Problem 40 - Calculus for Biology and Medicine

Question

We consider differential equations of the form $$ \frac{d \mathbf{x}}{d t}=A \mathbf{x}(t) $$ where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{ll} 0 & 2 \\ 3 & 7 \end{array}\right] $$

Step-by-Step Solution

Answer

The equilibrium \((0,0)\) is a saddle point because one eigenvalue is positive and one is negative.

Step 1: Calculate the Eigenvalues of A

To assess the stability, we need to calculate the eigenvalues of the matrix \( A \). For matrix \( A \):\[A = \begin{pmatrix} 0 & 2 \ 3 & 7 \end{pmatrix}\]The characteristic equation is given by the determinant \( \text{det}(A - \lambda I) = 0 \).\[\text{det}\begin{pmatrix} -\lambda & 2 \ 3 & 7-\lambda \end{pmatrix} = 0\]Calculate the determinant:\[ (-\lambda)(7-\lambda) - (3)(2) = \lambda^2 - 7\lambda - 6 = 0 \]Solve for \( \lambda \) using the quadratic formula: \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1, b = -7, c = -6 \).

Step 2: Solve the Characteristic Equation

Substituting the values into the quadratic formula:\[\lambda = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \times 1 \times (-6)}}{2 \times 1} = \frac{7 \pm \sqrt{49 + 24}}{2} = \frac{7 \pm \sqrt{73}}{2} \]The eigenvalues are real and distinct:\[\lambda_1 = \frac{7 + \sqrt{73}}{2}, \quad \lambda_2 = \frac{7 - \sqrt{73}}{2}\]

Step 3: Determine Stability and Nature of the Equilibrium

Based on the eigenvalues:- If both eigenvalues are positive, then the system is an unstable node or source.- If both eigenvalues are negative, then the system is a stable node or sink.- If one eigenvalue is positive and the other negative, then the system is a saddle point.Since \( \lambda_1 = \frac{7 + \sqrt{73}}{2} \) (positive) and \( \lambda_2 = \frac{7 - \sqrt{73}}{2} \) (negative), the equilibrium is a saddle point.

Key Concepts

Understanding Eigenvalues

Eigenvalues are critical in analyzing differential equations, particularly for understanding the dynamics of linear systems. When we have a matrix, like the one in our exercise, the eigenvalues correspond to the scalar values that transform a vector without changing its direction. Mathematically, eigenvalues are obtained from the matrix equation \( A\mathbf{v} = \lambda\mathbf{v} \), where \( \lambda \) is the eigenvalue and \( \mathbf{v} \) is the eigenvector.

• To find these eigenvalues, we solve the characteristic equation, which is derived from the matrix \( A \).
• In our problem, solving the quadratic equation \( \lambda^2 - 7\lambda - 6 = 0 \) resulted in the eigenvalues \( \lambda_1 = \frac{7 + \sqrt{73}}{2} \) and \( \lambda_2 = \frac{7 - \sqrt{73}}{2} \).

Eigenvalues tell us about the behavior of the system's solutions over time, which is crucial for analyzing equilibrium stability in dynamical systems.

Exploring Stability Analysis

Stability analysis utilizes eigenvalues to understand how solutions of differential equations behave near equilibrium points. It tells us whether small perturbations away from an equilibrium point will decay, stay constant, or grow over time. This is essential in many fields, including engineering and physics, where system stability can be crucial.

• Positive eigenvalues indicate that solutions move away from the equilibrium, suggesting instability (often termed a source).
• Negative eigenvalues imply stability, meaning solutions will return to equilibrium (commonly called a sink).
• If eigenvalues have different signs, the equilibrium is a saddle point, showing some solutions approach, while others diverge.

In our exercise, because we have one positive \( \lambda_1 \) and one negative \( \lambda_2 \), the equilibrium at \((0,0)\) is classified as a saddle point, indicating mixed stability characteristics.

Equilibrium Classification

Classifying equilibria is a fundamental part of understanding the overall behavior and stability of a system described by differential equations. This classification assists in visualizing and predicting how systems evolve over time.

• A **sink** is an equilibrium where all nearby trajectories converge, showing stable behavior. It's characterized by eigenvalues that are both negative.
• A **source** is an equilibrium with diverging trajectories, indicating instability. This occurs when both eigenvalues are positive.
• A **saddle point** shows a combination of stable and unstable behavior. Trajectories approach along some directions and diverge in others, as seen in our problem with one positive and one negative eigenvalue.

In practical terms, equilibrium classification helps in the design and control of systems to ensure desired behavior, such as ensuring mechanical structures remain stable under varying conditions. In our example, the mixed stability is a hallmark of a saddle point equilibrium.