Analysis of Stability, Bifurcation, and Chaos in Generalized Mackey-Glass Equations

TL;DR

This paper extends the Mackey-Glass equation with fractional derivatives, analyzing stability, bifurcation, and chaos, and introduces a control method for chaotic behavior in these generalized models.

Contribution

It introduces fractional-order generalizations of the Mackey-Glass equation and provides detailed stability, bifurcation, and chaos analysis, including a novel chaos control method.

Findings

Stable and chaotic regimes identified in the models

Parameter space divided into distinct stability regions

A control method for chaos is proposed

Abstract

Mackey-Glass equation arises in the leukemia model. We generalize this equation to include fractional-order derivatives in two directions. The first generalization contains one whereas the second contains two fractional derivatives. Such generalizations improve the model because the nonlocal operators viz. fractional derivatives are more suitable for the natural systems. We present the detailed stability and bifurcation analysis of the proposed models. We observe stable orbits, periodic oscillations, and chaos in these models. The parameter space is divided into a variety of regions, viz. stable region (delay independent), unstable region, single stable region, and stability/instability switch. Furthermore, we propose a control method for chaos in these general equations.