Lévy on-off intermittency

Intermittency of three-dimensional perturbations in a point-vortex model

“…Here we summarize only the relevant results. It is shown in [1] that in the case α < 1 and β = 1 which applies here, the system (10), with γ k interpreted as white Lévy noise, is unstable for all values of ν: since the mean value of γ k → +∞, viscosity ν, no matter how large, cannot stop the growth of A k . If, however, the attained values of γ k are truncated, so that a finite value of γ k exists, then there is a critical value of viscosity ν c above which all trajectories converge to zero A k → 0.…”

“…For a given value of β, the mean energy in the statistically stationary state can be measured from the time series. Thus, like every microcanonical temperature corresponds to an energy E according to (1), in the noisy gradient method every value of β corresponds to a mean energy E in steady state. The resulting mean energy as a function of temperature is shown in figure 2.…”

“…1 in [42]). In the same figure 5, we also show the energy of individual ergophages, 1 2 A 2 k , by dashed lines. Their sum is equal to the blue solid line.…”

“…While the role of long-time correlated noise in on-off intermittency has been considered before [72][73][74][75], the case of on-off intermittency with heavy-tailed noise has not previously been studied explicitly, to our knowledge. Our companion paper [1] is devoted to this topic. Here we summarize only the relevant results.…”

“…(ii) The passive-nonlinear regime of the model, where the 2-D flow evolves independently of the ergophage amplitudes, which saturate by non-linear self-interactions without affecting the 2-D flow. In this regime the system exhibits a new type of on-off intermittency that we name Lévy on-off intermittency, which we define and study in a companion paper [1]. We compute the bifurcation diagram for the mean and variance of the perturbation amplitude, as well as the probability density of the perturbation amplitude.…”