Melnikov functions and Bautin ideal

2001 ◽  
Vol 2 (1) ◽  
pp. 67-78 ◽  
Author(s):  
Robert Roussarie
Keyword(s):  
Melnikov Functions ◽  
Bautin Ideal

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 Bifurcations of a Homoclinic Orbit to Saddle-Center in Reversible Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Zhiqin Qiao ◽  
Yancong Xu
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Moving Frame ◽  
Reversible Systems ◽  
Reversible System ◽  
Melnikov Functions ◽  
Existence And Nonexistence ◽  
Local Moving Frame

The bifurcations near a primary homoclinic orbit to a saddle-center are investigated in a 4-dimensional reversible system. By establishing a new kind of local moving frame along the primary homoclinic orbit and using the Melnikov functions, the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding codimension 1 or codimension 3 surfaces, are obtained.

Ground States and Singular Ground States for Quasilinear Partial Differential Equations with Critical Exponent in the Perturbative Case

2004 ◽  
Vol 4 (1) ◽  
Author(s):  
Matteo Franca ◽  
Russell Johnson
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exponential Dichotomy ◽  
Invariant Manifolds ◽  
Ground States ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Quasilinear Partial Differential Equations ◽  
Melnikov Functions ◽  
The Family

AbstractWe study the structure of the family of radially symmetric ground states and singular ground states for certain elliptic partial differential equations with p- Laplacian. We use methods of Dynamical systems such as Melnikov functions, invariant manifolds, and exponential dichotomy.

Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-de Vries equation

1994 ◽  
Vol 445 (1923) ◽  
pp. 1-21 ◽  
Keyword(s):  
Lyapunov Exponent ◽  
Vries Equation ◽  
Period Doubling ◽  
Homoclinic Orbits ◽  
Dynamical Behaviour ◽  
Chaotic Behaviour ◽  
Subharmonic Bifurcations ◽  
Melnikov Functions ◽  
De Vries ◽  
Period Doubling Bifurcations

The dynamical behaviour of a reduction of the forced (and damped) Korteweg-de Vries equation is studied numerically. Chaos arising from subharmonic instability and homoclinic crossings are observed. Both period-doubling bifurcations and the Melnikov sequence of subharmonic bifurcations are found and lead to chaotic behaviour, here characterised by a positive Lyapunov exponent. Supporting theoretical analysis includes the construction of periodic solutions and homoclinic orbits, and their behaviour under perturbation using 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.

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