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

Deepa Gupta, Sachin Bhalekar

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.

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

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.

Paper Structure

This paper contains 17 sections, 11 theorems, 32 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linearization near the equilibrium point $x_*$
  4. Mackey-Glass equation
  5. Stability results
  6. Fractional order Mackey-Glass equation
  7. Stability analysis of $x_1^*=0$
  8. Stability analysis of $x_2^*=(\frac{q-p}{p})^{1/r}$
  9. Chaos in fractional order Mackey-Glass system
  10. Chaos control in fractional order Mackey-Glass equation
  11. Generalized Mackey-Glass equation
  12. Stability analysis of the equilibrium point $x_1^*=0$
  13. Stability analysis of the equilibrium point $x_2^*=(\frac{q-p}{p})^{1/r}$
  14. Different types of chaotic attractors in the generalized Mackey-Glass equation
  15. Chaos control
  16. ...and 2 more sections

Key Result

Theorem 2.1

bhalekar2016stabilitybhalekar2023can Consider the fractional order delay differential equation

Figures (7)

  • Figure 1: Bifurcation diagram for different values of parameters given in equation \ref{['eq5.1.2']}
  • Figure 2: Period doubling in fractional order Mackey-Glass equation \ref{['eq5.1..1.1.1']}
  • Figure 3: Chaos control in equation \ref{['eq5.1.1.4']} with $k=-0.9$, $p=1$, $q=2$, $r=10$ and $\alpha=0.9$
  • Figure 4: Two types of chaotic attractors in the generalized Mackey-Glass equation
  • Figure 5: Chaos in Section \ref{['sec5.5']} is controlled by taking control parameter in the permissible range
  • ...and 2 more figures

Theorems & Definitions (29)

  • Definition 2.1: Fractional Integral
  • Definition 2.2: Caputo Fractional Derivative
  • Definition 2.3: Equilibrium Point
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7: Single stable region (SSR)
  • Definition 2.8: Stability Switch (SS)
  • Definition 2.9: Instability switches (IS)
  • Theorem 2.1
  • ...and 19 more