Saddle-Node Bifurcation of Periodic Orbits for a Delay Differential Equation

TL;DR

This paper analyzes a delay differential equation with a nondecreasing feedback function, demonstrating a saddle-node bifurcation of large-amplitude periodic orbits that oscillate around two unstable fixed points as a parameter varies.

Contribution

It establishes the occurrence of saddle-node bifurcation of periodic orbits in a delay differential equation with specific nonlinear feedback, expanding understanding of bifurcation phenomena in such systems.

Findings

Identifies saddle-node bifurcation of periodic orbits as parameter K varies.

Shows periodic orbits oscillate around two unstable fixed points.

Demonstrates large amplitude of the periodic orbits.

Abstract

We consider the scalar delay differential equation $$ \dot{x}(t)=-x(t)+f_{K}(x(t-1)) $$ with a nondecreasing feedback function $f_{K}$ depending on a parameter $K$, and we verify that a saddle-node bifurcation of periodic orbits takes place as $K$ varies. The nonlinearity $f_{K}$ is chosen so that it has two unstable fixed points (hence the dynamical system has two unstable equilibria), and these fixed points remain bounded away from each other as $K$ changes. The generated periodic orbits are of large amplitude in the sense that they oscillate about both unstable fixed points of $f_{K}$.