scholarly journals The splitting of separatrices for analytic diffeomorphisms

1990 ◽  
Vol 10 (2) ◽  
pp. 295-318 ◽  
Author(s):  
E. Fontich ◽  
C. Simó
Keyword(s):  
Invariant Manifolds ◽  
Complex Singularities ◽  
Homoclinic Points ◽  
Unperturbed Problem ◽  
Splitting Of Separatrices ◽  
Maximum Separation ◽  
Exponentially Small

AbstractWe study families of diffeomorphisms close to the identity, which tend to it when the parameter goes to zero, and having homoclinic points. We consider the analytical case and we find that the maximum separation between the invariant manifolds, in a given region, is exponentially small with respect to the parameter. The exponent is related to the complex singularities of a flow which is taken as an unperturbed problem. Finally several examples are given.

Invariant manifolds for near identity differentiable maps and splitting of separatrices

1990 ◽  
Vol 10 (2) ◽  
pp. 319-346 ◽  
Author(s):  
E. Fontich ◽  
C. Simó
Keyword(s):  
Asymptotic Behaviour ◽  
Invariant Manifolds ◽  
Splitting Of Separatrices ◽  
Maximum Separation ◽  
Heteroclinic Points ◽  
Hyperbolic Points

AbstractWe consider families of differentiable diffeomorphisms with hyperbolic points, close to the identity, which tend to it when the parameter goes to zero.We study the asymptotic behaviour of the invariant manifolds. Then we consider the case when there are homo-heteroclinic points and we find that the maximum separation between the invariant manifolds is of the order of some power of the parameter which is related to the degree of differentiability.Finally the analogous case for flows is considered.

scholarly journals On the generic existence of homoclinic points

1987 ◽  
Vol 7 (4) ◽  
pp. 567-595 ◽  
Author(s):  
Fernando Oliveira
Keyword(s):  
Periodic Point ◽  
Invariant Manifolds ◽  
Compact Manifolds ◽  
Homoclinic Points ◽  
Symplectic Diffeomorphisms ◽  
Generic Existence ◽  
Area Preserving ◽  
Orientable Surfaces

AbstractThis work is concerned with the generic existence of homoclinic points for area preserving diffeomorphisms of compact orientable surfaces. We give a shorter proof of Pixton's theorem that shows that, Cr-generically, an area preserving diffeomorphism of the two sphere has the property that every hyperbolic periodic point has transverse homoclinic points. Then, we extend Pixton's result to the torus and investigate certain generic aspects of the accumulation of the invariant manifolds all over themselves in the case of symplectic diffeomorphisms of compact manifolds.

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.

Global dynamics of a planar mapping exhibiting orbits of periods 1, 2, 3 and no chaos

1990 ◽  
Vol 333 (1630) ◽  
pp. 209-259
Keyword(s):  
Invariant Manifolds ◽  
Periodic Points ◽  
Chaotic Behaviour ◽  
Global Dynamics ◽  
Geometrical Analysis ◽  
Small Oscillations ◽  
Self Similar ◽  
Planar Mapping ◽  
Exponentially Small

A geometrical analysis of the planar mapping A : ( x,y ) -> ( y+xy,x ) is presented. A complete global portrait of the invariant manifolds of A is found, primarily by deductive methods. The behaviour of some manifolds was initially investigated numerically, but theoretical explanations for the observations are given in every case. The most significant features of the mapping A are : that it has periodic points of periods 1, 2 and 3 only; that it possesses no chaotic behaviour; that it has sequences of abutting regions of self-similar structure, and that it exhibits heteroclinic behaviour manifesting itself as exponentially small oscillations in some of the invariant manifolds.

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