scholarly journals Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems

2000 ◽  
Vol 10 (4) ◽  
pp. 433-476 ◽  
Author(s):  
A. Delshams ◽  
P. Gutiérrez
Keyword(s):  
Hamiltonian Systems ◽  
Melnikov Method ◽  
Whiskered Tori

ANALYTICAL EXPRESSIONS FOR STABLE AND UNSTABLE MANIFOLDS IN HIGHER DEGREE OF FREEDOM HAMILTONIAN SYSTEMS

1996 ◽  
Vol 06 (11) ◽  
pp. 1997-2013 ◽  
Author(s):  
HARRY DANKOWICZ
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Initial Conditions ◽  
Saddle Points ◽  
Melnikov Method ◽  
Type Systems ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
Analytical Expressions

Perturbations of completely integrable Hamiltonian systems with three or more degrees of freedom are studied. In particular, the unperturbed systems are assumed to be separable into a product of simple oscillator-type systems and a system containing homo- or heteroclinic connections consisting of stable and unstable manifolds of saddle points. Under a perturbation, the manifolds persist but separate and may no longer intersect. In this paper we show how, with proper choices for initial conditions, one may solve the variational equations to obtain analytical expressions for orbits on the perturbed manifolds in the form of expansions in the small parameter characterizing the perturbation. The derivation also shows how the distance between the manifolds can be uniquely defined, and thus provides an alternative to the traditional higher dimensional Melnikov method. It is finally argued that the approximate knowledge of the shape and position of the perturbed manifolds could be utilized for the study of large-scale phase-space motions, such as those associated with Arnold diffusion. The theory is further illuminated in two example problems.

scholarly journals Exponentially Small Lower Bounds for the Splitting of Separatrices to Whiskered Tori with Frequencies of Constant Type

2014 ◽  
Vol 24 (08) ◽  
pp. 1440011 ◽  
Author(s):  
Amadeu Delshams ◽  
Marina Gonchenko ◽  
Pere Gutiérrez
Keyword(s):  
Lower Bounds ◽  
Invariant Manifolds ◽  
Irrational Number ◽  
Melnikov Method ◽  
Arithmetic Properties ◽  
Constant Type ◽  
Whiskered Tori ◽  
Vector Formula ◽  
Splitting Of Invariant Manifolds ◽  
Exponentially Small

We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a two-dimensional torus with a fast frequency vector [Formula: see text], with ω = (1, Ω) where Ω is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincaré–Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.

scholarly journals Horseshoes for autonomous Hamiltonian systems using the Melnikov integral

1988 ◽  
Vol 8 (8) ◽  
pp. 395-409 ◽  
Keyword(s):  
Hamiltonian Systems ◽  
Melnikov Method ◽  
Melnikov Function ◽  
Integrable Hamiltonian Systems ◽  
Completely Integrable ◽  
Perturbed System ◽  
Melnikov Integral ◽  
Completely Integrable Hamiltonian Systems

AbstractThis paper applies the Melnikov method to autonomous perturbations of completely integrable Hamiltonian systems. The forcing of the perturbed system is caused by internal oscillations which are not necessarily decoupled. A unified treatment is presented which relates some results of Holmes and Marsden with a result of Lerman and Umanskii. It is also shown that two forms of the Melnikov function by integrals are in fact equal.

scholarly journals Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds

2004 ◽  
Vol 11 (4) ◽  
pp. 785-826 ◽  
Author(s):  
Amadeu Delshams ◽  
◽  
Pere Gutiérrez ◽  
Tere M. Seara ◽  
◽  
...  
Keyword(s):  
Hamiltonian Systems ◽  
Upper Bounds ◽  
Exponentially Small Splitting ◽  
Whiskered Tori ◽  
Exponentially Small
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