Delay-induced blow-up in a planar oscillation model

TL;DR

This paper demonstrates that even minimal delays in a planar oscillation model can cause solutions to blow up in finite time, fundamentally altering the system's dynamics from stable cycles to unbounded growth.

Contribution

It proves that small delays induce finite-time blow-up in a system that otherwise has stable limit cycles, revealing the drastic impact of delay on system behavior.

Findings

Delay causes finite-time blow-up regardless of delay size.

The system has infinitely many periodic solutions with delay.

Non-delay system has only one stable limit cycle.

Abstract

In this paper we study a system of delay differential equations from the viewpoint of a finite time blow-up of the solution. We prove that the system admits a blow-up solution, no matter how small the length of the delay is. In the non-delay system every solution approaches to a stable unit circle in the plane, thus time delay induces blow-up of solutions, which we call "delay-induced blow-up" phenomenon. Furthermore, it is shown that the system has a family of infinitely many periodic solutions, while the non-delay system has only one stable limit cycle. The system studied in this paper is an example that arbitrary small delay can be responsible for a drastic change of the dynamics. We show numerical examples to illustrate our theoretical results.