Res. Assis. İsmail Güzel was a visiting researcher at Computational Mathematics, Science and Engineering, Michigan State University, USA with supervisor Dr. Elizabeth Munch thanks to TÜBİTAK 2214-A International Doctoral Research Fellowship Programme.
The study titled “Detecting bifurcations in dynamical systems with CROCKER plots” was published in Chaos: An Interdisciplinary Journal of Nonlinear Science by ITU Mathematics Engineering faculty member Res. Assis. İsmail Güzel and this article has been selected as Featured Article by the editor.
Topological Data Analysis (TDA), a new field in data science, deals with the shape of data, often using the method of persistent homology, and there are many studies that use TDA in applied fields. Among the most important of these applications are nonlinear time series analysis and the detection of bifurcations and chaos in dynamical systems. In the candidate's published paper, he investigated the bifurcation parameter and chaos states in dynamical systems and showed that they are determined by topological invariants computed with TDA. The method proposed in this paper is found to be faster and more efficient compared to older methods in the literature. This paper has been selected as a Featured Article by the editors of the relevant publication.
Bifurcations are qualitative changes in system response, such as transitions from regular to chaotic activity. Detecting these bifurcations becomes difficult when the system is characterized by differential equations with a high number of degrees of freedom, or when all that is known is a time series from a system observable. As a result, it is concentrated on a data-driven approach to bifurcation identification in which it is assessed features of embedded time series to identify if and when the system's behavior has altered. Topological data analysis (TDA) methods are employed, and shape is quantified using algebraic topology tools. The goal of this study is to demonstrate how the CROCKER plot, a visual tool for understanding evolving shapes, may be used to analyze and discover bifurcations. Furthermore, compressed forms of this information may be connected with system changes by taking the norm over columns of the resultant matrix. The study is carried out by empirically testing the theories on ten dynamical systems and comparing them to established bifurcation analysis methods.