Linear Systems Phase Portrait Quiz for University Students

📚 Quick Study Guide

• 🔢 Linear System: A system of differential equations that can be written in the form $\frac{d\vec{x}}{dt} = A\vec{x}$, where $A$ is a constant matrix.
• ➡️ Eigenvalues and Eigenvectors: Key to understanding phase portraits. Solve $\det(A - \lambda I) = 0$ for eigenvalues ($\lambda$) and then $(A - \lambda I)\vec{v} = \vec{0}$ for eigenvectors ($\vec{v}$).
• ✨ Node: Occurs when both eigenvalues are real and have the same sign. Stable if both negative, unstable if both positive.
• 🧲 Saddle Point: Occurs when eigenvalues are real and have opposite signs. Always unstable.
• 🌀 Spiral Point: Occurs when eigenvalues are complex conjugates. Stable if real part is negative, unstable if real part is positive.
• 🔄 Center: Occurs when eigenvalues are purely imaginary (real part is zero). Neutrally stable.
• 📝 Phase Portrait: A diagram showing typical trajectories in the phase plane (x, y), visualizing the behavior of solutions to the system.

Practice Quiz

1. What type of critical point is a stable node?
   1. A) Eigenvalues are real, distinct, and both positive.
   2. B) Eigenvalues are real, distinct, and both negative.
   3. C) Eigenvalues are complex with a positive real part.
   4. D) Eigenvalues are purely imaginary.
2. For the system $\frac{d\vec{x}}{dt} = A\vec{x}$, if $A$ has eigenvalues $\lambda_1 = 2$ and $\lambda_2 = -1$, what type of critical point is the origin?
   1. A) Stable Node
   2. B) Unstable Node
   3. C) Saddle Point
   4. D) Spiral Point
3. If the eigenvalues of matrix $A$ are $3i$ and $-3i$, what type of critical point does the linear system $\frac{d\vec{x}}{dt} = A\vec{x}$ have at the origin?
   1. A) Spiral Point
   2. B) Center
   3. C) Saddle Point
   4. D) Node
4. Which of the following is true for a stable spiral point?
   1. A) Eigenvalues are real and positive.
   2. B) Eigenvalues are real and negative.
   3. C) Eigenvalues are complex with a positive real part.
   4. D) Eigenvalues are complex with a negative real part.
5. Consider a system with eigenvalues $\lambda_1 = -2 + i$ and $\lambda_2 = -2 - i$. What is the nature of the critical point?
   1. A) Unstable Spiral Point
   2. B) Stable Spiral Point
   3. C) Center
   4. D) Saddle Point
6. What characteristic distinguishes a saddle point from other types of critical points?
   1. A) Trajectories spiral inward towards the origin.
   2. B) Trajectories move directly towards or away from the origin.
   3. C) Trajectories move towards the origin along some directions and away from it along others.
   4. D) Trajectories form closed loops around the origin.
7. For a matrix $A$, if both eigenvalues are positive real numbers, the critical point at the origin is a(n):
   1. A) Stable Node
   2. B) Unstable Node
   3. C) Saddle Point
   4. D) Stable Spiral Point