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Analysis of Observed Chaotic Data

When I encountered the idea of chaotic behavior in deterministic dynami­ cal systems, it gave me both great pause and great relief. The origin of the great relief was work I had done earlier on renormalization group properties of homogeneous, isotropic fluid turbulence. At the time I worked on that,...

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Bibliographic Details
Main Author: Abarbanel, Henry
Format: eBook
Language:English
Published: New York, NY Springer New York 1996, 1996
Edition:1st ed. 1996
Series:Institute for Nonlinear Science
Subjects:
Physics And Astronomy
Physics
Astronomy
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 5.3 Global Lyapunov Exponents
  • 5.4 Lyapunov Dimension
  • 5.5 Global Lyapunov Exponents from Data
  • 5.6 Local Lyapunov Exponents
  • 5.7 Local Lyapunov Exponents from Data
  • 5.8 A Few Remarks About Lyapunov Exponents
  • 6 Modeling Chaos
  • 6.1 Model Making in Chaos
  • 6.2 Local Models
  • 6.3 Global Models
  • 6.4 Phase Space Models for Dependent Dynamical Variables
  • 6.5 “Black Boxes” and Physics
  • 7 Signal Separation
  • 7.1 General Comments
  • 7.2 Full Knowledge of the Dynamics
  • 7.3 Knowing a Signal: Probabilistic Cleaning
  • 7.4 “Blind” Signal Separation
  • 7.5 A Few Remarks About Signal Separation
  • 8 Control and Chaos
  • 8.1 Parametric Control to Unstable Periodic Orbits
  • 8.2 Other Controls
  • 8.3 Examples of Control
  • 8.4 A Few (Irreverent) Remarks About Chaos and Control
  • 9 Synchronization of Chaotic Systems
  • 9.1Identical Systems
  • 9.2 Dissimilar Systems
  • 9.3 Mutual False Nearest Neighbors
  • 9.4 Predictability Tests for Generalized Synchronization
  • 1 Introduction
  • 1.1 Chatter in Machine Tools
  • 2 Reconstruction of Phase Space
  • 2.1 Observations of Regular and Chaotic Motions
  • 2.2 Chaos in Continuous and Discrete Time Dynamics
  • 2.3 Observed Chaos
  • 2.4 Embedding: Phase Space Reconstruction
  • 2.5 Reconstruction Demystified
  • 3 Choosing Time Delays
  • 3.1 Prescriptions for a Time Delay
  • 3.2 Chaos as an Information Source
  • 3.3 Average Mutual Information
  • 3.4 A Few Remarks About I(T)
  • 4 Choosing the Dimension of Reconstructed Phase Space
  • 4.1 Global Embedding Dimension dE
  • 4.2 Global False Nearest Neighbors
  • 4.3 A Few Remarks About Global False Nearest Neighbors
  • 4.4 False Strands
  • 4.5 Other Methods for Identifying dE
  • 4.6 The Local or Dynamical Dimension dL
  • 4.7 Forward and Backward Lyapunov Exponents
  • 4.8 Local False Neighbors
  • 4.9 A Few Remarks About Local False Nearest Neighbors
  • 5 Invariants of the Motion
  • 5.1 Invariant Characteristics of the Dynamics
  • 5.2 Fractal Dimensions
  • 9.5 A Few Remarks About Synchronization
  • 10 Other Example Systems
  • 10.1 Chaotic Laser Intensity Fluctuations
  • 10.2 Chaotic Volume Fluctuations of the Great Salt Lake
  • 10.3 Chaotic Motion in a Fluid Boundary Layer
  • 11 Estimating in Chaos: Cramér-Rao Bounds
  • 11.1 The State Estimation Problem
  • 11.2 The Cramér-Rao Bound
  • 11.3 Symmetric Linear Dynamics
  • 11.4 Arbitrary, Time-Invariant, Linear Systems
  • 11.5 Nonlinear, Chaotic Dynamics
  • 11.6 Connection with Chaotic Signal Separation
  • 11.7 Conclusions
  • 12 Summary and Conclusions
  • 12.1 The Toolkit-Present and Future
  • 12.2 Making ‘Physics’ out of Chaos-Present and Future
  • 12.3 Topics for the Next Edition
  • A.1 Information Theory and Nonlinear Systems
  • A.2 Stability and Instability
  • A.2.1 Lorenz Model
  • References

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