Delay-induced blow-up in a planar oscillation model

2018/03/21 by Alexey Eremin, Emiko Ishiwata, Eremin, Alexey +5 · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.1803.07815

arxiv created 2021/05/30 · arxiv updated 2021/06/01

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.

Cited by

• Threshold dynamics in time-delay systems: polynomial β-control in a pressing process and connections to blow-up

Related

• Stochastic Delay Differential Equations have blow-up solutions if and only if their instantaneous counterparts have them
• Delay Induced Oscillations in a Fundamental Power System Model
• Modeling Nonlinear Dynamical Systems with Delay-differential Equations
• Universal bifurcation scenarios in delay-differential equations with one delay
• Delay Effects on Amplitude Death, Oscillation Death, and Renewed Limit Cycle Behavior in Cyclically Coupled Oscillators