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Dynamics of the Uni...
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Gaidashev, Denis,1973-Uppsala universitet,Matematiska institutionen
(author)
Dynamics of the Universal Area-Preserving Map Associated with Period Doubling : Hyperbolic Sets
- Article/chapterEnglish2009
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2009-09-10
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IOP Publishing,2009
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LIBRIS-ID:oai:DiVA.org:uu-107533
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https://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-107533URI
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https://doi.org/10.1088/0951-7715/22/10/010DOI
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Language:English
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Summary in:English
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Subject category:ref swepub-contenttype
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Subject category:art swepub-publicationtype
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It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of R-2. A renormalization approach has been used in Eckmann et al (1982 Phys. Rev. A 26 720-2) and Eckmann et al (1984 Mem. Am. Math. Soc. 47 1-121) in a computer-assisted proof of existence of a 'universal' area-preserving map F-*-a map with orbits of all binary periods 2(k), k is an element of N. In this paper, we consider maps in some neighbourhood of F-* and study their dynamics. We first demonstrate that the map F* admits a 'bi-infinite heteroclinic tangle': a sequence of periodic points {z(k)}, k is an element of Z, vertical bar z(k vertical bar) ->(k ->infinity) 0, vertical bar z(k vertical bar) k ->(k ->infinity) infinity, (1) whose stable and unstable manifolds intersect transversally; and, for any N is an element of N, a compact invariant set on which F-* is homeomorphic to a topological Markov chain on the space of all two-sided sequences composed of N symbols. A corollary of these results is the existence of unbounded and oscillating orbits. We also show that the third iterate for all maps close to F* admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set: 0.7673 >= dim(H) (C-F) >= epsilon approximate to 0.00013 e(-7499).
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Johnson, Tomas,1979-Uppsala universitet,Matematiska institutionen(Swepub:uu)tomjo390
(author)
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Uppsala universitetMatematiska institutionen
(creator_code:org_t)
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In:Nonlinearity: IOP Publishing22:10, s. 2487-25200951-77151361-6544
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