A Unified Study on the Cyclicity of Period Annulus of the Reversible Quadratic Hamiltonian Systems

2004 ◽  
Vol 16 (2) ◽  
pp. 271-295 ◽  
Author(s):  
Chengzhi Li ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian Systems ◽  
Period Annulus ◽  
Unified Study ◽  
Quadratic Hamiltonian

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

2020 ◽  
Vol 30 (15) ◽  
pp. 2050230
Author(s):  
Jiaxin Wang ◽  
Liqin Zhao
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Period Annulus ◽  
Quadratic Hamiltonian

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

On asymptotically quadratic Hamiltonian systems

1983 ◽  
Vol 7 (8) ◽  
pp. 929-931 ◽  
Author(s):  
V. Benci ◽  
A. Capozzi ◽  
D. Fortunato
Keyword(s):  
Hamiltonian Systems ◽  
Quadratic Hamiltonian

Even homoclinic orbits for super quadratic Hamiltonian systems

2010 ◽  
Vol 33 (14) ◽  
pp. 1755-1761 ◽  
Author(s):  
Jian Ding ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Hamiltonian Systems ◽  
Homoclinic Orbits ◽  
Quadratic Hamiltonian
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