Levy on-off intermittency

We present an alternative form of intermittency, Levy on-off intermittency, which arises from multiplicative alpha-stable white noise close to an instability threshold. We study this problem in the linear and nonlinear regimes, both theoretically and numerically, for the case of a pitchfork bifurcation with fluctuating growth rate. We compute the stationary distribution analytically and numerically from the associated fractional Fokker-Planck equation in the Stratonovich interpretation. We characterize the system in the parameter space (alpha, beta) of the noise, with stability parameter alpha is an element of (0, 2) and skewness parameter beta is an element of [-1, 1]. Five regimes are identified in this parameter space, in addition to the well-studied Gaussian case alpha = 2. Three regimes are located at 1 < alpha < 2, where the noise has finite mean but infinite variance. They are differentiated by beta and all display a critical transition at the deterministic instability threshold, with on-off intermittency close to onset. Critical exponents are computed from the stationary distribution. Each regime is characterized by a specific form of the density and specific critical exponents, which differ starkly from the Gaussian case. A finite or infinite number of integer-order moments may converge, depending on parameters. Two more regimes are found at 0 < alpha <= 1. There, the mean of the noise diverges, and no critical transition occurs. In one case, the origin is always unstable, independently of the distance mu. from the deterministic threshold. In the other case, the origin is conversely always stable, independently of mu. We thus demonstrate that an instability subject to nonequilibrium, power-law-distributed fluctuations can display substantially different properties than for Gaussian thermal fluctuations, in terms of statistics and critical behavior.

Automated summary

In this study, Levy on-off intermittency arising from multiplicative alpha-stable white noise close to an instability threshold was investigated in linear and nonlinear regimes for a pitchfork bifurcation with fluctuating growth rate. Different parameter regimes were identified, with critical exponents computed from the stationary distribution. The properties of the system, influenced by nonequilibrium, power-law-distributed fluctuations, were found to be substantially different from Gaussian thermal fluctuations in terms of statistics and critical behavior.