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Sampled-Data Stabilization of Nonlinear Delay Systems with a Compact Absorbing Set and State Measurement Read chapter Read first chapter

2017 | OriginalPaper | Chapter

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

Authors : Firas A. Khasawneh, Elizabeth Munch

Published in: Time Delay Systems

Publisher: 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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previous chapter Numerical Stability Test of Linear Time-Delay Systems of Neutral Type
next chapter Stability and Control of Fractional Periodic Time-Delayed Systems
Footnotes
1
We are using homology with \({\mathbb Z}_2\) coefficients, hence \(H_k({\mathbb X})\) is always a vector space.
 
Literature
1.
Hale, J.K., Lunel, S.V.: Introduction to Functional Differential Equations. Springer, New York (1993)CrossRefMATH
2.
Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Functions. Wiley (1989)
3.
Walther, H.-O.: Topics in delay differential equations. Jahresbericht der Deutschen Mathematiker-Vereinigung 116(2), 87–114 (2014)MathSciNetCrossRefMATH
4.
Buckwar, E., Kuske, R., L’Esperance, B., Soo, T.: Noise-sensitivity in machine tool vibrations. Int. J. Bifurc. Chaos 16(08), 2407–2416 (2006)CrossRefMATH
5.
Kuske, R.: Multiple-scales approximation of a coherence resonance route to chatter. Comput. Sci. Eng. 8(3), 35–43 (2006)CrossRef
6.
Klamecki, B.E.: Enhancement of the low-level components of milling vibration signals by stochastic resonance. Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng. 218(1), 33–41 (2004)CrossRef
7.
Yilmaz, A., AL-Regib, E., Ni, J.: Machine tool chatter suppression by multi-level random spindle speed variation. J. Manuf. Sci. Eng. 124(2), 208–216 (2002)CrossRef
8.
Barrio, M., Burrage, K., Leier, A., Tian, T.: Oscillatory regulation of Hes1: discrete stochastic delay modelling and simulation. PLoS Comput. Biol. 2(9), e117 (2006)
9.
Tian, T, Burrage, K., Burrage, P.M., Carletti, M.: Stochastic delay differential equations for genetic regulatory networks. J. Comput. Appl. Math. 205(2), 696–707 (2007) (Special issue on evolutionary problems)
10.
Mackey, M.C., Nechaeva, I.G.: Solution moment stability in stochastic differential delay equations. Phys. Rev. E 52(4), 3366–3376 (1995)MathSciNetCrossRef
11.
Guillouzic, S., L’Heureux, I., Longtin, A.: Small delay approximation of stochastic delay differential equations. Phys. Rev. E 59, 3970–3982 (1999)CrossRefMATH
12.
Elbeyli, O., Sun, J.Q., Unal, G.: A semi-discretization method for delayed stochastic systems. Commun. Nonlinear Sci. Numer. Simul. 10(1), 85–94 (2005)MathSciNetCrossRefMATH
13.
Frank, T.D., Beek, P.J.: Stationary solutions of linear stochastic delay differential equations: applications to biological systems. Phys. Rev. E 64, 021917 (2001)CrossRef
14.
McKelvey, T., Guerin, G.: Non-parametric frequency response estimation using a local rational model. In: 16th IFAC Symposium on System Identification the International Federation of Automatic Control Brussels, Belgium, July 11–13, 2012, vol. 16(1), pp. 49–54 (2012)
15.
Kantz, H., Just, W., Baba, N., Gelfert, K., Riegert, A.: Fast chaos versus white noise: entropy analysis and a Fokker-Planck model for the slow dynamics. Phys. D Nonlinear Phenom. 187(1–4), 200–213 (2004)CrossRefMATH
16.
Fontanella, L., Granturco, M.: Spectral analysis in frequency and time domain for noisy time series. In: Bock, H.-H., Chiodi, M., Mineo, A. (eds.) Advances in Multivariate Data Analysis. Studies in classification, data analysis, and knowledge organization, pp. 67–79. Springer, Heidelberg (2004)CrossRef
