TY - JOUR

T1 - Probability density functions of some skew tent maps

AU - Billings, L.

AU - Bollt, E. M.

PY - 2001/1/2

Y1 - 2001/1/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0343462471&partnerID=8YFLogxK

U2 - 10.1016/S0960-0779(99)00204-0

DO - 10.1016/S0960-0779(99)00204-0

M3 - Article

AN - SCOPUS:0343462471

SN - 0960-0779

VL - 12

SP - 365

EP - 376

JO - Chaos, Solitons and Fractals

JF - Chaos, Solitons and Fractals

IS - 2

ER -