Periodic delay orbits and the polyfold implicit function theorem

Abstract

We consider differential delay equations of the form ${\partial }_{t}x(t)=X_t(x(t-\tau ))$ in ${\mathbb{R}}^n$, where $(X_t{)}_{t\in S^1}$ is a time-dependent family of smooth vector fields on ${\mathbb{R}}^n$ 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 delay. However, it seems difficult to prove this using the classical implicit function theorem, since the equation above 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 [HWZ09, HWZ17] to overcome this problem in a natural setup.

24 pages. Improved thanks to suggestions by the referee, the results remain unchanged

References in corpus (6)

• The shift map on Floer trajectory spaces
• Hamiltonian delay equations -- examples and a lower bound for the number of periodic solutions
• Counterexamples in Scale Calculus
• Global Bifurcation of Periodic Solutions in Delay Equations with Symmetric Monotone Feedback
• $C^1$-smooth dependence on initial conditions and delay: spaces of initial histories of Sobolev type, and differentiability of translation in $L^p$
• Nullity bounds for certain Hamiltonian delay equations

Cited by in corpus (1)

• Families of periodic delay orbits