Modeling of extreme events and bursting processes by the nonlinear stochastic differential equations

Many complex systems exhibit quiet periods separated by bursts, i.e., events of rapid evolution. Such systems often produce noise with the power-law characteristics, which can be modeled in terms of avalanches. We derive and analyze nonlinear stochastic differential equations simulating processes which exhibit, bursts and extreme events, characterized by power-law distributions, including 1/f noise, q-exponential and q-Gaussian distributions. The proposed model may simulate self-organized critical, long-memory and other systems revealing avalanches, bursts or clustering of events. The generated signal itself exhibits the power-law distributions of the signal intensity, 1/fβ noise, power-law auto correlations and second order structural (height-height correlation) functions. The proposed model reproduces 1/fβ noise and the processes in SOC and crackling systems and it is also related to the clustering Poisson process, 1/f noise in nanochannels, single-channel and ion channel currents, etc and may be used for simulation of long-range scaled processes exhibiting 1/f noise, power-law distributions and self-organization. (authors)