Problem 8 - Advanced Engineering Mathematics

Question

In Problems \(3-10\), without solving explicitly, classify the critical points of the given first-order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$ \frac{d x}{d t}=k(\alpha-x)(\beta-x)(\gamma-x), \alpha>\beta>\gamma $$

Step-by-Step Solution

Answer

The critical points \( x = \alpha \) and \( x = \gamma \) are asymptotically stable, \( x = \beta \) is unstable.

Step 1: Identify Critical Points

To find critical points, set \( \frac{d x}{d t} = 0 \). This gives \( k(\alpha-x)(\beta-x)(\gamma-x) = 0 \). The critical points are \( x = \alpha, \beta, \gamma \).

Step 2: Analyze the Stability - Critical Point \( x = \alpha \)

For \( x = \alpha \), the expression \((\alpha-x)\) changes sign as \( x \) crosses \( \alpha \) from positive to negative, indicating \( x_\alpha \) is stable if moving from right to left causes decreasing \( \frac{d x}{d t} \), hence \( x = \alpha \) is stable since \( \alpha > \beta > \gamma \) implies \((\beta-\alpha)(\gamma-\alpha) < 0 \).

Step 3: Analyze the Stability - Critical Point \( x = \beta \)

For \( x = \beta \), the sign of the derivative changes oppositely compared to \( x = \beta \), indicating instability. Since \( \alpha > \beta > \gamma \), \((\alpha-\beta)(\gamma-\beta) < 0\), \( x = \beta \) is unstable.

Step 4: Analyze the Stability - Critical Point \( x = \gamma \)

For \( x = \gamma \), the situation is similar to \( x = \alpha \). Since the sign changes when \( x \) crosses \( \gamma \) from negative to positive, \((\alpha-\gamma)(\beta-\gamma) > 0 \), making \( x = \gamma \) stable.

Key Concepts

Critical Points

Critical points are specific values of the variable in a differential equation where the rate of change is zero. For the autonomous differential equation \[\frac{dx}{dt} = k(\alpha-x)(\beta-x)(\gamma-x)\]we find the critical points by setting the derivative equal to zero:\[k(\alpha-x)(\beta-x)(\gamma-x) = 0\]This means that the critical points are those where any of the factors in the product is zero, leading us to the solutions:

• \(x = \alpha\)
• \(x = \beta\)
• \(x = \gamma\)

Each of these values represents a point where the velocity or rate of change is temporarily paused, allowing us to investigate the behavior of the system at these points more closely.

Stability Analysis

Stability analysis involves examining the nature of each critical point to determine whether it is stable or unstable. In simple terms, we assess how small deviations from these points affect the system.
The goal is to see if the system naturally returns to the critical point (stable), or if it diverges away from it (unstable). To assess this, we often use a qualitative approach. By looking at the derivatives' sign on either side of a critical point, we can decide if it acts as a point of attraction or repulsion.

• A stable critical point is one where small disturbances diminish over time, meaning the system returns to this point.
• An unstable critical point is one where small disturbances grow, causing the system to move away from this point.

Asymptotic Stability

Asymptotic stability is a stronger form of stability seen in differential equations. It means not only does the solution remain close to a critical point when slightly disturbed, but it also eventually moves closer to the critical point over time.
This concept is integral when examining the behavior of dynamic systems over long periods. In our example, any deviation from a critical point like \(x = \alpha\) would result in the system adjusting and eventually returning to this critical point as time progresses, indicating asymptotic behavior.

• If the sign of the derivative changes from positive to negative around a critical point, it often signals asymptotic stability.
• This condition is crucial when ensuring that biological populations, physical systems, or engineering models settle into a specific behavior long term.

Differential Equations

Differential equations are mathematical equations that involve the rates of change of variables. They are fundamental in describing how quantities evolve over time or space.
In our problem, we deal with an autonomous differential equation, which means the rate of change only depends on the current state, not on time explicitly. These equations are pivotal in modeling natural phenomena, engineering systems, and more. They help us predict the future behavior of a system from its current state.

• They can describe motion, growth rates, decay patterns, and many other dynamic processes.
• Autonomous differential equations are particularly useful when the system's rules are static over time, allowing us to focus on the variable's behavior solely.

Mastering these concepts enables one to better understand and control complex dynamic systems.