Power-law statistics from nonlinear stochastic differential equations driven by Lévy stable noise

•Nonlinear equations with Lévy stable noise can generate signals having 1/f spectrum.•We numerically investigate the width of 1/f region in the spectrum.•Such equations can lead both to sub-diffusion and supper-diffusion.

Anomalous diffusion occurring in complex dynamical systems can often be described by Langevin equations driven by Lévy stable noise. Nonlinear stochastic differential equations yielding power-law steady state distribution and generating signals with 1/f   power spectral density can be generalized by replacing the Gaussian noise with a more general Lévy stable noise. These nonlinear equations can generate signals exhibiting anomalous diffusion: either sub-diffusion or super-diffusion. In a special case when stability index is α=2,α=2, we retain the equations with the Gaussian noise. We investigate numerically the frequency range where the spectrum has 1/f form and demonstrate that this frequency range depends on power-law exponent in steady state distribution as well as on the index of stability α. We expect that this generalization may be useful for describing 1/f fluctuations in the complex systems exhibiting anomalous diffusion.