Mathematical Problems in Engineering
Volume 2004 (2004), Issue 3, Pages 305-321
doi:10.1155/S1024123X0440101X
A hybrid neural network model for the dynamics of the Kuramoto-Sivashinsky equation
Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
Received 2 January 2004; Revised 14 February 2004
Copyright © 2004 Nejib Smaoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A hybrid approach consisting of two neural networks is used to
model the oscillatory dynamical behavior of the
Kuramoto-Sivashinsky (KS) equation at a bifurcation parameter
α=84.25. This oscillatory behavior results from a fixed
point that occurs at α=72
having a shape of two-humped
curve that becomes unstable and undergoes a Hopf bifurcation at
α=83.75. First, Karhunen-Loève (KL) decomposition was
used to extract five coherent structures of the oscillatory
behavior capturing almost 100% of the energy. Based on the five
coherent structures, a system offive ordinary differential
equations (ODEs) whose dynamics is similar to the original
dynamics of the KS equation was derived via KL Galerkin
projection. Then, an autoassociative neural network was utilized
on the amplitudes of the ODEs system with the task of reducing
the dimension of the dynamical behavior to its intrinsic
dimension, and a feedforward neural network was usedto model
the dynamics at a future time. We show that by combining KL
decomposition and neural networks, a reduced dynamical model of
the KS equation is obtained.