Problem 51 - 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 complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{ll}-1 & 1 \\ -3 & 1\end{array}\right]\)

Step-by-Step Solution

Answer

The equilibrium (0,0) is a center and is neither stable nor unstable.

Step 1: Compute the Characteristic Polynomial

To find the eigenvalues of matrix \( A \), we need to compute the characteristic polynomial, which is given by \( \lambda^2 - \text{tr}(A)\lambda + \det(A) \). Here, \( \text{tr}(A) = a_{11} + a_{22} = -1 + 1 = 0 \) and \( \det(A) = (-1)(1) - (1)(-3) = -1 + 3 = 2 \). Thus, the characteristic polynomial is \( \lambda^2 + 2 = 0 \).

Step 2: Solve for Eigenvalues

Solve the characteristic equation \( \lambda^2 + 2 = 0 \) to find the eigenvalues. Rearranging gives \( \lambda^2 = -2 \), so the eigenvalues are \( \lambda = \pm i\sqrt{2} \), which are complex conjugates, specifically \( i\sqrt{2} \) and \( -i\sqrt{2} \).

Step 3: Analyze the Eigenvalues for Stability

Eigenvalues in the form \( \pm i\omega \) (purely imaginary) indicate a center. If the real parts are zero and there is no damping term, the equilibrium is a center. Thus, for this system with eigenvalues \( \pm i\sqrt{2} \), the equilibrium point is classified as a center point.

Key Concepts

Eigenvalues

Eigenvalues are critical in understanding the behavior of systems of differential equations. They are values of \( \lambda \) that satisfy the characteristic equation derived from the matrix \( A \), representing the linear differential equation \( \frac{d\mathbf{x}}{dt} = A\mathbf{x}(t) \). Eigenvalues provide insight into the system dynamics:

• Real eigenvalues lead to exponential growth or decay.
• Complex eigenvalues, like in our case \( i\sqrt{2} \) and \( -i\sqrt{2} \), suggest oscillatory behavior.
• The sign of the real part of eigenvalues indicates whether solutions grow or decay over time.

For our matrix \( A \), solving the characteristic polynomial \( \lambda^2 + 2 = 0 \) provides us with complex conjugate eigenvalues. These indicate that our system's solutions follow oscillatory paths, without tending to infinity or decaying to zero, as there are no real parts.

Stability Analysis

Stability analysis helps determine the long-term behavior of a differential equation's solutions. It tells us if small changes or perturbations to the system grow or vanish. This analysis is done using eigenvalues:

• Purely real eigenvalues indicate direct stability (negative means stable, positive means unstable).
• Purely imaginary eigenvalues, like \( \pm i\sqrt{2} \) in our exercise, indicate neutral stability or a center.

As in our example, when the eigenvalues are complex with zero real parts, the system exhibits neutral stability. The equilibrium point at (0,0) neither attracts nor repels nearby trajectories but instead causes them to circle around in a sustained oscillation, which is a hallmark of center stability.

Characteristic Polynomial

The characteristic polynomial is a fundamental tool in finding eigenvalues. It is derived from the matrix \( A \) of the system by calculating \( \det(\lambda I - A)\), where \( I \) is the identity matrix of the same size as \( A \). For a 2x2 matrix, solving for eigenvalues typically involves:

• Finding the trace of \( A \), denoted \( \text{tr}(A) = a_{11} + a_{22} \).
• Calculating the determinant \( \det(A) = a_{11}a_{22} - a_{12}a_{21} \).
• Formulating the polynomial \( \lambda^2 - \text{tr}(A)\lambda + \det(A) \).

In the example, solving \( \lambda^2 + 2 = 0 \) gives us the eigenvalues, which are critical in determining the system's oscillatory nature.The characteristic polynomial thus plays a crucial role in transitioning from the matrix representation to understanding the nature of the differential equation's solutions.