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Topological characterization and early detection of bifurcations and chaos in complex systems using persistent homology. (English) Zbl 1388.37026
Summary: Early detection of bifurcations and chaos and understanding their topological characteristics are essential for safe and reliable operation of various electrical, chemical, physical, and industrial processes. However, the presence of non-linearity and high-dimensionality in system behavior makes this analysis a challenging task. The existing methods for dynamical system analysis provide useful tools for anomaly detection (e.g., Bendixson-Dulac and Poincare-Bendixson criteria can detect the presence of limit cycles); however, they do not provide a detailed topological understanding about system evolution during bifurcations and chaos, such as the changes in the number of subcycles and their positions, lifetimes, and sizes. This paper addresses this research gap by using topological data analysis as a tool to study system evolution and develop a mathematical framework for detecting the topological changes in the underlying system using persistent homology. Using the proposed technique, topological features (e.g., number of relevant \(k\)-dimensional holes, etc.) are extracted from nonlinear time series data which are useful for deeper analysis of the system behavior and early detection of bifurcations and chaos. When applied to a Logistic map, a Duffing oscillator, and a real life Op-amp based Jerk circuit, these features are shown to accurately characterize the system dynamics and detect the onset of chaos.
©2017 American Institute of Physics

MSC:
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37G10 Bifurcations of singular points in dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37G35 Dynamical aspects of attractors and their bifurcations
34C28 Complex behavior and chaotic systems of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37M10 Time series analysis of dynamical systems
Software:
javaPlex; K2
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[1] Bandt, C.; Pompe, B., Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., 88, 174102, (2002)
[2] De Silva, V.; Carlsson, G. E., Topological estimation using witness complexes, 157-166, (2004)
[3] Donoho, D. L., De-noising by soft-thresholding, IEEE Trans. Inf. Theory, 41, 613-627, (1995) · Zbl 0820.62002
[4] Donoho, D. L.; Johnstone, J. M., Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81, 425-455, (1994) · Zbl 0815.62019
[5] Edelsbrunner, H.; Letscher, D.; Zomorodian, A., Topological persistence and simplification, Discrete Comput. Geom., 28, 511-533, (2002) · Zbl 1011.68152
[6] Garland, J.; Bradley, E.; Meiss, J. D., Exploring the topology of dynamical reconstructions, Phys. D, 334, 49-59, (2016) · Zbl 1415.37103
[7] Ghrist, R., Barcodes: The persistent topology of data, Bull. Am. Math. Soc., 45, 61-75, (2008) · Zbl 1391.55005
[8] Grassberger, P.; Procaccia, I., Measuring the strangeness of strange attractors, The Theory of Chaotic Attractors, 170-189, (2004), Springer · Zbl 0593.58024
[9] Gupta, S.; Ray, A., Pattern identification using lattice spin systems: A thermodynamic formalism, Appl. Phys. Lett., 91, 194105, (2007)
[10] Gupta, S.; Ray, A., Statistical mechanics of complex systems for pattern identification, J. Stat. Phys., 134, 337-364, (2009) · Zbl 1162.82007
[11] Hatcher, A., Algebraic Topology, (2002), Cambridge University Press · Zbl 1044.55001
[12] Jha, D. K.; Singh, D. S.; Gupta, S.; Ray, A., Fractal analysis of crack initiation in polycrystalline alloys using surface interferometry, EPL (Europhys. Lett.), 98, 44006, (2012)
[13] Jordan, D.; Smith, P., Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, (2007), Oxford University Press on Demand · Zbl 1130.34001
[14] Khalil, H. K.; Grizzle, J., Nonlinear Systems, (1996), Prentice Hall: Prentice Hall, New Jersey
[15] Maletić, S.; Zhao, Y.; Rajković, M., Persistent topological features of dynamical systems, Chaos, 26, 053105, (2016) · Zbl 1361.37068
[16] Marwan, N.; Romano, M. C.; Thiel, M.; Kurths, J., Recurrence plots for the analysis of complex systems, Phys. Rep., 438, 237-329, (2007)
[17] Mees, A. I., Dynamics of Feedback Systems, (1981), John Wiley & Sons, Inc. · Zbl 0454.93003
[18] Munkres, J. R., Elements of Algebraic Topology, (1984), Addison-Wesley Menlo Park · Zbl 0673.55001
[19] Pereira, C. M.; de Mello, R. F., Persistent homology for time series and spatial data clustering, Expert Syst. Appl., 42, 6026-6038, (2015)
[20] Sauer, T. D.; Tempkin, J. A.; Yorke, J. A., Spurious lyapunov exponents in attractor reconstruction, Phys. Rev. Lett., 81, 4341, (1998)
[21] Takens, F., Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, Warwick 1980, 366-381, (1981), Springer · Zbl 0513.58032
[22] Tausz, A.; Vejdemo-Johansson, M.; Adams, H., JavaPlex: A research software package for persistent (co)homology, 129-136, (2014) · Zbl 1402.65186
[23] Tchitnga, R.; Nguazon, T.; Fotso, P. H. L.; Gallas, J. A., Chaos in a single op-amp-based jerk circuit: Experiments and simulations, IEEE Trans. Circuits Syst., 63, 239-243, (2016)
[24] Zomorodian, A.; Carlsson, G., Computing persistent homology, Discrete Comput. Geom., 33, 249-274, (2005) · Zbl 1069.55003
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