Abstract
Spatio-temporal analysis of a time series from a complex dynamical system often requires reconstruction of the state-space attractor from observations of a single state variable. The standard approach takes advantage of the Takens delay embedding theorem to obtain the reconstruction. We investigate here a modification which makes use of nonlinear spectral graph techniques for learning the underlying manifold from high-dimensional data. Specifically, we examine how well diffusion maps and locally-linear embedding recover system dynamics and their sensitivity to parameters. Analysis is conducted using individual observations of the chaotic Lorenz and Hénon attractors. We show that manifold embeddings, given selected parameter choices, can improve forecasting capability for chaotic time series.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Becks, F. Hilker, H. Malchow, K. Jurgens, H. Arndt, Experimental demonstration of chaos in a microbial food web. Nature Lett. 435(30), 1226–1229 (2005)
E. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)
F. Moon, Fractal boundary for chaos in a two-state mechanical oscillator. Phys. Rev. Lett. 53(10), 962–964 (1984)
F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence: Springer Lecture Notes in Mathematics, ed. by D. Rand, L. Young (Springer, Berlin, 1981), pp. 366–381
A. Fraser, H. Swinney, Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33(2), 1134–1140 (1986)
M. Kennel, R. Brown, H. Abarbanel, Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45(6), 3403–3411 (1992)
T. Sauer, J. Yorke, How many delay coordinates do you need? Int. J. Bifurcation Chaos 3(3), 737–744 (1993)
D. Broomhead, G. King, Extracting qualitative dynamics from experimental data. Physica D 20(23), 217236 (1986)
R. Vautard, M. Ghil, Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series. Physica D 35, 395–424 (1989)
N. Golyandina, V. Nekrutkin, A. Zhigljavsky, Analysis of Time Series Structure: SSA and Related Techniques (CRC, Boka Raton, 2001)
J. Lee, M. Verleysen, Nonlinear Dimensionality Reduction (Springer, New York, 2007)
M. Belkin, P. Niyogi, Semi-supervised learning on riemannian manifolds. Mach. Learn. 56, 209–239 (2004)
D. Giannakis, A. Majda, Nonlinear laplacian spectral analysis for time series with intermittency and low-frequency variability. Proc. Nat. Acad. Sci. 109(7), 2222–2227 (2012)
H. Suetani, S. Akaho, A ransac-based isomap for filiform manifolds in nonlinear dynamical systems -an application to chaos in a dripping faucet, in ICANN, pp. 277–284 (2011)
R. Coifman, S. Lafon, Nonlinear dimensionality reduction using locally linear embedding. Appl. Comput. Harmon. Anal. 21(22), 530 (2006)
S. Roweis, L. Saul, Nonlinear dimensionality reduction using locally linear embedding. Science 290(22)
M. Hénon, A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50(1), 69–77 (1976)
J. Eckmann, D. Ruelle, Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617 (1985)
P. Grassberger, I. Procaccia, Measuring the strangeness of strange attractors. Physica D 9(1–2), 189–208 (1983)
J. Kaplan, J. Yorke, Chaotic behavior of multidimensional difference equations, in Lecture Notes in Mathematics, vol. 730, ed. by H. Peitgen, W. Walther (Springer, Berlin, 1979), pp. 204–227
M. Sano, Y. Sawada, Measurement of the lyapunov spectrum from a chaotic time series. Phys. Rev. Lett. 55(10), 1082–1085 (1985)
L. Pecora, T. Carroll, Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data. Chaos 6(3), 432–439 (1196)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Overbey, L.A., Olson, C.C. (2014). Investigating the Use of Manifold Embedding for Attractor Reconstruction from Time Series. In: In, V., Palacios, A., Longhini, P. (eds) International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012). Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-02925-2_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-02925-2_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02924-5
Online ISBN: 978-3-319-02925-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)