Stability and Hopf bifurcation for differential equations with state depend delay
by
Moulay Lhassan Hbid
Dept Of Maths. Faculty of Sciences, B.P. 2390, Marrakech, Morocco

We consider the following class of differential equations with state dependend delay:

(d/(dt))x(t)=f(x(t-r(xt)) (1) Exponential asymptotic stability will be discussed and a Hopf bifurcation results will be presented in the case where the state dependent delay is close to zero.

We are first concerned by the exponential asymptotic stability of solutions of equation (1). Our approach is based on semigroups properties. Using the Desh and Schappacher Theorem [1], we show that the equilibrium is stable. The exponential asymptotic stability is proved via the approximation of Crandall-Liggett type.

To prove the existence of periodic solutions, we transform this equation into a perturbed constant delay equation, and using the Hopf bifurcation result and the Poincaré procedure for this last equation, we prove the existence of a branch of periodic solutions for the state dependent delay equation, bifurcating from r≡0.

[1] W. Desh and W. Schappacher, Linearized stability for nonlinear semigroup in differential equation in Banach space. (A. Favani and E. Obreacht. Edts), pp 61-73, Lectures notes in math, Vol 1223, Springer Verlag, New York/ Berlin, (1996).

Date received: February 23, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calu-90.