Some stability results for the fractional differential equations with two delays

This paper establishes delay-independent stability conditions for a nonlinear fractional differential equation with two discrete delays and a delay-dependent coefficient through linearization, characteristic equations, and bifurcation theory, with results validated by numerical simulations and stability diagrams.

The Gist

Imagine you are trying to keep a broom balanced on your hand. This is a classic control problem: you have to react to the broom's tilt to keep it upright.

Now, imagine two things make this much harder:

1. Memory: Your brain doesn't just react to the current tilt; it remembers how the broom was tilted a moment ago, and that memory influences how you move your hand. This is the "Fractional" part of the math.
2. Lag: There is a delay between when you see the broom tilt and when your hand actually moves. Worse, there are two different delays happening at once (like seeing the tilt, waiting, reacting, and then waiting again for the effect to show). This is the "Two Delays" part.

This paper is like a safety manual for engineers and biologists who are trying to build systems (like robot arms, drug delivery systems, or even modeling how blood platelets are made) that have these tricky "memory" and "lag" features.

Here is a breakdown of what the authors found, using simple analogies:

1. The Setup: The "Platelet" Problem

The authors are studying a specific equation that models how things grow or change over time. They used it to model platelet production (the cells that help your blood clot).

• The Scenario: Your body produces platelets. But the signal to produce them takes time to travel (delay). Also, the strength of that signal depends on how long the delay was (a "delay-dependent coefficient").
• The Goal: They want to know: Will the number of platelets stay steady and healthy, or will they go crazy (oscillate wildly) or disappear?

2. The Two Scenarios They Tested

The researchers looked at two main situations, like testing a car on a straight road versus a winding mountain pass.

Case A: One Delay is Zero (The Straight Road)

First, they pretended one of the delays didn't exist. It's like saying, "Okay, the signal travels instantly, but the second part of the process is slow."

• The Finding: They drew a map (a stability diagram) showing exactly when the system stays calm and when it goes wild.
• The Surprise: Sometimes, even if the system looks stable at the start, adding more delay can suddenly make it unstable. It's like driving a car: at low speeds, you are safe. But if you hit a specific speed (a "critical delay"), the car might start swerving uncontrollably.
• The Rule: They found specific "safe zones" based on the numbers in the equation. If your numbers fall in the "Green Zone," the system is safe no matter how long the delay is. If they fall in the "Red Zone," it's unstable no matter what.

Case B: Both Delays are Active (The Winding Mountain)

Then, they let both delays exist. This is the real-world scenario.

• The Finding: They discovered that the relationship between the "strength" of the feedback (how hard the system tries to correct itself) and the "memory" of the system is crucial.
• The "Switch": They found a critical tipping point. If the feedback strength ($k$) is too high compared to the natural decay ($\gamma$), the system becomes unstable. It's like a thermostat that is too sensitive: if the room gets slightly cold, the heater blasts on full power, overheating the room, which makes the AC blast on, freezing the room, and so on.
• The Math Magic: They calculated a specific "danger number" ($k^*$). If your system's feedback is stronger than this number, the system will crash (become unstable) regardless of how you tune the delays.

3. The "Fractional" Twist

Why is this different from normal math?

• Normal Math (Integer Order): Imagine a car that reacts only to the road right in front of it.
• Fractional Math: Imagine a car that reacts to the road right in front AND remembers the bumps from the last 10 seconds. This "memory" makes the system behave differently. It can be more stable or more chaotic depending on how the memory is weighted.

4. What This Means for Real Life

The authors validated their math with computer simulations (drawing graphs that look like squiggly lines).

• If you are a Biologist: This helps you understand why some biological systems (like immune responses or hormone levels) might suddenly start oscillating (getting sick) even if the inputs seem normal. It tells you which parameters to tweak to keep the system healthy.
• If you are an Engineer: If you are building a robot or a control system with sensors that have lag, this paper gives you a checklist. It tells you: "Don't set your feedback gain higher than X, or your robot will start shaking violently."

Summary in a Nutshell

Think of the system as a tightrope walker.

• The Delays are the wind gusts that hit the walker a few seconds after they move.
• The Fractional Order is the walker's muscle memory.
• The Paper tells us exactly how strong the wind can get and how much memory the walker needs before they fall off the rope.

They found that if the "wind" (delay) and "memory" interact in a certain way, the walker can stay balanced forever. But if the feedback is too strong or the delays hit a specific "sweet spot," the walker will inevitably lose their balance. The paper provides the exact formulas to calculate that safe zone.

Technical Summary

Here is a detailed technical summary of the paper "Some stability results for the fractional differential equations with two delays" by Pragati Dutta and Sachin Bhalekar.

1. Problem Statement

The paper addresses the stability analysis of a specific class of nonlinear fractional delay differential equations (FDDEs) featuring two discrete delays ($\tau_1, \tau_2$) and a delay-dependent coefficient.

The governing equation is:
$$D^\alpha x(t) = -\gamma x(t) + g(x(t - \tau_1)) - e^{-\gamma \tau_2}g(x(t - \tau_1 - \tau_2))$$
where:

• $D^\alpha$ is the Caputo fractional derivative of order $0 < \alpha \leq 1$.
• $\gamma \in \mathbb{R}$ is a parameter.
• $g \in C^1(\mathbb{R})$ is a nonlinear function with $g(0)=0$.
• The term $e^{-\gamma \tau_2}$ introduces a coefficient that depends explicitly on the delay $\tau_2$.

