scholarly journals Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems a)

2020 ◽  
pp. 640-646
Author(s):  
Philip J. Holmes ◽  
Jerrold E. Marsden
Keyword(s):  
Hamiltonian Systems ◽  
Arnold Diffusion ◽  
Integrable Hamiltonian Systems ◽  
Melnikov's Method ◽  
Melnikov’S Method

scholarly journals Perturbed generalised Hamiltonian systems and some advection models

1998 ◽  
Vol 57 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Jibin Li ◽  
J.R. Christie ◽  
K. Gopalsamy
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Flow Patterns ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Existence Of Periodic Solutions

In this paper, it is shown that the theory of perturbed generalised Hamiltonian systems provides an effective method for understanding the description of flow patterns of some three-dimensional flows. Firstly, theorems for the persistence of periodic solutions of three-dimensional generalised Hamiltonian systems under perturbation are given by developing Melnikov's method. Then, three different systems of three-dimensional steady fluid flows are discussed and the existence or non-existence of periodic solutions of these systems is proved.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

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