Stationary states of non-linear oscillators driven by Lévy noise

• Published in: Chemical physics Vol. 284; no. 1; pp. 233 - 251
• Main Authors: Chechkin, A., Gonchar, V., Klafter, J., Metzler, R., Tanatarov, L.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-11-2002
• Subjects: Fractional derivative; Fractional kinetic equation; Lévy flights; Lévy stable noise; Non-linear oscillator; Lévy flights; Non-linear oscillator; Fractional kinetic equation; Lévy stable noise; Fractional derivative
• ISSN: 0301-0104
• DOI: 10.1016/S0301-0104(02)00551-7

We study the probability density function in the stationary state of non-linear oscillators which are subject to Lévy stable noise and confined within symmetric potentials of the general form
U(x)∝x
2m+2/(2m+2),
m=0,1,2,…
. For
m⩾1, the probability density functions display a distinct bimodal character and have power-law tails which decay faster than those of the noise probability density. This is in contrast to the Lévy harmonic oscillator
m=0. For the particular case of an anharmonic Lévy oscillator with
U(
x)=
ax
2/2+
bx
4/4,
a>0, we find a turnover from unimodality to bimodality at stationarity.