Periodic delay orbits and the polyfold implicit function theorem - Albers, Peter; Seifert, Irene (Commentarii mathematici Helvetici, 2022)
We consider differential delay equations of the form partial derivative(t)x(t) = X-t(x(t-tau)) in R-n, where (X-t)(t is an element of S)1 is a time-dependent family of smooth vector fields on R-n and tau is a delay parameter. If there is a (suitably non-degenerate) periodic solution x(0) of this equation for tau = 0, that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by tau. However, it seems difficult to prove this using the classical implicit function theorem, since the equation above, considered as an operator, is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer-Wysocki-Zehnder (2009, 2021) to overcome this problem in a natural setup.