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Updated: Jun 16, 2025

Gain-compensation Methodology for a Sinusoidal Scan of a Galvanometer Mirror in Proportional-Integral-Differential Control Using Pre-emphasis Techniques

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Phase and amplitude responses for delay equations using harmonic balance.

R Nicks¹, R Allen¹, S Coombes¹

¹School of Mathematical Sciences, <a href="https://ror.org/01ee9ar58">University of Nottingham</a>, Nottingham NG7 2RD, United Kingdom.

Physical Review. E

|August 20, 2024

Summary

This summary is machine-generated.

Researchers developed a new method to analyze oscillations in delay-differential equations (DDEs). This framework uses harmonic balance to construct phase and amplitude response functions for understanding DDEs under external forcing.

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Area of Science:

Mathematical modeling and dynamical systems
Nonlinear dynamics and oscillations
Applied mathematics and theoretical physics

Background:

Delay-induced oscillations are prevalent in natural systems and are frequently modeled using delay-differential equations (DDEs).
Phase-amplitude reductions have proven successful for ordinary differential equations exhibiting limit cycle oscillations.
There is a growing need for analogous reduction techniques for DDEs to analyze their behavior under external forcing.

Purpose of the Study:

To develop a novel framework for constructing phase and amplitude response functions for DDEs.
To adapt and extend phase-amplitude reduction techniques to delay-differential equations.
To provide tools for understanding the response of DDEs to external perturbations.

Main Methods:

The study utilizes the method of harmonic balance for the construction of response functions.
The proposed framework integrates Floquet theory with harmonic balance.
Development of a systematic approach for phase and amplitude reduction in DDEs.

Main Results:

A robust framework for constructing phase and amplitude response functions for DDEs has been successfully developed.
The method enables a deeper understanding of how DDEs respond to external forcing.
Demonstration of the utility of harmonic balance in the context of DDE reductions.

Conclusions:

The developed framework offers a powerful new tool for analyzing oscillations in DDEs.
This work extends the applicability of phase-amplitude reduction techniques to systems with delays.
The findings are significant for fields relying on DDEs to model natural phenomena.