Bifurcation set and distribution of limit cycles for a class of cubic Hamiltonian system with higher-order perturbed terms

2000 ◽  
Vol 11 (14) ◽  
pp. 2293-2304 ◽  
Author(s):  
Hongjun Cao ◽  
Zhengrong Liu ◽  
Zhujun Jing
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Higher Order ◽  
Bifurcation Set

scholarly journals Perturbation of a Period Annulus with a Unique Two-Saddle Cycle in Higher Order Hamiltonian

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Hongying Zhu ◽  
Sumin Yang ◽  
Xiaochun Hu ◽  
Weihua Huang
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Outer Boundary ◽  
Higher Order ◽  
Heteroclinic Cycle ◽  
Period Annulus

In this paper, we study the number of limit cycles emerging from the period annulus by perturbing the Hamiltonian system x˙=y,y˙=x(x2-1)(x2+1)(x2+2). The period annulus has a heteroclinic cycle connecting two hyperbolic saddles as the outer boundary. It is proved that there exist at most 4 and at least 3 limit cycles emerging from the period annulus, and 3 limit cycles are near the boundaries.

Limit Cycles of Higher Order Nonlinear Autonomous Systems

1971 ◽  
Vol 291 (3) ◽  
pp. 195-210 ◽  
Author(s):  
R.K. Jonnada ◽  
C.N. Weygandt
Keyword(s):  
Limit Cycles ◽  
Autonomous Systems ◽  
Higher Order

scholarly journals Limit Cycles of a Class of Perturbed Differential Systems via the First-Order Averaging Method

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Amor Menaceur ◽  
Salah Mahmoud Boulaaras ◽  
Amar Makhlouf ◽  
Karthikeyan Rajagobal ◽  
Mohamed Abdalla
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Averaging Method ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems

By means of the averaging method of the first order, we introduce the maximum number of limit cycles which can be bifurcated from the periodic orbits of a Hamiltonian system. Besides, the perturbation has been used for a particular class of the polynomial differential systems.

scholarly journals The number of limit cycles for a class of quintic Hamiltonian systems under quintic perturbations

2002 ◽  
Vol 73 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Guowei Chen ◽  
Yongbin Wu ◽  
Xinan Yang
Keyword(s):  
Hopf Bifurcation ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Homoclinic Bifurcation

AbstractThe Hopf bifurcation and homoclinic bifurcation of the quintic Hamiltonian system is analyzed under quintic perturbations by using unfolding theory in this paper. We show that a quintic system can have at least 29 limit cycles.

scholarly journals Higher Order Hamiltonian Systems with Generalized Legendre Transformation

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 163
Author(s):  
Dana Smetanová
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Higher Order ◽  
Second Order ◽  
Legendre Transformation ◽  
Lagrange Equations ◽  
Hamilton Equations ◽  
Strongly Regular

The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition.

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.

SMALL LIMIT CYCLES BIFURCATING FROM Z4-EQUIVARIANT NEAR-HAMILTONIAN SYSTEM OF DEGREES 9 AND 7

2012 ◽  
Vol 22 (11) ◽  
pp. 1250272 ◽  
Author(s):  
XIANBO SUN ◽  
JUNMIN YANG
Keyword(s):  
Hopf Bifurcation ◽  
Hamiltonian System ◽  
Limit Cycles ◽  
Bifurcation Theory

In this paper, we study the number and distribution of small limit cycles of some Z4-equivariant near-Hamiltonian system of degree 9. Using the methods of Hopf bifurcation theory, we find that this system can have 64 small limit cycles. The configuration of 64 small limit cycles of the system is also illustrated in Fig. 1. When we let some parameters be zero, then we find that there can be 40 small limit cycles in a seventh system.

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