Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 829484, 14 pages
Research Article

Effects of Time Delay and Noise on Asymptotic Stability in Human Quiet Standing Model

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China

Received 25 December 2009; Revised 12 May 2010; Accepted 2 September 2010

Academic Editor: Carlo Cattani

Copyright © 2010 Caihong Wang and Jian Xu. 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.


A human quiet standing stability is discussed in this paper. The model under consideration is proposed to be a delayed differential equation (DDE) with multiplicative white noise perturbation. The method of the center manifold is generalized to reduce a delayed differential equation to a two-dimensional ordinary differential equation, to study delay-induced instability or Hopf bifurcation. Then, the stochastic average method is employed to obtain the Itô equation. Thus, the top Lyapunov exponent is calculated and the necessary and sufficient condition of the asymptotic stability in views of probability one is obtained. The results show that the exponent is related to not only the strength of noise but also the delay, namely, the reaction speed of brain. The effect of the strength of noise on the human quiet standing losing stability is weak for a small delay. With the delay increasing, such effect becomes stronger and stronger. A small change in the strength of noise may destabilize the quiet standing for a large delay. It implies that a person with slow reaction is easy to lose the stability of his/her quiet standing.