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

Cooke, Kenneth; Huang, Wenzhang

doi:10.1090/S0002-9939-96-03437-5

On the problem of linearization for state-dependent delay differential equations
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by Kenneth L. Cooke and Wenzhang Huang PDF

Proc. Amer. Math. Soc. 124 (1996), 1417-1426 Request permission

Abstract:

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

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