Date of Award


Document Type


Degree Name

Master of Science (MS)


College of Science and Mathematics


Mathematical Sciences

Thesis Sponsor/Dissertation Chair/Project Chair

Lora Billings

Committee Member

Roman Zaritski

Committee Member

Philip A. Yecko


Various methods are described in this thesis that will approximate probability density functions (PDF), also known as invariant measures for chaotic maps. When studying discrete dynamical systems, the measure of a map is important in describing its behavior because it captures the statistics of long term simulations. Being invariant means that this distribution remains the same, no matter when you observe it. The main goal is to understand how to obtain these PDFs for various one-dimensional maps using alternative methods to time series data.

The methods studied in this thesis include the Z-matrix and the Frobenius-Perron Operator, which are analytic methods used to calculate the invariant measure. A more measure-theoretic approach was also studied known as Markov partitioning. The Markov approach involves partitioning the domain into disjoint sets and measuring the transport between them. In all methods studied, a transition matrix was derived and the eigenvector associated to the eigenvalue of one corresponds to the invariant measure of the map, when properly normalized. The methods are demonstrated using a set of piecewise linear maps, the logistic map, and the double wigwam map from [11].

Matrix analysis is used in conjunction with computer programming to investigate the behavior of one-dimensional dynamical systems. The theory behind the Markov Transport Matrix and the use of programming are described. A transition/transport matrix is a square matrix that describes the probabilities of moving from one state to another in a dynamical system. Programs were created throughout the research to create this transition matrix for non-uniform and uniform mesh sizes. The programs were designed following the theory in [11] to create transition matrices and find the PDF without knowing the partition. The well-known result given by this transition matrix is the PDF (invariant measure). Using Maple or the power method program created we can find the left eigenvector of the transition matrix and in return will have found the invariant measure. We also extend this by finding the right eigenvector associated with the second largest eigenvalue to find the “almost invariant” measure.

The tools described help categorize information about the original noise-less system. Identifying the invariant and “almost invariant” regions are especially important when noise is added because then there will exist a transition between the two sets. The methods used in this paper come from analytic measure-theoretic approaches rather than time-series data approximations.

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