Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

Noise-induced statistically stable oscillations in a deterministically divergent nonlinear dynamical system

Pages: 137 – 157



Katarína Bodová (Department of Applied Mathematics & Statistics, Comenius University, Bratislava, Slovakia)

Charles Doering (R.)


Inspired by partial differential equation models of homogeneous convection possessing heteroclinic connections to infinity, we study a two dimensional system of ordinary differential equations whose solutions diverge exponentially for almost all initial conditions. Random perturbations of the dynamical system destabilize the divergences resulting in stochastic oscillations. Stochastic Lyapunov function methods are used to prove the existence of a statistically stationary state. A novel Monte-Carlo method is implemented to measure the extreme statistics associated with the stochastic oscillations, and a WKB analysis at low noise amplitude is carried out to corroborate the simulations.


nonlinear dynamical systems, stochastic dynamical systems, Monte-Carlo methods, low-noise asymptotic analysis

2010 Mathematics Subject Classification

60G40, 60H10, 60H30, 65C05

Published 14 October 2011