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Correlation dimension as a measure of geophysical log chaos
Published in Vladimir Litvinenko, Topical Issues of Rational Use of Natural Resources 2019, 2019
Grassberger and Procaccia gave the algorithm of calculating correlation dimension based on a correlation sum (correlation integral) (Grassberger, Procaccia 1983). Mathematically it can be written: C(R)=limN→∞1N(N−1)∑i=1N∑j=1,j≠iNΘR−xi−xj
Bifurcation and Chaos in Discrete Models: An Introductory Survey
Published in Wilfrid Perruquetti, Jean-Pierre Barbot, Chaos in Automatic Control, 2018
In the presence of only experimental data, in the form of time series, a fundamental problem is that of discriminating the chaos from the noise, or extracting a deterministic phenomenon from the random noise. For such a purpose the power spectrum technique has limited efficiency. The notion of dimension, which can be done in several ways, is preferred. Thus one has the Kolgomorov’s capacity dimension Dc, an improvement of which is the information dimension DI defined from the information entropy [61]. The measure of correlation between points of a chaotic attractor can be made by the correlation integral, from which the correlation dimension Dco is defined. In general Dco ≤ DI < Dc. From an experimental signal, if Dco is lesser than or equal to the phase space dimension, it is likely that one has a deterministic origin. The advantage of the correlation dimension Dco lies in its determination which is easier to obtain than the Dc and DI ones. The notion of generalized information dimension includes the aforementioned three dimensions as particular cases [74], and leads to a thermodynamic analogy. In presence of time series of only one variable, an important problem is that of the phase space reconstitution. A method is presented in Grassberger and Procaccia [61], the possibilities of which are limited in presence of noise. An interesting approach for the determination of the phase space, and the distinction of a chaotic signal from a random one, is given in Sugihara and May [157] and May [96].
Spatio-temporal dynamics of jerky flow in high-entropy alloy at extremely low temperature
Published in Philosophical Magazine, 2021
Z. Pu, Z. C. Xie, R. Sarmah, Y. Chen, C. Lu, G. Ananthakrishna, L. H. Dai
When an irregular time series of the kind seen in our experiments is suspected to be of deterministic origin, the true collective dynamics operating in a higher dimensional space can be unfolded by reconstructing the attractor using an embedding technique [56,60]. Consider a time series of length M recorded in units of given by . A dimensional attractor can then be reconstructed by defining the vectors ; ], where τ is a suitable delay time. Then, Grassberger-Procaccia (GP) algorithm [61] defines the correlation integral as the fraction of the pairs of points and whose distance is less than a specified value r. Then, the self-similar nature of is given by in the limit of small r. Then, the correlation dimension is defined byThe limit is seldom reached. In practice, one looks for the convergence of the slope to a finite value in the plot of ln verses ln r (over a fair range) as increases. The value of is identified as the correlation dimension of the attractor, a characteristic feature of chaotic dynamics.