37(2):267~280 |
| Page view： 6455
Net amount： 4113 |
Authors： |
Feng LIANG; Maoan HAN |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (No.11271261), the Natural Science Foundation
of Anhui Province (No.1308085MA08), and the Doctoral Program
Foundation (2012) of Anhui Normal University. |
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| Abstract： |
In this paper, the authors consider limit cycle bifurcations for a
kind of non-smooth polynomial differential systems by perturbing a
piecewise linear Hamiltonian system with a center at the origin and
a heteroclinic loop around the origin. When the degree of perturbing
polynomial terms is $n~(n\geq1),$ it is obtained that $n$ limit
cycles can appear near the origin and the heteroclinic loop
respectively by using the first Melnikov function of piecewise
near-Hamiltonian systems, and that there are at most
$n+[\frac{n+1}{2}]$ limit cycles bifurcating from the periodic
annulus between the center and the heteroclinic loop up to the first
order in $\varepsilon.$ Especially, for $n=1,2,3$ and $4$, a precise
result on the maximal number of zeros of the first Melnikov function
is derived. |
Keywords： |
Limit cycle, Heteroclinic loop, Melnikov function, Chebyshev system,
Bifurcation, Piecewise smooth system |
Classification： |
34C05, 34C07, 37G15 |
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