Ordinary differential equations are considered consisting of two equations with
nonlinear coupling where the linear parts of the two equations have equilibria which are,
respectively, a saddle and a center. Perturbation terms are added which correspond to damping and
forcing. A reduced equation is obtained from the hyperbolic equation by setting to zero the
variable from the center equation. Melnikov theory is used to obtain a transverse homoclinic
solution, and hence chaos, in the reduced equation. Conditions are then established such that the
chaos for the reduced equation is shadowed by chaos for the full equation. The resonant case is also
studied when the chaos of the full system is not detected from the reduced equation. The techniques
make use of exponential dichotomies.
Publié le : 2004-03-14
Classification:
ordinary differential equations,
homoclinic solutions,
bifurcations,
center manifold,
34C37,
37C29,
37D45
@article{1080056162,
author = {Fe\v ckan, Michal and Gruendler, Joseph},
title = {The Existence of Chaos for Ordinary Differential Equations with a Center Manifold},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {11},
number = {1},
year = {2004},
pages = { 77-94},
language = {en},
url = {http://dml.mathdoc.fr/item/1080056162}
}
Fečkan, Michal; Gruendler, Joseph. The Existence of Chaos for Ordinary Differential Equations with a Center Manifold. Bull. Belg. Math. Soc. Simon Stevin, Tome 11 (2004) no. 1, pp. 77-94. http://gdmltest.u-ga.fr/item/1080056162/