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

TL;DR

This paper demonstrates that near the verge of Hopf instability, delay differential equations can be effectively approximated by simpler non-delay stochastic differential equations, even with nonlinearities and noise.

Contribution

It extends previous work by relaxing Lipschitz conditions and including multiplicative noise, providing a new approximation method for delay equations at instability.

Findings

Approximation error remains small near instability

Includes cases with nonlinear and noise perturbations

Provides a framework for non-Lipschitz coefficients

Abstract

We consider linear delay differential equations at the verge of Hopf instability, i.e. a pair of roots of the characteristic equation are on the imaginary axis of the complex plane and all other roots have negative real parts. When nonlinear and noise perturbations are present, we show that the error in approximating the dynamics of the delay system by certain two dimensional stochastic differential equation without delay is small (in an appropriately defined sense). Two cases are considered: (i) linear perturbations and multiplicative noise (ii) cubic perturbations and additive noise. The two-dimensional system without-delay is related to the projection of the delay equation onto the space spanned by the eigenfunctions corresponding to the imaginary roots of the characteristic equation. A part of this article is an attempt to relax the Lipschitz restriction imposed on the coefficients in doi:10.1142/S0219493716500131 Also, the multiplicative noise case was not considered there. Examples without rigorous proofs are worked in arXiv:1403.3029