Stability of simple periodic solutions of neutral functional differential equations

Z. Shao, Millersville University, Millersville, U.S.A.
Y. Lu, Bloomsburg University, Bloomsburg, U.S.A.

E. J. Qualitative Theory of Diff. Equ., No. 1. (2003), pp. 1-11.

Abstract: We study the stability property of a simple periodic solution of an autonomous neutral functional differential equation (NFDE) of the form
$${d\over dt} D(x_t) = f (x_t).$$
A new proof based on local integral manifold theory and the implicit function theorem is given for the classical result that a simple periodic orbit of the equation above is asymptotically orbitally stable with asymptotic phase. The technique used overcomes the difficulty that the solution operator of a NFDE does not smooth as $t$ increases.

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