Up to second order Melnikov functions for general piecewise Hamiltonian systems with nonregular separation line

2021 ◽  
Vol 285 ◽  
pp. 583-606
Author(s):  
Peixing Yang ◽  
Yuan Yang ◽  
Jiang Yu
Keyword(s):  
Hamiltonian Systems ◽  
Second Order ◽  
Separation Line ◽  
Melnikov Functions

scholarly journals Second Order Melnikov Functions of Piecewise Hamiltonian Systems

2020 ◽  
Vol 30 (01) ◽  
pp. 2050016
Author(s):  
Peixing Yang ◽  
Jean-Pierre Françoise ◽  
Jiang Yu
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Second Order ◽  
Quadratic Polynomial ◽  
First Order ◽  
Melnikov Functions

In this paper, we consider the general perturbations of piecewise Hamiltonian systems. A formula for the second order Melnikov functions is derived when the first order Melnikov functions vanish. As an application, we can improve an upper bound of the number of bifurcated limit cycles of a piecewise Hamiltonian system with quadratic polynomial perturbations.

Bifurcation of a Kind of Piecewise Smooth Generalized Abel Equation via First and Second Order Analyses

2020 ◽  
Vol 30 (16) ◽  
pp. 2050247
Author(s):  
Jianfeng Huang ◽  
Zhixiang Peng
Keyword(s):  
Limit Cycles ◽  
Second Order ◽  
Piecewise Smooth ◽  
Abel Equation ◽  
Order Analysis ◽  
Separation Line ◽  
Melnikov Functions

In this paper, we consider the problem of estimating the number of nontrivial limit cycles for a kind of piecewise trigonometrical smooth generalized Abel equation with the separation line [Formula: see text]. Under the first and second order analyses, we show that the first two order Melnikov functions of the equation share a same structure which can be studied by an ECT-system. Furthermore, let [Formula: see text] be the maximum number of nontrivial limit cycles of the equation bifurcating from the periodic annulus up to [Formula: see text]th order analysis. We prove that [Formula: see text] and [Formula: see text] (resp., [Formula: see text] and [Formula: see text]) when [Formula: see text] is even (resp., odd).

Nontrivial homoclinic orbits for second-order singular and periodic Hamiltonian systems

1999 ◽  
Vol 44 (2) ◽  
pp. 123-129 ◽  
Author(s):  
Chengyue Li ◽  
Tianyou Fan ◽  
Mingsheng Tong
Keyword(s):  
Hamiltonian Systems ◽  
Homoclinic Orbits ◽  
Second Order
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