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Quantifying Chaos: Practical Estimation of the Correlation Dimension


Theiler, James Patrick (1988) Quantifying Chaos: Practical Estimation of the Correlation Dimension. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ct5h-nx62.


Be it a physical object or a mathematical model, a nonlinear dynamical system can display complicated aperiodic behavior, or "chaos." In many cases, this chaos is associated with motion on a strange attractor in the system's phase space. And the dimension of the strange attractor indicates the effective number of degrees of freedom in the dynamical system.

In this thesis, we investigate numerical issues involved with estimating the dimension of a strange attractor from a finite time series of measurements on the dynamical system.

Of the various definitions of dimension, we argue that the correlation dimension is the most efficiently calculable and we remark further that it is the most commonly calculated. We are concerned with the practical problems that arise in attempting to compute the correlation dimension. We deal with geometrical effects (due to the inexact self-similarity of the attractor), dynamical effects (due to the nonindependence of points generated by the dynamical system that defines the attractor), and statistical effects (due to the finite number of points that sample the attractor). We propose a modification of the standard algorithm, which eliminates a specific effect due to autocorrelation, and a new implementation of the correlation algorithm, which is computationally efficient.

Finally, we apply the algorithm to chaotic data from the Caltech tokamak and the Texas tokamak (TEXT); we conclude that plasma turbulence is not a low-dimensional phenomenon.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Physics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Corngold, Noel Robert
Thesis Committee:
  • Corngold, Noel Robert (chair)
  • Fox, Geoffrey C.
  • Cross, Michael Clifford
  • Caughey, Thomas Kirk
  • Gould, Roy Walter
Defense Date:11 September 1987
Record Number:CaltechTHESIS:03192013-154809803
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7530
Deposited By: Benjamin Perez
Deposited On:19 Mar 2013 22:59
Last Modified:16 Apr 2021 22:27

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