Modeling scaled processes and 1/fβ noise using nonlinear stochastic differential equations

Abstract

We present and analyze stochastic nonlinear differential equationsgenerating signals with the power-law distributions of the signal intensity,1/fβ noise, power-law autocorrelations and second-order structural (height–height correlation)functions. Analytical expressions for such characteristics are derived and acomparison with numerical calculations is presented. The numerical calculationsreveal links between the proposed model and models where signals consist ofbursts characterized by power-law distributions of burst size, burst duration andinterburst time, as in the case of avalanches in self-organized critical models andthe extreme event return times in long-term memory processes. The approachpresented may be useful for modeling long-range scaled processes exhibiting1/f noise and power-law distributions.