Transversal connecting orbits from shadowing
Resource
Numerische Mathematik 106 (3): 427-469
Journal
Numerische Mathematik
Pages
427-469
Date Issued
2007
Date
2007
Author(s)
Coomes, B. A.
Kocak, H.
Palmer, K. J.
Abstract
A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in ℝ is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits. © Springer-Verlag 2007.
Type
journal article
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