Bistability, Bifurcations and Chaos in the Mackey-Glass Equation

TL;DR

This paper investigates complex dynamics like bifurcations, chaos, and bistability in the Mackey-Glass delay differential equation using numerical bifurcation analysis and simulations.

Contribution

It identifies new bifurcation structures such as cusp bifurcations and mechanisms for chaos destruction in the Mackey-Glass system.

Findings

Discovery of cusp bifurcation leading to bistability

Identification of chaos destruction via interior and boundary crises

Observation of subcritical period-doubling sequences

Abstract

Numerical bifurcation analysis, and in particular two-parameter continuation, is used in consort with numerical simulation to reveal complicated dynamics in the Mackey-Glass equation for moderate values of the delay close to the onset of chaos. In particular a cusp bifurcation of periodic orbits and resulting branches of folds of periodic orbits effectively partition the parameter space into regions where different behaviours are seen. The cusp bifurcation leads directly to bistability between periodic orbits, and subsequently to bistability between a periodic orbit and a chaotic attractor. This leads to two different mechanisms by which the chaotic attractor is destroyed in a global bifurcation with a periodic orbit in either an interior crisis or a boundary crisis. In another part of parameter space a sequence of subcritical period-doublings is found to give rise to bistability between a periodic orbit and a chaotic attractor. Torus bifurcations, and a codimension-two fold-flip bifurcation are also identified, and Lyapunov exponent computations are used to determine chaotic regions and attractor dimension.