#### Title

Finite Element Maximum Entropy Method for Approximating Absolutely Continuous Invariant Measures

#### Date of Award

Summer 2017

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Committee Chair

Dr. Jiu Ding

#### Committee Chair Department

Mathematics

#### Committee Member 2

Dr. Ching-Shyang Chen

#### Committee Member 2 Department

Mathematics

#### Committee Member 3

Dr. James V. Lambers

#### Committee Member 3 Department

Mathematics

#### Committee Member 4

Dr. Haiyan Tian

#### Committee Member 4 Department

Mathematics

#### Abstract

In a chaotic dynamical system, the eventual behavior of iterates of initial points of a map is unpredictable even though the map is deterministic. A system which is chaotic in a deterministic point of view may be regular in a statistical viewpoint. The statistical viewpoint requires the study of absolutely continuous invariant measure (ACIM) of a map with respect to the Lebesgue measure. An invariant density of the Frobenius-Perron (F-P) operator associated with a nonsingular map is employed to evaluate an ACIM of the map. The ACIM is a key factor for studying the eventual behavior of iterates of almost all initial points of the map. It is difficult to obtain an invariant density of the F-P operator in an exact mathematical form except for some simple maps. Different numerical schemes have been developed to approximate such densities. The maximum entropy principle gives a criterion to select a least-biased density among all densities satisfying a system of moment equations. In this principle, a least-biased density maximizes the Boltzmann entropy. In this dissertation, piecewise quadratic functions and quadratic splines are used in the maximum entropy method to calculate the L^{1} errors between the exact and the approximate invariant densities of the F-P operator associated with nonsingular maps defined from [0;1] to itself. The numerical results are supported by rigorous mathematical proofs. The L^{1} errors between the exact and approximate invariant densities of the Markov operator associated with Markov type position dependent random maps, defined from [0;1] to itself, are calculated by using the piecewise linear polynomials maximum entropy method.

#### Recommended Citation

Upadhyay, Tulsi, "Finite Element Maximum Entropy Method for Approximating Absolutely Continuous Invariant Measures" (2017). *Dissertations*. 1434.

http://aquila.usm.edu/dissertations/1434