1/f noise and anomalous scaling in L\'evy noise-driven on-off intermittency

Abstract

On-off intermittency occurs in nonequilibrium physical systems close to bifurcation points and is characterised by an aperiodic switching between a large-amplitude "on" state and a small-amplitude "off" state. L\'evy on-off intermittency is a recently introduced generalisation of on-off intermittency to multiplicative L\'evy noise, which depends on a stability parameter $\alpha$ and a skewness parameter $\beta$. Here, we derive two novel results on L\'evy on-off intermittency by leveraging known exact results on the first-passage time statistics of L\'evy flights. First, we compute anomalous critical exponents explicitly as a function of arbitrary L\'evy noise parameters $(\alpha,\beta)$ for the first time, by a heuristic method, complementing previous results. The predictions are verified using numerical solutions of the fractional Fokker-Planck equation. Second, we derive the power spectrum $S(f)$ of L\'evy on-off intermittency and show that it displays a power law $S(f)\propto f^\kappa$ at low frequencies $f$, where $\kappa\in (-1,0)$ depends on the noise parameters $\alpha,\beta$. An explicit expression for $\kappa$ is obtained in terms of $(\alpha,\beta)$. The predictions are verified using long time series realisations of L\'evy on-off intermittency. Our findings help shed light on instabilities subject to non-equilibrium, power-law-distributed fluctuations, emphasizing that their properties can differ starkly from the case of Gaussian fluctuations.