Persistent homology of the logistic map: an exploration of chaos - Neville, Rachel, author; Shipman, Patrick, advisor; Peterson, Chris, committee member; Thompson, Stephen, committee member (Colorado State University. Libraries, 2014)
Given a discrete sampling of points, how can one reconstruct the underlying geometric object? Further, the question arises how can one discern between noise and sampling distortion and important topological features. Algebraic and topological techniques used computationally can prove to be powerful and currently unconventional tools to understand the "shape" of data. In recent years, persistent homology has been explored as a computational way to capture information regarding the longevity of topological features of discrete data sets. In this project, the persistent homology of functions is explored specifically as a way of examining features of functions. Persistent homology tracks the longevity of connected components of level sets in a persistence diagram. By connecting points generated by a discrete time dynamical system with line segments, this data can be viewed as a (piece-wise linear) function, persistent homology is used to track features of the data. This provides a novel and useful tool for computationally examining dynamical systems. The logistic map is one of the simplest examples of a nonlinear map that displays periodic behavior for some parameter values, but for others, displays chaotic behavior. When the persistence diagram is generated for an orbit of the logistic map, all of the points surprisingly lie approximately on a line. This is not true for a general sequence. This pattern arises not only after stability has been reached in the periodic case, but also as points approach stability for parameters in the periodic regime but also perhaps more surprisingly, for parameter values that lie in the chaotic regime as well. In fact, the slope of this line is fairly similar as the parameter values are varied. This arises from the order in which the points pair to form the persistence diagram and a scaling factor seen in the periodic regime of a class of maps (including the logistic map). It is interesting that the effects of this scaling are still seen in the chaotic regime. This pattern not only arises for the logistic map, but for other unimodal maps and other higher dimensional systems that are "close" to these maps such as the Lorenz system.
Keywords: chaos; unimodal; persistent homology; logistic map