Modeling scaled processes and 1/f^b noise by the nonlinear stochastic differential equations

TL;DR

This paper introduces nonlinear stochastic differential equations that generate signals with 1/f^b noise, power-law distributions, and long-range correlations, providing analytical and numerical insights into such complex systems.

Contribution

The paper develops a new class of nonlinear stochastic differential equations that model 1/f^b noise and power-law behaviors, linking them to avalanche and SOC models.

Findings

Derived analytical expressions for signal characteristics.

Numerical simulations confirm links to avalanche models.

Model captures long-range correlations and power-law distributions.

Abstract

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