Problem 33 - 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 equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{ll}-4 & 2 \\ -5 & 3\end{array}\right]\)

Step-by-Step Solution

Answer

The equilibrium at \((0,0)\) is a saddle point and is unstable.

Step 1: Calculate the Characteristic Polynomial of A

To determine the eigenvalues, we first need to find the characteristic polynomial of the matrix \(A\). The characteristic polynomial is given by \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix and \(\lambda\) are the eigenvalues. For the matrix \(A\):\[A - \lambda I = \begin{bmatrix}-4 - \lambda & 2 \-5 & 3 - \lambda\end{bmatrix}\]The determinant is \[\det(A - \lambda I) = (-4 - \lambda)(3 - \lambda) - (2)(-5)\]Expanding this, we calculate the characteristic polynomial to be \[\lambda^2 + \lambda - 2 = 0.\]

Step 2: Solve the Characteristic Polynomial

Solve the quadratic equation \(\lambda^2 + \lambda - 2 = 0\) to find the eigenvalues:\[\lambda = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]where \(a = 1\), \(b = 1\), and \(c = -2\).Substitute into the formula:\[\lambda = \frac{{-1 \pm \sqrt{{1 + 8}}}}{2} = \frac{{-1 \pm 3}}{2}.\]Thus, the eigenvalues are \(\lambda_1 = 1\) and \(\lambda_2 = -2\).

Step 3: Analyze the Stability of the Equilibrium based on Eigenvalues

Using the eigenvalues found, \(\lambda_1 = 1\) and \(\lambda_2 = -2\), we analyze the stability:1. A positive eigenvalue (\(\lambda_1 = 1\)) indicates an unstable direction (away from the equilibrium).2. A negative eigenvalue (\(\lambda_2 = -2\)) indicates a stable direction (toward the equilibrium).With one positive and one negative eigenvalue, the equilibrium point \((0,0)\) is a saddle point, signaling instability.

Key Concepts

Understanding Eigenvalues

Eigenvalues play a crucial role in understanding the behavior of dynamical systems described by differential equations. When analyzing a system of linear differential equations, matrices are often used to simplify the representation of the system. Each matrix has associated eigenvalues, which give insights into the system's properties and behavior.

For the matrix \(A\), eigenvalues are found by solving the characteristic equation derived from the determinant \(\det(A - \lambda I) = 0\), with \(\lambda\) representing the eigenvalues and \(I\) the identity matrix. The characteristic polynomial can be a powerful tool for modeling the evolution of a system over time. Calculating these eigenvalues helps us predict how the system will behave under various initial conditions.

• Eigenvalues indicative of behavior: If all eigenvalues are negative, the system tends to a stable point. If any are positive, the system has instability in some directions.
• Simple example: In our case, solving the characteristic polynomial gives us eigenvalues \(\lambda_1 = 1\) and \(\lambda_2 = -2\), which indicates mixed behavior.

Stability Analysis

Stability analysis examines how a system responds to small disturbances around an equilibrium point. It is a vital step when determining whether a system will return to equilibrium or move away from it after being slightly altered.

In this process, eigenvalues of the matrix \(A\) are analyzed:

• If any eigenvalue has a positive real part, it indicates instability in that direction, meaning the system will not return to its equilibrium state. It may even diverge further.
• If all eigenvalues have negative real parts, the system is stable and will tend to return to equilibrium when slightly disturbed.

For our given matrix \(A\), we found eigenvalues \(\lambda_1 = 1\) and \(\lambda_2 = -2\). The positive eigenvalue \(\lambda_1 = 1\) indicates that the system is unstable in that direction since a small disturbance will lead the trajectory away from the equilibrium point.

Equilibrium Classification

Classifying equilibria according to their stability characteristics helps in predicting long-term behavior of dynamical systems. We describe equilibrium points in terms of their response to perturbations using terms such as 'sink', 'source', or 'saddle point'.

Here's how these classifications work:

• Sink: All eigenvalues with negative real parts. The system moves towards equilibrium, indicating stability.
• Source: Any eigenvalue with a positive real part. Moves away from equilibrium, indicating instability.
• Saddle Point: A mix of positive and negative eigenvalues. Shows instability due to the presence of divergent trajectories despite there being stable directions.

The system described by the matrix \(A\), with eigenvalues \(\lambda_1 = 1\) (positive) and \(\lambda_2 = -2\) (negative), is categorized as a saddle point. This indicates an unstable equilibrium, where the presence of both diverging and converging directions creates an overall unstable system behavior.