Preprint of the project: SFB 343: Discrete structures in mathematics

Topics in numerical linear algebra and discrete dynamical systems

96-024 Wolf-Jürgen Beyn, Jan-Martin Kleinkauf.
Numerical approximation of Homoclinic Chaos

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. 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.