scholarly journals The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos

2018 ◽  
Vol 38 (4) ◽  
pp. 1955-1981 ◽  
Author(s):  
Radosław Kurek ◽  
◽  
Paweł Lubowiecki ◽  
Henryk Żołądek ◽  
Keyword(s):  
Splitting Of Separatrices

Splitting of Separatrices in the Rapidly Forced Duffing Oscillator

1993 ◽  
Author(s):  
Alexander F. Vakakis
Keyword(s):  
Asymptotic Approximation ◽  
Duffing Oscillator ◽  
Perturbation Parameter ◽  
Negative Stiffness ◽  
Stable And Unstable Manifolds ◽  
Analytic Approximations ◽  
Melnikov Integral ◽  
Unstable Manifolds ◽  
Splitting Of Separatrices ◽  
Splitting Distance

Abstract The splitting of the stable and unstable manifolds of the rapidly forced Duffing oscillator with negative stiffness is investigated. The method used relies on the computation of analytic approximations for the orbits on the perturbed manifolds, and the asymptotic approximation of these orbits by successive integrations by parts. It is shown, that the splitting of the manifolds becomes exponentially small as the perturbation parameter tends to zero, and that the estimate for the splitting distance given by the Melnikov Integral dominates over high order corrections.

Splitting of separatrices for standard and semistandard mappings

1989 ◽  
Vol 40 (2) ◽  
pp. 235-248 ◽  
Author(s):  
V.F. Lazutkin ◽  
I.G. Schachmannski ◽  
M.B. Tabanov
Keyword(s):  
Splitting Of Separatrices

scholarly journals On a galoisian approach to the splitting of separatrices

1999 ◽  
Vol 8 (1) ◽  
pp. 125-141 ◽  
Author(s):  
Juan J. Morales-Ruiz ◽  
Josep Maria Peris
Keyword(s):  
Splitting Of Separatrices

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.

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