Bifurcation, bimodality, and finite variance in confined Lévy flights

Abstract

We investigate the statistical behavior of Lévy flights confined in a symmetric, quartic potential well U(x) proportional, variant x(4). At stationarity, the probability density function features a distinct bimodal shape and decays with power-law tails which are steep enough to give rise to a finite variance, in contrast to free Lévy flights. From a delta-initial condition, a bifurcation of the unimodal state is observed at t(c)>0. The nonlinear oscillator with potential U(x)=ax(2)/2+bx(4)/4, a,b>0, shows a crossover from unimodal to bimodal behavior at stationarity, depending on the ratio a/b.

Lévy Flight Superdiffusion: An Introduction

“…This gives rise to a confined superdiffused motion, characterized by a bimodal stationary probability density, as previously reported in Refs. [Chechkin et al, 2002a[Chechkin et al, , 2003a[Chechkin et al, , 2004[Chechkin et al, , 2006. Here we analyze the SPD as a function of a dimensionless parameter β, which is the ratio between the noise intensity D and the steepness γ of the potential profile.…”

Fluctuation-driven directed transport in the presence of Lévy flights

“…The role of Lévy flights in confining potentials has been addressed in the literature before. Examples include the study of the decay properties of the probability distribution function, the study of unimodal-multimodal bifurcations during relaxation [8], and the barrier crossing Kramers problem [9]. However, despite the widespread recognition of the importance of Lévy processes, the effect of Lévy noise in ratchet potentials has not been addressed.…”

Lévy Flights in a Steep Potential Well

“…Thus, for a potential of the form U (x) = ax 2 + bx 4 with a, b > 0, a turnover can be tuned from the properties of the solution of the harmonic problem (mono-modal, diverging variance) to a finite variance and bimodal solution by varying the ratio b : a [35]. If such bimodality occurs, it results from a bifurcation at a critical time t c [35]. A typical result is shown in figure 1, for the quartic case m = 1 and Lévy index α = 1.2: from an initial δ-peak, eventually a bimodal distribution emerges.…”