Bifurcation, bimodality, and finite variance in confined Levy flights

We investigate the statistical behavior of Levy flights confined in a symmetric, quartic potential well U(x)proportional tox(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 Levy 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.