Publication Abstracts

Smith 1988

Smith, L.A., 1988: Lacunarity and Chaos in Nature. Ph.D. thesis. Columbia University.

The general solution of self-similar scaling equations is seen to contain an oscillatory component. The lacunarity of a self-similar set is reflected in this oscillatory component of the set's scaling statistics. Strange attractors found in nonlinear dynamical systems often possess self-similar structure. After introducing some new results regarding a strange attractor of astrophysical interest and considering the effects of using digital computers to study chaos, we demonstrate that the standard method of characterizing the global structure of these sets is vulnerable to errors arising from a misinterpretation of the oscillatory component. A new method of describing strange attractors by a spectrum of local scaling exponents computed about unstable periodic orbits is proposed. In addition to being computationally tractable, this approach utilizes the lacunarity oscillation to compute an accurate estimate of the residual error. The lacunarity oscillation exists both for self-similar physical distributions and strange attractors. The role of this oscillation in the structure functions of fully developed fluid turbulence is exhibited through the beta model. The physical structure of the observable matter distribution in the universe is also considered. A new example of Hamiltonian chaos which is of geophysical interest is introduced. Specifically, a model is developed to simulate the motion of slowly settling tracers in a time-dependent flow. Here the time development of the length of a material line of tracer particles is shown to be quite complicated even when the background flow field is laminar. A class of two-dimensional maps that simulate this type of flow is developed. Finally, the concept of a strange accumulator is introduced to describe the strong inhomogeneities observed to form in the probability distribution of a single chaotic trajectory.

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BibTeX Citation

  author={Smith, L. A.},
  title={Lacunarity and Chaos in Nature},
  school={Columbia University},
  address={New York, N.Y.},

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RIS Citation

ID  - sm05000n
AU  - Smith, L. A.
PY  - 1988
BT  - Lacunarity and Chaos in Nature
PB  - Columbia University
CY  - New York, N.Y.
ER  -

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