International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 2, Pages 311-328

Dimensional reduction of nonlinear time delay systems

M. S. Fofana

Integrative Manufacturing Control and Dynamics, Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester 01609-2280, MA, USA

Received 17 October 2002; Revised 15 June 2003

Copyright © 2005 M. S. Fofana. 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.


Whenever there is a time delay in a dynamical system, the study of stability becomes an infinite-dimensional problem. The centre manifold theorem, together with the classical Hopf bifurcation, is the most valuable approach for simplifying the infinite-dimensional problem without the assumption of small time delay. This dimensional reduction is illustrated in this paper with the delay versions of the Duffing and van der Pol equations. For both nonlinear delay equations, transcendental characteristic equations of linearized stability are examined through Hopf bifurcation. The infinite-dimensional nonlinear solutions of the delay equations are decomposed into stable and centre subspaces, whose respective dimensions are determined by the linearized stability of the transcendental equations. Linear semigroups, infinitesimal generators, and their adjoint forms with bilinear pairings are the additional candidates for the infinite-dimensional reduction.