DelayAndPeriodicity

TL;DR

This paper investigates the generic properties of delay systems, revealing the existence of multiple periodic solutions that reappear with varying delays and analyzing their stability through spectral properties.

Contribution

It provides a comprehensive analysis of periodic solutions in delay systems, highlighting their reappearance, coexistence, and stability characteristics as delay varies.

Findings

Delay systems have families of periodic solutions that reappear at infinitely many delay times.

Increasing delay leads to overlapping solution families and coexistence of multiple solutions.

The spectrum of characteristic multipliers splits into pseudo-continuous and strongly unstable parts, affecting stability.

Abstract

Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions. We also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers. We show that the spectrum of multipliers can be splitted into two parts: pseudo-continuous and strongly unstable. The pseudo-continuous part of the spectrum mediates destabilization of periodic solutions.