Probability density functions of some skew tent maps

Lora Billings, E. M. Bollt

Research output: Contribution to journalArticle

33 Citations (Scopus)

Abstract

We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval Fa,b. We show that Fa,b is Markov for a dense set of parameters in the chaotic region, and we exactly find the probability density function (pdf), for any of these maps. It is well known, that when a sequence of transformations has a uniform limit F, and the corresponding sequence of invariant pdfs has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdfs are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.

Original languageEnglish
Pages (from-to)365-376
Number of pages12
JournalChaos, Solitons and Fractals
Volume12
Issue number2
DOIs
StatePublished - 2 Jan 2001

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Tent Map
Probability density function
Skew
Invariant
Unimodal Map
Weak Limit
Convergent Sequence
Piecewise Linear
Lyapunov Exponent
Two Parameters
Integrate
Interval
Family

Cite this

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Probability density functions of some skew tent maps. / Billings, Lora; Bollt, E. M.

In: Chaos, Solitons and Fractals, Vol. 12, No. 2, 02.01.2001, p. 365-376.

Research output: Contribution to journalArticle

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