## Master's Theses

### Stationary Density Computation of the Frobenius-Perron Operators Based on the Dirac Delta Function

Spring 5-2011

Masters Thesis

#### Degree Name

Master of Science (MS)

Mathematics

Jiu Ding

Mathematics

Joseph Kolibal

Mathematics

Haiyan Tian

Mathematics

#### Abstract

The statistical study of chaotic dynamical systems has received a great deal of attention in the past several decades. As a branch of applied mathematics, its application has been found in various fields in science and engineering, while the theory and methods for the existence and computation of absolutely invariant measures have played an important role in this field. In this study, we focus on the computation of a nontrivial fixed point of Frobenius-Perron operators (F-P operators).

Let S: [0,1] → [0,1] be a piecewise monotonic mapping, and let PS : [0,1] → [0,1] be the Frobenius-Perron operators associated with S, which is defined by

PSf(x) = d/dx ∫S-1([0,x]) fdm, x ∈ [0,1] a.e.,

where m is the Lebesgue measure of [0,1]. Suppose that PS: [0,1] → [0,1] has a stationary density f*. By using Ulam's method, which he proposed based on a probability argument, approximating the fixed density function f* can be constructed by piecewise constant functions with respect to a partition of [0,1]. From another argument, we propose a different form for the definition of the Frobenius-Penon operator by combining the properties of the Dirac delta function. We can prove that the two definitions for Frobenius-Perron operators are equivalent. Then, we find that by approximating the Dirac delta function, we can exactly obtain the famous Ulam's method again. For the computation of fixed density functions we use the quasi- Monte Carlo method. We partition [0,1] into n subintervals, and for each subinterval we take N equal distance test points. Numerical results are given for several one dimensional test mappings.

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