# Unbounded sequences of stable limit cycles in the delayed Duffing equation: an exact analysis

## TL;DR

This paper provides an exact analysis demonstrating that the delayed Duffing equation has an infinite sequence of stable, rapidly oscillating periodic solutions when the delay parameter satisfies a specific inequality.

## Contribution

It offers the first exact proof of unbounded stable limit cycles in the delayed Duffing equation, contrasting with previous approximate methods.

## Key findings

- Infinite sequence of stable limit cycles identified
- Stable solutions occur for delays satisfying T^2<1.5π^2
- Analysis is exact, not approximate

## Abstract

The delayed Duffing equation $\ddot{x}(t)+x(t-T)+x^3(t)=0$ is shown to possess an infinite and unbounded sequence of rapidly oscillating, asymptotically stable periodic solutions, for fixed delays such that $T^2<\tfrac{3}{2}\pi^2$. In contrast to several previous works which involved approximate solutions, the treatment here is exact.