Journal of Numerical Mathematics and Stochastics,2 (1) : 01-11, 2010 © JNM@S http://www.jnmas.org/jnmas2-1.pdf Euclidean Press, LLC Online: ISSN 2151-2302 Numerical Solution of a Stochastic Lorenz Attractor M. ZAHRI Department of Mathematics, Faculty of Science, Taibah-University, Al Madinah Almunawwarah, KSA, E-mail: zahri@gmx.net Abstract. The aim of this paper is to extend the -deterministic- Lorenz attractor to a stochastic system and to numerically solve it. We propose and implement the Milstein scheme for solving multidimensional nonlinear Itô stochastic differential systems, with particular emphasis on the Lorenz attractor. In order to assure the first convergence order of the Milstein scheme, we use Fourier series to approximate the double Itô integrals. Furthermore, numerical behaviors of the stochastic Lorenz attractor solutions are presented and analyzed. Key words : System of Stochastic Differential Equations, Milstein Scheme, Random Lorenz Attractor, Random Dynamical System. AMS Subject Classifications: 35R60, 60H15 1. Introduction The Lorenz attractor invokes a solution of a three dimensional deterministic system of differential equations given in [4]. It was first studied by Edward N. Lorenz, a meteorologist, around 1963. The system was derived from a simplified model of convection in the earth’s atmosphere. It also arises naturally in lasers and dynamos models. Its beautiful three dimensional plots are most commonly expressed as a solution of the three coupled non-linear differential equations, which are also popular in the field of Chaos. The equations describe the flow of fluid in a box which is heated along the bottom. This model was intended to simulate medium-scale atmospheric convection. Lorenz simplified in his model some of the Navier-Stokes equations in the area of fluid dynamics, and obtained the following non-linear three dimensional ordinary differential system: ∂ t x  pyt − xt, 1 ∂ t y  rxt − yt − xtzt, ∂ t z  xtyt − czt, 1