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An increasingly important problem in physics concerns scale invariance symmetry in diverse complex systems, often characterized by heteroscedastic dynamics. We investigate the nature of the relationship between the heteroscedastic and fractal aspects of the dynamics of complex systems, by analyzing the sensitivity to heteroscedasticity of the scaling properties of weakly nonstationary time series. By using multifractal detrended fluctuation analysis, we study the singularity spectra of currency exchange rate fluctuations, after partially or completely eliminating n-point correlations via data shuffling techniques. We conclude that heteroscedasticity can significantly increase multifractality and interpret these findings in the context of selforganizing and adaptive complex systems.

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An important open problem concerns the physical origin of long-range correlations, multifractality and fat-tailed distributions observed in heteroscedastic time series associated with complex systems. Financial stylized facts provides one useful example usually not explained by traditional economic models. We investigate the behavior of an agent-based model consisting of N agents which interact with each other via fixed rules. We show that fat-tailed distributions, longrange correlations, heteroscedasticity and multifractality arise as N becomes large. Our findings suggest that such stylized facts can in principle arise as emergent properties.