# On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems

## TL;DR

This paper introduces a numerical method to estimate the domain of attraction in nonlinear time-delay systems, combining initial function selection, simulations, and bifurcation analysis for improved bounds.

## Contribution

It presents a novel methodology for estimating attraction domains in nonlinear time-delay systems using a combination of numerical simulations and bifurcation analysis.

## Key findings

- Numerical upper bounds on attraction domains can be refined iteratively.
- Bifurcation analysis improves the estimation of the domain of attraction.
- Application to a time-delayed swing equation demonstrates the method's effectiveness.

## Abstract

For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects.