Numerical approximation of homoclinic chaos

Beyn W-J, Kleinkauf JM (1997)
In: Numerical Algorithms. Numerical Algorithms, 14(1-3). BALTZER SCI PUBL BV: 25-53.

Konferenzbeitrag | Veröffentlicht | Englisch
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Beyn, Wolf-JürgenUniBi; Kleinkauf, J. M.
Abstract / Bemerkung
Transversal homoclinic orbits of maps are known to generate a Canter 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 Canter 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.
multi-humped homoclinic orbits; numerical methods; homoclinic points for maps; chaos; dynamical systems
Titel des Konferenzbandes
Numerical Algorithms
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Beyn W-J, Kleinkauf JM. Numerical approximation of homoclinic chaos. In: Numerical Algorithms. Numerical Algorithms. Vol 14. BALTZER SCI PUBL BV; 1997: 25-53.
Beyn, W. - J., & Kleinkauf, J. M. (1997). Numerical approximation of homoclinic chaos. Numerical Algorithms, Numerical Algorithms, 14, 25-53. BALTZER SCI PUBL BV. doi:10.1023/A:1019196426363
Beyn, W. - J., and Kleinkauf, J. M. (1997). “Numerical approximation of homoclinic chaos” in Numerical Algorithms Numerical Algorithms, vol. 14, (BALTZER SCI PUBL BV), 25-53.
Beyn, W.-J., & Kleinkauf, J.M., 1997. Numerical approximation of homoclinic chaos. In Numerical Algorithms. Numerical Algorithms. no.14 BALTZER SCI PUBL BV, pp. 25-53.
W.-J. Beyn and J.M. Kleinkauf, “Numerical approximation of homoclinic chaos”, Numerical Algorithms, Numerical Algorithms, vol. 14, BALTZER SCI PUBL BV, 1997, pp.25-53.
Beyn, W.-J., Kleinkauf, J.M.: Numerical approximation of homoclinic chaos. Numerical Algorithms. Numerical Algorithms. 14, p. 25-53. BALTZER SCI PUBL BV (1997).
Beyn, Wolf-Jürgen, and Kleinkauf, J. M. “Numerical approximation of homoclinic chaos”. Numerical Algorithms. BALTZER SCI PUBL BV, 1997.Vol. 14. Numerical Algorithms. 25-53.


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