Problem 64 - Calculus for Biology and Medicine

Question

Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium. $$ A=\left[\begin{array}{rr} -2 & -7 \\ 1 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The equilibrium at the origin is a saddle point and is unstable.

Step 1: Define the Problem

The given problem requires us to determine the stability and type of the equilibrium point at the origin for a linear system characterized by the matrix \( A = \begin{pmatrix} -2 & -7 \ 1 & 2 \end{pmatrix} \). The stability is assessed using eigenvalues.

Step 2: Find the Characteristic Equation

The characteristic equation of a matrix \( A \) is found using \( \det(A - \lambda I) = 0 \), where \( \lambda \) is an eigenvalue and \( I \) is the identity matrix. So, we calculate \( \det\begin{pmatrix} -2-\lambda & -7 \ 1 & 2-\lambda \end{pmatrix} = 0 \).

Step 3: Calculate the Determinant

Calculate the determinant \( (-2-\lambda)(2-\lambda) - (-7)\cdot 1 = \lambda^2 - 0\lambda - 3 \). So, the characteristic equation is \( \lambda^2 - 3 = 0 \).

Step 4: Solve for Eigenvalues

Solve the equation \( \lambda^2 - 3 = 0 \) to find the eigenvalues. This gives us \( \lambda = \pm \sqrt{3} \).

Step 5: Determine Stability and Type

Since the eigenvalues \( \lambda = \sqrt{3} \) and \( \lambda = -\sqrt{3} \) have positive and negative real parts, the equilibrium point at the origin is a saddle point, which is unstable.

Key Concepts

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It is fundamental in understanding systems of linear equations, which can be represented in matrix form. Here are some key points about linear algebra in this context:

• Matrix representation: Systems of linear equations can be expressed as matrices, which makes them easier to solve and analyze. The matrix associated with the exercise is a 2x2 matrix.
• Determinants and inverses: Determinants aid in solving equations and finding inverses. These play crucial roles in stability analysis as they help in characterizing equilibrium points.
• Transformations: In linear algebra, matrices represent linear transformations, mapping vectors from one space to another. Understanding these transformations helps in visualizing changes in the system over time.

Linear algebra is particularly useful in stability analysis, where equilibrium points of a system and the influences of small perturbations are studied.

eigenvalues

Eigenvalues are important components in linear algebra that help analyze various properties of a system. They emerge when solving the characteristic equation, a key feature in understanding stability.

• Definition: An eigenvalue is a number associated with a matrix that provides insight into the system's behavior. It represents how a vector is stretched or shrunk during a transformation.
• Stability analysis: Eigenvalues help in classifying the stability of an equilibrium point. Positive eigenvalues indicate instability while negative ones suggest stability. If your system has eigenvalues of both signs, it results in a saddle point, which is unstable.
• Real and complex eigenvalues: Eigenvalues can be real or complex numbers. Real eigenvalues directly relate to exponential growth or decay in the system.

The eigenvalues for any given matrix provide crucial insights into both the short-term and long-term behavior of its associated system.

saddle point

A saddle point is a type of equilibrium point in a dynamical system. It is characterized by its unique stability property, where the trajectories around the point have both directions of attraction and repulsion.

• Characteristics: At a saddle point, the system will exhibit both stable and unstable behavior, depending on the direction of analysis.
• Influence of eigenvalues: Saddle points typically occur when a system's equilibrium has both positive and negative eigenvalues, leading to mixed stability.
• Visualization: Picture a saddle on a horse; it curves upwards in one direction and downwards in the other. This illustrates how solutions approach and diverge from the point.

Understanding saddle points is critical as they highlight the complex nature of dynamical systems and are often associated with instability.

characteristic equation

The characteristic equation is a foundational tool in linear algebra for determining the eigenvalues of a matrix. It lays the groundwork for understanding the dynamic properties of systems.

• Definition: Formed by setting the determinant of matrix \( A - \lambda I \) equal to zero, where \( \lambda \) represents the eigenvalues and \( I \) is the identity matrix.
• Calculation: For a 2x2 matrix, like the one in the exercise, the characteristic equation is found through simple determinant expansion.
• Application: Solving the characteristic equation yields the eigenvalues, which are then used to analyze stability and other dynamic characteristics of the system.

The characteristic equation provides a pathway from matrix representation to understanding key features such as stability and oscillation tendencies of the respective system.