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RIGOROUS COMPUTATIONS OF HOMOCLINIC TANGENCIES ZIN ARAI AND KONSTANTIN MISCHAIKOW
 

Summary: RIGOROUS COMPUTATIONS OF HOMOCLINIC TANGENCIES
ZIN ARAI AND KONSTANTIN MISCHAIKOW
Abstract. In this paper, we propose a rigorous computational method for detecting homoclinic
tangencies and structurally unstable connecting orbits. It is a combination of several tools and
algorithms, including the interval arithmetic, the subdivision algorithm, the Conley index theory,
and the computational homology theory. As an example we prove the existence of generic homoclinic
tangencies in the H´enon family.
Key words. homoclinic tangency, connecting orbit, Conley index, computational homology
AMS subject classifications. 37B30, 37G25, 37M20
1. Introduction. In this paper, we present a method for proving the existence
of homoclinic tangencies and structurally unstable connecting orbits. More precisely
we are interested in proving the existence of generic tangencies in a one-parameter
family of maps; that is, a quadratic tangency that unfolds generically in the family.
The importance of the generic homoclinic tangency comes from the fact that it implies
the occurrence of the Newhouse phenomena [17] and strange attractors [12].
To explain how the method works, we apply it to the H´enon family
Ha,b : R2
R2
(1.1)
(x, y) (a - x2

  

Source: Arai, Zin - Department of Mathematics, Kyoto University

 

Collections: Mathematics