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Quasi-periodic perturbation of unimodal maps exhibiting an attracting 3-cycle
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2012 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, no 3, p. 683-741Article in journal (Refereed) Published
Abstract [en]

We study a class of smooth maps Phi : T x [0, 1]. T x [0, 1] of the form theta bar right arrow theta + omega x bar right arrow c(theta)h(x) where h : [0, 1] --> [0, 1] is a unimodal map exhibiting an attracting periodic point of prime period 3, and omega is irrational (T = R/Z). We show that the following phenomenon can occur for certain h and c : T --> R: There exists a single measurable function psi : T --> [0, 1] whose graph attracts (exponentially fast) a.e. (theta, x) is an element of T x [0, 1] under forward iterations of the map Phi. Moreover, the graph of psi is dense in a cylinder M subset of T x [0, 1]. Furthermore, for every integer n >= 1 there exists n distinct repelling continuous curves Gamma(k) : (theta, phi(k)(theta))(theta is an element of T), all lying in M, such that Phi(Gamma(k)) = Gamma(k+1) (k < n) and Phi(Gamma(n)) = Gamma(1). We give concrete examples where both c(theta) and h(x) are real-analytic, but in the analysis we only need that they are C-1. In our setting the function c(theta) will be very close to 1 for all theta outside a tiny interval; on the interval c(theta) > 1 makes a small bump. Thus we cause the perturbation of h by rare quasi-periodic kicking.

Place, publisher, year, edition, pages
2012. Vol. 25, no 3, p. 683-741
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-91605DOI: 10.1088/0951-7715/25/3/683ISI: 000300607900007Scopus ID: 2-s2.0-84857554638OAI: oai:DiVA.org:kth-91605DiVA, id: diva2:513080
Note
QC 20120330Available from: 2012-03-30 Created: 2012-03-19 Last updated: 2017-12-07Bibliographically approved

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