Perturbations of linear delay differential equations at the verge of instability

2014/03/31 by Nishanth Lingala, N. Sri Namachchivaya · 1 citation
Computer Science · Decision Sciences · Mathematics · Physics and Astronomy · #Applied mathematics #Characteristic equation #Complex plane #Computer science #Delay differential equation #Differential equation #Geometry #Instability #Linear differential equation #Linear stability #Mathematical analysis #Mathematics #Nonlinear Dynamics and Pattern Formation #Partial differential equation #Physics #Plane (geometry) #Probabilistic and Robust Engineering Design #Stability (learning theory) #Stochastic differential equation #math.PR #msc:34K06 #msc:34K27 #msc:34K33 #msc:34K50 #stochastic dynamics and bifurcation

paper · pdf · doi:10.1103/physreve.93.062104

published as Phys. Rev. E 93, 062104 (2016) · Essential ideas for scalar systems are in arXiv:1311.4532. In multidimensional case it is easier to work with complexifications. Here results are summarized without proof

arxiv created 2016/02/19 · openalex publication_date 2016/06/02 · arxiv updated 2016/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e., a pair of roots of the characteristic equation lies on the imaginary axis of the complex plane and all other roots have negative real parts. It is shown that when small noise perturbations are present, the probability distribution of the dynamics can be approximated by the probability distribution of a certain one-dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDEs are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.

Citations

• Nonlinear and additive white noise perturbations of linear delay differential equations at the verge of instability: an averaging approach

Cited by

• Approximation of delay differential equations at the verge of instability by equations without delay