Problem 44 - 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. $$ A=\left[\begin{array}{rr} -1 & -5 \\ 4 & -3 \end{array}\right] $$

Step-by-Step Solution

Answer

The equilibrium \((0,0)\) is a stable spiral.

Step 1: Determine Eigenvalues

First, find the eigenvalues of the matrix \( A \). The eigenvalues are found from the characteristic equation \( \det(A - \lambda I) = 0 \), where \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix.

Step 2: Set Up the Characteristic Equation

For the matrix \( A = \begin{bmatrix} -1 & -5 \ 4 & -3 \end{bmatrix} \), the characteristic equation is \( \det \begin{bmatrix} -1 - \lambda & -5 \ 4 & -3 - \lambda \end{bmatrix} = 0 \).

Step 3: Calculate the Determinant

The determinant is calculated as: \[ (-1-\lambda)(-3-\lambda) - (-5)(4) = \lambda^2 + 4\lambda + 11 \]This is the characteristic polynomial.

Step 4: Solve the Characteristic Equation

Solve the quadratic equation \( \lambda^2 + 4\lambda + 11 = 0 \) using the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 4, c = 11 \).

Step 5: Apply the Quadratic Formula

Substitute into the quadratic formula: \[ \lambda = \frac{-4 \pm \sqrt{16 - 44}}{2} = \frac{-4 \pm \sqrt{-28}}{2} \]\[ \lambda = -2 \pm i \sqrt{7} \].

Step 6: Analyze the Eigenvalues

The eigenvalues are complex \( \lambda = -2 \pm i \sqrt{7} \). The real part of the eigenvalues is \( -2 \).

Step 7: Determine Stability

Since the real part of the eigenvalues is negative, the equilibrium at \((0,0)\) is stable. The presence of complex eigenvalues indicates spiral behavior.

Step 8: Classify the Equilibrium

The equilibrium is a stable spiral since the real parts of the eigenvalues are negative.

Key Concepts

Stability Analysis

Understanding stability in differential equations helps us determine how systems behave over time. In simple terms, stability analysis involves examining whether solutions of a system converge to a point (equilibrium) as time progresses. For a matrix system like \( \frac{d \mathbf{x}}{d t}=A \mathbf{x}(t) \), stability is often assessed by checking the real parts of its eigenvalues.

• If all real parts are negative, solutions tend to approach the equilibrium, indicating stability.
• If any real part is positive, the equilibrium is unstable since solutions diverge.

Analyzing eigenvalues, particularly their real parts, is crucial in determining overall system behavior.

Eigenvalues

Eigenvalues are critical in analyzing linear transformations represented by matrices. Essentially, an eigenvalue \( \lambda \) relates to a special set of vectors known as eigenvectors, which, when the matrix is applied, only get scaled but not a directionally altered. To find the eigenvalues of a matrix \( A \), one uses the characteristic equation formed as: \[ \det(A - \lambda I) = 0 \]Eigenvalues tell us how the dynamical system evolves, particularly around an equilibrium point.

Equilibrium Classification

Once we calculate eigenvalues, their nature helps in classifying equilibrium points of differential equations. Here's how classification generally works:

• Stable focus or spiral: Occurs if complex eigenvalues have negative real parts. The system returns to equilibrium in a spiraling fashion.
• Unstable focus or spiral: If complex eigenvalues have positive real parts, the system spirals out of equilibrium.
• Center: Occurs when eigenvalues are purely imaginary with zero real parts, indicating neutral stability with circular solution paths around the equilibrium.

Understanding these classifications provides insights into how solutions behave around equilibrium points.

Complex Eigenvalues

Complex eigenvalues arise when solving the characteristic equation leads to square roots of negative numbers. They are typically expressed as \( \alpha \pm \beta i \), where \( \alpha \) is the real part and \( \beta \) is the imaginary part.

• The real part \( \alpha \) determines whether solutions grow or decay over time.
• The imaginary part \( \beta \) relates to oscillatory behavior, indicating how solutions spiral around equilibrium.

In stability analysis, complex eigenvalues often signal rotational behavior, manifesting as spirals in the solution graph.

Characteristic Equation

The characteristic equation comes into play when determining eigenvalues. Each matrix has an associated characteristic polynomial derived from the determinant equation:\[ \det(A - \lambda I) = 0 \]This polynomial is typically quadratic for 2x2 matrices, as in the given example problem. Solving this equation reveals the eigenvalues, guiding us on the system's dynamic nature. The quadratic formula, \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is used to solve it, which can yield real or complex solutions. Determining these eigenvalues helps in analyzing system behavior and classification of equilibrium.