Utilizing Topological Data Analysis for Studying Signals of Time-Delay Systems

  • Firas A. KhasawnehEmail author
  • Elizabeth Munch
Part of the Advances in Delays and Dynamics book series (ADVSDD, volume 7)


This chapter describes a new approach for studying the stability of stochastic delay equations by investigating their time series using topological data analysis (TDA). The approach is illustrated utilizing two stochastic delay equations. The first model equation is the stochastic version of Hayes equation—a scalar autonomous delay equation—where the noise is an additive term. The second model equation is the stochastic version of Mathieu’s equation—a time-periodic delay equation. In the latter, noise is added via a multiplicative term in the time-periodic coefficient. The time series is generated using Euler–Maruyama method and a corresponding point cloud is obtained using the Takens’ embedding. The point cloud is then analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems that are described using constant as well as time-periodic coefficients. The presented approach can be used for signals generated from both numerical simulation and experiments. It can be used as a tool to study the stability of stochastic delay equations for which there are currently a limited number of analysis tools.


Point Cloud Stability Boundary Stochastic Resonance Stability Diagram Spectral Element Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors would like to acknowledge financial support from the U.S. National Science Foundation through grants CMMI-1562012 and CMMI-1562459.


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Authors and Affiliations

  1. 1.Mechanical EngineeringSUNY Polytechnic InstituteUticaUSA
  2. 2.Department of Mathematics and StatisticsUniversity at Albany – SUNYAlbanyUSA

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