Numerical approximation of homoclinic chaos |
| |
Authors: | Beyn W-J Kleinkauf J-M |
| |
Institution: | 1.Fakult?t für Mathematik, Universit?t Bielefeld, Postfach 10 01 31, D-33501, Bielefeld, Germany ; |
| |
Abstract: | Transversal homoclinic orbits of maps are known to generate a Cantor set on which a power of the map conjugates to the Bernoulli
shift on two symbols. This conjugacy may be regarded as a coding map, which for example assigns to a homoclinic symbol sequence
a point in the Cantor set that lies on a homoclinic orbit of the map with a prescribed number of humps. In this paper we develop
a numerical method for evaluating the conjugacy at periodic and homoclinic symbol sequences in a systematic way. The approach
combines our previous method for computing the primary homoclinic orbit with the constructive proof of Smale's theorem given
by Palmer. It is shown that the resulting nonlinear systems are well conditioned uniformly with respect to the characteristic
length of the symbol sequence and that Newton's method converges uniformly too when started at a proper pseudo orbit. For
the homoclinic symbol sequences an error analysis is given. The method works in arbitrary dimensions and it is illustrated
by examples.
This revised version was published online in June 2006 with corrections to the Cover Date. |
| |
Keywords: | dynamical systems numerical methods homoclinic points for maps multihumped homoclinic orbits chaos 58F13 58F15 58F08 |
本文献已被 SpringerLink 等数据库收录! |
|