Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators

TL;DR

This paper demonstrates that the normal form for triple-zero nilpotent bifurcation in a class of delayed nonlinear oscillators can be fully realized at any order through appropriate nonlinearities, aiding understanding of complex bifurcation behaviors.

Contribution

It shows the realizability of the nonlinear normal form for triple-zero bifurcation in delayed oscillators, extending the theoretical understanding of such systems.

Findings

Normal form can be fully realized at any order.

Realization depends on appropriate nonlinearities.

Applicable to a class of second-order delay-differential equations.

Abstract

The effects of delayed feedback terms on nonlinear oscillators has been extensively studied, and have important applications in many areas of science and engineering. We study a particular class of second-order delay-differential equations near a point of triple-zero nilpotent bifurcation. Using center manifold and normal form reduction, we show that the three-dimensional nonlinear normal form for the triple-zero bifurcation can be fully realized at any given order for appropriate choices of nonlinearities in the original delay-differential equation.