On the problem of linearization for state-dependent delay differential equations

Abstract

The local stability of the equilibrium for a general class of state-dependent delay equations of the form \[ x ˙ ( t ) = f ( x t , ∫ − r 0 0 d η ( s ) g ( x t ( − τ ( x t ) + s ) ) ) \dot x(t)=f\left (x_t, \int ^0_{-r_0}\,d\eta (s)g(x_t(-\tau (x_t)+s))\right ) \] has been studied under natural and minimal hypotheses. In particular, it has been shown that generically the behavior of the state-dependent delay τ \tau (except the value of τ ) \tau ) near an equilibrium has no effect on the stability, and that the local linearization method can be applied by treating the delay τ \tau as a constant value at the equilibrium.