Test Questions on Finding and Classifying Critical Points in Autonomous Systems

📚 Quick Study Guide

• 🧭 Autonomous System: A system of differential equations of the form $\frac{dx}{dt} = f(x, y)$, $\frac{dy}{dt} = g(x, y)$, where $f$ and $g$ do not explicitly depend on $t$.
• 📍 Critical Points (Equilibrium Points): Points $(x_0, y_0)$ where $f(x_0, y_0) = 0$ and $g(x_0, y_0) = 0$. These are points where the system is at rest.
• 🔍 Finding Critical Points: Solve the system of equations $f(x, y) = 0$ and $g(x, y) = 0$ simultaneously.
• 📊 Jacobian Matrix: The Jacobian matrix is given by $J = \begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{bmatrix}$. Evaluate the Jacobian at each critical point.
• 📈 Eigenvalues: Find the eigenvalues $\lambda$ of the Jacobian matrix by solving the characteristic equation $\det(J - \lambda I) = 0$, where $I$ is the identity matrix.
• 📉 Classification of Critical Points:
  • 💫 Stable Node: Both eigenvalues are real, negative, and distinct.
  • 🌟 Unstable Node: Both eigenvalues are real, positive, and distinct.
  • 🧲 Saddle Point: Eigenvalues are real with opposite signs.
  • 🌀 Stable Spiral: Eigenvalues are complex with negative real parts.
  • 🌪️ Unstable Spiral: Eigenvalues are complex with positive real parts.
  • 🔄 Center: Eigenvalues are purely imaginary.

Practice Quiz

1. Question 1: Consider the system $\frac{dx}{dt} = x - y$ and $\frac{dy}{dt} = x^2 - 4$. Which of the following is a critical point of this system?
   1. (2, 2)
   2. (0, 0)
   3. (-2, 2)
   4. (2, -2)
2. Question 2: Given the system $\frac{dx}{dt} = x(2 - x - y)$ and $\frac{dy}{dt} = y(y - x)$, what are the critical points?
   1. (0, 0), (2, 0), (1, 1)
   2. (0, 0), (0, 2), (1, 1)
   3. (0, 0), (2, 2), (1, 1)
   4. (0, 0), (2, 0), (0, 2)
3. Question 3: Suppose a system has a Jacobian matrix at a critical point with eigenvalues $\lambda_1 = 3$ and $\lambda_2 = -1$. What type of critical point is it?
   1. Stable Node
   2. Unstable Node
   3. Saddle Point
   4. Stable Spiral
4. Question 4: For a system with eigenvalues $\lambda_1 = -2 + 3i$ and $\lambda_2 = -2 - 3i$ at a critical point, what type of critical point is it?
   1. Stable Node
   2. Unstable Spiral
   3. Center
   4. Stable Spiral
5. Question 5: A system has a Jacobian with eigenvalues $\lambda_1 = 2i$ and $\lambda_2 = -2i$. What type of critical point is it?
   1. Stable Spiral
   2. Unstable Spiral
   3. Center
   4. Saddle Point
6. Question 6: Consider the system $\frac{dx}{dt} = y$ and $\frac{dy}{dt} = -x$. Find the eigenvalues of the Jacobian matrix at the critical point (0,0).
   1. $\lambda = \pm 1$
   2. $\lambda = \pm i$
   3. $\lambda = \pm 2i$
   4. $\lambda = \pm 2$
7. Question 7: Given the system $\frac{dx}{dt} = x - y^2$ and $\frac{dy}{dt} = x - 4$, find a critical point of the system.
   1. (4, 2)
   2. (2, 4)
   3. (0, 0)
   4. (1, 1)