/pdf/determining-lyapunov-exponents-from-a-time-series-3bxxq6d0wt.pdf

Determining Lyapunov exponents from a time series

Measuring the Strangeness of Strange Attractors

The infinite number of generalized dimensions of fractals and strange attractors

/pdf/the-dimension-of-chaotic-attractors-365qdkiktc.pdf

The dimension of chaotic attractors

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.

/pdf/estimating-fractal-dimension-4bfbt20bh2.pdf

Estimating fractal dimension

An algorithm proposed by Takens, which can determine the capacity (generalized dimensionality) of a dynamical system from the time series of a single observable, is tested numerically for several intrinsically stochastic models. The algorithm is found to converge too slowly (if at all) to be useful for the analysis of experimental data.

T. Pignataro

Papers

Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors