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 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.

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

2004 ◽  
Vol 128 (9) ◽  
pp. 749-760 ◽  
Author(s):  
Ahmed Jebrane ◽  
Pavao Mardešić ◽  
Michèle Pelletier
Keyword(s):  
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.

scholarly journals Higher-order Melnikov functions for single-DOF mechanical oscillators: theoretical treatment and applications

2004 ◽  
Vol 2004 (2) ◽  
pp. 145-168 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Recursive Formula ◽  
Higher Order ◽  
Theoretical Treatment ◽  
Stable And Unstable Manifolds ◽  
Single Degree Of Freedom ◽  
Melnikov Functions ◽  
Higher Order Terms ◽  
Melnikov Analysis ◽  
Mechanical Oscillators ◽  
Abstract Framework

A Melnikov analysis of single-degree-of-freedom (DOF) oscillators is performed by taking into account the first (classical) and higher-order Melnikov functions, by considering Poincaré sections nonorthogonal to the flux, and by explicitly determining both the distance between perturbed and unperturbed manifolds (“one-half” Melnikov functions) and the distance between perturbed stable and unstable manifolds (“full” Melnikov function). The analysis is developed in an abstract framework, and a recursive formula for computing the Melnikov functions is obtained. These results are then applied to various mechanical systems. Softening versus hardening stiffness and homoclinic versus heteroclinic bifurcations are considered, and the influence of higher-order terms is investigated in depth. It is shown that the classical (first-order) Melnikov analysis is practically inaccurate at least for small and large excitation frequencies, in correspondence to degenerate homo/heteroclinic bifurcations, and in the case of generic periodic excitations.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Quantitative EPMA of nitrogen : A tricky element in the electron-probe microanalyzer

1992 ◽  
Vol 50 (2) ◽  
pp. 1622-1623
Author(s):  
G.F. Bastin ◽  
H.J.M. Heijligers
Keyword(s):  
Background Subtraction ◽  
Electron Probe ◽  
Detection System ◽  
Higher Order ◽  
Light Elements ◽  
Strong Suppression ◽  
Electron Probe Microanalyzer ◽  
Quantitative Epma ◽  
Peak Count ◽  
Lead Stearate

Among the ultra-light elements B, C, N, and O nitrogen is the most difficult element to deal with in the electron probe microanalyzer. This is mainly caused by the severe absorption that N-Kα radiation suffers in carbon which is abundantly present in the detection system (lead-stearate crystal, carbonaceous counter window). As a result the peak-to-background ratios for N-Kα measured with a conventional lead-stearate crystal can attain values well below unity in many binary nitrides . An additional complication can be caused by the presence of interfering higher-order reflections from the metal partner in the nitride specimen; notorious examples are elements such as Zr and Nb. In nitrides containing these elements is is virtually impossible to carry out an accurate background subtraction which becomes increasingly important with lower and lower peak-to-background ratios. The use of a synthetic multilayer crystal such as W/Si (2d-spacing 59.8 Å) can bring significant improvements in terms of both higher peak count rates as well as a strong suppression of higher-order reflections.

Effects of Higher-Order Laue Zone reflections on structure images

1993 ◽  
Vol 51 ◽  
pp. 450-451
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Wave Vector ◽  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Image Plane ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Higher Order ◽  
Lattice Image ◽  
Lattice Vector ◽  
Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

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