📚 Topic Summary
Phase plane analysis is a graphical method used to determine the qualitative behavior of solutions to systems of differential equations. For homogeneous systems, which are of the form $\dot{x} = Ax$ where $A$ is a constant matrix, the phase plane provides a visual representation of the system's trajectories. The stability of the system, such as whether solutions converge to or diverge from the origin, can be inferred from the nature of these trajectories and the eigenvalues of matrix $A$. Analyzing the eigenvalues of $A$ helps classify the critical points (equilibrium points) as stable nodes, unstable nodes, saddle points, stable spirals, unstable spirals, or centers.
Understanding the behavior near these critical points is crucial for determining the overall stability of the system. This involves classifying the eigenvalues (real, complex, positive, negative) and examining the eigenvectors to sketch the phase portrait accurately.
🧠 Part A: Vocabulary
Match each term with its correct definition:
| Term |
Definition |
| 1. Stable Node |
A. A critical point where trajectories move away from the origin. |
| 2. Unstable Node |
B. A critical point where trajectories spiral inwards towards the origin. |
| 3. Saddle Point |
C. A critical point where trajectories move towards the origin. |
| 4. Stable Spiral |
D. A critical point where trajectories are tangent to a line, with trajectories approaching or receding from the critical point. |
| 5. Center |
E. A critical point with trajectories both approaching and receding from it. |
✏️ Part B: Fill in the Blanks
Complete the following paragraph using the words: eigenvalues, phase plane, stability, trajectories, homogeneous.
The analysis of a __________ system often involves studying the __________ to understand the system's __________. By examining the __________ of the matrix $A$, one can determine the nature of the __________ and thus predict the long-term behavior of the system.
🤔 Part C: Critical Thinking
Consider a system $\dot{x} = Ax$. How does the presence of repeated real __________ affect the classification of the critical point and the overall __________ of the system? Explain with examples.