Nonlinear Systems Group                                                 @ University of Illinois at Urbana-Champaign

Director: N. Sri. Namachchivaya 306 Talbot Laboratory, 104 South Wright Street Urbana, IL 61801 Phone: (217) 244-0683, Fax: (217) 244-0720 E-mail: navam@uiuc.edu Department of Aerospace Engineering Created: 11/16/2003 Last Updated: 11/16/2003			Nonstandard reduction of the noisy Duffing-van der Pol equation The purpose of this work is to improve the understanding of the solution of the noisy {\it Duffing-van der Pol} equation. We achieve this by developing rigorous methods to replace, in some limiting regime, the original complicated system by a {\it simpler, constructive, and rational} approximation -- a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional Hamiltonian system. We achieve the model-reduction through {\it stochastic averaging} and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations. N. Sri Namachchivaya, Richard Sowers and Lalit Vedula, ``Nonstandard Reduction of Noisy Duffing-van der Pol Equation", Dynamical Systems: An International Journal, Vol. 16(3), 2001, pp. 223-245.