The DOI matches its Crossref record; 1 of the 1 references it cites were found in a registry or an index.
Estimating fractal dimension
Abstract
Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor.
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Citations
Cites
Detecting strange attractors in turbulence
Takens, F. (1981) · Lecture Notes in Mathematics · doi:10.1007/bfb0091924 · Cited by 10,459
Cited by
Improved Surrogate Data for Nonlinearity Tests
Schreiber, T., & Schmitz, A. (1996) · Physical Review Letters · doi:10.1103/physrevlett.77.635 · Cited by 1,506
Causality detection based on information-theoretic approaches in time series analysis
HLAVACKOVASCHINDLER, K., PALUS, M., VEJMELKA, M., & BHATTACHARYA, J. (2007) · Physics Reports · doi:10.1016/j.physrep.2006.12.004 · Cited by 800
Chaos in Ecology: Is Mother Nature a Strange Attractor?
Hastings, A., Hom, C. L., Ellner, S., Turchin, P., & Godfray, H. C. J. (1993) · Annual Review of Ecology and Systematics · doi:10.1146/annurev.es.24.110193.000245 · Cited by 554
Approximate Entropy as a diagnostic tool for machine health monitoring
Yan, R., & Gao, R. X. (2007) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2006.02.009 · Cited by 454
A new hyperchaotic map and its application in an image encryption scheme
Boriga, R., Dăscălescu, A. C., & Priescu, I. (2014) · Signal Processing: Image Communication · doi:10.1016/j.image.2014.04.001 · Cited by 101
Metadata
- Authors
- Theiler, J.
- Venue
- Journal of the Optical Society of America A
- Publisher
- Optica Publishing Group
- Published
- 1990-06-01
- Volume
- 7
- Issue
- 6
- Pages
- 1055
- ISSN
- 1084-7529 (ISSN-L)
- 1520-8532
- OpenAlex
- W2164284840
- Cited by
- 947
- References
- 1
- FWCI
- 29.88
- Type
- journal article
- Language
- en
- Open access
- closed
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Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
