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Some stability results for the fractional differential equations with two delays

Pragati Dutta, Sachin Bhalekar

Abstract

This paper investigates the stability properties of a nonlinear fractional differential equation with two discrete delays and a delay-dependent coefficient. Such equations arise in various biological and control systems where temporal delays influence feedback mechanisms. In the first case, we set one of the delays to zero and analyzed the resulting system. We then extended the study to a more general case where both delays are allowed to vary. We derive delay-independent stability conditions using linearization, characteristic equations, and bifurcation theory, along with complete theoretical proofs. The results are validated through numerical simulations and stability diagrams.

Some stability results for the fractional differential equations with two delays

Abstract

This paper investigates the stability properties of a nonlinear fractional differential equation with two discrete delays and a delay-dependent coefficient. Such equations arise in various biological and control systems where temporal delays influence feedback mechanisms. In the first case, we set one of the delays to zero and analyzed the resulting system. We then extended the study to a more general case where both delays are allowed to vary. We derive delay-independent stability conditions using linearization, characteristic equations, and bifurcation theory, along with complete theoretical proofs. The results are validated through numerical simulations and stability diagrams.

Paper Structure

This paper contains 9 sections, 4 theorems, 48 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Main Model
  4. Case 1: $\tau_1 = 0$
  5. Illustrative Examples for Case 1
  6. Case 2: $\tau_1>0, \;\tau_2\ge0$
  7. Illustrative Examples for Case 2
  8. Illustrative Examples for a Nonlinear Function $g(x)$
  9. Conclusion

Key Result

Theorem 2.1

bhalekar2016stability Consider the scalar FDE with a single discrete delay: Then, the zero equilibrium $x_*=0$ has the following stability behavior: Notation: Here, SSR means Single Stable Region.

Figures (22)

  • Figure 1: Stability regions of Equation (\ref{['eq1']}).
  • Figure 2: Analysis for the region $0<\gamma<2k$
  • Figure 3: Region: $0<\gamma<2k$
  • Figure 4: Stability regions for Case 1
  • Figure 5: $\alpha=0.4,\; k=-0.1,\; \gamma=-0.2,\; \tau=1.5$
  • ...and 17 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof