Solved Examples: Eigenvalue Analysis for Equilibrium Point Stability

📚 Quick Study Guide

• 🔢 Equilibrium Point: A point where the rate of change of the system is zero.
• 📈 Stability: Refers to the behavior of the system near the equilibrium point. It can be stable (returns to equilibrium), unstable (moves away), or neutrally stable.
• 📊 Eigenvalue Analysis: A method to determine stability by analyzing the eigenvalues of the Jacobian matrix evaluated at the equilibrium point.
• ✒️ Jacobian Matrix: A matrix containing the partial derivatives of the system's equations.
• 🔑 Eigenvalues: Scalars, $\lambda$, associated with eigenvectors of the Jacobian matrix, obtained by solving the characteristic equation: $det(J - \lambda I) = 0$, where $J$ is the Jacobian and $I$ is the identity matrix.
• ✅ Stability Criteria:
  • Stable Node/Spiral: All eigenvalues have negative real parts.
  • Unstable Node/Spiral: At least one eigenvalue has a positive real part.
  • Saddle Node: Has both positive and negative real eigenvalues.
  • Center: Purely imaginary eigenvalues (stability is neutral or requires further investigation).

Practice Quiz

1. The eigenvalues of the Jacobian matrix at an equilibrium point are -2 and -3. What is the stability of the equilibrium point?

   1. Unstable Node
   2. Stable Node
   3. Saddle Point
   4. Center

2. The eigenvalues of the Jacobian matrix at an equilibrium point are 1 and -1. What is the stability of the equilibrium point?

   1. Unstable Node
   2. Stable Node
   3. Saddle Point
   4. Center

3. The eigenvalues of the Jacobian matrix at an equilibrium point are 2 + i and 2 - i. What is the stability of the equilibrium point?

   1. Stable Spiral
   2. Unstable Spiral
   3. Center
   4. Saddle Point

4. The eigenvalues of the Jacobian matrix at an equilibrium point are -0.5 + i and -0.5 - i. What is the stability of the equilibrium point?

   1. Stable Spiral
   2. Unstable Spiral
   3. Center
   4. Saddle Point

5. The eigenvalues of the Jacobian matrix at an equilibrium point are i and -i. What is the stability of the equilibrium point?

   1. Stable Spiral
   2. Unstable Spiral
   3. Center
   4. Stable Node

6. The Jacobian matrix at an equilibrium point is $\begin{bmatrix} -3 & 0 \\ 0 & -2 \end{bmatrix}$. What are the eigenvalues?

   1. -3 and -2
   2. 3 and 2
   3. -3 and 2
   4. 3 and -2

7. The Jacobian matrix at an equilibrium point is $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$. What are the eigenvalues?

   1. 1 and -1
   2. i and -i
   3. 1 and i
   4. -1 and -i