17.
Yao, Z., Mei, D., Chen, Z.: On-line chatter detection and identification based on wavelet and support vector machine. J. Mater. Process. Technol. 210(5), 713–719 (2010)CrossRef
18.
Buckwar, E.: Introduction to the numerical analysis of stochastic delay differential equations. J. Comput. Appl. Math. 125(1–2), 297–307 (2000) (Numerical Analysis 2000. Vol. VI: Ordinary Differential Equations and Integral Equations)
19.
Buckwar, E., Winkler, R.: Multi-step maruyama methods for stochastic delay differential equations. Stoch. Anal. Appl. 25(5), 933–959 (2007)MathSciNetCrossRefMATH
20.
Mao, X., Sabanis, S.: Numerical solutions of stochastic differential delay equations under local lipschitz condition. J. Comput. Appl. Math. 151(1), 215–227 (2003)MathSciNetCrossRefMATH
21.
Yuan, C., Glover, W.: Approximate solutions of stochastic differential delay equations with markovian switching. J. Comput. Appl. Math. 194(2), 207–226 (2006)MathSciNetCrossRefMATH
22.
Yaozhong, H., Mohammed, S.-E.A., Yan, F.: Discrete-time approximations of stochastic delay equations: the milstein scheme. Ann. Probab. 32(1A), 265–314 (2004)MathSciNetCrossRefMATH
23.
Baker, C.T.H., Buckwar, E.: Exponential stability in p-th mean of solutions, and of convergent euler-type solutions, of stochastic delay differential equations. J. Comput. Appl. Math. 184(2), 404–427 (2005)MathSciNetCrossRefMATH
24.
Luo, J.: A note on exponential stability in pth mean of solutions of stochastic delay differential equations. J. Comput. Appl. Math. 198(1), 143–148 (2007)MathSciNetCrossRefMATH
25.
Mao, X.: Exponential stability of equidistant euler-maruyama approximations of stochastic differential delay equations. J. Comput. Appl. Math. 200(1), 297–316 (2007)MathSciNetCrossRefMATH
26.
Khasawneh, F.A., Munch, E.: Exploring equilibria in stochastic delay differential equations using persistent homology. In: Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, August 17–20, 2014, Buffalo, NY, USA (2014). Paper no. DETC2014/VIB-35655
27.
Khasawneh, F.A., Munch, E.: Stability determination in turning using persistent homology and time series analysis. In: Proceedings of the ASME 2014 International Mechanical Engineering Congress & Exposition, November 14–20, 2014, Montreal, Canada (2014). Paper no. IMECE2014-40221
28.
Khasawneh, F.A., Munch, E.: Chatter detection in turning using persistent homology. Mech. Syst. Signal Process. 70–71, 527–541 (2016)CrossRef
29.
Takens, F.: Detecting strange attractors in turbulence. In: Rand, D., Young, L.-S. (eds.) Dynamical Systems and Turbulence. Warwick 1980. Lecture Notes in Mathematics, vol. 898, pp. 366–381. Springer, Heidelberg (1981)
30.
de Silva, V., Skraba, P., Vejdemo-Johansson, M.: Topological analysis of recurrent systems. In: NIPS 2012 Workshop on Algebraic Topology and Machine Learning (2012)
31.
Berwald, J., Gidea, M., Vejdemo-Johansson, M.: Automatic recognition and tagging of topologically different regimes in dynamical systems (2013). arXiv:​1312.​2482
32.
Brown, K.A., Knudson, K.P.: Nonlinear statistics of human speech data. Int. J. Bifurc. Chaos 19(07), 2307–2319 (2009)MathSciNetCrossRef
33.
Emrani, S., Gentimis, T., Krim, H.: Persistent homology of delay embeddings and its application to wheeze detection. IEEE Signal Process. Lett. 21(4), 459–463 (2014)CrossRef
34.
Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Mileyko, Y.: Lipschitz functions have \(l_p\)-stable persistence. Found. Comput. Math. 10(2), 127–139 (2010)MathSciNetCrossRefMATH
