Solved Examples of Sketching Phase Portraits for Various Eigenvalues

📚 Quick Study Guide

• 🔢 Eigenvalues: These are special numbers associated with a matrix that help describe the behavior of linear transformations. In the context of phase portraits, eigenvalues determine the stability and nature of the critical point (equilibrium).
• 📈 Real Eigenvalues:
  • 🌱 Both positive: Unstable node (trajectories move away from the origin).
  • 🍂 Both negative: Stable node (trajectories move towards the origin).
  • 💫 One positive, one negative: Saddle point (trajectories move towards the origin along one eigenvector and away along the other).
• 🌀 Complex Eigenvalues:
  • spiraling Trajectories: $a \pm bi$ (where $a$ and $b$ are real numbers, and $i$ is the imaginary unit).
  • If $a > 0$: Unstable spiral (trajectories spiral away from the origin).
  • If $a < 0$: Stable spiral (trajectories spiral towards the origin).
  • If $a = 0$: Center (trajectories form closed loops around the origin).
• 🧭 Eigenvectors: These are the directions along which the linear transformation acts by scaling. They help determine the orientation of trajectories in the phase portrait, especially for nodes and saddle points.
• ✏️ Sketching Tips:
  • Find the eigenvalues and eigenvectors.
  • Determine the type of critical point (node, saddle, spiral, center).
  • Sketch the eigenvectors (if applicable).
  • Sketch representative trajectories, paying attention to the stability of the critical point.

Practice Quiz

1. What type of critical point is formed when both eigenvalues are positive?

   1. A. Stable Node
   2. B. Unstable Node
   3. C. Saddle Point
   4. D. Stable Spiral

2. What type of critical point arises when one eigenvalue is positive and the other is negative?

   1. A. Stable Node
   2. B. Unstable Node
   3. C. Saddle Point
   4. D. Center

3. For complex eigenvalues of the form $a + bi$ with $a < 0$, what type of critical point is formed?

   1. A. Stable Spiral
   2. B. Unstable Spiral
   3. C. Center
   4. D. Saddle Point

4. If the eigenvalues are purely imaginary (i.e., of the form $bi$), the critical point is a:

   1. A. Stable Spiral
   2. B. Unstable Spiral
   3. C. Center
   4. D. Saddle Point

5. Which of the following best describes an unstable node?

   1. A. Trajectories move towards the origin.
   2. B. Trajectories spiral towards the origin.
   3. C. Trajectories move away from the origin.
   4. D. Trajectories form closed loops around the origin.

6. What role do eigenvectors play in sketching phase portraits for nodes and saddle points?

   1. A. They determine the stability of the critical point.
   2. B. They define the direction of spiraling trajectories.
   3. C. They indicate the orientation of trajectories near the critical point.
   4. D. They have no role in sketching phase portraits.

7. What is the first step in sketching a phase portrait?

   1. A. Sketching trajectories
   2. B. Finding the eigenvectors
   3. C. Finding the eigenvalues
   4. D. Determining the stability