scholarly journals THE NUMBER OF LIMIT CYCLES FROM A QUARTIC CENTER BY THE HIGHER-ORDER MELNIKOV FUNCTIONS

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Xia Liu ◽  
◽  
Keyword(s):  
Limit Cycles ◽  
Higher Order ◽  
Melnikov Functions

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.

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

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.

A note on higher order Melnikov functions

2005 ◽  
Vol 6 (2) ◽  
pp. 273-287 ◽  
Author(s):  
Ahmed Jebrane ◽  
Henryk Żolłądek
Keyword(s):  
Higher Order ◽  
Melnikov Functions

scholarly journals A generalization of Françoise's algorithm for calculating higher order Melnikov functions

2002 ◽  
Vol 126 (9) ◽  
pp. 705-732 ◽  
Author(s):  
Ahmed Jebrane ◽  
Pavao Mardešić ◽  
Michèle Pelletier
Keyword(s):  
Higher Order ◽  
Melnikov Functions

scholarly journals Limit Cycles in the Equation of Whirling Pendulum With Piecewise Smooth Perturbations

2021 ◽  
Author(s):  
Jihua Yang
Keyword(s):  
Limit Cycles ◽  
Upper Bounds ◽  
Piecewise Smooth ◽  
Chebyshev System ◽  
Exact Bound ◽  
First Order ◽  
Pendulum Equation ◽  
Melnikov Functions ◽  
Switching Line ◽  
Special Case

Abstract This paper deals with the problem of limit cycles for the whirling pendulum equation ẋ = y, ẏ = sin x(cos x-r) under piecewise smooth perturbations of polynomials of cos x, sin x and y of degree n with the switching line x = 0. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained by using the Picard-Fuchs equations which the generating functions of the associated first order Melnikov functions satisfy. Further, the exact bound of a special case is given by using the Chebyshev system.

scholarly journals Higher order stroboscopic averaged functions: a general relationship with Melnikov functions

2021 ◽  
pp. 1-9
Author(s):  
Douglas Novaes
Keyword(s):  
Differential Equation ◽  
Averaging Method ◽  
Higher Order ◽  
Research Literature ◽  
General Relationship ◽  
The Other ◽  
Asymptotic Estimates ◽  
Melnikov Functions ◽  
Averaging Functions ◽  
Identity Transformations

In the research literature, one can find distinct notions for higher order averaged functions of regularly perturbed non-autonomous T-periodic differential equations of the kind x′=ε F(t,x,ε ). By one hand, the classical (stroboscopic) averaging method provides asymptotic estimates for its solutions in terms of some uniquely defined functions gi's, called averaged functions, which are obtained through near-identity stroboscopic transformations and by solving homological equations. On the other hand, a Melnikov procedure is employed to obtain bifurcation functions fi's which controls in some sense the existence of isolated T-periodic solutions of the differential equation above. In the research literature, the bifurcation functions fi's are sometimes likewise called averaged functions, nevertheless, they also receive the name of Poincaré–Pontryagin–Melnikov functions or just Melnikov functions. While it is known that f1=Tg1, a general relationship between gi and fi is not known so far for i≥ 2. Here, such a general relationship between these two distinct notions of averaged functions is provided, which allows the computation of the stroboscopic averaged functions of any order avoiding the necessity of dealing with near-identity transformations and homological equations. In addition, an Appendix is provided with implemented Mathematica algorithms for computing both higher order averaging functions.

Sign in / Sign up

Export Citation Format

Share Document