High order Melnikov method: Pendulums

2022 ◽  
Vol 312 ◽  
pp. 176-208
Author(s):  
Ali Oksasoglu ◽  
Qiudong Wang
Keyword(s):  
Melnikov Method ◽  
High Order

High-Order Melnikov Method for Time-Periodic Equations

2017 ◽  
Vol 17 (4) ◽  
pp. 793-818 ◽  
Author(s):  
Fengjuan Chen ◽  
Qiudong Wang
Keyword(s):  
Explicit Formula ◽  
Melnikov Method ◽  
Melnikov Function ◽  
High Order ◽  
New Method ◽  
Integral Formulas ◽  
Multiple Integrals ◽  
Time Periodic

AbstractThis paper discusses a high-order Melnikov method for periodically perturbed equations. We introduce a new method to compute {M_{k}(t_{0})} for all {k\geq 0}, among which {M_{0}(t_{0})} is the traditional Melnikov function, and {M_{1}(t_{0}),M_{2}(t_{0}),\ldots\,} are its high-order correspondences. We prove that, for all {k\geq 0}, {M_{k}(t_{0})} is a sum of certain multiple integrals, the integrand of which we can explicitly compute. In particular, we obtain explicit integral formulas for {M_{0}(t_{0})} and {M_{1}(t_{0})}. We also study a concrete equation for which the explicit formula of {M_{1}(t_{0})} is used to prove the existence of a transversal homoclinic intersection in the case of {M_{0}(t_{0})\equiv 0}.

High order Melnikov method: Theory and application

2019 ◽  
Vol 267 (2) ◽  
pp. 1095-1128
Author(s):  
Fengjuan Chen ◽  
Qiudong Wang
Keyword(s):  
Melnikov Method ◽  
High Order

scholarly journals A High-Order Melnikov Method for Heteroclinic Orbits in Planar Vector Fields and Heteroclinic Persisting Perturbations

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yi Zhong
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Heteroclinic Orbit ◽  
Melnikov Method ◽  
High Order ◽  
Heteroclinic Orbits ◽  
Van Der Pol ◽  
Planar Vector Fields ◽  
Planar Vector

This work extends the high-order Melnikov method established by FJ Chen and QD Wang to heteroclinic orbits, and it is used to prove, under a certain class of perturbations, the heteroclinic orbit in a planar vector field that remains unbroken. Perturbations which have this property together form the heteroclinic persisting space. The Van der Pol system is analysed as an application.

Determination of the Burgers vector of a dislocation in a Fe-Mn alloy by weak-beam image in HVEM

1978 ◽  
Vol 36 (1) ◽  
pp. 330-331
Author(s):  
Y. Ishida ◽  
H. Ishida ◽  
K. Kohra ◽  
H. Ichinose
Keyword(s):  
High Order ◽  
Simulation Technique ◽  
The Other ◽  
Image Simulation ◽  
Diffraction Vector ◽  
Burgers Vector ◽  
Weak Beam ◽  
Beam Image ◽  
Edge Component

IntroductionA simple and accurate technique to determine the Burgers vector of a dislocation has become feasible with the advent of HVEM. The conventional image vanishing technique(1) using Bragg conditions with the diffraction vector perpendicular to the Burgers vector suffers from various drawbacks; The dislocation image appears even when the g.b = 0 criterion is satisfied, if the edge component of the dislocation is large. On the other hand, the image disappears for certain high order diffractions even when g.b ≠ 0. Furthermore, the determination of the magnitude of the Burgers vector is not easy with the criterion. Recent image simulation technique is free from the ambiguities but require too many parameters for the computation. The weak-beam “fringe counting” technique investigated in the present study is immune from the problems. Even the magnitude of the Burgers vector is determined from the number of the terminating thickness fringes at the exit of the dislocation in wedge shaped foil surfaces.

The usefulness of high index low symmetry zone axes for holz line analysis

1987 ◽  
Vol 45 ◽  
pp. 384-385
Author(s):  
C. M. Sung ◽  
D. B. Williams
Keyword(s):  
Lattice Parameter ◽  
High Order ◽  
Zone Axis ◽  
High Index ◽  
High Symmetry ◽  
Second Phases ◽  
Line Patterns ◽  
Low Symmetry ◽  
Line Analysis

Researchers have tended to use high symmetry zone axes (e.g. <111> <114>) for High Order Laue Zone (HOLZ) line analysis since Jones et al reported the origin of HOLZ lines and described some of their applications. But it is not always easy to find HOLZ lines from a specific high symmetry zone axis during microscope operation, especially from second phases on a scale of tens of nanometers. Therefore it would be very convenient if we can use HOLZ lines from low symmetry zone axes and simulate these patterns in order to measure lattice parameter changes through HOLZ line shifts. HOLZ patterns of high index low symmetry zone axes are shown in Fig. 1, which were obtained from pure Al at -186°C using a double tilt cooling holder. Their corresponding simulated HOLZ line patterns are shown along with ten other low symmetry orientations in Fig. 2. The simulations were based upon kinematical diffraction conditions.

Are HOLZ lines kinematic in off-zone-axis orientation?

1993 ◽  
Vol 51 ◽  
pp. 692-693
Author(s):  
J. M. Zuo ◽  
A. L. Weickenmeier ◽  
R. Holmestad ◽  
J. C. H. Spence
Keyword(s):  
Structure Determination ◽  
Standard Procedure ◽  
High Order ◽  
Zone Axis ◽  
Great Promise ◽  
Line Position ◽  
Measured Intensity ◽  
Constant Intensity ◽  
Axis Orientation ◽  
Diffraction Condition

The application of high order reflections in a weak diffraction condition off the zone axis center, including those in high order laue zones (HOLZ), holds great promise for structure determination using convergent beam electron diffraction (CBED). It is believed that in this case the intensities of high order reflections are kinematic or two-beam like. Hence, the measured intensity can be related to the structure factor amplitude. Then the standard procedure of structure determination in crystallography may be used for solving unknown structures. The dynamic effect on HOLZ line position and intensity in a strongly diffracting zone axis is well known. In a weak diffraction condition, the HOLZ line position may be approximated by the kinematic position, however, it is not clear whether this is also true for HOLZ intensities. The HOLZ lines, as they appear in CBED patterns, do show strong intensity variations along the line especially near the crossing of two lines, rather than constant intensity along the Bragg condition as predicted by kinematic or two beam theory.

Relativistic high-order harmonic generation

2003 ◽  
Vol 50 (3-4) ◽  
pp. 375-386
Author(s):  
D. B. MilosÕeviĆ ◽  
W. Becker
Keyword(s):  
Harmonic Generation ◽  
High Order ◽  
High Order Harmonic Generation ◽  
High Order Harmonic
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