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Measuring and locating zones of chaos and irregularity. (English) Zbl 1395.37051
Summary: The new measures computed here are the spectral detrended fluctuation analysis (sDFA) and spectral multi-taper method (sMTM). sDFA applies the standard detrended fluctuation analysis (DFA) algorithm to power spectra. sMTM exploits the minute increases in the broadband response, typical of chaotic spectra approaching optimal values. The authors chose the Brusselator, Lorenz, and Duffing as the proposed models to measure and locate chaos and severe irregularity. Their series of chaotic parametric responses in short time-series is advantageous. Where cycles have only a limited number of slow oscillations such as for systems biology and medicine. It is difficult to create, locate, or monitor chaos. From 50 linearly increasing starting points applied to the chaos target function (CTF); the mean percentage increases in Kolmogorov-Sinai entropy (KS-Entropy) for the proposed chosen models; and \(p\)-values when the models were compared statistically by Kruskal-Wallis and ANOVA1 test with distributions assumed normal are Duffing (CTF: 31%: \(p<0.03\)); Lorenz (CTF: 2%: \(p<0.03\)), and Brusselator (CTF: 8%: \(p<0.01\)). Principal component analysis (PCA) is applied to assess the significance of the objective functions for tuning the chaotic response. From PCA the conclusion is that CTF is the most beneficial objective function overall delivering the highest increases in mean KS-Entropy.
MSC:
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
34C28 Complex behavior and chaotic systems of ordinary differential equations
65P20 Numerical chaos
Software:
sapa
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