Modeling scaled processes and 1/fβnoise using nonlinear stochastic differential equations

We present and analyze stochastic nonlinear differential equations generating signals with the power-law distributions of the signal intensity, 1/f β noise, power-law autocorrelations and second-order structural (height-height correlation) functions.Analytical expressions for such characteristics are derived and a comparison with numerical calculations is presented.The numerical calculations reveal links between the proposed model and models where signals consist of bursts characterized by power-law distributions of burst size, burst duration and interburst time, as in the case of avalanches in self-organized critical models and the extreme event return times in long-term memory processes.The approach presented may be useful for modeling long-range scaled processes exhibiting 1/f noise and power-law distributions.