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2017 | OriginalPaper | Buchkapitel

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

verfasst von : Firas A. Khasawneh, Elizabeth Munch

Erschienen in: Time Delay Systems

Verlag: Springer International Publishing

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Abstract

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.

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Vorheriges Kapitel Numerical Stability Test of Linear Time-Delay Systems of Neutral Type

Nächstes Kapitel Stability and Control of Fractional Periodic Time-Delayed Systems

We are using homology with \({\mathbb Z}_2\) coefficients, hence \(H_k({\mathbb X})\) is always a vector space.

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Titel: Utilizing Topological Data Analysis for Studying Signals of Time-Delay Systems
verfasst von: Firas A. Khasawneh
Elizabeth Munch
Verlag: Springer International Publishing
Buch: Time Delay Systems
Print ISBN: 978-3-319-53425-1

Electronic ISBN: 978-3-319-53426-8

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-53426-8_7

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