Characterization of Nonlinear Systems

Tentative Outline (References here)

• Introduction and Review (e.g. Abraham and Shaw, 1992).
  • Models of dynamical processes: Maps, flows
  • Steady-state behavior: limit sets and attractors.
• Phase-Space Reconstruction
  • Extract information from a single measured quantity.
  • Method of Delays. (Gershenfeld, 1986; Takens, 1980; Nokes, 1991)
  • Choose a good delay
    • Average mutual information
  • Determine the embedding dimension, estimate the number of active states.
    • Singular system analysis (Broomhead and King, 1986; Abarbanel, 1996)
    • False nearest neighbors (Abarbanel, 1996)
  • Put data into a form accessible to many techniques below.
  • Nonsmoothness problems (Feeny and Liang, 1997).
• Proper Orthogonal Decomposition (Berkooz et al., 1993)
  • Estimate the number of active states (Berkooz et al., 1993; Cusumano, 1993)
  • Spatial coherence or modal information in systems with many degrees of freedom.
  • POMs and linear normal modes (Feeny and Kappagantu, 1998; and Feeny, 1997)
  • POMs and nonlinear normal modes (Feeny and Kappagantu, 1998; Feeny, 1997)
• Fractals--Quantify the geometry of behavior (Mandelbrot, 1983)
  • Hausdorf dimension, limit capacity (Feder, 1988)
  • Information dimension, correlation dimension (Grassberger and Procaccia, 1983; Gershenfeld, 1988)
  • Generalized dimension (Grassberger, 1983; Hentschel and Procaccia, 1983)
  • Multifractals (Halsey et al., 1986; Feder, 1988; Gould and Tobochnik, 1990)
    • dynamics examples (Halsey et al., 1986; Jensen et al., 1985)
    • box counting examples (Feeny, 2000, and figures)
  • Algorithms for generalized dimensions and multifractals
    • box-counting (Molteno, 1993; Feeny, 2000)
    • direct computation
    • correlation dimension (Grassberger, 1983; Pawelzik and Schuster, 1987)
    • comparison of algorithms by Theiler (1990), Ueda (1995).
  • Dynamics on fractals (Barnsley, 1988)
    • symbol dynamics
    • iterated function systems
• Self-Organized Criticality (Bak, 1996)
  • Large dimensional dynamics with spatial and temporal complexity. Unpredictable.
  • Earthquakes, avalanches, and the Gutenberg-Richter law
• Lyapunov Exponents (Abarbanel, 1996; Oseledec, 1968)
  • Quantify predictability--local expansion and contraction.
  • Lyapunov exponents from time series data
  • Lyapunov exponents from a simulated set of ODEs.
• "Skeletons" and Knots (Tufillaro, 1992)
  • Extract unstable periodic orbits (UPOs) from data.
  • Template analysis
  • local dynamics
  • application of UPOs to characterization tools.
• System Identification (harmonic balance, Hilbert transform, higher-order spectra)
  • Seek the form of a model.
  • Estimate parameters in a model.
• Wavelets
  • Analysis of transient data.
  • Application to fractals.