# Control by time delayed feedback near a Hopf bifurcation point

## TL;DR

This paper investigates stabilizing rotating waves near a Hopf bifurcation using time delayed feedback control, analyzing stability through Floquet multipliers and bifurcation theory, and extending control schemes to neutral type equations.

## Contribution

It introduces a broader analysis framework for stability of periodic orbits under delayed feedback, including an extension to neutral type equations and a simple method to determine bifurcation direction.

## Key findings

- Stability conditions for periodic orbits are derived.
- Extension of Pyragas control to neutral type equations.
- Method to determine bifurcation direction via root tendency.

## Abstract

In this paper we study the stabilization of rotating waves using time delayed feedback control. It is our aim to put some recent results in a broader context by discussing two different methods to determine the stability of the target periodic orbit in the controlled system: 1) by directly studying the Floquet multipliers and 2) by use of the Hopf bifurcation theorem. We also propose an extension of the Pyragas control scheme for which the controlled system becomes a functional differential equation of neutral type. Using the observation that we are able to determine the direction of bifurcation by a relatively simple calculation of the root tendency, we find stability conditions for the periodic orbit as a solution of the neutral type equation.