Problem 59 - Calculus for Biology and Medicine

Question

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{rr}-1 & -1 \\ 5 & -3\end{array}\right]\)

Step-by-Step Solution

Answer

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

Step 1: Define the Equilibrium Point

The given differential equation is \( \frac{dx}{dt} = Ax(t) \). The equilibrium point for a linear system occurs when \( Ax = 0 \). Here, the point given is \( (0, 0) \) which is the standard equilibrium for linear systems of the form \( \frac{dx}{dt} = Ax(t) \).

Step 2: Determine the Eigenvalues of Matrix A

To analyze stability, first calculate the eigenvalues of matrix \( A \). The characteristic equation of the matrix \( A = \begin{bmatrix} -1 & -1 \ 5 & -3 \end{bmatrix} \) is \( \text{det}(A - \lambda I) = 0 \). Compute:\[ \begin{vmatrix} -1-\lambda & -1 \ 5 & -3-\lambda \end{vmatrix} = (-1-\lambda)(-3-\lambda) - (-1)(5) = \lambda^2 + 4\lambda + 8 = 0 \].

Step 3: Solve the Characteristic Equation

The characteristic equation is \( \lambda^2 + 4\lambda + 8 = 0 \). Solve this quadratic equation using the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 4 \), \( c = 8 \). Compute:\[ \lambda = \frac{-4 \pm \sqrt{16 - 32}}{2} = \frac{-4 \pm \sqrt{-16}}{2} = \frac{-4 \pm 4i}{2} = -2 \pm 2i \].

Step 4: Analyze the Stability using Eigenvalues

The eigenvalues of \( A \) are \( -2 + 2i \) and \( -2 - 2i \). Since both eigenvalues have negative real parts (\( -2 \)), the equilibrium point \( (0, 0) \) is stable. Specifically, because the eigenvalues are complex with negative real parts, the equilibrium is classified as a stable focus (or a spiral sink).

Key Concepts

Stability Analysis

Stability analysis is a vital technique used to determine whether equilibrium points in differential systems tend to return to their equilibrium state after a perturbation. For a system defined by the differential equation \( \frac{dx}{dt} = Ax(t) \), stability analysis helps predict whether the system will stabilize, oscillate, or diverge over time.

To perform stability analysis, one of the most effective tools is eigenvalue calculation of the matrix \( A \). If the real parts of all eigenvalues are negative, the equilibrium point is stable, indicating it is an attractor. Conversely, if any eigenvalue has a positive real part, the system will be unstable, and the equilibrium point will repel perturbations. By distinguishing these scenarios, stability analysis provides insight into the behavior of complex systems.

Eigenvalues

Eigenvalues are crucial in the analysis of linear systems and understanding their dynamic behavior. Derived from a mathematical matrix, eigenvalues reveal how the system evolves over time and influence the stability of equilibrium points. When computing eigenvalues for the matrix \( A \), one typically solves the characteristic equation

• \( \text{det}(A - \lambda I) = 0 \)

where \( \lambda \) represents the eigenvalues and \( I \) is the identity matrix. This equation sets the stage for discovering the eigenvalues that dictate the response of the system to initial conditions. Imagine the eigenvalues as keys that unlock the secrets to the system's future behavior, capturing whether response patterns will remain steady, cyclical, or diverge indefinitely.

Equilibrium Points

Equilibrium points are states in a system where the variables remain constant over time, provided no external changes occur. In the context of differential equations, these are points where the rate of change is zero, i.e., \( Ax = 0 \).

At the equilibrium point, the system is perfectly balanced; it represents a state where forces within a model counteract each other. The equilibrium at \( (0,0) \) signifies that without disturbances, the system remains at rest.

Identifying these points is crucial for analyzing if and how a system reacts to changes. By scrutinizing the stability of these points, one can predict how the system reacts to perturbations, much like observing a ball at the bottom of a bowl, oscillating but returning back to its point of rest.

Matrix Analysis

Matrix analysis forms the backbone of evaluating linear systems in differential equations. Through this lens, matrix \( A \) shapes how the system behaves over time by transforming vectors in the state space.

Conducting an analysis involves several components:

• Understanding the structure of the matrix.
• Calculating eigenvalues and eigenvectors.
• Interpreting these in the context of system response and stability.

Within matrix analysis, each element and arrangement within a matrix can influence system dynamics. Structural properties like trace and determinant provide additional insights into the behavior and stability of dynamical systems.

By examining the intrinsic properties of matrix \( A \), matrix analysis allows us to extract insights about the system's temporal evolution, providing a viewpoint on how different constituent parts interact. It's like having a map that reveals not just where we've been, but where the system is potentially headed.