35.
Deckard, A., Anafi, R.C., Hogenesch, J.B., Haase, S.B., Harer, J.: Design and analysis of large-scale biological rhythm studies: a comparison of algorithms for detecting periodic signals in biological data. Bioinformatics 29(24), 3174–3180 (2013)CrossRef
36.
Perea, J.A., Deckard, A., Harer, J., Haase, S.: Sw1pers: sliding windows and 1-persistence scoring; discovering periodicity in gene expression time series data (2013) (Preprint)
37.
Perea, J.A., Harer, J.: Sliding windows and persistence: an application of topological methods to signal analysis. Found. Comput. Math. 15(3), 799–838 (2015)MathSciNetCrossRefMATH
38.
Alexander, Z.: A topology-based approach for nonlinear time series with applications in computer performance analysis. Ph.D. thesis, University of Colorado at Boulder (2012)
39.
Alexander, Z., Meiss, J.D., Bradley, E., Garland, J.: Iterated function system models in data analysis: detection and separation. Chaos Interdiscip. J. Nonlinear Sci. 22(2) (2012)
40.
Vejdemo-Johansson, M., Pokorny, F.T., Skraba, P., Kragic, D.: Cohomological learning of periodic motion. Appl. Algebr. Eng. Commun. Comput. 1–22 (2015)
41.
Robinson, M.: Topological Signal Processing. Springer (2014)
42.
de Silva, V., Morozov, D., Vejdemo-Johansson, M.: Persistent cohomology and circular coordinates. Discret. Comput. Geom. 45(4), 737–759 (2011)MathSciNetCrossRefMATH
43.
Carlsson, G.: Topological pattern recognition for point cloud data. Acta Numer. 23, 289–368 (2014). Survey
44.
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, 2000, pp. 454 –463 (2000)
45.
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society (2010)
46.
47.
Hatcher, A.: Algebraic Topology. Cambridge University Press (2002)
48.
Munkres, J.R.: Elements of Algebraic Topology. Addison Wesley (1993)
49.
50.
Milosavljevic, N., Morozov, D., Skraba, P.: Zigzag persistent homology in matrix multiplication time. In: Proceedings of the 27th Annual Symposium on Computational Geometry (2011)
51.
Otter, N., Porter, M.A., Tillmann, U., Grindrod, P., Harrington, H.A.: A roadmap for the computation of persistent homology (2015). arXiv:​ 1506.​08903
52.
Insperger, T., Stépán, G.: Semi-discretization for time-delay systems: stability and engineering applications. In: Stability Charts for Fundamental Delay-Differential Equations, pp. 13–37. Springer New York, New York (2011)
53.
Oksendal, B.: Stochastic Differential Equations, 6th edn. Springer (2007)
54.
Khasawneh, F.A., Mann, B.P.: A spectral element approach for the stability of delay systems. Int. J. Numer. Methods Eng. 87(6), 566–592 (2011)MathSciNetCrossRefMATH
55.
Klosek, M., Kuske, R.: Multiscale analysis of stochastic delay differential equations. Multiscale Model. Simul. 3(3), 706–729 (2005)MathSciNetCrossRefMATH
56.
57.
Mathieu, E.: Memoire sur le mouvement vibratorie d’une membrane de forme elliptique. J. Math. 13, 137–203 (1868)MATH
58.
Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)MathSciNetCrossRefMATH
59.
Harer, J., Slaczedek, J., Bendich, P.: Ripscollapse: Discrete Morse Theory and Fast Computation of One-Dimensional Persistence. Manuscript, Duke University (2014)
Metadata
Title
Utilizing Topological Data Analysis for Studying Signals of Time-Delay Systems
Authors
Firas A. Khasawneh
Elizabeth Munch
Copyright Year
2017
Book
Time Delay Systems

Print ISBN: 978-3-319-53425-1

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

Copyright Year: 2017

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

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