Expansions in the delay of quasi-periodic solutions for state dependent delay equations

TL;DR

This paper develops a formal perturbation method to construct quasi-periodic solutions for state-dependent delay differential equations with small delays, providing a way to approximate true solutions near these formal series.

Contribution

It introduces a novel perturbation approach using power series expansions to find quasi-periodic solutions in SDDEs affected by small parameters.

Findings

Constructed formal power series solutions for SDDEs

Proved existence of true solutions near approximate series

Provided a framework for cataloguing SDDE solutions

Abstract

We consider several models of State Dependent Delay Differential Equations (SDDEs), in which the delay is affected by a small parameter. This is a very singular perturbation since the nature of the equation changes. Under some conditions, we construct formal power series, which solve the SDDEs order by order. These series are quasi-periodic functions of time. This is very similar to the Lindstedt procedure in celestial mechanics. Truncations of these power series can be taken as input for a-posteriori theorems, that show that near the approximate solutions there are true solutions. In this way, we hope that one can construct a catalogue of solutions for SDDEs, bypassing the need of a systematic theory of existence and uniqueness for all initial conditions.