Nonlinear stochastic models of 1/f noise and power-law distributions

TL;DR

This paper derives nonlinear stochastic differential equations from a point process model to describe signals with 1/f^{eta} noise and power-law distributions, useful for modeling complex systems with self-organization.

Contribution

It introduces a new derivation of nonlinear stochastic equations for 1/f^{eta} noise and power-law distributions based on a generalized point process model.

Findings

Numerical solutions demonstrate 1/f^{eta} noise behavior.

The model captures power-law distributions in signal intensity.

Applicable to systems with long-range memory and self-organization.

Abstract

Starting from the developed generalized point process model of $1/f$ noise (B. Kaulakys et al, Phys. Rev. E 71 (2005) 051105; cond-mat/0504025) we derive the nonlinear stochastic differential equations for the signal exhibiting 1/f^{\beta}$ noise and $1/x^{\lambda}$ distribution density of the signal intensity with different values of $\beta$ and $\lambda$. The processes with $1/f^{\beta}$ are demonstrated by the numerical solution of the derived equations with the appropriate restriction of the diffusion of the signal in some finite interval. The proposed consideration may be used for modeling and analysis of stochastic processes in different systems with the power-law distributions, long-range memory or with the elements of self-organization.