Context: Such equations model biological and control systems where memory effects (fractional order) and temporal lags (delays) interact. A specific application mentioned is the production of platelets. While single-delay FDDEs are well-studied, systems with multiple delays and delay-dependent coefficients remain under-explored in the literature.

2. Methodology

The authors employ a combination of linearization, spectral analysis, and bifurcation theory to derive stability conditions.

1. Linearization: The system is linearized around the trivial equilibrium point $x^* = 0$. Assuming $g'(0) = k$, the nonlinear equation is approximated by a linear FDDE:
   $$D^\alpha x(t) = -\gamma x(t) + kx(t - \tau_1) - ke^{-\gamma \tau_2}x(t - \tau_1 - \tau_2)$$
2. Characteristic Equation: Applying the Laplace transform, the authors derive the characteristic equation:
   $$\lambda^\alpha = -\gamma + ke^{-\lambda \tau_1} - ke^{-\gamma \tau_2}e^{-\lambda(\tau_1 + \tau_2)}$$
3. Case Analysis: The study is divided into two primary cases:
   • Case 1: One delay is zero ($\tau_1 = 0, \tau_2 = \tau$). This reduces the system to a scalar equation with a single delay but a delay-dependent coefficient.
   • Case 2: Both delays are non-zero ($\tau_1 > 0, \tau_2 \geq 0$).
4. Stability Criteria:
   • Delay-Independent Stability: Conditions where the equilibrium is stable for all delay values.
   • Delay-Dependent Stability: Conditions where stability depends on the magnitude of the delays, involving the calculation of critical delay values ($\tau_{cr}$) where Hopf bifurcations occur.
   • The analysis utilizes theorems regarding the location of roots in the complex plane (specifically checking for roots crossing the imaginary axis) and compares the real and imaginary parts of the characteristic equation.

3. Key Contributions

• Novel Model Formulation: The paper analyzes a fractional differential equation with two delays and a delay-dependent coefficient, a combination not extensively covered in existing literature.
• Derivation of Stability Regions: The authors map out stability regions in the $(k, \gamma)$ parameter plane for both cases. They distinguish between:
  • Delay-Independent Stability: Regions where the system is stable regardless of delay magnitude.
  • Delay-Independent Instability: Regions where the system is unstable regardless of delay magnitude.
  • Delay-Dependent Stability: Regions where stability switches occur as delays vary.
• Critical Thresholds: Explicit formulas are derived for critical delay values ($\tau_1^*, \tau_2^*$) and critical parameter values ($k^*$) that determine the transition between stable and unstable states.
• Extension to Nonlinear Functions: The theoretical results are validated for specific nonlinear functions, such as $g(x) = k \sin(x)$, demonstrating the applicability of the linearized stability conditions to the original nonlinear system.

4. Key Results

Case 1: Single Delay ($\tau_1 = 0$)

The stability of the equilibrium $x^*=0$ depends on the relationship between parameters $k$ and $\gamma$:

• Unstable Regions:
  • If $k, \gamma < 0$ (Third Quadrant), the system is unstable for all $\tau \geq 0$.
  • If $k > 0, \gamma < 0$ (Fourth Quadrant) and $\tau$ is small, the system is unstable.
• Stable Regions:
  • If $\gamma > 2k > 0$ or $\gamma > 0, k < 0$ (Second Quadrant), the system is asymptotically stable for all $\tau \geq 0$ (Delay-Independent Stability).
• Delay-Dependent Behavior:
  • In the region $0 < \gamma < 2k$, stability depends on the delay magnitude. The system can switch between stable and unstable states as$\tau$increases, crossing critical values$\tau_1^(\tau)$and$\tau_2^$.
  • A "switch" phenomenon (Unstable $\to$ Stable $\to$ Unstable) is identified in specific parameter ranges.

Case 2: Two Delays ($\tau_1 > 0, \tau_2 \geq 0$)

• Delay-Independent Stability:
  • The equilibrium is stable for all delays if $\gamma > 2k > 0$ OR $\gamma > -2k > 0$.
• Delay-Independent Instability:
  • If $k < 0$ and $\gamma < 0$ (Third Quadrant), the system is unstable for all delays.
  • For integer order ($\alpha=1$), instability occurs in the Fourth Quadrant ($k>0, \gamma<0$).
• Critical Parameter $k^*$:
  • In the Fourth Quadrant ($k>0, \gamma<0$), the authors derive a critical value $k^*$ (dependent on $\alpha, \tau_1, \tau_2$). If $k > k^*$, the system is guaranteed to be unstable.

5. Significance and Validation

• Validation: The theoretical findings are rigorously validated through numerical simulations. The paper presents numerous time-series plots (Figures 5–22) showing convergence to zero (stability) or divergence (instability) for various parameter sets, confirming the derived stability diagrams.
• Biological Relevance: The results provide a theoretical framework for understanding how memory (fractional order) and multiple time lags affect feedback mechanisms in biological systems (e.g., hematopoiesis/platelet production).
• Control Implications: The identification of delay-independent stability regions offers robust control strategies where system stability can be guaranteed without precise knowledge of delay magnitudes. Conversely, understanding delay-dependent switches is crucial for avoiding instability in systems where delays are unavoidable.

Conclusion

This paper fills a gap in the literature by providing a comprehensive stability analysis for fractional differential equations with two delays and delay-dependent coefficients. It successfully extends existing single-delay theories to more complex, realistic scenarios, offering explicit conditions for stability and instability that are crucial for modeling and controlling complex dynamical systems.