Suppression of oscillations by Levy noise

Abstract

We find analytical solution of pair of stochastic equations with arbitrary forces and multiplicative Lévy noises in a steady-state nonequilibrium case. This solution shows that Lévy flights suppress always a quasi-periodical motion related to the limit cycle. We prove that difference between stochastic systems driven by Lévy and Gaussian noises is that the Lévy variation $\Delta L\sim (\Delta t{)}^{1/\alpha }$ with the exponent $\alpha <2$ is much less than the Gaussian one $\Delta W\sim (\Delta t{)}^{1/2}$ in the $\Delta t\to 0$ limit. Moreover, this difference is shown to remove the problem of the calculus choice because related addition to the physical force is of order $(\Delta t{)}^{2/\alpha }\ll \Delta t$.

18 pages, 1 figure. Submitted to Fluctuation and Noise Letters

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• Predator-prey cycles from resonant amplification of demographic stochasticity
• Phase transitions induced by noise cross-correlations
• Phase transitions driven by Lévy stable noise: exact solutions and stability analysis of nonlinear fractional Fokker-Planck equations