Delay Terms in the Slow Flow

TL;DR

This paper compares two analytical approaches for studying Hopf bifurcations in delayed self-feedback nonlinear systems, assessing the accuracy of simplifying delayed differential equations to ordinary differential equations.

Contribution

It introduces and compares two methods for analyzing slow flows with delay, evaluating the validity of replacing delays with non-delayed variables.

Findings

Approach I simplifies delay differential equations to ODEs.

Approach II retains delay variables for more accurate analysis.

Comparison shows the conditions under which delay approximation is valid.

Abstract

This work concerns the dynamics of nonlinear systems that are subjected to delayed self-feedback. Perturbation methods applied to such systems give rise to slow flows which characteristically contain delayed variables. We consider two approaches to analyzing Hopf bifurcations in such slow flows. In one approach, which we refer to as approach I, we follow many researchers in replacing the delayed variables in the slow flow with non-delayed variables, thereby reducing the DDE slow flow to an ODE. In a second approach, which we refer to as approach II, we keep the delayed variables in the slow flow. By comparing these two approaches we are able to assess the accuracy of making the simplifying assumption which replaces the DDE slow flow by an ODE. We apply this comparison to two examples, Duffing and van der Pol equations with self-feedback.