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.

Limit Cycle Bifurcations by Perturbing a Hamiltonian System with a 3-Polycycle Having a Cusp of Order One or Two

2018 ◽  
Vol 28 (03) ◽  
pp. 1850038
Author(s):  
Marzieh Mousavi ◽  
Hamid R. Z. Zangeneh
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Perturbed Systems

In this paper, we study the asymptotic expansion of the first order Melnikov function near a 3-polycycle connecting a cusp (of order one or two) to two hyperbolic saddles for a near-Hamiltonian system in the plane. The formulas for the first coefficients of the expansion are given as well as the method of bifurcation of limit cycles. Then we use the results to study two Hamiltonian systems with this 3-polycycle and determine the number and distribution of limit cycles that can bifurcate from the perturbed systems. Moreover, a sharp upper bound for the number of limit cycles bifurcated from the whole periodic annulus is found when there is a cusp of order one.

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.

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

2012 ◽  
Vol 22 (12) ◽  
pp. 1250296 ◽  
Author(s):  
MAOAN HAN
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Asymptotic Expansions ◽  
Melnikov Function ◽  
Heteroclinic Loop ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Center

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

Limit Cycle Bifurcations from an Order-3 Nilpotent Center of Cubic Hamiltonian Systems Perturbed by Cubic Polynomials

2020 ◽  
Vol 30 (09) ◽  
pp. 2050126
Author(s):  
Li Zhang ◽  
Chenchen Wang ◽  
Zhaoping Hu
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
First Order ◽  
Cubic Polynomials ◽  
Nilpotent Center

From [Han et al., 2009a] we know that the highest order of the nilpotent center of cubic Hamiltonian system is [Formula: see text]. In this paper, perturbing the Hamiltonian system which has a nilpotent center of order [Formula: see text] at the origin by cubic polynomials, we study the number of limit cycles of the corresponding cubic near-Hamiltonian systems near the origin. We prove that we can find seven and at most seven limit cycles near the origin by the first-order Melnikov function.

Bifurcation of Limit Cycles in a Near-Hamiltonian System with a Cusp of Order Two and a Saddle

2016 ◽  
Vol 26 (11) ◽  
pp. 1650180 ◽  
Author(s):  
Ali Bakhshalizadeh ◽  
Hamid R. Z. Zangeneh ◽  
Rasool Kazemi
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
First Order ◽  
Period Annulus

In this paper, the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp of order two and a hyperbolic saddle for a planar near-Hamiltonian system is given. Next, we consider the limit cycle bifurcations of a hyper-elliptic Liénard system with this kind of heteroclinic loop and study the least upper bound of limit cycles bifurcated from the period annulus inside the heteroclinic loop, from the heteroclinic loop itself and the center. We find that at most three limit cycles can be bifurcated from the period annulus, also we present different distributions of bifurcated limit cycles.

Limit Cycle Bifurcations by Perturbing a Hamiltonian System with a Cuspidal Loop of Order m

2015 ◽  
Vol 25 (06) ◽  
pp. 1550083 ◽  
Author(s):  
Yanqing Xiong
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Liénard System ◽  
First Order ◽  
Lienard System

This paper is concerned with the expansion of the first-order Melnikov function for general Hamiltonian systems with a cuspidal loop having order m. Some criteria and formulas are derived, which can be used to obtain first-order coefficients in the expansion. In particular, we deduce the first-order coefficients for the case m = 3 and give the corresponding conditions of existing several limit cycles. As an application, we study a Liénard system of type (n, 9) and prove that it can have 14 limit cycles near a cuspidal loop of order 3 for n = 8.

scholarly journals Limit Cycle Bifurcations by Perturbing a Compound Loop with a Cusp and a Nilpotent Saddle

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Huanhuan Tian ◽  
Maoan Han
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Polynomial System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Melnikov Functions ◽  
Nilpotent Saddle

We study the expansions of the first order Melnikov functions for general near-Hamiltonian systems near a compound loop with a cusp and a nilpotent saddle. We also obtain formulas for the first coefficients appearing in the expansions and then establish a bifurcation theorem on the number of limit cycles. As an application example, we give a lower bound of the maximal number of limit cycles for a polynomial system of Liénard type.

The Artificial Hamiltonian, First Integrals, and Closed-Form Solutions of Dynamical Systems for Epidemics

2018 ◽  
Vol 73 (4) ◽  
pp. 323-330 ◽  
Author(s):  
Rehana Naz ◽  
Imran Naeem
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian System ◽  
Closed Form ◽  
Hamiltonian Systems ◽  
Physical Structure ◽  
First Integrals ◽  
Second Order ◽  
Hamiltonian Structures ◽  
Closed Form Solutions ◽  
First Order

AbstractThe non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

scholarly journals Limit Cycle Bifurcations Near a Cuspidal Loop

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1425
Author(s):  
Pan Liu ◽  
Maoan Han
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation

In this paper, we study limit cycle bifurcation near a cuspidal loop for a general near-Hamiltonian system by using expansions of the first order Melnikov functions. We give a method to compute more coefficients of the expansions to find more limit cycles near the cuspidal loop. As an application example, we considered a polynomial near-Hamiltonian system and found 12 limit cycles near the cuspidal loop and the center.

EIGHT LIMIT CYCLES AROUND A CENTER IN QUADRATIC HAMILTONIAN SYSTEM WITH THIRD-ORDER PERTURBATION

2013 ◽  
Vol 23 (01) ◽  
pp. 1350005 ◽  
Author(s):  
PEI YU ◽  
MAOAN HAN
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Higher Order ◽  
Degree Polynomial ◽  
Third Order ◽  
First Order ◽  
Melnikov Functions ◽  
Quadratic Hamiltonian ◽  
Focus Value

In this paper, we show that generic planar quadratic Hamiltonian systems with third degree polynomial perturbation can have eight small-amplitude limit cycles around a center. We use higher-order focus value computation to prove this result, which is equivalent to the computation of higher-order Melnikov functions. Previous results have shown, based on first-order and higher-order Melnikov functions, that planar quadratic Hamiltonian systems with third degree polynomial perturbation can have five or seven small-amplitude limit cycles around a center. The result given in this paper is a further improvement.

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