Complexity Analysis of Chaos and Other Fluctuating Phenomena

Abstract: The refined composite multiscale-entropy algorithm was applied to the time-dependent behavior of the Weierstrass functions, colored noise, and Logistic map to provide fresh insight into the dynamics of these fluctuating phenomena. For the Weierstrass function, the complexity of fluctuations was found to increase with respect to the fractional dimension, D, of the graph. Additionally, the sample-entropy curves increased in an exponential fashion with increasing D. This increase in the complexity was found to correspond to a rising amount of irregularities in the oscillations. In terms of the colored noise, the complexity of the fluctuations was found to be highest for the 1/f noise (f is the frequency of the generated noise), which is in agreement with findings in the literature. Moreover, the sample-entropy curves exhibited a decreasing trend for noise when the spectral exponent, \beta, was less than 1 and obeyed an increasing trend when \beta > 1. Importantly, a direct relationship was observed between the power-law exponents for the curves and the spectral exponents of the noise. For the logistic map, a correspondence was observed between the complexity maps and its bifurcation diagrams. Specifically, the map of the sample-entropy curves was negligible when the bifurcation parameter, R, varied between 3 - 3.5. Beyond these values, the curves attained non-zero values that increased with increasing R, in general.